%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUM155-1 : 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 : n015.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 08:08:18 AM UTC 2026
% Result : Unsatisfiable 122.47s 31.66s
% Output : Refutation 111.92s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 162
% Syntax : Number of formulae : 683 ( 199 unt; 0 typ; 0 def)
% Number of atoms : 1622 ( 635 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 5992 ( 833 ~; 699 |; 0 &;4460 @)
% ( 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 : 83 ( 80 usr; 22 con; 0-3 aty)
% Number of variables : 929 ( 0 ^; 929 !; 0 ?; 929 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(x_decl,type,
x: $i ).
thf(ordinal_numbers_decl,type,
ordinal_numbers: $i ).
thf(y_decl,type,
y: $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(function_decl,type,
function: $i > $o ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(symmetrization_of_decl,type,
symmetrization_of: $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(section_decl,type,
section: $i > $i > $i > $o ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(transitive_decl,type,
transitive: $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(flip_decl,type,
flip: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(irreflexive_decl,type,
irreflexive: $i > $i > $o ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(choice_decl,type,
choice: $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $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(power_class_decl,type,
power_class: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(rest_of_decl,type,
rest_of: $i > $i ).
thf(connected_decl,type,
connected: $i > $i > $o ).
thf(not_well_ordering_decl,type,
not_well_ordering: $i > $i > $i ).
thf(well_ordering_decl,type,
well_ordering: $i > $i > $o ).
thf(rest_relation_decl,type,
rest_relation: $i ).
thf(application_function_decl,type,
application_function: $i ).
thf(asymmetric_decl,type,
asymmetric: $i > $i > $o ).
thf(null_class_decl,type,
null_class: $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(segment_decl,type,
segment: $i > $i > $i > $i ).
thf(least_decl,type,
least: $i > $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(compose_class_decl,type,
compose_class: $i > $i ).
thf(union_of_range_map_decl,type,
union_of_range_map: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(recursion_decl,type,
recursion: $i > $i > $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(add_relation_decl,type,
add_relation: $i ).
thf(ordinal_multiply_decl,type,
ordinal_multiply: $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(ordinal_add_decl,type,
ordinal_add: $i > $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(singleton_relation_decl,type,
singleton_relation: $i ).
thf(recursion_equation_functions_decl,type,
recursion_equation_functions: $i > $i ).
thf(kind_1_ordinals_decl,type,
kind_1_ordinals: $i ).
thf(limit_ordinals_decl,type,
limit_ordinals: $i ).
thf(integer_of_decl,type,
integer_of: $i > $i ).
thf(78,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(313,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(314,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[313]) ).
thf(28,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(212,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(213,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[212]) ).
thf(1020,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)],[314,213]) ).
thf(1021,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1020:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1081,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1021]) ).
thf(1500,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[314,1081]) ).
thf(1501,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1500:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(162,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(509,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(510,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[509]) ).
thf(511,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[510]) ).
thf(3,negated_conjecture,
y @ ( x @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_no_ordinal_between_3) ).
thf(165,plain,
y @ ( x @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(143,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(462,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(463,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[462]) ).
thf(541,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[165,463]) ).
thf(542,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[541]) ).
thf(543,plain,
~ ( y @ complement @ ( x @ member ) ),
inference(pattern_uni,[status(thm)],[542:[bind(A,$thf( x )),bind(B,$thf( y ))]]) ).
thf(587,plain,
( ( ( y @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,543]) ).
thf(588,plain,
( ( y @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[587]) ).
thf(598,plain,
( ( ( y @ complement )
!= y )
| ( x != x ) ),
inference(simp,[status(thm)],[588]) ).
thf(601,plain,
( ( y @ complement )
!= y ),
inference(simp,[status(thm)],[598]) ).
thf(1777,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,601]) ).
thf(1778,plain,
( ( null_class != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1777:[bind(A,$thf( y @ complement ))]]) ).
thf(11352,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1778]) ).
thf(11353,plain,
! [A: $i] :
( ( A != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[11352:[]]) ).
thf(11504,plain,
( ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[11353]) ).
thf(87,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(333,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(134,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',integer_function2) ).
thf(443,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(444,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[443]) ).
thf(445,plain,
! [A: $i] :
( ( omega @ ( A @ member ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(lifteq,[status(thm)],[444]) ).
thf(539,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,463]) ).
thf(540,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[539]) ).
thf(545,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= y )
| ( x != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[540]) ).
thf(546,plain,
! [A: $i] :
( ( ( A @ complement )
!= y )
| ~ ( A @ ( x @ member ) ) ),
inference(simp,[status(thm)],[545]) ).
thf(5776,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= y )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,546]) ).
thf(5777,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= y )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[5776]) ).
thf(5778,plain,
( ( ( omega @ complement )
!= y )
| ( ( x @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[5777:[bind(A,$thf( x )),bind(B,$thf( omega ))]]) ).
thf(141,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(459,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(1,negated_conjecture,
ordinal_numbers @ ( x @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_no_ordinal_between_1) ).
thf(163,plain,
ordinal_numbers @ ( x @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(519,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[163,463]) ).
thf(520,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[519]) ).
thf(521,plain,
~ ( ordinal_numbers @ complement @ ( x @ member ) ),
inference(pattern_uni,[status(thm)],[520:[bind(A,$thf( x )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(574,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,521]) ).
thf(575,plain,
( ( universal_class @ ( omega @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[574]) ).
thf(582,plain,
( ( ( ordinal_numbers @ complement )
!= universal_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[575]) ).
thf(31,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(217,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)],[31]) ).
thf(1864,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[511,521]) ).
thf(1865,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1864]) ).
thf(2053,plain,
! [A: $i] :
( ( A
!= ( ordinal_numbers @ complement ) )
| ( ( A @ regular )
!= x )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1865]) ).
thf(2062,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= x )
| ( ( ordinal_numbers @ complement )
= null_class ) ),
inference(simp,[status(thm)],[2053]) ).
thf(76,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(308,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)],[76]) ).
thf(871,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)],[213,213]) ).
thf(872,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[871:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(915,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[872]) ).
thf(1358,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[915,213]) ).
thf(1359,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1358:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1438,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1359]) ).
thf(2966,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1438,915]) ).
thf(2967,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2966:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(3108,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[2967]) ).
thf(29,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(214,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(215,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[214]) ).
thf(114,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(398,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(1169,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( universal_class @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[215,398]) ).
thf(1170,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1169:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(517,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[163,463]) ).
thf(518,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[517]) ).
thf(523,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= ordinal_numbers )
| ( x != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[518]) ).
thf(527,plain,
! [A: $i] :
( ( ( A @ complement )
!= ordinal_numbers )
| ~ ( A @ ( x @ member ) ) ),
inference(simp,[status(thm)],[523]) ).
thf(1173,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,527]) ).
thf(1174,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1173]) ).
thf(1195,plain,
! [B: $i,A: $i] :
( ( universal_class != B )
| ( ( A @ singleton )
!= x )
| ( ( B @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1174]) ).
thf(1205,plain,
! [A: $i] :
( ( ( A @ singleton )
!= x )
| ( ( universal_class @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1195]) ).
thf(1590,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,333]) ).
thf(1591,plain,
( ( null_class @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1590:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(107,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(379,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)],[107]) ).
thf(57,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(269,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)],[57]) ).
thf(24,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(202,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(962,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,543]) ).
thf(963,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[962]) ).
thf(970,plain,
! [A: $i] :
( ( ( y @ complement )
!= omega )
| ( A != x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[963]) ).
thf(977,plain,
( ( ( y @ complement )
!= omega )
| ( ( x @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[970]) ).
thf(6,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(169,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(170,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[169]) ).
thf(109,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(385,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)],[109]) ).
thf(570,plain,
( ( ( ordinal_numbers @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,521]) ).
thf(571,plain,
( ( ordinal_numbers @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[570]) ).
thf(581,plain,
( ( ( ordinal_numbers @ complement )
!= y )
| ( x != x ) ),
inference(simp,[status(thm)],[571]) ).
thf(584,plain,
( ( ordinal_numbers @ complement )
!= y ),
inference(simp,[status(thm)],[581]) ).
thf(1780,plain,
! [A: $i] :
( ( A
!= ( ordinal_numbers @ complement ) )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,584]) ).
thf(1781,plain,
( ( null_class != y )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1780:[bind(A,$thf( ordinal_numbers @ complement ))]]) ).
thf(12850,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1781,521]) ).
thf(12851,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12850]) ).
thf(12887,plain,
( ( ( ordinal_numbers @ complement )
!= ( ordinal_numbers @ complement ) )
| ( ( ordinal_numbers @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12851]) ).
thf(12939,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12887]) ).
thf(158,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(500,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(2,negated_conjecture,
ordinal_numbers @ ( y @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_no_ordinal_between_2) ).
thf(164,plain,
ordinal_numbers @ ( y @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(514,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[164,463]) ).
thf(515,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[514]) ).
thf(516,plain,
~ ( ordinal_numbers @ complement @ ( y @ member ) ),
inference(pattern_uni,[status(thm)],[515:[bind(A,$thf( y )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(530,plain,
( ( ( ordinal_numbers @ complement @ ( y @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,516]) ).
thf(531,plain,
( ( ordinal_numbers @ complement @ ( y @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) ),
inference(simp,[status(thm)],[530]) ).
thf(534,plain,
( ( ( ordinal_numbers @ complement )
!= ordinal_numbers )
| ( y != y ) ),
inference(simp,[status(thm)],[531]) ).
thf(536,plain,
( ( ordinal_numbers @ complement )
!= ordinal_numbers ),
inference(simp,[status(thm)],[534]) ).
thf(1616,plain,
! [A: $i] :
( ( A
!= ( ordinal_numbers @ complement ) )
| ( null_class != ordinal_numbers )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,536]) ).
thf(1617,plain,
( ( null_class != ordinal_numbers )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1616:[bind(A,$thf( ordinal_numbers @ complement ))]]) ).
thf(7928,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != ordinal_numbers )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1617]) ).
thf(7929,plain,
! [A: $i] :
( ( A != ordinal_numbers )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7928:[]]) ).
thf(8057,plain,
( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( ordinal_numbers @ ( ordinal_numbers @ regular @ member ) ) ),
inference(simp,[status(thm)],[7929]) ).
thf(4,negated_conjecture,
x @ successor @ ( y @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_no_ordinal_between_4) ).
thf(166,plain,
x @ successor @ ( y @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(815,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)],[398,463]) ).
thf(816,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)],[815]) ).
thf(817,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[816:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(828,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[817]) ).
thf(1171,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ~ ( universal_class @ complement @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[215,828]) ).
thf(1172,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1171:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(1217,plain,
! [A: $i] :
( ( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( x @ successor @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,1172]) ).
thf(1218,plain,
! [A: $i] :
( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( x @ successor @ ( y @ member ) ) ),
inference(simp,[status(thm)],[1217]) ).
thf(1233,plain,
! [A: $i] :
( ( ( universal_class @ complement )
!= ( x @ successor ) )
| ( ( A @ singleton )
!= y ) ),
inference(simp,[status(thm)],[1218]) ).
thf(91,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(340,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)],[91]) ).
thf(72,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(301,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(1639,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,301]) ).
thf(1640,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1639:[bind(A,$thf( successor_relation ))]]) ).
thf(27,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(209,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(210,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[209]) ).
thf(211,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[210]) ).
thf(9116,plain,
! [B: $i,A: $i] :
( ( A
!= ( y @ complement ) )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[211,601]) ).
thf(9117,plain,
! [A: $i] :
( ( A != y )
| ~ ( y @ complement @ ( A @ subclass ) )
| ~ ( A @ ( y @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[9116:[bind(A,$thf( y @ complement )),bind(B,$thf( B ))]]) ).
thf(10594,plain,
( ~ ( y @ complement @ ( y @ subclass ) )
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[9117]) ).
thf(8,axiom,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetrization) ).
thf(173,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(174,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(lifteq,[status(thm)],[173]) ).
thf(15,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(185,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)],[15]) ).
thf(122,axiom,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',limit_ordinals) ).
thf(416,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(417,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(lifteq,[status(thm)],[416]) ).
thf(51,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_of_range_map1) ).
thf(257,plain,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(1649,plain,
! [A: $i] :
( ( A != union_of_range_map )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,257]) ).
thf(1650,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( union_of_range_map @ ( union_of_range_map @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1649:[bind(A,$thf( union_of_range_map ))]]) ).
thf(562,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[459,463]) ).
thf(563,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[562]) ).
thf(564,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[563:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(614,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,564]) ).
thf(615,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[614]) ).
thf(618,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[615]) ).
thf(623,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[618]) ).
thf(8684,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[211,623]) ).
thf(8685,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[8684:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(10410,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[8685]) ).
thf(71,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(300,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(18170,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[10410,300]) ).
thf(18171,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[18170]) ).
thf(81,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(320,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(558,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,516]) ).
thf(559,plain,
( ( universal_class @ ( omega @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[558]) ).
thf(565,plain,
( ( ( ordinal_numbers @ complement )
!= universal_class )
| ( omega != y ) ),
inference(simp,[status(thm)],[559]) ).
thf(1365,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[213,915]) ).
thf(1366,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1365:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(110,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(389,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(12856,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1781,516]) ).
thf(12857,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12856]) ).
thf(12909,plain,
( ( ( ordinal_numbers @ complement )
!= ( ordinal_numbers @ complement ) )
| ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12857]) ).
thf(12943,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12909]) ).
thf(14,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(183,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(184,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[183]) ).
thf(160,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(504,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)],[160]) ).
thf(610,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,564]) ).
thf(611,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[610]) ).
thf(619,plain,
( ( ( universal_class @ complement )
!= y )
| ( omega != x ) ),
inference(simp,[status(thm)],[611]) ).
thf(98,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(357,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)],[98]) ).
thf(142,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(460,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(189,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(83,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(324,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(325,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[324]) ).
thf(37,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(228,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)],[37]) ).
thf(136,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(448,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)],[136]) ).
thf(79,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(315,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(316,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[315]) ).
thf(1211,plain,
! [A: $i] :
( ( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,1172]) ).
thf(1212,plain,
! [A: $i] :
( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[1211]) ).
thf(1232,plain,
! [A: $i] :
( ( ( universal_class @ complement )
!= y )
| ( ( A @ singleton )
!= x ) ),
inference(simp,[status(thm)],[1212]) ).
thf(11,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(179,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(321,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[320]) ).
thf(105,axiom,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',kind_1_ordinals) ).
thf(375,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(376,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(lifteq,[status(thm)],[375]) ).
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(485,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(41,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(238,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(94,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(348,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)],[94]) ).
thf(96,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(353,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)],[96]) ).
thf(54,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(263,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(18188,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[184,18171]) ).
thf(18189,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[18188]) ).
thf(18247,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[18189]) ).
thf(18280,plain,
( ( omega != universal_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[18247]) ).
thf(18319,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[263,18280]) ).
thf(18320,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[18319]) ).
thf(18324,plain,
( ( ( universal_class @ complement )
!= omega )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[18320]) ).
thf(150,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',integer_function1) ).
thf(480,plain,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(7977,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true
| ( null_class != ordinal_numbers ) ),
inference(paramod_ordered,[status(thm)],[1617,516]) ).
thf(7978,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[7977]) ).
thf(8005,plain,
( ( ( ordinal_numbers @ complement )
!= ( ordinal_numbers @ complement ) )
| ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[7978]) ).
thf(8051,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[8005]) ).
thf(8066,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( A != ordinal_numbers )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,8051]) ).
thf(8067,plain,
! [A: $i] :
( ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( A != ordinal_numbers )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8066:[]]) ).
thf(8109,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( ordinal_numbers @ ( ordinal_numbers @ regular @ member ) ) ),
inference(simp,[status(thm)],[8067]) ).
thf(30,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_of1) ).
thf(216,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(74,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(304,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(305,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(lifteq,[status(thm)],[304]) ).
thf(19164,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ union ) )
!= ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) ) )
| ( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[305,376]) ).
thf(19165,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[19164:[bind(A,$thf( null_class @ singleton )),bind(B,$thf( ordinal_numbers @ ( successor_relation @ image ) ))]]) ).
thf(34,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(221,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)],[34]) ).
thf(113,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(396,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)],[113]) ).
thf(61,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(277,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)],[61]) ).
thf(278,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
inference(lifteq,[status(thm)],[277]) ).
thf(2286,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ singleton @ ( add_relation @ image ) @ sum_class @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
inference(rewrite,[status(thm)],[278,321]) ).
thf(593,plain,
( ( ( y @ complement @ ( x @ member ) )
!= ( ordinal_numbers @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[163,543]) ).
thf(594,plain,
( ( y @ complement @ ( x @ member ) )
!= ( ordinal_numbers @ ( x @ member ) ) ),
inference(simp,[status(thm)],[593]) ).
thf(597,plain,
( ( ( y @ complement )
!= ordinal_numbers )
| ( x != x ) ),
inference(simp,[status(thm)],[594]) ).
thf(600,plain,
( ( y @ complement )
!= ordinal_numbers ),
inference(simp,[status(thm)],[597]) ).
thf(1958,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( null_class != ordinal_numbers )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,600]) ).
thf(1959,plain,
( ( null_class != ordinal_numbers )
| ( y @ complement @ ( y @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1958:[bind(A,$thf( y @ complement ))]]) ).
thf(15554,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class != ordinal_numbers ) ),
inference(paramod_ordered,[status(thm)],[1959,543]) ).
thf(15555,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[15554]) ).
thf(15581,plain,
( ( ( y @ complement )
!= ( y @ complement ) )
| ( ( y @ complement @ regular )
!= x )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[15555]) ).
thf(15618,plain,
( ( ( y @ complement @ regular )
!= x )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[15581]) ).
thf(2038,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[511,543]) ).
thf(2039,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[2038]) ).
thf(2049,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( ( A @ regular )
!= x )
| ( A = null_class ) ),
inference(simp,[status(thm)],[2039]) ).
thf(2058,plain,
( ( ( y @ complement @ regular )
!= x )
| ( ( y @ complement )
= null_class ) ),
inference(simp,[status(thm)],[2049]) ).
thf(127,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(428,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)],[127]) ).
thf(1179,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,516]) ).
thf(1180,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[1179]) ).
thf(1201,plain,
! [A: $i] :
( ( ( ordinal_numbers @ complement )
!= universal_class )
| ( ( A @ singleton )
!= y ) ),
inference(simp,[status(thm)],[1180]) ).
thf(47,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(250,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(251,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[250]) ).
thf(229,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[228]) ).
thf(17298,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[251,229]) ).
thf(67,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(292,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(46,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(248,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(132,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(439,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(512,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( y @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,463]) ).
thf(513,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ ( y @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[512]) ).
thf(526,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= ordinal_numbers )
| ( y != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[513]) ).
thf(529,plain,
! [A: $i] :
( ( ( A @ complement )
!= ordinal_numbers )
| ~ ( A @ ( y @ member ) ) ),
inference(simp,[status(thm)],[526]) ).
thf(23,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(200,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(201,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[200]) ).
thf(1192,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,543]) ).
thf(1193,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[1192]) ).
thf(1199,plain,
! [A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( ( A @ singleton )
!= x ) ),
inference(simp,[status(thm)],[1193]) ).
thf(108,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(381,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)],[108]) ).
thf(33,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(220,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(1177,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( B @ ( y @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,529]) ).
thf(1178,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( B @ ( y @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1177]) ).
thf(1196,plain,
! [B: $i,A: $i] :
( ( universal_class != B )
| ( ( A @ singleton )
!= y )
| ( ( B @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1178]) ).
thf(1206,plain,
! [A: $i] :
( ( ( A @ singleton )
!= y )
| ( ( universal_class @ complement )
!= ordinal_numbers ) ),
inference(simp,[status(thm)],[1196]) ).
thf(39,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(233,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)],[39]) ).
thf(138,axiom,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions1) ).
thf(452,plain,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(88,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(334,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(73,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(302,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)],[73]) ).
thf(53,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(260,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)],[53]) ).
thf(69,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(296,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(297,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[296]) ).
thf(60,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(275,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)],[60]) ).
thf(18,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps1) ).
thf(191,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(192,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[191]) ).
thf(119,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(407,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)],[119]) ).
thf(145,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(467,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(48,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(252,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(21,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps2) ).
thf(195,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(196,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[195]) ).
thf(197,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ maps ) ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[196]) ).
thf(100,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(361,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)],[100]) ).
thf(130,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(435,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(140,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(456,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)],[140]) ).
thf(35,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(224,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)],[35]) ).
thf(949,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,516]) ).
thf(950,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[949]) ).
thf(967,plain,
! [A: $i] :
( ( ( ordinal_numbers @ complement )
!= omega )
| ( A != y )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[950]) ).
thf(974,plain,
( ( ( ordinal_numbers @ complement )
!= omega )
| ( ( y @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[967]) ).
thf(12864,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1781,463]) ).
thf(12865,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[12864]) ).
thf(12866,plain,
( ~ ( ordinal_numbers @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[12865:[bind(A,$thf( ordinal_numbers @ complement @ regular )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(80,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(317,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(1387,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[314,915]) ).
thf(1388,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1387:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1445,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1388]) ).
thf(4501,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1445,915]) ).
thf(4502,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[4501:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(4668,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[4502]) ).
thf(15445,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != ordinal_numbers )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1959]) ).
thf(15446,plain,
! [A: $i] :
( ( A != ordinal_numbers )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[15445:[]]) ).
thf(15616,plain,
( ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( ordinal_numbers @ ( ordinal_numbers @ regular @ member ) ) ),
inference(simp,[status(thm)],[15446]) ).
thf(117,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(403,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)],[117]) ).
thf(11433,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1778,463]) ).
thf(11434,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[11433]) ).
thf(11435,plain,
( ~ ( y @ ( y @ complement @ regular @ member ) )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[11434:[bind(A,$thf( y @ complement @ regular )),bind(B,$thf( y ))]]) ).
thf(124,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(420,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)],[124]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(328,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(225,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[224]) ).
thf(19,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(193,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(148,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(476,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)],[148]) ).
thf(551,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[166,463]) ).
thf(552,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[551]) ).
thf(553,plain,
~ ( x @ successor @ complement @ ( y @ member ) ),
inference(pattern_uni,[status(thm)],[552:[bind(A,$thf( y )),bind(B,$thf( x @ successor ))]]) ).
thf(630,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,553]) ).
thf(631,plain,
( ( universal_class @ ( omega @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[630]) ).
thf(637,plain,
( ( ( x @ successor @ complement )
!= universal_class )
| ( omega != y ) ),
inference(simp,[status(thm)],[631]) ).
thf(12779,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,1781]) ).
thf(12780,plain,
! [A: $i] :
( ( A != y )
| ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[12779:[]]) ).
thf(12942,plain,
( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[12780]) ).
thf(2977,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[915,1438]) ).
thf(2978,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2977:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(612,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( x @ successor @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,564]) ).
thf(613,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( x @ successor @ ( y @ member ) ) ),
inference(simp,[status(thm)],[612]) ).
thf(622,plain,
( ( ( universal_class @ complement )
!= ( x @ successor ) )
| ( omega != y ) ),
inference(simp,[status(thm)],[613]) ).
thf(112,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(393,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)],[112]) ).
thf(159,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(502,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(503,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[502]) ).
thf(92,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(343,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)],[92]) ).
thf(44,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',section1) ).
thf(244,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(56,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(266,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)],[56]) ).
thf(26,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(205,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(77,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(311,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)],[77]) ).
thf(70,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(298,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(126,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(425,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)],[126]) ).
thf(128,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(430,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)],[128]) ).
thf(55,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(264,plain,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(62,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(279,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)],[62]) ).
thf(12,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(180,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(951,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,553]) ).
thf(952,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[951]) ).
thf(968,plain,
! [A: $i] :
( ( ( x @ successor @ complement )
!= omega )
| ( A != y )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[952]) ).
thf(975,plain,
( ( ( x @ successor @ complement )
!= omega )
| ( ( y @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[968]) ).
thf(9,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(175,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(176,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[175]) ).
thf(121,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(413,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)],[121]) ).
thf(63,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(281,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)],[63]) ).
thf(25,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(203,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)],[25]) ).
thf(204,plain,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
inference(lifteq,[status(thm)],[203]) ).
thf(111,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(391,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)],[111]) ).
thf(18781,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( y @ complement @ subclass ) ) )
| ~ ( y @ complement @ ( y @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[300,10594]) ).
thf(18782,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( y @ complement @ subclass ) ) )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[18781]) ).
thf(18811,plain,
! [A: $i] :
( ( universal_class != y )
| ( A
!= ( y @ complement ) )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[18782]) ).
thf(18880,plain,
( ( universal_class != y )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[18811]) ).
thf(1785,plain,
! [A: $i] :
( ( A
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,176]) ).
thf(1786,plain,
( ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( subset_relation @ inverse @ ( subset_relation @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1785:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(157,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(498,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(1634,plain,
! [A: $i] :
( ( A
!= ( kind_1_ordinals @ complement ) )
| ( ( ordinal_numbers @ ( null_class @ intersection ) )
= limit_ordinals )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,417]) ).
thf(1635,plain,
( ( ( ordinal_numbers @ ( null_class @ intersection ) )
= limit_ordinals )
| ( kind_1_ordinals @ complement @ ( kind_1_ordinals @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1634:[bind(A,$thf( kind_1_ordinals @ complement ))]]) ).
thf(89,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(335,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(40,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(236,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)],[40]) ).
thf(13,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(181,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)],[13]) ).
thf(16,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(187,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)],[16]) ).
thf(7,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(171,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(172,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[171]) ).
thf(32,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_relation1) ).
thf(219,plain,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(38,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(230,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)],[38]) ).
thf(560,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,463]) ).
thf(561,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[560]) ).
thf(566,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[561]) ).
thf(567,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[566]) ).
thf(303,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)],[302]) ).
thf(106,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(377,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)],[106]) ).
thf(1588,plain,
! [A: $i] :
( ( A != domain_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,333]) ).
thf(1589,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( domain_relation @ ( domain_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1588:[bind(A,$thf( domain_relation ))]]) ).
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(271,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(206,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[205]) ).
thf(207,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[206]) ).
thf(208,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[207]) ).
thf(591,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,543]) ).
thf(592,plain,
( ( universal_class @ ( omega @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[591]) ).
thf(599,plain,
( ( ( y @ complement )
!= universal_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[592]) ).
thf(1360,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[915,213]) ).
thf(1361,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1360:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1439,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1361]) ).
thf(155,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(494,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(616,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( ordinal_numbers @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[163,564]) ).
thf(617,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( ordinal_numbers @ ( x @ member ) ) ),
inference(simp,[status(thm)],[616]) ).
thf(621,plain,
( ( ( universal_class @ complement )
!= ordinal_numbers )
| ( omega != x ) ),
inference(simp,[status(thm)],[617]) ).
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(483,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(42,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(240,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)],[42]) ).
thf(19746,plain,
! [A: $i] :
( ( ( ordinal_numbers @ ( successor_relation @ image ) @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
!= ( A @ complement ) )
| ( ( kind_1_ordinals @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[19165,213]) ).
thf(19747,plain,
( ( kind_1_ordinals @ ( element_relation @ image ) @ complement )
= ( ordinal_numbers @ ( successor_relation @ image ) @ complement @ ( null_class @ singleton @ complement @ intersection ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[19746:[bind(A,$thf( ordinal_numbers @ ( successor_relation @ image ) @ complement @ ( null_class @ singleton @ complement @ intersection ) ))]]) ).
thf(22,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(198,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(86,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(330,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)],[86]) ).
thf(93,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(345,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)],[93]) ).
thf(102,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(367,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)],[102]) ).
thf(1175,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,521]) ).
thf(1176,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[1175]) ).
thf(1198,plain,
! [A: $i] :
( ( ( ordinal_numbers @ complement )
!= universal_class )
| ( ( A @ singleton )
!= x ) ),
inference(simp,[status(thm)],[1176]) ).
thf(115,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(399,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)],[115]) ).
thf(75,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(306,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)],[75]) ).
thf(36,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(226,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)],[36]) ).
thf(11449,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1778,543]) ).
thf(11450,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[11449]) ).
thf(11490,plain,
( ( ( y @ complement )
!= ( y @ complement ) )
| ( ( y @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[11450]) ).
thf(11507,plain,
( ( ( y @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[11490]) ).
thf(5,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(167,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(154,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(490,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)],[154]) ).
thf(50,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(255,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(97,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(355,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)],[97]) ).
thf(20148,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ( universal_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[300,18880]) ).
thf(20149,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[20148]) ).
thf(20170,plain,
! [A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( A != y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[20149]) ).
thf(20196,plain,
( ( ( y @ complement )
!= universal_class )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[20170]) ).
thf(20,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(194,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(153,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(487,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)],[153]) ).
thf(1769,plain,
! [A: $i] :
( ( A
!= ( x @ successor ) )
| ( null_class @ ( y @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,166]) ).
thf(1770,plain,
( ( null_class @ ( y @ member ) )
| ( x @ successor @ ( x @ successor @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1769:[bind(A,$thf( x @ successor ))]]) ).
thf(626,plain,
( ( ( x @ successor @ complement @ ( y @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,553]) ).
thf(627,plain,
( ( x @ successor @ complement @ ( y @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) ),
inference(simp,[status(thm)],[626]) ).
thf(638,plain,
( ( ( x @ successor @ complement )
!= ordinal_numbers )
| ( y != y ) ),
inference(simp,[status(thm)],[627]) ).
thf(640,plain,
( ( x @ successor @ complement )
!= ordinal_numbers ),
inference(simp,[status(thm)],[638]) ).
thf(947,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,521]) ).
thf(948,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[947]) ).
thf(964,plain,
! [A: $i] :
( ( ( ordinal_numbers @ complement )
!= omega )
| ( A != x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[948]) ).
thf(971,plain,
( ( ( ordinal_numbers @ complement )
!= omega )
| ( ( x @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[964]) ).
thf(1471,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1081,213]) ).
thf(1472,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1471:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1566,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1472]) ).
thf(4516,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1081,1445]) ).
thf(4517,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[4516:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1636,plain,
! [A: $i] :
( ( A != kind_1_ordinals )
| ( ( ordinal_numbers @ ( null_class @ complement @ intersection ) )
= limit_ordinals )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,417]) ).
thf(1637,plain,
( ( ( ordinal_numbers @ ( null_class @ complement @ intersection ) )
= limit_ordinals )
| ( kind_1_ordinals @ ( kind_1_ordinals @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1636:[bind(A,$thf( kind_1_ordinals ))]]) ).
thf(10,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(177,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(133,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(441,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)],[133]) ).
thf(103,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(370,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(146,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(470,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)],[146]) ).
thf(99,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(359,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(1903,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[511,516]) ).
thf(1904,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( y @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1903]) ).
thf(2048,plain,
! [A: $i] :
( ( A
!= ( ordinal_numbers @ complement ) )
| ( ( A @ regular )
!= y )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1904]) ).
thf(2056,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= y )
| ( ( ordinal_numbers @ complement )
= null_class ) ),
inference(simp,[status(thm)],[2048]) ).
thf(336,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[335]) ).
thf(135,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(446,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)],[135]) ).
thf(589,plain,
( ( ( x @ successor @ ( y @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,543]) ).
thf(590,plain,
( ( x @ successor @ ( y @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[589]) ).
thf(595,plain,
( ( ( y @ complement )
!= ( x @ successor ) )
| ( y != x ) ),
inference(simp,[status(thm)],[590]) ).
thf(137,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(450,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)],[137]) ).
thf(547,plain,
( ( ( ordinal_numbers @ complement @ ( y @ member ) )
!= ( x @ successor @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,516]) ).
thf(548,plain,
( ( ordinal_numbers @ complement @ ( y @ member ) )
!= ( x @ successor @ ( y @ member ) ) ),
inference(simp,[status(thm)],[547]) ).
thf(554,plain,
( ( ( ordinal_numbers @ complement )
!= ( x @ successor ) )
| ( y != y ) ),
inference(simp,[status(thm)],[548]) ).
thf(556,plain,
( ( ordinal_numbers @ complement )
!= ( x @ successor ) ),
inference(simp,[status(thm)],[554]) ).
thf(218,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( B @ ( B @ ( A @ not_well_ordering ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
inference(cnf,[status(esa)],[217]) ).
thf(329,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[328]) ).
thf(2290,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[329,201]) ).
thf(2302,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( ordinal_numbers @ ( successor_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[2290,376]) ).
thf(2303,plain,
( ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[2302:[bind(A,$thf( successor_relation )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(17301,plain,
( ( successor_relation @ ( universal_class @ ( ordinal_numbers @ cross_product ) @ intersection ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(rewrite,[status(thm)],[2303,229]) ).
thf(95,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(351,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)],[95]) ).
thf(116,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(401,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)],[116]) ).
thf(123,axiom,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers2) ).
thf(418,plain,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(944,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,527]) ).
thf(945,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( x @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[944]) ).
thf(946,plain,
( ( ( omega @ complement )
!= ordinal_numbers )
| ( ( x @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[945:[bind(A,$thf( x )),bind(B,$thf( omega ))]]) ).
thf(52,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(258,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(66,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(290,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(280,plain,
! [B: $i,A: $i] :
( ( B @ ( union_of_range_map @ ( successor_relation @ ( A @ recursion ) ) @ apply ) )
= ( B @ ( A @ ordinal_add ) ) ),
inference(lifteq,[status(thm)],[279]) ).
thf(19014,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( union_of_range_map @ ( successor_relation @ ( A @ recursion ) ) @ image ) @ sum_class )
= ( B @ ( A @ ordinal_add ) ) ),
inference(rewrite,[status(thm)],[280,321]) ).
thf(149,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(478,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(234,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( A @ least ) @ ( C @ ( A @ segment ) ) )
= null_class )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(cnf,[status(esa)],[233]) ).
thf(235,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) )
| ( ( C @ ( A @ least ) @ ( C @ ( A @ segment ) ) )
= null_class ) ),
inference(lifteq,[status(thm)],[234]) ).
thf(45,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(246,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(65,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(287,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)],[65]) ).
thf(608,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,564]) ).
thf(609,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( ordinal_numbers @ ( y @ member ) ) ),
inference(simp,[status(thm)],[608]) ).
thf(620,plain,
( ( ( universal_class @ complement )
!= ordinal_numbers )
| ( omega != y ) ),
inference(simp,[status(thm)],[609]) ).
thf(129,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(432,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(168,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[167]) ).
thf(118,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(405,plain,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(190,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[189]) ).
thf(125,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(422,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)],[125]) ).
thf(59,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(273,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)],[59]) ).
thf(104,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(373,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(120,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(410,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)],[120]) ).
thf(2304,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( C @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2290,213]) ).
thf(2305,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2304:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2326,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2305]) ).
thf(17297,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(rewrite,[status(thm)],[2326,229]) ).
thf(84,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering1) ).
thf(326,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(68,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(294,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)],[68]) ).
thf(15538,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != ordinal_numbers ) ),
inference(paramod_ordered,[status(thm)],[1959,463]) ).
thf(15539,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[15538]) ).
thf(15540,plain,
( ~ ( y @ ( y @ complement @ regular @ member ) )
| ( null_class != ordinal_numbers ) ),
inference(pattern_uni,[status(thm)],[15539:[bind(A,$thf( y @ complement @ regular )),bind(B,$thf( y ))]]) ).
thf(161,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(506,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)],[161]) ).
thf(131,axiom,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers1) ).
thf(437,plain,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(1181,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1170,553]) ).
thf(1182,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( x @ successor @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[1181]) ).
thf(1197,plain,
! [A: $i] :
( ( ( x @ successor @ complement )
!= universal_class )
| ( ( A @ singleton )
!= y ) ),
inference(simp,[status(thm)],[1182]) ).
thf(624,plain,
( ( ( x @ successor @ complement @ ( y @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,553]) ).
thf(625,plain,
( ( x @ successor @ complement @ ( y @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[624]) ).
thf(635,plain,
( ( ( x @ successor @ complement )
!= y )
| ( y != x ) ),
inference(simp,[status(thm)],[625]) ).
thf(64,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(283,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(239,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[238]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(242,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(147,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(474,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(144,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(464,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)],[144]) ).
thf(7985,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != ordinal_numbers ) ),
inference(paramod_ordered,[status(thm)],[1617,463]) ).
thf(7986,plain,
! [B: $i,A: $i] :
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[7985]) ).
thf(7987,plain,
( ~ ( ordinal_numbers @ ( ordinal_numbers @ complement @ regular @ member ) )
| ( null_class != ordinal_numbers ) ),
inference(pattern_uni,[status(thm)],[7986:[bind(A,$thf( ordinal_numbers @ complement @ regular )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(178,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[177]) ).
thf(82,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(322,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(90,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(337,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)],[90]) ).
thf(7971,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class != ordinal_numbers ) ),
inference(paramod_ordered,[status(thm)],[1617,521]) ).
thf(7972,plain,
( ( ( ordinal_numbers @ complement @ ( ordinal_numbers @ complement @ regular @ member ) )
!= ( ordinal_numbers @ complement @ ( x @ member ) ) )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[7971]) ).
thf(8029,plain,
( ( ( ordinal_numbers @ complement )
!= ( ordinal_numbers @ complement ) )
| ( ( ordinal_numbers @ complement @ regular )
!= x )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[7972]) ).
thf(8055,plain,
( ( ( ordinal_numbers @ complement @ regular )
!= x )
| ( null_class != ordinal_numbers ) ),
inference(simp,[status(thm)],[8029]) ).
thf(628,plain,
( ( ( x @ successor @ complement @ ( y @ member ) )
!= ( x @ successor @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,553]) ).
thf(629,plain,
( ( x @ successor @ complement @ ( y @ member ) )
!= ( x @ successor @ ( y @ member ) ) ),
inference(simp,[status(thm)],[628]) ).
thf(634,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( y != y ) ),
inference(simp,[status(thm)],[629]) ).
thf(639,plain,
( ( x @ successor @ complement )
!= ( x @ successor ) ),
inference(simp,[status(thm)],[634]) ).
thf(1086,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( y @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[445,529]) ).
thf(1087,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( y @ member ) ) )
| ( ( B @ complement )
!= ordinal_numbers )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1086]) ).
thf(1088,plain,
( ( ( omega @ complement )
!= ordinal_numbers )
| ( ( y @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[1087:[bind(A,$thf( y )),bind(B,$thf( omega ))]]) ).
thf(101,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(364,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(231,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[230]) ).
thf(232,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[231]) ).
thf(49,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(254,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(227,plain,
! [B: $i,A: $i] :
( ( A @ symmetrization_of @ ( identity_relation @ union ) @ ( B @ ( B @ cross_product ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
inference(cnf,[status(esa)],[226]) ).
thf(156,axiom,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions2) ).
thf(496,plain,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(139,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(454,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(35880,plain,
$false,
inference(e,[status(thm)],[1501,11504,333,5778,582,217,2062,308,3108,1205,1591,379,511,269,202,977,170,385,12939,500,8057,1233,340,1640,10594,174,185,417,1650,301,18171,257,320,565,1366,389,12943,184,504,619,357,460,189,416,325,228,448,316,1778,1232,164,179,443,321,376,623,211,485,238,348,353,18324,480,8109,216,19165,221,396,2286,15618,2058,428,1201,527,17298,292,248,439,529,1438,546,201,1199,381,220,1206,233,452,334,302,260,297,275,165,192,407,467,229,252,197,361,435,456,224,324,974,12866,317,4668,15616,403,11435,169,420,328,225,193,476,637,12942,2978,212,622,393,173,503,343,244,266,205,311,1170,398,298,425,430,1959,166,18280,264,279,375,601,180,296,584,975,176,191,413,281,204,391,18880,1786,445,313,498,1635,335,236,181,187,172,219,230,567,509,303,377,1589,271,208,599,1439,494,213,621,483,240,915,19747,198,330,462,345,367,521,1198,399,1445,306,226,11507,167,490,255,1081,516,355,553,20196,194,209,487,1770,640,971,1566,4517,263,1637,177,441,370,502,463,470,359,2056,336,600,175,314,446,595,1617,250,450,556,203,1781,218,17301,304,351,401,171,418,946,258,290,19014,478,536,235,246,214,287,300,620,315,432,168,405,190,183,422,273,373,410,17297,564,305,326,294,15540,506,195,437,1197,828,635,283,239,242,1172,474,8051,464,7987,459,178,277,163,200,322,337,543,215,8055,639,1088,364,232,254,227,496,454]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : NUM155-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.07 % 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
% 0.12/0.37 % Computer : n015.cluster.edu
% 0.12/0.37 % Model : x86_64 x86_64
% 0.12/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.37 % Memory : 8046.5625MB
% 0.12/0.37 % OS : Linux 6.8.0-71-generic
% 0.12/0.37 % CPULimit : 300
% 0.12/0.37 % WCLimit : 300
% 0.12/0.37 % DateTime : Sat Sep 26 02:11:12 UTC 2026
% 0.12/0.37 % CPUTime :
% 0.12/0.37 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
% 0.94/0.94 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.62/1.23 % [INFO] Parsing done (288ms).
% 1.62/1.25 % [INFO] Running in sequential loop mode.
% 2.35/1.63 % [INFO] eprover registered as external prover.
% 2.35/1.64 % [INFO] Scanning for conjecture ...
% 2.54/1.76 % [INFO] Found a conjecture (or negated_conjecture) and 158 axioms. Running axiom selection ...
% 2.71/1.89 % [INFO] Axiom selection finished. Selected 158 axioms (removed 0 axioms).
% 3.40/2.05 % [INFO] Problem is propositional (TPTP CNF).
% 3.40/2.06 % [INFO] Type checking passed.
% 3.40/2.06 % [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 ...
% 122.47/31.65 % External prover 'e' found a proof!
% 122.47/31.65 % [INFO] Killing All external provers ...
% 122.47/31.65 % Time passed: 31141ms (effective reasoning time: 30398ms)
% 122.47/31.66 % 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
% 122.47/31.66 % No. of inferences in proof: 683
% 122.47/31.66 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 31141 ms resp. 30398 ms w/o parsing
% 111.92/31.94 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 111.92/31.95 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------