%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUM030-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:08:08 AM UTC 2026
% Result : Unsatisfiable 103.94s 28.86s
% Output : Refutation 105.36s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 160
% Syntax : Number of formulae : 624 ( 186 unt; 0 typ; 0 def)
% Number of atoms : 1471 ( 565 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 5812 ( 734 ~; 651 |; 0 &;4427 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 21 con; 0-3 aty)
% Number of variables : 919 ( 0 ^; 919 !; 0 ?; 919 :)
% Comments :
%------------------------------------------------------------------------------
thf(symmetrization_of_decl,type,
symmetrization_of: $i > $i ).
thf(x_decl,type,
x: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(recursion_equation_functions_decl,type,
recursion_equation_functions: $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(rest_of_decl,type,
rest_of: $i > $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(ordinal_numbers_decl,type,
ordinal_numbers: $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(operation_decl,type,
operation: $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(union_of_range_map_decl,type,
union_of_range_map: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(null_class_decl,type,
null_class: $i ).
thf(choice_decl,type,
choice: $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(kind_1_ordinals_decl,type,
kind_1_ordinals: $i ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(compose_class_decl,type,
compose_class: $i > $i ).
thf(application_function_decl,type,
application_function: $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(rest_relation_decl,type,
rest_relation: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(limit_ordinals_decl,type,
limit_ordinals: $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(well_ordering_decl,type,
well_ordering: $i > $i > $o ).
thf(recursion_decl,type,
recursion: $i > $i > $i > $i ).
thf(add_relation_decl,type,
add_relation: $i ).
thf(ordinal_multiply_decl,type,
ordinal_multiply: $i > $i > $i ).
thf(ordinal_add_decl,type,
ordinal_add: $i > $i > $i ).
thf(integer_of_decl,type,
integer_of: $i > $i ).
thf(segment_decl,type,
segment: $i > $i > $i > $i ).
thf(not_well_ordering_decl,type,
not_well_ordering: $i > $i > $i ).
thf(connected_decl,type,
connected: $i > $i > $o ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(singleton_relation_decl,type,
singleton_relation: $i ).
thf(least_decl,type,
least: $i > $i > $i ).
thf(section_decl,type,
section: $i > $i > $i > $o ).
thf(transitive_decl,type,
transitive: $i > $i > $o ).
thf(single_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(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(asymmetric_decl,type,
asymmetric: $i > $i > $o ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(irreflexive_decl,type,
irreflexive: $i > $i > $o ).
thf(14,axiom,
! [B: $i,A: $i] :
( ( ( A @ range_of @ sum_class )
= B )
| ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_range_map2) ).
thf(191,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)],[14]) ).
thf(192,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of @ sum_class )
= B )
| ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[191]) ).
thf(193,plain,
! [B: $i,A: $i] :
( ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ range_of @ sum_class )
= B ) ),
inference(lifteq,[status(thm)],[192]) ).
thf(109,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(404,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(405,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[404]) ).
thf(5509,plain,
! [B: $i,A: $i] :
( ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ inverse @ domain_of @ sum_class )
= B ) ),
inference(rewrite,[status(thm)],[193,405]) ).
thf(90,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',well_ordering1) ).
thf(365,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(32,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(234,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(117,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_relation1) ).
thf(422,plain,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(2,negated_conjecture,
~ ( x @ ( x @ inverse @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_symmetrization_property3_2) ).
thf(162,plain,
~ ( x @ ( x @ inverse @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(164,plain,
~ ( x @ ( x @ inverse @ subclass ) ),
inference(polarity_switch,[status(thm)],[162]) ).
thf(553,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[422,164]) ).
thf(554,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[553]) ).
thf(555,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= rest_relation ) ),
inference(simp,[status(thm)],[554]) ).
thf(235,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[234]) ).
thf(71,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(319,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(18,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair2) ).
thf(202,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(203,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[202]) ).
thf(12382,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[319,203]) ).
thf(12383,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[12382]) ).
thf(12384,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[12383:[bind(A,$thf( omega ))]]) ).
thf(12459,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(simp,[status(thm)],[12384]) ).
thf(12549,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) ) )
| ( A @ singleton @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[235,12459]) ).
thf(12550,plain,
omega @ singleton @ ( omega @ member ),
inference(pattern_uni,[status(thm)],[12549:[bind(A,$thf( omega )),bind(B,$thf( omega ))]]) ).
thf(73,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(322,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(323,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[322]) ).
thf(512,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[319,323]) ).
thf(513,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[512]) ).
thf(514,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[513:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(12609,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12550,514]) ).
thf(12610,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[12609]) ).
thf(12677,plain,
( ( ( universal_class @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[12610]) ).
thf(12699,plain,
( ( universal_class @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[12677]) ).
thf(146,axiom,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetrization) ).
thf(479,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(31,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(232,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(233,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[232]) ).
thf(546,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[319,514]) ).
thf(547,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[546]) ).
thf(548,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[547]) ).
thf(549,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[548]) ).
thf(1102,plain,
! [A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( universal_class @ complement ) )
| ( ( A @ power_class )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[233,549]) ).
thf(1105,plain,
! [A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) )
!= universal_class )
| ( ( A @ power_class )
!= universal_class ) ),
inference(simp,[status(thm)],[1102]) ).
thf(510,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[319,323]) ).
thf(511,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[510]) ).
thf(516,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[511]) ).
thf(519,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[516]) ).
thf(12627,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12550,519]) ).
thf(12628,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[12627]) ).
thf(12629,plain,
( ( omega @ singleton @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[12628:[bind(A,$thf( omega @ singleton ))]]) ).
thf(140,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible4) ).
thf(468,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)],[140]) ).
thf(147,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ domain_of @ ( A @ maps ) ) )
| ~ ( B @ ( A @ range_of @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps4) ).
thf(481,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)],[147]) ).
thf(84,axiom,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_can_define_singleton) ).
thf(351,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(352,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[351]) ).
thf(51,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range) ).
thf(276,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)],[51]) ).
thf(66,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn2) ).
thf(308,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)],[66]) ).
thf(95,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(378,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(379,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[378]) ).
thf(126,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',section1) ).
thf(440,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(48,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(269,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)],[48]) ).
thf(111,axiom,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn3) ).
thf(408,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)],[111]) ).
thf(5,axiom,
! [B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
= A )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions4) ).
thf(170,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)],[5]) ).
thf(98,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function1) ).
thf(385,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(82,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(346,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(347,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[346]) ).
thf(348,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[347]) ).
thf(1,negated_conjecture,
( ( x @ symmetrization_of )
= x ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_symmetrization_property3_1) ).
thf(161,plain,
( ( x @ symmetrization_of )
= x ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(163,plain,
( ( x @ symmetrization_of )
= x ),
inference(lifteq,[status(thm)],[161]) ).
thf(1844,plain,
! [A: $i] :
( ( A
!= ( x @ symmetrization_of ) )
| ( null_class = x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,163]) ).
thf(1845,plain,
( ( null_class = x )
| ( x @ symmetrization_of @ ( x @ symmetrization_of @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1844:[bind(A,$thf( x @ symmetrization_of ))]]) ).
thf(2439,plain,
( ( null_class = x )
| ( x @ ( x @ regular @ member ) ) ),
inference(rewrite,[status(thm)],[1845,163]) ).
thf(112,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(410,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(411,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[410]) ).
thf(412,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[411]) ).
thf(413,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[412]) ).
thf(559,plain,
! [A: $i] :
( ( ( A @ ( A @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[413,164]) ).
thf(560,plain,
! [A: $i] :
( ( A @ ( A @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[559]) ).
thf(561,plain,
! [A: $i] :
( ( A != x )
| ( A
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[560]) ).
thf(562,plain,
( ( x @ inverse )
!= x ),
inference(simp,[status(thm)],[561]) ).
thf(2007,plain,
! [A: $i] :
( ( A
!= ( x @ inverse ) )
| ( null_class != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,562]) ).
thf(2008,plain,
( ( null_class != x )
| ( x @ inverse @ ( x @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2007:[bind(A,$thf( x @ inverse ))]]) ).
thf(2903,plain,
! [B: $i,A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2008,323]) ).
thf(2904,plain,
! [B: $i,A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2903]) ).
thf(2905,plain,
( ~ ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[2904:[bind(A,$thf( x @ inverse @ regular )),bind(B,$thf( x @ inverse ))]]) ).
thf(3386,plain,
( ( ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
!= ( x @ inverse @ ( x @ inverse @ regular @ member ) ) )
| ~ $true
| ( null_class != x )
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2008,2905]) ).
thf(3387,plain,
( ( ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
!= ( x @ inverse @ ( x @ inverse @ regular @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3386]) ).
thf(3393,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( ( x @ inverse @ regular )
!= ( x @ inverse @ regular ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3387]) ).
thf(3407,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3393]) ).
thf(3427,plain,
( ( null_class != null_class )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( x != x )
| ( x @ ( x @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2439,3407]) ).
thf(3428,plain,
( ( null_class != null_class )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[3427]) ).
thf(3429,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3428:[]]) ).
thf(2879,plain,
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2008,514]) ).
thf(2880,plain,
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2879]) ).
thf(2933,plain,
( ( ( universal_class @ complement )
!= ( x @ inverse ) )
| ( ( x @ inverse @ regular )
!= omega )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2880]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(199,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(200,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[199]) ).
thf(201,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[200]) ).
thf(11428,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( A @ ( A @ ( choice @ apply ) @ member ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[319,201]) ).
thf(11429,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[11428]) ).
thf(11430,plain,
( ( omega @ ( omega @ ( choice @ apply ) @ member ) )
| ( omega = null_class ) ),
inference(pattern_uni,[status(thm)],[11429:[bind(A,$thf( omega ))]]) ).
thf(49,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(272,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(12157,plain,
( ( omega != omega )
| ( null_class @ inductive )
| ( omega @ ( omega @ ( choice @ apply ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[11430,272]) ).
thf(12158,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ ( choice @ apply ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[12157:[]]) ).
thf(12489,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12459,514]) ).
thf(12490,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[12489]) ).
thf(12559,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[12490]) ).
thf(12585,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) ),
inference(simp,[status(thm)],[12559]) ).
thf(29,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(229,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(1341,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)],[235,229]) ).
thf(1342,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1341:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(107,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip1) ).
thf(402,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(902,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[402,164]) ).
thf(903,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[902]) ).
thf(913,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= x )
| ( ( A @ flip )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[903]) ).
thf(11,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(184,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(185,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[184]) ).
thf(114,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class1) ).
thf(417,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(72,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(320,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(171,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
= A )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[170]) ).
thf(172,plain,
! [B: $i,A: $i] :
( ~ ( B @ recursion_equation_functions @ ( A @ member ) )
| ( ( A @ rest_of @ ( B @ compose ) )
= A ) ),
inference(lifteq,[status(thm)],[171]) ).
thf(138,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(463,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(1204,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[463,164]) ).
thf(1205,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[1204]) ).
thf(1216,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( B @ ( A @ compose ) )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[1205]) ).
thf(1220,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
!= ( D @ ( C @ compose ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( A
!= ( x @ inverse ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[172,1216]) ).
thf(1221,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( B
!= ( x @ inverse ) )
| ~ ( A @ recursion_equation_functions @ ( B @ member ) ) ),
inference(pattern_uni,[status(thm)],[1220:[bind(A,$thf( E )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E @ rest_of ))]]) ).
thf(1230,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ~ ( A @ recursion_equation_functions @ ( x @ inverse @ member ) ) ),
inference(simp,[status(thm)],[1221]) ).
thf(1250,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions @ ( x @ inverse @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[319,1230]) ).
thf(1251,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions @ ( x @ inverse @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x ) ),
inference(simp,[status(thm)],[1250]) ).
thf(1254,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions )
!= universal_class )
| ( ( x @ inverse )
!= omega )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x ) ),
inference(simp,[status(thm)],[1251]) ).
thf(42,axiom,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_ordinals) ).
thf(257,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(101,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/sandbox2/benchmark/theBenchmark.p',section3) ).
thf(389,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)],[101]) ).
thf(158,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(504,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(81,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(344,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(87,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(357,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(15,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(194,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)],[15]) ).
thf(195,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(cnf,[status(esa)],[194]) ).
thf(196,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( 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 ) ) ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(13,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(189,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(604,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[385,164]) ).
thf(605,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[604]) ).
thf(606,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= composition_function ) ),
inference(simp,[status(thm)],[605]) ).
thf(55,axiom,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_relation2) ).
thf(284,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(1082,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)],[233,233]) ).
thf(1083,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1082:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1115,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1083]) ).
thf(57,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(289,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(130,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps3) ).
thf(448,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(70,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ apply ) )
= C )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn3) ).
thf(316,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)],[70]) ).
thf(44,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(261,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)],[44]) ).
thf(24,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ ( C @ compose ) )
!= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition3) ).
thf(216,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)],[24]) ).
thf(2236,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,319]) ).
thf(2237,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2236:[bind(A,$thf( omega ))]]) ).
thf(144,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(475,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)],[144]) ).
thf(8,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip2) ).
thf(178,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)],[8]) ).
thf(179,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[178]) ).
thf(94,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).
thf(376,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(115,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_of1) ).
thf(419,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(621,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[419,164]) ).
thf(622,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[621]) ).
thf(623,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( A @ rest_of )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[622]) ).
thf(149,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cantor_class) ).
thf(485,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(34,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(238,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)],[34]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair3) ).
thf(353,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(480,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(lifteq,[status(thm)],[479]) ).
thf(105,axiom,
! [A: $i] :
( ( A = ordinal_numbers )
| ( ordinal_numbers @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers4) ).
thf(397,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)],[105]) ).
thf(59,axiom,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_multiplication) ).
thf(293,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)],[59]) ).
thf(46,axiom,
! [B: $i,A: $i] :
( ( composition_function @ ( B @ ( A @ compose ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function3) ).
thf(265,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)],[46]) ).
thf(159,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn1) ).
thf(506,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(507,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[506]) ).
thf(68,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(312,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(20,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[189]) ).
thf(3013,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[190,405]) ).
thf(3020,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ power_class @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( C @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3013,1115]) ).
thf(3021,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3020:[bind(A,$thf( element_relation )),bind(B,$thf( D @ power_class )),bind(C,$thf( D ))]]) ).
thf(3068,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3021]) ).
thf(53,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetric_difference) ).
thf(280,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)],[53]) ).
thf(39,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product4) ).
thf(248,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)],[39]) ).
thf(96,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(380,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(1914,plain,
! [A: $i] :
( ( A
!= ( x @ inverse ) )
| ~ ( x @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,164]) ).
thf(1915,plain,
( ~ ( x @ ( null_class @ subclass ) )
| ( x @ inverse @ ( x @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1914:[bind(A,$thf( x @ inverse ))]]) ).
thf(25,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(220,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(16,axiom,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions2) ).
thf(197,plain,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(198,plain,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[197]) ).
thf(9284,plain,
! [B: $i,A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( B @ recursion_equation_functions @ ( A @ member ) ) )
| ( A @ function )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,198]) ).
thf(9285,plain,
! [B: $i,A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( B @ recursion_equation_functions @ ( A @ member ) ) )
| ( A @ function )
| ( null_class = x ) ),
inference(simp,[status(thm)],[9284]) ).
thf(9305,plain,
! [B: $i,A: $i] :
( ( ( B @ recursion_equation_functions )
!= x )
| ( ( x @ regular )
!= A )
| ( A @ function )
| ( null_class = x ) ),
inference(simp,[status(thm)],[9285]) ).
thf(9323,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions )
!= x )
| ( x @ regular @ function )
| ( null_class = x ) ),
inference(simp,[status(thm)],[9305]) ).
thf(133,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/sandbox2/benchmark/theBenchmark.p',transitive1) ).
thf(452,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)],[133]) ).
thf(63,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction1) ).
thf(302,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(61,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',integer_function2) ).
thf(297,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(121,axiom,
! [B: $i,A: $i] :
( ( A @ symmetrization_of @ ( identity_relation @ union ) @ ( B @ ( B @ cross_product ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected1) ).
thf(429,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)],[121]) ).
thf(3,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(165,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(83,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(349,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(119,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(424,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(3022,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( C @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3013,1115]) ).
thf(3023,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3022:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3069,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[3023]) ).
thf(2468,plain,
! [B: $i,A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,323]) ).
thf(2469,plain,
! [B: $i,A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2468]) ).
thf(2470,plain,
( ~ ( x @ complement @ ( x @ regular @ member ) )
| ( null_class = x ) ),
inference(pattern_uni,[status(thm)],[2469:[bind(A,$thf( x @ regular )),bind(B,$thf( x ))]]) ).
thf(2534,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( x @ ( x @ regular @ member ) ) )
| ~ $true
| ( null_class = x )
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,2470]) ).
thf(2535,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( x @ ( x @ regular @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2534]) ).
thf(2561,plain,
( ( ( x @ complement )
!= x )
| ( ( x @ regular )
!= ( x @ regular ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2535]) ).
thf(2565,plain,
( ( ( x @ complement )
!= x )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2561]) ).
thf(124,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible3) ).
thf(435,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)],[124]) ).
thf(76,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/sandbox2/benchmark/theBenchmark.p',well_ordering8) ).
thf(329,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)],[76]) ).
thf(137,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_members) ).
thf(461,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexive2) ).
thf(493,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(74,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation2) ).
thf(324,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)],[74]) ).
thf(27,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( A @ ( C @ restrict ) ) )
= B )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_of3) ).
thf(224,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)],[27]) ).
thf(148,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(483,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(484,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[483]) ).
thf(135,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation4) ).
thf(456,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)],[135]) ).
thf(78,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(335,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(336,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[335]) ).
thf(1133,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)],[336,233]) ).
thf(1134,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1133:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1175,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1134]) ).
thf(1739,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)],[336,1175]) ).
thf(1740,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1739:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(2456,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,519]) ).
thf(2457,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2456]) ).
thf(2496,plain,
! [A: $i] :
( ( x != A )
| ( ( x @ regular )
!= omega )
| ( ( A @ complement )
!= universal_class )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2457]) ).
thf(2498,plain,
( ( ( x @ regular )
!= omega )
| ( ( x @ complement )
!= universal_class )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2496]) ).
thf(2893,plain,
! [A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2008,519]) ).
thf(2894,plain,
! [A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2893]) ).
thf(2936,plain,
! [A: $i] :
( ( ( x @ inverse )
!= A )
| ( ( x @ inverse @ regular )
!= omega )
| ( ( A @ complement )
!= universal_class )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2894]) ).
thf(2946,plain,
( ( ( x @ inverse @ regular )
!= omega )
| ( ( x @ inverse @ complement )
!= universal_class )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2936]) ).
thf(108,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate1) ).
thf(403,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(12,axiom,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation2) ).
thf(186,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(187,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[186]) ).
thf(188,plain,
! [B: $i,A: $i] :
( ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[187]) ).
thf(4,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate2) ).
thf(168,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(169,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[168]) ).
thf(116,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( B @ ( B @ ( A @ not_well_ordering ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',well_ordering7) ).
thf(420,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)],[116]) ).
thf(156,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ asymmetric ) )
| ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asymmetric2) ).
thf(499,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)],[156]) ).
thf(141,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(471,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(23,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation3) ).
thf(212,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)],[23]) ).
thf(151,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(489,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(12646,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[12550,323]) ).
thf(12647,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[12646]) ).
thf(12648,plain,
~ ( omega @ singleton @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[12647:[bind(A,$thf( omega )),bind(B,$thf( omega @ singleton ))]]) ).
thf(12877,plain,
( ( ( x @ ( x @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,12648]) ).
thf(12878,plain,
( ( ( x @ ( x @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[12877]) ).
thf(12917,plain,
( ( ( omega @ singleton @ complement )
!= x )
| ( ( x @ regular )
!= omega )
| ( null_class = x ) ),
inference(simp,[status(thm)],[12878]) ).
thf(103,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( B @ ( C @ ( A @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',section2) ).
thf(393,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)],[103]) ).
thf(12904,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12459,12648]) ).
thf(12905,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[12904]) ).
thf(12931,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[12905]) ).
thf(12955,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) ),
inference(simp,[status(thm)],[12931]) ).
thf(6,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function2) ).
thf(173,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)],[6]) ).
thf(881,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)],[229,323]) ).
thf(882,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)],[881]) ).
thf(883,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[882:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(895,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[883]) ).
thf(1343,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)],[235,895]) ).
thf(1344,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1343:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(2473,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ complement @ ( A @ singleton @ member ) ) )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,1344]) ).
thf(2474,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ complement @ ( A @ singleton @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2473]) ).
thf(2497,plain,
! [A: $i] :
( ( ( universal_class @ complement )
!= x )
| ( ( x @ regular )
!= ( A @ singleton ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2474]) ).
thf(37,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction2) ).
thf(244,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(19,axiom,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',kind_1_ordinals) ).
thf(204,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(205,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(lifteq,[status(thm)],[204]) ).
thf(160,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn2) ).
thf(508,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(132,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_range_map1) ).
thf(451,plain,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(556,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[451,164]) ).
thf(557,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[556]) ).
thf(558,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= union_of_range_map ) ),
inference(simp,[status(thm)],[557]) ).
thf(45,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_definition) ).
thf(263,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(264,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[263]) ).
thf(9,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(180,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(128,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(444,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(1633,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)],[1115,233]) ).
thf(1634,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1633:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1698,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)],[1634]) ).
thf(1958,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,484]) ).
thf(1959,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1958:[bind(A,$thf( subset_relation ))]]) ).
thf(7,axiom,
! [B: $i,A: $i] :
( ( ordinal_numbers @ ( A @ domain_of @ member ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions3) ).
thf(176,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)],[7]) ).
thf(118,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn1) ).
thf(423,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(12856,plain,
( ( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12550,12648]) ).
thf(12857,plain,
( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[12856]) ).
thf(12938,plain,
( ( ( omega @ singleton @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[12857]) ).
thf(12958,plain,
( ( omega @ singleton @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[12938]) ).
thf(58,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(291,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)],[58]) ).
thf(12526,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[12459,323]) ).
thf(12527,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[12526]) ).
thf(12528,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[12527:[bind(A,$thf( E )),bind(B,$thf( omega )),bind(C,$thf( E @ ( omega @ unordered_pair ) ))]]) ).
thf(12599,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(simp,[status(thm)],[12528]) ).
thf(14849,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12550,12599]) ).
thf(14850,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[14849]) ).
thf(14944,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[14850]) ).
thf(14971,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[14944]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(259,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(21,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(209,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(2160,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,209]) ).
thf(2161,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2160:[bind(A,$thf( choice ))]]) ).
thf(102,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(391,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(12507,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[12459,519]) ).
thf(12508,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[12507]) ).
thf(12509,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[12508:[bind(A,$thf( D )),bind(B,$thf( D @ ( omega @ unordered_pair ) ))]]) ).
thf(12592,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(simp,[status(thm)],[12509]) ).
thf(97,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/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions5) ).
thf(382,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)],[97]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(236,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(181,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[180]) ).
thf(298,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[297]) ).
thf(299,plain,
! [A: $i] :
( ( omega @ ( A @ member ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(lifteq,[status(thm)],[298]) ).
thf(1238,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ recursion_equation_functions @ ( x @ inverse @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[299,1230]) ).
thf(1239,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ recursion_equation_functions @ ( x @ inverse @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1238]) ).
thf(1255,plain,
! [B: $i,A: $i] :
( ( ( B @ recursion_equation_functions )
!= omega )
| ( A
!= ( x @ inverse ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1239]) ).
thf(1258,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions )
!= omega )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1255]) ).
thf(5430,plain,
! [B: $i,A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[1915,323]) ).
thf(5431,plain,
! [B: $i,A: $i] :
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[5430]) ).
thf(5432,plain,
( ~ ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[5431:[bind(A,$thf( x @ inverse @ regular )),bind(B,$thf( x @ inverse ))]]) ).
thf(110,axiom,
! [A: $i] :
( ( ordinal_numbers @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers3) ).
thf(406,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)],[110]) ).
thf(30,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(230,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)],[30]) ).
thf(89,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( ( B @ ( A @ not_well_ordering ) )
!= null_class )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',well_ordering6) ).
thf(362,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)],[89]) ).
thf(509,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[508]) ).
thf(123,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(433,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(434,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[433]) ).
thf(3014,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)],[3013,233]) ).
thf(3015,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3014:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3066,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3015]) ).
thf(207,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[206]) ).
thf(208,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ maps ) ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[207]) ).
thf(145,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_relation) ).
thf(477,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)],[145]) ).
thf(100,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/sandbox2/benchmark/theBenchmark.p',well_ordering5) ).
thf(387,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)],[100]) ).
thf(607,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[423,164]) ).
thf(608,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[607]) ).
thf(609,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= application_function ) ),
inference(simp,[status(thm)],[608]) ).
thf(131,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(450,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(624,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[450,164]) ).
thf(625,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[624]) ).
thf(626,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( A @ compose_class )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[625]) ).
thf(345,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(lifteq,[status(thm)],[344]) ).
thf(91,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ compose ) )
= B )
| ~ ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition2) ).
thf(367,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)],[91]) ).
thf(99,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(386,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(550,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[386,164]) ).
thf(551,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[550]) ).
thf(552,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= element_relation ) ),
inference(simp,[status(thm)],[551]) ).
thf(3033,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( ordinal_numbers @ ( successor_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[3013,205]) ).
thf(3034,plain,
( ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[3033:[bind(A,$thf( successor_relation )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(35,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(240,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(40,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible2) ).
thf(251,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)],[40]) ).
thf(1636,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)],[1115,233]) ).
thf(1637,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1636:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1699,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1637]) ).
thf(52,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain) ).
thf(278,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)],[52]) ).
thf(65,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ segment ) ) )
= ( C @ singleton @ ( B @ ( A @ restrict ) ) @ domain_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',segment) ).
thf(306,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)],[65]) ).
thf(47,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(267,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(122,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ connected ) )
| ~ ( B @ symmetrization_of @ ( identity_relation @ union ) @ ( A @ ( A @ cross_product ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected2) ).
thf(431,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)],[122]) ).
thf(166,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[165]) ).
thf(167,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) ) ),
inference(lifteq,[status(thm)],[166]) ).
thf(142,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(473,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
= null_class )
| ~ ( B @ ( A @ asymmetric ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',asymmetric1) ).
thf(426,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)],[120]) ).
thf(93,axiom,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation3) ).
thf(374,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(12155,plain,
( ( omega != omega )
| ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ ( choice @ apply ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[11430,319]) ).
thf(12156,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ ( choice @ apply ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[12155:[]]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(355,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)],[86]) ).
thf(150,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(487,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(2539,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( x @ complement @ ( x @ regular @ member ) ) )
| ~ $true
| ( null_class = x )
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[299,2470]) ).
thf(2540,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( x @ complement @ ( x @ regular @ member ) ) )
| ( null_class = x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[2539]) ).
thf(2551,plain,
! [A: $i] :
( ( ( x @ complement )
!= omega )
| ( A
!= ( x @ regular ) )
| ( null_class = x )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[2540]) ).
thf(2567,plain,
( ( ( x @ complement )
!= omega )
| ( null_class = x )
| ( ( x @ regular @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[2551]) ).
thf(60,axiom,
! [B: $i,A: $i] :
( ( B @ ( union_of_range_map @ ( successor_relation @ ( A @ recursion ) ) @ apply ) )
= ( B @ ( A @ ordinal_add ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_addition) ).
thf(295,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)],[60]) ).
thf(3435,plain,
( ( null_class != null_class )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( x != x )
| ~ ( x @ complement @ ( x @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2470,3407]) ).
thf(3436,plain,
( ( null_class != null_class )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ~ ( x @ complement @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[3435]) ).
thf(3437,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ~ ( x @ complement @ ( x @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3436:[]]) ).
thf(9209,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( x @ ( x @ regular @ member ) ) )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ~ $true
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) ) ),
inference(paramod_ordered,[status(thm)],[3429,3437]) ).
thf(9210,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( x @ ( x @ regular @ member ) ) )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[9209]) ).
thf(9241,plain,
( ( ( x @ complement )
!= x )
| ( ( x @ regular )
!= ( x @ regular ) )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[9210]) ).
thf(9253,plain,
( ( ( x @ complement )
!= x )
| ( ( x @ inverse @ complement )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[9241]) ).
thf(54,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ ( A @ member ) )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_of2) ).
thf(282,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)],[54]) ).
thf(113,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(414,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(75,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(327,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(67,axiom,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions1) ).
thf(310,plain,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(92,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/sandbox2/benchmark/theBenchmark.p',union_of_range_map3) ).
thf(370,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)],[92]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexive1) ).
thf(502,plain,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(2444,plain,
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[2439,514]) ).
thf(2445,plain,
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2444]) ).
thf(2486,plain,
( ( ( universal_class @ complement )
!= x )
| ( ( x @ regular )
!= omega )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2445]) ).
thf(88,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',integer_function1) ).
thf(359,plain,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(177,plain,
! [B: $i,A: $i] :
( ( ordinal_numbers @ ( A @ domain_of @ member ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[176]) ).
thf(10,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(182,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(2503,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( x @ complement @ ( x @ regular @ member ) ) )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[1342,2470]) ).
thf(2504,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( x @ complement @ ( x @ regular @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2503]) ).
thf(2558,plain,
! [A: $i] :
( ( ( x @ complement )
!= universal_class )
| ( ( A @ singleton )
!= ( x @ regular ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[2504]) ).
thf(1711,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)],[1175,233]) ).
thf(1712,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1711:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1797,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1712]) ).
thf(69,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(314,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(129,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(446,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(154,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(495,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(593,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[495,164]) ).
thf(594,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[593]) ).
thf(595,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= domain_relation ) ),
inference(simp,[status(thm)],[594]) ).
thf(174,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[173]) ).
thf(175,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ compose ) )
= C ) ),
inference(lifteq,[status(thm)],[174]) ).
thf(521,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[473,164]) ).
thf(522,plain,
! [A: $i] :
( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[521]) ).
thf(523,plain,
! [A: $i] :
( ( universal_class != x )
| ( A
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[522]) ).
thf(524,plain,
universal_class != x,
inference(simp,[status(thm)],[523]) ).
thf(143,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(474,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(586,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[474,164]) ).
thf(587,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[586]) ).
thf(588,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( ( x @ inverse )
!= successor_relation ) ),
inference(simp,[status(thm)],[587]) ).
thf(2272,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,272]) ).
thf(2273,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2272:[bind(A,$thf( omega ))]]) ).
thf(64,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product3) ).
thf(304,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)],[64]) ).
thf(258,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(lifteq,[status(thm)],[257]) ).
thf(1660,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)],[336,1115]) ).
thf(1661,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1660:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1706,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)],[1661]) ).
thf(478,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)],[477]) ).
thf(56,axiom,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers1) ).
thf(287,plain,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(38,axiom,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_relation3) ).
thf(246,plain,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(62,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(300,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)],[62]) ).
thf(1641,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)],[233,1115]) ).
thf(1642,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1641:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1956,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,484]) ).
thf(1957,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1956:[bind(A,$thf( subset_relation ))]]) ).
thf(183,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[182]) ).
thf(50,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(273,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)],[50]) ).
thf(5381,plain,
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[1915,514]) ).
thf(5382,plain,
( ( ( x @ inverse @ ( x @ inverse @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[5381]) ).
thf(5477,plain,
( ( ( universal_class @ complement )
!= ( x @ inverse ) )
| ( ( x @ inverse @ regular )
!= omega )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[5382]) ).
thf(3144,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3066,233]) ).
thf(3145,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3144:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(22,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product1) ).
thf(210,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)],[22]) ).
thf(36,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate3) ).
thf(242,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)],[36]) ).
thf(294,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
inference(lifteq,[status(thm)],[293]) ).
thf(80,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(341,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(125,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/sandbox2/benchmark/theBenchmark.p',well_ordering2) ).
thf(437,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)],[125]) ).
thf(106,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(400,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(917,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[403,164]) ).
thf(918,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( x @ ( x @ inverse @ subclass ) ) ),
inference(simp,[status(thm)],[917]) ).
thf(928,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= x )
| ( ( A @ rotate )
!= ( x @ inverse ) ) ),
inference(simp,[status(thm)],[918]) ).
thf(268,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[267]) ).
thf(104,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/sandbox2/benchmark/theBenchmark.p',transitive2) ).
thf(395,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)],[104]) ).
thf(139,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(464,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation2) ).
thf(332,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(136,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class2) ).
thf(459,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(3040,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ diagonalise @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( C @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3013,1175]) ).
thf(3041,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3040:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(3076,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[3041]) ).
thf(127,axiom,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers2) ).
thf(442,plain,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(486,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[485]) ).
thf(79,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/sandbox2/benchmark/theBenchmark.p',rest_of4) ).
thf(337,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)],[79]) ).
thf(8809,plain,
( ( ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
!= ( x @ inverse @ ( x @ inverse @ regular @ member ) ) )
| ~ $true
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[1915,5432]) ).
thf(8810,plain,
( ( ( x @ inverse @ complement @ ( x @ inverse @ regular @ member ) )
!= ( x @ inverse @ ( x @ inverse @ regular @ member ) ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[8809]) ).
thf(8915,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ( ( x @ inverse @ regular )
!= ( x @ inverse @ regular ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[8810]) ).
thf(8940,plain,
( ( ( x @ inverse @ complement )
!= ( x @ inverse ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[8915]) ).
thf(2205,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[348,386]) ).
thf(2206,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( element_relation @ ( element_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2205:[bind(A,$thf( element_relation ))]]) ).
thf(12674,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true
| ( null_class = x ) ),
inference(paramod_ordered,[status(thm)],[12550,2470]) ).
thf(12675,plain,
( ( ( x @ complement @ ( x @ regular @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ( null_class = x ) ),
inference(simp,[status(thm)],[12674]) ).
thf(12694,plain,
( ( ( x @ complement )
!= ( omega @ singleton ) )
| ( ( x @ regular )
!= omega )
| ( null_class = x ) ),
inference(simp,[status(thm)],[12675]) ).
thf(152,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(491,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(26,axiom,
! [C: $i,B: $i,A: $i] :
( ( application_function @ ( B @ ( A @ apply ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn4) ).
thf(222,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)],[26]) ).
thf(41,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(254,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)],[41]) ).
thf(28,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product2) ).
thf(227,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)],[28]) ).
thf(155,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/sandbox2/benchmark/theBenchmark.p',well_ordering4) ).
thf(496,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)],[155]) ).
thf(134,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/sandbox2/benchmark/theBenchmark.p',well_ordering3) ).
thf(454,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)],[134]) ).
thf(21644,plain,
$false,
inference(e,[status(thm)],[5509,365,234,555,12699,479,1105,12629,468,481,352,276,308,379,440,269,202,408,170,385,3429,2933,12158,12585,1342,913,404,185,417,320,1254,257,389,184,504,344,357,196,189,606,284,1115,289,448,316,261,216,2237,475,164,179,376,623,485,238,514,348,353,480,397,293,265,507,312,206,3068,280,248,380,201,1915,220,2905,9323,233,452,302,297,429,165,349,12550,424,3069,197,2565,435,329,461,493,324,224,229,484,456,1740,519,2498,2946,403,188,169,420,499,471,3407,212,489,12917,393,12955,173,2497,244,205,508,558,264,161,180,444,1698,1959,562,176,423,12958,291,14971,204,259,413,191,402,2161,391,12592,382,335,236,181,1258,5432,406,172,419,230,1175,362,451,509,434,3066,208,477,387,609,323,483,895,626,1344,198,345,367,552,3034,240,251,2439,1699,278,306,299,267,431,167,473,426,162,12599,374,12156,355,194,209,487,2567,295,9253,282,414,12648,327,1230,263,310,199,370,502,463,2486,359,177,182,336,2558,1797,314,446,595,175,386,450,203,524,495,1216,588,2273,319,304,258,351,186,378,433,1706,478,272,346,549,287,235,246,300,1642,1957,410,168,405,2008,183,273,422,5477,3145,210,242,294,341,437,474,12459,400,928,506,268,395,464,332,459,163,178,3076,442,11430,3013,322,486,3437,337,8940,2206,12694,491,222,2470,232,254,227,496,454]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : NUM030-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.13 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.23/5.72 % Computer : n020.cluster.edu
% 0.23/5.72 % Model : x86_64 x86_64
% 0.23/5.72 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.23/5.72 % Memory : 8046.5625MB
% 0.23/5.72 % OS : Linux 6.8.0-71-generic
% 0.23/5.72 % CPULimit : 300
% 0.23/5.72 % WCLimit : 300
% 0.23/5.73 % DateTime : Sat Sep 26 01:55:26 UTC 2026
% 0.23/5.73 % CPUTime :
% 0.23/5.73 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.39/6.59 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.24/6.98 % [INFO] Parsing done (387ms).
% 2.24/7.00 % [INFO] Running in sequential loop mode.
% 3.47/7.58 % [INFO] eprover registered as external prover.
% 3.47/7.59 % [INFO] Scanning for conjecture ...
% 3.96/7.74 % [INFO] Found a conjecture (or negated_conjecture) and 158 axioms. Running axiom selection ...
% 4.38/7.95 % [INFO] Axiom selection finished. Selected 158 axioms (removed 0 axioms).
% 5.34/8.16 % [INFO] Problem is propositional (TPTP CNF).
% 5.34/8.18 % [INFO] Type checking passed.
% 5.34/8.19 % [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 ...
% 103.94/28.85 % External prover 'e' found a proof!
% 103.94/28.85 % [INFO] Killing All external provers ...
% 103.94/28.85 % Time passed: 22917ms (effective reasoning time: 21843ms)
% 103.94/28.86 % 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
% 103.94/28.86 % No. of inferences in proof: 624
% 103.94/28.86 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 22917 ms resp. 21843 ms w/o parsing
% 105.36/29.14 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 105.36/29.15 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------