%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUM069-1 : 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 : n009.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 08:08:11 AM UTC 2026
% Result : Unsatisfiable 187.89s 48.61s
% Output : Refutation 176.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 162
% Syntax : Number of formulae : 520 ( 162 unt; 0 typ; 0 def)
% Number of atoms : 1165 ( 400 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 4548 ( 603 ~; 505 |; 0 &;3440 @)
% ( 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 : 84 ( 81 usr; 23 con; 0-3 aty)
% Number of variables : 812 ( 0 ^; 812 !; 0 ?; 812 :)
% Comments :
%------------------------------------------------------------------------------
thf(well_ordering_decl,type,
well_ordering: $i > $i > $o ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(y_decl,type,
y: $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(u_decl,type,
u: $i ).
thf(v_decl,type,
v: $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(section_decl,type,
section: $i > $i > $i > $o ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(transitive_decl,type,
transitive: $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(ordinal_numbers_decl,type,
ordinal_numbers: $i ).
thf(kind_1_ordinals_decl,type,
kind_1_ordinals: $i ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(complement_decl,type,
complement: $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(irreflexive_decl,type,
irreflexive: $i > $i > $o ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(single_valued1_decl,type,
single_valued1: $i > $i ).
thf(single_valued2_decl,type,
single_valued2: $i > $i ).
thf(single_valued3_decl,type,
single_valued3: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(asymmetric_decl,type,
asymmetric: $i > $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(limit_ordinals_decl,type,
limit_ordinals: $i ).
thf(second_decl,type,
second: $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(recursion_decl,type,
recursion: $i > $i > $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(add_relation_decl,type,
add_relation: $i ).
thf(union_of_range_map_decl,type,
union_of_range_map: $i ).
thf(ordinal_multiply_decl,type,
ordinal_multiply: $i > $i > $i ).
thf(ordinal_add_decl,type,
ordinal_add: $i > $i > $i ).
thf(symmetrization_of_decl,type,
symmetrization_of: $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(segment_decl,type,
segment: $i > $i > $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(recursion_equation_functions_decl,type,
recursion_equation_functions: $i > $i ).
thf(rest_of_decl,type,
rest_of: $i > $i ).
thf(least_decl,type,
least: $i > $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(compose_class_decl,type,
compose_class: $i > $i ).
thf(application_function_decl,type,
application_function: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(connected_decl,type,
connected: $i > $i > $o ).
thf(not_well_ordering_decl,type,
not_well_ordering: $i > $i > $i ).
thf(rest_relation_decl,type,
rest_relation: $i ).
thf(integer_of_decl,type,
integer_of: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(singleton_relation_decl,type,
singleton_relation: $i ).
thf(97,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(365,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)],[97]) ).
thf(150,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(479,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(87,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(347,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(146,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(468,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(151,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(481,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)],[151]) ).
thf(82,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(333,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(42,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(248,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(249,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[248]) ).
thf(6,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(169,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)],[6]) ).
thf(170,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( B @ ( C @ ( A @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(cnf,[status(esa)],[169]) ).
thf(138,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(452,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(142,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(458,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(459,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[458]) ).
thf(563,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[452,459]) ).
thf(564,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[563]) ).
thf(565,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[564:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(572,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[452,565]) ).
thf(573,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[572]) ).
thf(579,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[573]) ).
thf(582,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[579]) ).
thf(71,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(308,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)],[71]) ).
thf(137,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(449,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)],[137]) ).
thf(162,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(509,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(510,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[509]) ).
thf(511,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[510]) ).
thf(52,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(269,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)],[52]) ).
thf(28,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(217,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(55,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(275,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(276,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[275]) ).
thf(53,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(271,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(272,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[271]) ).
thf(12,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(182,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(183,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[182]) ).
thf(5312,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[272,183]) ).
thf(94,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(360,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(361,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[360]) ).
thf(5341,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)],[5312,361]) ).
thf(5342,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[5341:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(5405,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[5342]) ).
thf(105,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(379,plain,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(107,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(384,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)],[107]) ).
thf(4,negated_conjecture,
u @ ( v @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_to_well_ordering_property3_4) ).
thf(166,plain,
u @ ( v @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(531,plain,
! [B: $i,A: $i] :
( ( ( u @ ( v @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[166,459]) ).
thf(532,plain,
! [B: $i,A: $i] :
( ( ( u @ ( v @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[531]) ).
thf(533,plain,
~ ( u @ complement @ ( v @ member ) ),
inference(pattern_uni,[status(thm)],[532:[bind(A,$thf( v )),bind(B,$thf( u ))]]) ).
thf(159,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(500,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)],[159]) ).
thf(91,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(356,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(3,negated_conjecture,
v @ ( u @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_to_well_ordering_property3_3) ).
thf(165,plain,
v @ ( u @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(512,plain,
! [B: $i,A: $i] :
( ( ( v @ ( u @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,459]) ).
thf(513,plain,
! [B: $i,A: $i] :
( ( ( v @ ( u @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[512]) ).
thf(518,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= v )
| ( u != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[513]) ).
thf(521,plain,
! [A: $i] :
( ( ( A @ complement )
!= v )
| ~ ( A @ ( u @ member ) ) ),
inference(simp,[status(thm)],[518]) ).
thf(690,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( u @ member ) ) )
| ( ( C @ complement )
!= v )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[356,521]) ).
thf(691,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( u @ member ) ) )
| ( ( C @ complement )
!= v ) ),
inference(simp,[status(thm)],[690]) ).
thf(711,plain,
! [C: $i,B: $i,A: $i] :
( ( universal_class != C )
| ( ( B @ ( A @ unordered_pair ) )
!= u )
| ( ( C @ complement )
!= v ) ),
inference(simp,[status(thm)],[691]) ).
thf(715,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= u )
| ( ( universal_class @ complement )
!= v ) ),
inference(simp,[status(thm)],[711]) ).
thf(14,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(185,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)],[14]) ).
thf(61,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(287,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(288,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[287]) ).
thf(67,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(301,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(194,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(195,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[194]) ).
thf(688,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( u @ complement @ ( v @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[356,533]) ).
thf(689,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( u @ complement @ ( v @ member ) ) ),
inference(simp,[status(thm)],[688]) ).
thf(708,plain,
! [B: $i,A: $i] :
( ( ( u @ complement )
!= universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= v ) ),
inference(simp,[status(thm)],[689]) ).
thf(1481,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( u @ complement )
!= universal_class )
| ( ( A @ singleton )
!= v ) ),
inference(paramod_ordered,[status(thm)],[195,708]) ).
thf(1482,plain,
! [A: $i] :
( ( ( u @ complement )
!= universal_class )
| ( ( A @ singleton )
!= v ) ),
inference(pattern_uni,[status(thm)],[1481:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(8,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(173,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(174,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[173]) ).
thf(116,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(404,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)],[116]) ).
thf(109,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(389,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)],[109]) ).
thf(13,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(184,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(160,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(504,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(92,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(357,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)],[92]) ).
thf(18,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(196,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(143,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(460,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)],[143]) ).
thf(59,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(284,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(121,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(416,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)],[121]) ).
thf(62,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(289,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)],[62]) ).
thf(48,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(260,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(261,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[260]) ).
thf(98,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_relation1) ).
thf(367,plain,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(16,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(191,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(192,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[191]) ).
thf(193,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[192]) ).
thf(2195,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[193,582]) ).
thf(2196,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2195:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(3543,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[2196]) ).
thf(4151,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[3543,284]) ).
thf(4152,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[4151]) ).
thf(4193,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[367,4152]) ).
thf(4194,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[4193]) ).
thf(4205,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( rest_relation != universal_class ) ),
inference(simp,[status(thm)],[4194]) ).
thf(148,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(475,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(10,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).
thf(177,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(178,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[177]) ).
thf(179,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ maps ) ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(134,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(443,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)],[134]) ).
thf(76,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(321,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)],[76]) ).
thf(25,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(211,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(153,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',integer_function1) ).
thf(485,plain,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(156,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(492,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)],[156]) ).
thf(2,negated_conjecture,
y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_to_well_ordering_property3_2) ).
thf(164,plain,
y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(590,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( A @ ( u @ member ) ) )
| ( ( A @ complement )
!= v )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,521]) ).
thf(591,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( A @ ( u @ member ) ) )
| ( ( A @ complement )
!= v ) ),
inference(simp,[status(thm)],[590]) ).
thf(599,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) )
!= A )
| ( ( v @ ( u @ ordered_pair ) )
!= u )
| ( ( A @ complement )
!= v ) ),
inference(simp,[status(thm)],[591]) ).
thf(602,plain,
( ( ( v @ ( u @ ordered_pair ) )
!= u )
| ( ( y @ ( y @ cross_product ) @ complement )
!= v ) ),
inference(simp,[status(thm)],[599]) ).
thf(112,axiom,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers2) ).
thf(397,plain,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(30,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(221,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)],[30]) ).
thf(139,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(453,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(40,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(243,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)],[40]) ).
thf(127,axiom,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_numbers1) ).
thf(428,plain,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(50,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(265,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(73,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(312,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)],[73]) ).
thf(22,axiom,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_ordinals) ).
thf(205,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(206,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(lifteq,[status(thm)],[205]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(167,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(168,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[167]) ).
thf(4197,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[168,4152]) ).
thf(4198,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[4197]) ).
thf(4214,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( universal_class != omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[4198]) ).
thf(4230,plain,
( ( universal_class != omega )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[4214]) ).
thf(57,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(280,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(63,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(292,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)],[63]) ).
thf(132,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(439,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(106,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(381,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)],[106]) ).
thf(1469,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= u )
| ( ( universal_class @ complement )
!= v ) ),
inference(paramod_ordered,[status(thm)],[195,715]) ).
thf(1470,plain,
! [A: $i] :
( ( ( A @ singleton )
!= u )
| ( ( universal_class @ complement )
!= v ) ),
inference(pattern_uni,[status(thm)],[1469:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(514,plain,
! [B: $i,A: $i] :
( ( ( v @ ( u @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[165,459]) ).
thf(515,plain,
! [B: $i,A: $i] :
( ( ( v @ ( u @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[514]) ).
thf(516,plain,
~ ( v @ complement @ ( u @ member ) ),
inference(pattern_uni,[status(thm)],[515:[bind(A,$thf( u )),bind(B,$thf( v ))]]) ).
thf(527,plain,
( ( ( u @ ( v @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,516]) ).
thf(528,plain,
( ( u @ ( v @ member ) )
!= ( v @ complement @ ( u @ member ) ) ),
inference(simp,[status(thm)],[527]) ).
thf(534,plain,
( ( ( v @ complement )
!= u )
| ( v != u ) ),
inference(simp,[status(thm)],[528]) ).
thf(529,plain,
! [B: $i,A: $i] :
( ( ( u @ ( v @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,459]) ).
thf(530,plain,
! [B: $i,A: $i] :
( ( ( u @ ( v @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[529]) ).
thf(535,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= u )
| ( v != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[530]) ).
thf(536,plain,
! [A: $i] :
( ( ( A @ complement )
!= u )
| ~ ( A @ ( v @ member ) ) ),
inference(simp,[status(thm)],[535]) ).
thf(698,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( v @ member ) ) )
| ( ( C @ complement )
!= u )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[356,536]) ).
thf(699,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( v @ member ) ) )
| ( ( C @ complement )
!= u ) ),
inference(simp,[status(thm)],[698]) ).
thf(710,plain,
! [C: $i,B: $i,A: $i] :
( ( universal_class != C )
| ( ( B @ ( A @ unordered_pair ) )
!= v )
| ( ( C @ complement )
!= u ) ),
inference(simp,[status(thm)],[699]) ).
thf(714,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= v )
| ( ( universal_class @ complement )
!= u ) ),
inference(simp,[status(thm)],[710]) ).
thf(36,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(233,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)],[36]) ).
thf(110,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(392,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(118,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(407,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)],[118]) ).
thf(1199,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)],[361,361]) ).
thf(1200,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1199:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1255,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1200]) ).
thf(68,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(302,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)],[68]) ).
thf(88,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(349,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)],[88]) ).
thf(44,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(252,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(197,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[196]) ).
thf(80,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(329,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(141,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(456,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)],[141]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions2) ).
thf(324,plain,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(34,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(229,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)],[34]) ).
thf(557,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( u @ complement @ ( v @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[452,533]) ).
thf(558,plain,
( ( universal_class @ ( omega @ member ) )
!= ( u @ complement @ ( v @ member ) ) ),
inference(simp,[status(thm)],[557]) ).
thf(566,plain,
( ( ( u @ complement )
!= universal_class )
| ( omega != v ) ),
inference(simp,[status(thm)],[558]) ).
thf(129,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',integer_function2) ).
thf(432,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(433,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[432]) ).
thf(434,plain,
! [A: $i] :
( ( omega @ ( A @ member ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(lifteq,[status(thm)],[433]) ).
thf(1553,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( u @ member ) ) )
| ( ( B @ complement )
!= v )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[434,521]) ).
thf(1554,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( u @ member ) ) )
| ( ( B @ complement )
!= v )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1553]) ).
thf(1555,plain,
( ( ( omega @ complement )
!= v )
| ( ( u @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[1554:[bind(A,$thf( u )),bind(B,$thf( omega ))]]) ).
thf(130,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(435,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)],[130]) ).
thf(60,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(285,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(31,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(224,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(570,plain,
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,565]) ).
thf(571,plain,
( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[570]) ).
thf(578,plain,
( ( ( y @ ( y @ cross_product ) )
!= ( universal_class @ complement ) )
| ( ( v @ ( u @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[571]) ).
thf(393,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[392]) ).
thf(1258,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[393,261]) ).
thf(1259,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[1258:[bind(A,$thf( subset_relation ))]]) ).
thf(115,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_range_map1) ).
thf(403,plain,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(123,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(420,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(154,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(488,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(85,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(339,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)],[85]) ).
thf(546,plain,
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,516]) ).
thf(547,plain,
( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( v @ complement @ ( u @ member ) ) ),
inference(simp,[status(thm)],[546]) ).
thf(554,plain,
( ( ( y @ ( y @ cross_product ) )
!= ( v @ complement ) )
| ( ( v @ ( u @ ordered_pair ) )
!= u ) ),
inference(simp,[status(thm)],[547]) ).
thf(476,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[475]) ).
thf(523,plain,
( ( ( v @ complement @ ( u @ member ) )
!= ( v @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,516]) ).
thf(524,plain,
( ( v @ complement @ ( u @ member ) )
!= ( v @ ( u @ member ) ) ),
inference(simp,[status(thm)],[523]) ).
thf(525,plain,
( ( ( v @ complement )
!= v )
| ( u != u ) ),
inference(simp,[status(thm)],[524]) ).
thf(526,plain,
( ( v @ complement )
!= v ),
inference(simp,[status(thm)],[525]) ).
thf(696,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[356,516]) ).
thf(697,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( v @ complement @ ( u @ member ) ) ),
inference(simp,[status(thm)],[696]) ).
thf(706,plain,
! [B: $i,A: $i] :
( ( ( v @ complement )
!= universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= u ) ),
inference(simp,[status(thm)],[697]) ).
thf(79,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(328,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(147,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(471,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)],[147]) ).
thf(46,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(256,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)],[46]) ).
thf(32,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(225,plain,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(4275,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[511,184]) ).
thf(4276,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4275:[bind(A,$thf( choice ))]]) ).
thf(1477,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( v @ complement )
!= universal_class )
| ( ( A @ singleton )
!= u ) ),
inference(paramod_ordered,[status(thm)],[195,706]) ).
thf(1478,plain,
! [A: $i] :
( ( ( v @ complement )
!= universal_class )
| ( ( A @ singleton )
!= u ) ),
inference(pattern_uni,[status(thm)],[1477:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(561,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[452,459]) ).
thf(562,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[561]) ).
thf(568,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[562]) ).
thf(569,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[568]) ).
thf(101,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(371,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)],[101]) ).
thf(266,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[265]) ).
thf(537,plain,
( ( ( u @ complement @ ( v @ member ) )
!= ( v @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,533]) ).
thf(538,plain,
( ( u @ complement @ ( v @ member ) )
!= ( v @ ( u @ member ) ) ),
inference(simp,[status(thm)],[537]) ).
thf(541,plain,
( ( ( u @ complement )
!= v )
| ( v != u ) ),
inference(simp,[status(thm)],[538]) ).
thf(86,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(343,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)],[86]) ).
thf(38,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(237,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)],[38]) ).
thf(66,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(298,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)],[66]) ).
thf(125,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(425,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(128,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(430,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)],[128]) ).
thf(11,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(180,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(103,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(375,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)],[103]) ).
thf(9,axiom,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',kind_1_ordinals) ).
thf(175,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(176,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(lifteq,[status(thm)],[175]) ).
thf(124,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(423,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(47,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(258,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(259,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[258]) ).
thf(1205,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)],[259,361]) ).
thf(1206,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1205:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1256,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1206]) ).
thf(120,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(413,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)],[120]) ).
thf(114,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(401,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(402,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[401]) ).
thf(135,axiom,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions1) ).
thf(445,plain,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(83,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(335,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)],[83]) ).
thf(181,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
inference(cnf,[status(esa)],[180]) ).
thf(15,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(187,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(117,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(406,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(7,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(171,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)],[7]) ).
thf(172,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ transitive ) )
| ~ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[171]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(1572,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( v @ member ) ) )
| ( ( B @ complement )
!= u )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[434,536]) ).
thf(1573,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( v @ member ) ) )
| ( ( B @ complement )
!= u )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1572]) ).
thf(1574,plain,
( ( ( omega @ complement )
!= u )
| ( ( v @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[1573:[bind(A,$thf( v )),bind(B,$thf( omega ))]]) ).
thf(104,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(377,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)],[104]) ).
thf(75,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(318,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)],[75]) ).
thf(95,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(362,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(559,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[452,516]) ).
thf(560,plain,
( ( universal_class @ ( omega @ member ) )
!= ( v @ complement @ ( u @ member ) ) ),
inference(simp,[status(thm)],[559]) ).
thf(567,plain,
( ( ( v @ complement )
!= universal_class )
| ( omega != u ) ),
inference(simp,[status(thm)],[560]) ).
thf(149,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(477,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(108,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(387,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)],[108]) ).
thf(550,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[164,459]) ).
thf(551,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[550]) ).
thf(552,plain,
~ ( y @ ( y @ cross_product ) @ complement @ ( v @ ( u @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[551:[bind(A,$thf( v @ ( u @ ordered_pair ) )),bind(B,$thf( y @ ( y @ cross_product ) ))]]) ).
thf(626,plain,
( ( ( y @ ( y @ cross_product ) @ complement @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[452,552]) ).
thf(627,plain,
( ( y @ ( y @ cross_product ) @ complement @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[626]) ).
thf(631,plain,
( ( ( y @ ( y @ cross_product ) @ complement )
!= universal_class )
| ( ( v @ ( u @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[627]) ).
thf(26,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(213,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(152,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(483,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(198,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(330,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(113,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(399,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)],[113]) ).
thf(1473,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= v )
| ( ( universal_class @ complement )
!= u ) ),
inference(paramod_ordered,[status(thm)],[195,714]) ).
thf(1474,plain,
! [A: $i] :
( ( ( A @ singleton )
!= v )
| ( ( universal_class @ complement )
!= u ) ),
inference(pattern_uni,[status(thm)],[1473:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(51,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(267,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(111,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(394,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)],[111]) ).
thf(70,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(306,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)],[70]) ).
thf(155,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(490,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(45,axiom,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetrization) ).
thf(254,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(255,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(lifteq,[status(thm)],[254]) ).
thf(1661,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( B @ union ) )
= ( B @ symmetrization_of ) ) ),
inference(paramod_ordered,[status(thm)],[393,255]) ).
thf(1662,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(pattern_uni,[status(thm)],[1661:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(126,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(426,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(24,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',section1) ).
thf(209,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(544,plain,
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( u @ complement @ ( v @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,533]) ).
thf(545,plain,
( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( u @ complement @ ( v @ member ) ) ),
inference(simp,[status(thm)],[544]) ).
thf(553,plain,
( ( ( y @ ( y @ cross_product ) )
!= ( u @ complement ) )
| ( ( v @ ( u @ ordered_pair ) )
!= v ) ),
inference(simp,[status(thm)],[545]) ).
thf(605,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( A @ ( v @ member ) ) )
| ( ( A @ complement )
!= u )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,536]) ).
thf(606,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( A @ ( v @ member ) ) )
| ( ( A @ complement )
!= u ) ),
inference(simp,[status(thm)],[605]) ).
thf(614,plain,
! [A: $i] :
( ( ( y @ ( y @ cross_product ) )
!= A )
| ( ( v @ ( u @ ordered_pair ) )
!= v )
| ( ( A @ complement )
!= u ) ),
inference(simp,[status(thm)],[606]) ).
thf(617,plain,
( ( ( v @ ( u @ ordered_pair ) )
!= v )
| ( ( y @ ( y @ cross_product ) @ complement )
!= u ) ),
inference(simp,[status(thm)],[614]) ).
thf(702,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)],[356,459]) ).
thf(703,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)],[702]) ).
thf(704,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[703:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(712,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[704]) ).
thf(574,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( v @ ( u @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,565]) ).
thf(575,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( v @ ( u @ member ) ) ),
inference(simp,[status(thm)],[574]) ).
thf(580,plain,
( ( ( universal_class @ complement )
!= v )
| ( omega != u ) ),
inference(simp,[status(thm)],[575]) ).
thf(65,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(295,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)],[65]) ).
thf(58,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(282,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)],[58]) ).
thf(72,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(310,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(199,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[198]) ).
thf(133,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(441,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)],[133]) ).
thf(144,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(463,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)],[144]) ).
thf(93,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(359,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(158,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(497,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)],[158]) ).
thf(4239,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[224,4230]) ).
thf(4240,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[4239]) ).
thf(4257,plain,
( ( ( universal_class @ complement )
!= omega )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[4240]) ).
thf(21,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(203,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)],[21]) ).
thf(576,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( u @ ( v @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,565]) ).
thf(577,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( u @ ( v @ member ) ) ),
inference(simp,[status(thm)],[576]) ).
thf(581,plain,
( ( ( universal_class @ complement )
!= u )
| ( omega != v ) ),
inference(simp,[status(thm)],[577]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',well_ordering1) ).
thf(495,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(99,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(368,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(43,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(250,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(145,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(465,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)],[145]) ).
thf(35,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(231,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)],[35]) ).
thf(1568,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[434,516]) ).
thf(1569,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( v @ complement @ ( u @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1568]) ).
thf(1583,plain,
! [A: $i] :
( ( ( v @ complement )
!= omega )
| ( A != u )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1569]) ).
thf(1590,plain,
( ( ( v @ complement )
!= omega )
| ( ( u @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1583]) ).
thf(1475,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)],[195,356]) ).
thf(1476,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1475:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(69,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(304,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)],[69]) ).
thf(89,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(351,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)],[89]) ).
thf(186,plain,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
inference(lifteq,[status(thm)],[185]) ).
thf(620,plain,
( ( ( y @ ( y @ cross_product ) @ complement @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[164,552]) ).
thf(621,plain,
( ( y @ ( y @ cross_product ) @ complement @ ( v @ ( u @ ordered_pair ) @ member ) )
!= ( y @ ( y @ cross_product ) @ ( v @ ( u @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[620]) ).
thf(628,plain,
( ( ( y @ ( y @ cross_product ) @ complement )
!= ( y @ ( y @ cross_product ) ) )
| ( ( v @ ( u @ ordered_pair ) )
!= ( v @ ( u @ ordered_pair ) ) ) ),
inference(simp,[status(thm)],[621]) ).
thf(632,plain,
( ( y @ ( y @ cross_product ) @ complement )
!= ( y @ ( y @ cross_product ) ) ),
inference(simp,[status(thm)],[628]) ).
thf(122,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(418,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)],[122]) ).
thf(23,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(207,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(478,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[477]) ).
thf(37,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(235,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)],[37]) ).
thf(41,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(246,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(136,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(447,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(74,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(315,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(119,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(410,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)],[119]) ).
thf(1556,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( u @ complement @ ( v @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[434,533]) ).
thf(1557,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( u @ complement @ ( v @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1556]) ).
thf(1586,plain,
! [A: $i] :
( ( ( u @ complement )
!= omega )
| ( A != v )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1557]) ).
thf(1593,plain,
( ( ( u @ complement )
!= omega )
| ( ( v @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[1586]) ).
thf(188,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[187]) ).
thf(189,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[188]) ).
thf(190,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[189]) ).
thf(5370,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( ordinal_numbers @ ( successor_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[5312,176]) ).
thf(5371,plain,
( ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[5370:[bind(A,$thf( successor_relation )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(54,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).
thf(273,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(1479,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)],[195,712]) ).
thf(1480,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1479:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(102,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(373,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)],[102]) ).
thf(49,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(262,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(78,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(326,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(39,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(239,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(64,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(294,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( A @ recursion_equation_functions @ ( B @ member ) )
| ( ( B @ rest_of @ ( A @ compose ) )
!= B )
| ~ ( ordinal_numbers @ ( B @ domain_of @ member ) )
| ~ ( B @ function )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',recursion_equation_functions5) ).
thf(506,plain,
! [B: $i,A: $i] :
( ( A @ recursion_equation_functions @ ( B @ member ) )
| ( ( B @ rest_of @ ( A @ compose ) )
!= B )
| ~ ( ordinal_numbers @ ( B @ domain_of @ member ) )
| ~ ( B @ function )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(131,axiom,
! [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(437,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)],[131]) ).
thf(464,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)],[463]) ).
thf(56,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(277,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)],[56]) ).
thf(100,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(369,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)],[100]) ).
thf(1,negated_conjecture,
y @ ( element_relation @ well_ordering ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_to_well_ordering_property3_1) ).
thf(163,plain,
y @ ( element_relation @ well_ordering ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(20,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(200,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)],[20]) ).
thf(84,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(337,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)],[84]) ).
thf(90,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(354,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)],[90]) ).
thf(539,plain,
( ( ( u @ complement @ ( v @ member ) )
!= ( u @ ( v @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[166,533]) ).
thf(540,plain,
( ( u @ complement @ ( v @ member ) )
!= ( u @ ( v @ member ) ) ),
inference(simp,[status(thm)],[539]) ).
thf(542,plain,
( ( ( u @ complement )
!= u )
| ( v != v ) ),
inference(simp,[status(thm)],[540]) ).
thf(543,plain,
( ( u @ complement )
!= u ),
inference(simp,[status(thm)],[542]) ).
thf(27,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(215,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(140,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(454,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(96,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rest_of1) ).
thf(364,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(33,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(227,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)],[33]) ).
thf(5493,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[393,5371]) ).
thf(5494,plain,
( ( universal_class @ ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[5493:[bind(A,$thf( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) ))]]) ).
thf(57506,plain,
$false,
inference(e,[status(thm)],[365,479,347,468,481,333,249,170,582,308,449,511,269,217,276,5405,379,384,533,500,715,185,288,301,565,1482,174,404,389,184,504,357,196,460,284,416,289,261,4205,475,179,443,321,211,485,492,164,602,397,221,453,243,428,265,312,206,4230,280,292,248,439,381,1470,534,714,233,452,392,407,275,165,1255,302,260,349,252,197,361,329,456,324,229,566,1555,435,285,224,578,1259,403,356,169,420,488,339,554,193,476,526,706,328,471,256,225,4276,393,1478,173,569,4152,371,266,360,205,541,343,237,298,425,430,166,180,375,176,423,1256,259,413,191,402,445,335,181,187,406,172,219,1574,377,318,271,362,567,509,434,477,387,631,213,483,198,330,367,552,521,399,1474,267,167,394,306,490,1662,426,255,209,553,458,194,617,516,712,580,295,282,310,199,441,463,359,177,182,175,497,4257,203,581,495,368,250,465,231,1590,1476,304,258,351,186,632,401,171,418,207,478,536,287,235,246,447,315,432,410,1593,190,183,5371,273,1480,373,168,262,326,239,294,506,195,437,464,459,277,369,163,200,337,354,543,215,454,364,708,254,227,5312,5494]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : NUM069-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.12 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.24/0.49 % Computer : n009.cluster.edu
% 0.24/0.49 % Model : x86_64 x86_64
% 0.24/0.49 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.24/0.49 % Memory : 8046.5625MB
% 0.24/0.49 % OS : Linux 6.8.0-71-generic
% 0.24/0.49 % CPULimit : 300
% 0.24/0.49 % WCLimit : 300
% 0.24/0.49 % DateTime : Sat Sep 26 01:59:00 UTC 2026
% 0.24/0.50 % CPUTime :
% 0.24/0.50 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.39/1.36 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.63/1.80 % [INFO] Parsing done (434ms).
% 2.63/1.82 % [INFO] Running in sequential loop mode.
% 3.76/2.40 % [INFO] eprover registered as external prover.
% 3.76/2.42 % [INFO] Scanning for conjecture ...
% 4.17/2.61 % [INFO] Found a conjecture (or negated_conjecture) and 158 axioms. Running axiom selection ...
% 4.66/2.80 % [INFO] Axiom selection finished. Selected 158 axioms (removed 0 axioms).
% 4.99/3.01 % [INFO] Problem is propositional (TPTP CNF).
% 4.99/3.03 % [INFO] Type checking passed.
% 4.99/3.04 % [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 ...
% 187.89/48.60 % External prover 'e' found a proof!
% 187.89/48.60 % [INFO] Killing All external provers ...
% 187.89/48.60 % Time passed: 47904ms (effective reasoning time: 46775ms)
% 187.89/48.61 % 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
% 187.89/48.61 % No. of inferences in proof: 520
% 187.89/48.61 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 47904 ms resp. 46775 ms w/o parsing
% 176.04/48.82 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 176.04/48.83 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------