%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET518-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n001.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 08:30:13 AM UTC 2026
% Result : Unsatisfiable 188.33s 47.33s
% Output : Refutation 178.29s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 114
% Syntax : Number of formulae : 637 ( 186 unt; 0 typ; 0 def)
% Number of atoms : 1554 ( 647 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 6032 ( 826 ~; 687 |; 0 &;4519 @)
% ( 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 : 61 ( 58 usr; 16 con; 0-3 aty)
% Number of variables : 835 ( 0 ^; 835 !; 0 ?; 835 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(x_decl,type,
x: $i ).
thf(y_decl,type,
y: $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(function_decl,type,
function: $i > $o ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(choice_decl,type,
choice: $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(single_valued1_decl,type,
single_valued1: $i > $i ).
thf(single_valued2_decl,type,
single_valued2: $i > $i ).
thf(single_valued3_decl,type,
single_valued3: $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(application_function_decl,type,
application_function: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(compose_class_decl,type,
compose_class: $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(second_decl,type,
second: $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(singleton_relation_decl,type,
singleton_relation: $i ).
thf(42,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(196,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(18,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(144,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[144]) ).
thf(146,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[145]) ).
thf(2,negated_conjecture,
x @ ( y @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_no_cycles_length_2_2) ).
thf(116,plain,
x @ ( y @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(96,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(311,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(312,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[311]) ).
thf(362,plain,
! [B: $i,A: $i] :
( ( ( x @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[116,312]) ).
thf(363,plain,
! [B: $i,A: $i] :
( ( ( x @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[362]) ).
thf(364,plain,
~ ( x @ complement @ ( y @ member ) ),
inference(pattern_uni,[status(thm)],[363:[bind(A,$thf( y )),bind(B,$thf( x ))]]) ).
thf(394,plain,
( ( ( x @ complement @ ( y @ member ) )
!= ( x @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[116,364]) ).
thf(395,plain,
( ( x @ complement @ ( y @ member ) )
!= ( x @ ( y @ member ) ) ),
inference(simp,[status(thm)],[394]) ).
thf(398,plain,
( ( ( x @ complement )
!= x )
| ( y != y ) ),
inference(simp,[status(thm)],[395]) ).
thf(399,plain,
( ( x @ complement )
!= x ),
inference(simp,[status(thm)],[398]) ).
thf(2404,plain,
! [B: $i,A: $i] :
( ( A
!= ( x @ complement ) )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[146,399]) ).
thf(2405,plain,
! [A: $i] :
( ( A != x )
| ~ ( x @ complement @ ( A @ subclass ) )
| ~ ( A @ ( x @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2404:[bind(A,$thf( x @ complement )),bind(B,$thf( B ))]]) ).
thf(3076,plain,
( ~ ( x @ complement @ ( x @ subclass ) )
| ~ ( x @ ( x @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[2405]) ).
thf(4339,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( x @ complement @ subclass ) ) )
| ~ ( x @ complement @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,3076]) ).
thf(4340,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( x @ complement @ subclass ) ) )
| ~ ( x @ complement @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[4339]) ).
thf(4449,plain,
! [A: $i] :
( ( universal_class != x )
| ( A
!= ( x @ complement ) )
| ~ ( x @ complement @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[4340]) ).
thf(4477,plain,
( ( universal_class != x )
| ~ ( x @ complement @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[4449]) ).
thf(4486,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ complement @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,4477]) ).
thf(4487,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ complement @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4486]) ).
thf(4542,plain,
! [A: $i] :
( ( ( x @ complement )
!= universal_class )
| ( A != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4487]) ).
thf(4560,plain,
( ( ( x @ complement )
!= universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4542]) ).
thf(113,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(349,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(350,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[349]) ).
thf(351,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[350]) ).
thf(1,negated_conjecture,
y @ ( x @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_no_cycles_length_2_1) ).
thf(115,plain,
y @ ( x @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(357,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[115,312]) ).
thf(358,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[357]) ).
thf(359,plain,
~ ( y @ complement @ ( x @ member ) ),
inference(pattern_uni,[status(thm)],[358:[bind(A,$thf( x )),bind(B,$thf( y ))]]) ).
thf(373,plain,
( ( ( y @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[115,359]) ).
thf(374,plain,
( ( y @ complement @ ( x @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[373]) ).
thf(377,plain,
( ( ( y @ complement )
!= y )
| ( x != x ) ),
inference(simp,[status(thm)],[374]) ).
thf(379,plain,
( ( y @ complement )
!= y ),
inference(simp,[status(thm)],[377]) ).
thf(1097,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,379]) ).
thf(1098,plain,
( ( null_class != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1097:[bind(A,$thf( y @ complement ))]]) ).
thf(6542,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1098,312]) ).
thf(6543,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[6542]) ).
thf(6544,plain,
( ~ ( y @ ( y @ complement @ regular @ member ) )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[6543:[bind(A,$thf( y @ complement @ regular )),bind(B,$thf( y ))]]) ).
thf(62,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(237,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(1230,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,237]) ).
thf(1231,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1230:[bind(A,$thf( omega ))]]) ).
thf(1835,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1231,364]) ).
thf(1836,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1835]) ).
thf(1840,plain,
( ( ( x @ complement )
!= omega )
| ( ( omega @ regular )
!= y )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1836]) ).
thf(114,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(352,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)],[114]) ).
thf(16,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(138,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)],[16]) ).
thf(56,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(224,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(225,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[224]) ).
thf(3,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(117,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(118,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[117]) ).
thf(1606,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[225,118]) ).
thf(49,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(209,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(210,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[209]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(147,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(148,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[147]) ).
thf(631,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)],[210,148]) ).
thf(632,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[631:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(701,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[632]) ).
thf(1618,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ diagonalise @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( C @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1606,701]) ).
thf(1619,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1618:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(1649,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1619]) ).
thf(23,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(154,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)],[23]) ).
thf(155,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[154]) ).
thf(6669,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[1649,155]) ).
thf(94,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(308,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(384,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[308,312]) ).
thf(385,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[384]) ).
thf(386,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[385:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(404,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( x @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[116,386]) ).
thf(405,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( x @ ( y @ member ) ) ),
inference(simp,[status(thm)],[404]) ).
thf(408,plain,
( ( ( universal_class @ complement )
!= x )
| ( omega != y ) ),
inference(simp,[status(thm)],[405]) ).
thf(79,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(276,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(28,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(165,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(166,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[165]) ).
thf(8988,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[166,118]) ).
thf(46,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(202,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)],[46]) ).
thf(4,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(119,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(120,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[119]) ).
thf(68,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(247,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)],[68]) ).
thf(59,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(232,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(1425,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,232]) ).
thf(1426,plain,
( ( null_class @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1425:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(109,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(340,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)],[109]) ).
thf(33,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(174,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)],[33]) ).
thf(1267,plain,
! [A: $i] :
( ( A
!= ( x @ complement ) )
| ( null_class != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,399]) ).
thf(1268,plain,
( ( null_class != x )
| ( x @ complement @ ( x @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1267:[bind(A,$thf( x @ complement ))]]) ).
thf(365,plain,
! [B: $i,A: $i] :
( ~ $true
| ( ( B @ complement @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[312]) ).
thf(367,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ( A != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[365]) ).
thf(372,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[367]) ).
thf(8814,plain,
! [B: $i,A: $i] :
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[1268,372]) ).
thf(8815,plain,
! [B: $i,A: $i] :
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( null_class != x ) ),
inference(simp,[status(thm)],[8814]) ).
thf(8816,plain,
( ( ( x @ complement @ complement )
!= ( x @ complement ) )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[8815:[bind(A,$thf( x @ complement @ regular )),bind(B,$thf( x @ complement ))]]) ).
thf(25,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(159,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(160,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[159]) ).
thf(40,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(192,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(193,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[192]) ).
thf(1045,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( B @ compose ) @ not_subclass_element ) @ first )
= ( B @ single_valued1 ) ) ),
inference(paramod_ordered,[status(thm)],[160,193]) ).
thf(1046,plain,
! [A: $i] :
( ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(pattern_uni,[status(thm)],[1045:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(50,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(211,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(212,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[211]) ).
thf(1555,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ inverse ) )
| ( ( B @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( B @ domain_of @ intersection ) )
= ( B @ cantor ) ) ),
inference(paramod_ordered,[status(thm)],[160,212]) ).
thf(1556,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[1555:[bind(A,$thf( element_relation ))]]) ).
thf(1601,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(simp,[status(thm)],[1556]) ).
thf(91,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(301,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)],[91]) ).
thf(15,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(137,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(121,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(122,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[121]) ).
thf(434,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( A @ one_to_one )
| ~ $true
| ~ ( A @ inverse @ function ) ),
inference(paramod_ordered,[status(thm)],[137,122]) ).
thf(435,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( A @ one_to_one )
| ~ ( A @ inverse @ function ) ),
inference(simp,[status(thm)],[434]) ).
thf(436,plain,
( ( choice @ one_to_one )
| ~ ( choice @ inverse @ function ) ),
inference(pattern_uni,[status(thm)],[435:[bind(A,$thf( choice ))]]) ).
thf(72,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(257,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)],[72]) ).
thf(382,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[308,312]) ).
thf(383,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[382]) ).
thf(388,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[383]) ).
thf(389,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[388]) ).
thf(20,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(149,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(150,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[149]) ).
thf(82,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(282,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(750,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)],[150,282]) ).
thf(751,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[750:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
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(177,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(178,plain,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
inference(lifteq,[status(thm)],[177]) ).
thf(12713,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( A @ ( C @ singleton @ ( B @ cross_product ) @ intersection ) @ not_subclass_element ) @ first ) ),
inference(rewrite,[status(thm)],[178,155]) ).
thf(7,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(125,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(111,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(344,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(39,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(190,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(191,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[190]) ).
thf(16718,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[237,191]) ).
thf(16719,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[16718]) ).
thf(16720,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[16719:[bind(A,$thf( omega ))]]) ).
thf(402,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[308,386]) ).
thf(403,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[402]) ).
thf(407,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[403]) ).
thf(409,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[407]) ).
thf(2454,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[146,409]) ).
thf(2455,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2454:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(3099,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[2455]) ).
thf(10337,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[3099,196]) ).
thf(10338,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[10337]) ).
thf(16797,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[16720,10338]) ).
thf(16798,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[16797]) ).
thf(16856,plain,
( ( ( universal_class @ complement )
!= omega )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[16798]) ).
thf(4663,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ image ) )
= ( A @ ( universal_class @ ( B @ cross_product ) @ intersection ) @ inverse @ domain_of ) ),
inference(rewrite,[status(thm)],[1606,155]) ).
thf(17097,plain,
( ( ( universal_class @ complement )
!= omega )
| ( ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of )
!= universal_class ) ),
inference(rewrite,[status(thm)],[16856,4663]) ).
thf(8817,plain,
! [B: $i,A: $i] :
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[1268,312]) ).
thf(8818,plain,
! [B: $i,A: $i] :
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[8817]) ).
thf(8819,plain,
( ~ ( x @ ( x @ complement @ regular @ member ) )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[8818:[bind(A,$thf( x @ complement @ regular )),bind(B,$thf( x ))]]) ).
thf(103,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(325,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(22,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(152,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(86,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(289,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)],[86]) ).
thf(74,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(261,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(355,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[115,312]) ).
thf(356,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[355]) ).
thf(369,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= y )
| ( x != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[356]) ).
thf(371,plain,
! [A: $i] :
( ( ( A @ complement )
!= y )
| ~ ( A @ ( x @ member ) ) ),
inference(simp,[status(thm)],[369]) ).
thf(1819,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( A @ ( x @ member ) ) )
| ( ( A @ complement )
!= y )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1231,371]) ).
thf(1820,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( A @ ( x @ member ) ) )
| ( ( A @ complement )
!= y )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1819]) ).
thf(1841,plain,
! [A: $i] :
( ( omega != A )
| ( ( omega @ regular )
!= x )
| ( ( A @ complement )
!= y )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1820]) ).
thf(1847,plain,
( ( ( omega @ regular )
!= x )
| ( ( omega @ complement )
!= y )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1841]) ).
thf(16953,plain,
omega @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ),
inference(rewrite,[status(thm)],[16720,4663]) ).
thf(390,plain,
( ( ( x @ complement @ ( y @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[115,364]) ).
thf(391,plain,
( ( x @ complement @ ( y @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[390]) ).
thf(396,plain,
( ( ( x @ complement )
!= y )
| ( y != x ) ),
inference(simp,[status(thm)],[391]) ).
thf(2070,plain,
! [B: $i,A: $i] :
( ( A != y )
| ( ( x @ complement )
!= y )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[146,396]) ).
thf(2071,plain,
! [A: $i] :
( ( ( x @ complement )
!= y )
| ( A != x )
| ~ ( y @ ( A @ subclass ) )
| ~ ( A @ ( y @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2070:[bind(A,$thf( y ))]]) ).
thf(2944,plain,
( ( ( x @ complement )
!= y )
| ~ ( y @ ( x @ subclass ) )
| ~ ( x @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[2071]) ).
thf(21751,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( y @ subclass ) ) )
| ( ( x @ complement )
!= y )
| ~ ( y @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,2944]) ).
thf(21752,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( y @ subclass ) ) )
| ( ( x @ complement )
!= y )
| ~ ( y @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[21751]) ).
thf(21892,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != y )
| ( ( x @ complement )
!= y )
| ~ ( y @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[21752]) ).
thf(21946,plain,
( ( universal_class != x )
| ( ( x @ complement )
!= y )
| ~ ( y @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[21892]) ).
thf(21965,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( x @ subclass ) ) )
| ( universal_class != x )
| ( ( x @ complement )
!= y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,21946]) ).
thf(21966,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( x @ subclass ) ) )
| ( universal_class != x )
| ( ( x @ complement )
!= y ) ),
inference(simp,[status(thm)],[21965]) ).
thf(22066,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != x )
| ( universal_class != x )
| ( ( x @ complement )
!= y ) ),
inference(simp,[status(thm)],[21966]) ).
thf(22089,plain,
( ( universal_class != y )
| ( universal_class != x )
| ( ( x @ complement )
!= y ) ),
inference(simp,[status(thm)],[22066]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(216,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(8335,plain,
( ( ( y @ ( y @ complement @ regular @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[308,6544]) ).
thf(8336,plain,
( ( ( y @ ( y @ complement @ regular @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[8335]) ).
thf(8394,plain,
( ( universal_class != y )
| ( ( y @ complement @ regular )
!= omega )
| ( null_class != y ) ),
inference(simp,[status(thm)],[8336]) ).
thf(27,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(163,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(164,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[163]) ).
thf(35,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(179,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)],[35]) ).
thf(101,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(321,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(63,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(238,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(392,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[308,364]) ).
thf(393,plain,
( ( universal_class @ ( omega @ member ) )
!= ( x @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[392]) ).
thf(397,plain,
( ( ( x @ complement )
!= universal_class )
| ( omega != y ) ),
inference(simp,[status(thm)],[393]) ).
thf(81,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(280,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)],[81]) ).
thf(11,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(132,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(12,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).
thf(133,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(47,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(204,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)],[47]) ).
thf(205,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[204]) ).
thf(206,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[205]) ).
thf(9750,plain,
( ( ( x @ ( x @ complement @ regular @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[115,8819]) ).
thf(9751,plain,
( ( ( x @ ( x @ complement @ regular @ member ) )
!= ( y @ ( x @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[9750]) ).
thf(9822,plain,
( ( y != x )
| ( ( x @ complement @ regular )
!= x )
| ( null_class != x ) ),
inference(simp,[status(thm)],[9751]) ).
thf(66,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(243,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(87,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(292,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)],[87]) ).
thf(60,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(233,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)],[60]) ).
thf(16809,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,16720]) ).
thf(16810,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16809:[bind(A,$thf( successor_relation ))]]) ).
thf(17257,plain,
( ( omega @ ( null_class @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(rewrite,[status(thm)],[16810,4663]) ).
thf(45,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(200,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(201,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[200]) ).
thf(54,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(220,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(43,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(197,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(1162,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,197]) ).
thf(1163,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1162:[bind(A,$thf( successor_relation ))]]) ).
thf(16972,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[160,16953]) ).
thf(16973,plain,
omega @ ( universal_class @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[16972:[bind(A,$thf( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) ))]]) ).
thf(1318,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[351,359]) ).
thf(1319,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1318]) ).
thf(1431,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( ( A @ regular )
!= x )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1319]) ).
thf(1502,plain,
( ( ( y @ complement @ regular )
!= x )
| ( ( y @ complement )
= null_class ) ),
inference(simp,[status(thm)],[1431]) ).
thf(16813,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,16720]) ).
thf(16814,plain,
( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16813:[bind(A,$thf( omega ))]]) ).
thf(22092,plain,
( ( null_class @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(rewrite,[status(thm)],[16814,4663]) ).
thf(89,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(297,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(30,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(169,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(170,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[169]) ).
thf(4661,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[170,155]) ).
thf(58,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).
thf(229,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(70,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(252,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)],[70]) ).
thf(466,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( x @ member ) ) )
| ( ( C @ complement )
!= y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[282,371]) ).
thf(467,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( x @ member ) ) )
| ( ( C @ complement )
!= y ) ),
inference(simp,[status(thm)],[466]) ).
thf(487,plain,
! [C: $i,B: $i,A: $i] :
( ( universal_class != C )
| ( ( B @ ( A @ unordered_pair ) )
!= x )
| ( ( C @ complement )
!= y ) ),
inference(simp,[status(thm)],[467]) ).
thf(493,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= x )
| ( ( universal_class @ complement )
!= y ) ),
inference(simp,[status(thm)],[487]) ).
thf(480,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[282,359]) ).
thf(481,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[480]) ).
thf(491,plain,
! [B: $i,A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= x ) ),
inference(simp,[status(thm)],[481]) ).
thf(756,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( y @ complement )
!= universal_class )
| ( ( A @ singleton )
!= x ) ),
inference(paramod_ordered,[status(thm)],[150,491]) ).
thf(757,plain,
! [A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( ( A @ singleton )
!= x ) ),
inference(pattern_uni,[status(thm)],[756:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(8339,plain,
( ( ( y @ ( y @ complement @ regular @ member ) )
!= ( x @ ( y @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[116,6544]) ).
thf(8340,plain,
( ( ( y @ ( y @ complement @ regular @ member ) )
!= ( x @ ( y @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[8339]) ).
thf(8403,plain,
( ( y != x )
| ( ( y @ complement @ regular )
!= y )
| ( null_class != y ) ),
inference(simp,[status(thm)],[8340]) ).
thf(84,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(285,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)],[84]) ).
thf(6588,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,1098]) ).
thf(6589,plain,
! [A: $i] :
( ( A != y )
| ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[6588:[]]) ).
thf(6656,plain,
( ( y @ complement @ ( y @ complement @ regular @ member ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[6589]) ).
thf(99,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(317,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(24,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(156,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)],[24]) ).
thf(8,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(127,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(128,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[127]) ).
thf(1226,plain,
! [A: $i] :
( ( A
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,128]) ).
thf(1227,plain,
( ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( subset_relation @ inverse @ ( subset_relation @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1226:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(104,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(328,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(1607,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)],[1606,148]) ).
thf(1608,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[1607:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(1646,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[1608]) ).
thf(1686,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[701,1646]) ).
thf(1687,plain,
! [A: $i] :
( ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1686:[bind(A,$thf( E )),bind(B,$thf( E @ diagonalise @ ( element_relation @ image ) ))]]) ).
thf(1788,plain,
! [A: $i] :
( ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1687]) ).
thf(22533,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[1788,4663,155]) ).
thf(475,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)],[282,312]) ).
thf(476,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)],[475]) ).
thf(477,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[476:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(497,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[477]) ).
thf(752,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)],[150,497]) ).
thf(753,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[752:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(6539,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1098,372]) ).
thf(6540,plain,
! [B: $i,A: $i] :
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( null_class != y ) ),
inference(simp,[status(thm)],[6539]) ).
thf(6541,plain,
( ( ( y @ complement @ complement )
!= ( y @ complement ) )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[6540:[bind(A,$thf( y @ complement @ regular )),bind(B,$thf( y @ complement ))]]) ).
thf(1423,plain,
! [A: $i] :
( ( A != domain_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,232]) ).
thf(1424,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( domain_relation @ ( domain_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1423:[bind(A,$thf( domain_relation ))]]) ).
thf(512,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)],[148,148]) ).
thf(513,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[512:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(552,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[513]) ).
thf(966,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[701,552]) ).
thf(967,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[966:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1039,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[967]) ).
thf(20456,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ diagonalise @ ( element_relation @ image ) @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( element_relation @ ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[1039,4663]) ).
thf(32,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(172,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(173,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[172]) ).
thf(9,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(129,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(175,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(cnf,[status(esa)],[174]) ).
thf(176,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[175]) ).
thf(11610,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[176,118]) ).
thf(134,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[133]) ).
thf(244,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[243]) ).
thf(76,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(266,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)],[76]) ).
thf(1134,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,132]) ).
thf(1135,plain,
( ( null_class @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1134:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(75,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(264,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(26,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(161,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)],[26]) ).
thf(10353,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,10338]) ).
thf(10354,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[10353]) ).
thf(10420,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[10354]) ).
thf(4662,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(rewrite,[status(thm)],[1646,155]) ).
thf(1128,plain,
! [A: $i] :
( ( A != omega )
| ( ( x @ complement )
!= universal_class )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,397]) ).
thf(1129,plain,
( ( ( x @ complement )
!= universal_class )
| ( null_class != y )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1128:[bind(A,$thf( omega ))]]) ).
thf(18882,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( ( x @ complement )
!= universal_class )
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1129,372]) ).
thf(18883,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( ( x @ complement )
!= universal_class )
| ( null_class != y ) ),
inference(simp,[status(thm)],[18882]) ).
thf(18884,plain,
( ( ( omega @ complement )
!= omega )
| ( ( x @ complement )
!= universal_class )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[18883:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(73,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(259,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(180,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second ) ),
inference(lifteq,[status(thm)],[179]) ).
thf(13358,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( A @ ( C @ ( B @ singleton @ cross_product ) @ intersection ) @ not_subclass_element ) @ second ) ),
inference(rewrite,[status(thm)],[180,155]) ).
thf(97,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(313,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(6572,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1098,359]) ).
thf(6573,plain,
( ( ( y @ complement @ ( y @ complement @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[6572]) ).
thf(6639,plain,
( ( ( y @ complement )
!= ( y @ complement ) )
| ( ( y @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[6573]) ).
thf(6667,plain,
( ( ( y @ complement @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[6639]) ).
thf(107,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(335,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)],[107]) ).
thf(36,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(181,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)],[36]) ).
thf(9752,plain,
( ( ( x @ ( x @ complement @ regular @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[308,8819]) ).
thf(9753,plain,
( ( ( x @ ( x @ complement @ regular @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[9752]) ).
thf(9808,plain,
( ( universal_class != x )
| ( ( x @ complement @ regular )
!= omega )
| ( null_class != x ) ),
inference(simp,[status(thm)],[9753]) ).
thf(1089,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,308]) ).
thf(1090,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1089:[bind(A,$thf( omega ))]]) ).
thf(38,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(187,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)],[38]) ).
thf(400,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( y @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[115,386]) ).
thf(401,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( y @ ( x @ member ) ) ),
inference(simp,[status(thm)],[400]) ).
thf(406,plain,
( ( ( universal_class @ complement )
!= y )
| ( omega != x ) ),
inference(simp,[status(thm)],[401]) ).
thf(29,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(167,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(168,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[167]) ).
thf(9838,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[137,168]) ).
thf(9839,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[9838]) ).
thf(9840,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[9839:[bind(A,$thf( choice ))]]) ).
thf(67,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(245,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(246,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[245]) ).
thf(1047,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( B @ compose ) @ not_subclass_element ) @ second )
= ( B @ single_valued2 ) ) ),
inference(paramod_ordered,[status(thm)],[160,246]) ).
thf(1048,plain,
! [A: $i] :
( ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(pattern_uni,[status(thm)],[1047:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(566,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[160,128]) ).
thf(567,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[566:[bind(A,$thf( subset_relation ))]]) ).
thf(16811,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,16720]) ).
thf(16812,plain,
( ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16811:[bind(A,$thf( omega ))]]) ).
thf(18743,plain,
( ( omega @ ( successor_relation @ ( universal_class @ ( null_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(rewrite,[status(thm)],[16812,4663]) ).
thf(92,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(303,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)],[92]) ).
thf(9889,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[160,9840]) ).
thf(9890,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[9889:[bind(A,$thf( choice ))]]) ).
thf(1966,plain,
! [B: $i,A: $i] :
( ( A
!= ( y @ complement ) )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[146,379]) ).
thf(1967,plain,
! [A: $i] :
( ( A != y )
| ~ ( y @ complement @ ( A @ subclass ) )
| ~ ( A @ ( y @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1966:[bind(A,$thf( y @ complement )),bind(B,$thf( B ))]]) ).
thf(3334,plain,
( ~ ( y @ complement @ ( y @ subclass ) )
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[1967]) ).
thf(12296,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ~ $true
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[196,3334]) ).
thf(12297,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[12296]) ).
thf(12438,plain,
! [A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( A != y )
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[12297]) ).
thf(12464,plain,
( ( ( y @ complement )
!= universal_class )
| ~ ( y @ ( y @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[12438]) ).
thf(102,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(323,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(8889,plain,
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[1268,364]) ).
thf(8890,plain,
( ( ( x @ complement @ ( x @ complement @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[8889]) ).
thf(8912,plain,
( ( ( x @ complement )
!= ( x @ complement ) )
| ( ( x @ complement @ regular )
!= y )
| ( null_class != x ) ),
inference(simp,[status(thm)],[8890]) ).
thf(8932,plain,
( ( ( x @ complement @ regular )
!= y )
| ( null_class != x ) ),
inference(simp,[status(thm)],[8912]) ).
thf(8948,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( x @ complement @ regular )
!= y )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,8932]) ).
thf(8949,plain,
! [A: $i] :
( ( ( x @ complement @ regular )
!= y )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8948:[]]) ).
thf(8983,plain,
( ( ( x @ complement @ regular )
!= y )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[8949]) ).
thf(17,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(140,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(51,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(213,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(44,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(198,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)],[44]) ).
thf(105,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(330,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(380,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[308,359]) ).
thf(381,plain,
( ( universal_class @ ( omega @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[380]) ).
thf(387,plain,
( ( ( y @ complement )
!= universal_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[381]) ).
thf(130,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[129]) ).
thf(80,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(278,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)],[80]) ).
thf(964,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[701,552]) ).
thf(965,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[964:[bind(A,$thf( E )),bind(B,$thf( E @ diagonalise @ ( element_relation @ image ) ))]]) ).
thf(1038,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[965]) ).
thf(19866,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( element_relation @ ( universal_class @ ( A @ diagonalise @ ( element_relation @ image ) @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[1038,4663]) ).
thf(13,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(135,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(90,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(299,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)],[90]) ).
thf(10351,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[132,10338]) ).
thf(10352,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[10351]) ).
thf(10424,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[10352]) ).
thf(57,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(226,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)],[57]) ).
thf(65,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(241,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(8855,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( x @ complement @ ( x @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,1268]) ).
thf(8856,plain,
! [A: $i] :
( ( A != x )
| ( x @ complement @ ( x @ complement @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8855:[]]) ).
thf(8922,plain,
( ( x @ complement @ ( x @ complement @ regular @ member ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[8856]) ).
thf(71,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(255,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)],[71]) ).
thf(748,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= x )
| ( ( universal_class @ complement )
!= y ) ),
inference(paramod_ordered,[status(thm)],[150,493]) ).
thf(749,plain,
! [A: $i] :
( ( ( A @ singleton )
!= x )
| ( ( universal_class @ complement )
!= y ) ),
inference(pattern_uni,[status(thm)],[748:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(41,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(194,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(110,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(342,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)],[110]) ).
thf(6,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(123,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(1100,plain,
! [A: $i] :
( ( A != omega )
| ( ( universal_class @ complement )
!= x )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,408]) ).
thf(1101,plain,
( ( ( universal_class @ complement )
!= x )
| ( null_class != y )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1100:[bind(A,$thf( omega ))]]) ).
thf(17391,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( ( universal_class @ complement )
!= x )
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[1101,372]) ).
thf(17392,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( ( universal_class @ complement )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[17391]) ).
thf(17393,plain,
( ( ( omega @ complement )
!= omega )
| ( ( universal_class @ complement )
!= x )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[17392:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(88,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(295,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(8132,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( y @ complement @ regular )
!= x )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,6667]) ).
thf(8133,plain,
! [A: $i] :
( ( ( y @ complement @ regular )
!= x )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8132:[]]) ).
thf(8163,plain,
( ( ( y @ complement @ regular )
!= x )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[8133]) ).
thf(1421,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[351,364]) ).
thf(1422,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1421]) ).
thf(1429,plain,
! [A: $i] :
( ( A
!= ( x @ complement ) )
| ( ( A @ regular )
!= y )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1422]) ).
thf(1499,plain,
( ( ( x @ complement @ regular )
!= y )
| ( ( x @ complement )
= null_class ) ),
inference(simp,[status(thm)],[1429]) ).
thf(199,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)],[198]) ).
thf(360,plain,
! [B: $i,A: $i] :
( ( ( x @ ( y @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[116,312]) ).
thf(361,plain,
! [B: $i,A: $i] :
( ( ( x @ ( y @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[360]) ).
thf(368,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= x )
| ( y != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[361]) ).
thf(370,plain,
! [A: $i] :
( ( ( A @ complement )
!= x )
| ~ ( A @ ( y @ member ) ) ),
inference(simp,[status(thm)],[368]) ).
thf(108,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(338,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(1698,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1646,552]) ).
thf(1699,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1698:[bind(A,$thf( H )),bind(B,$thf( universal_class @ ( H @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of ))]]) ).
thf(1791,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1699]) ).
thf(23138,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( element_relation @ ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[1791,4663]) ).
thf(182,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[181]) ).
thf(864,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[552,552]) ).
thf(865,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[864:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(925,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[865]) ).
thf(18007,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ ( element_relation @ image ) @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( element_relation @ ( universal_class @ ( A @ power_class @ ( element_relation @ image ) @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[925,4663]) ).
thf(1829,plain,
! [A: $i] :
( ( A
!= ( omega @ regular ) )
| ( null_class @ inductive )
| ( omega @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,1231]) ).
thf(1830,plain,
( ( null_class @ inductive )
| ( omega @ ( null_class @ member ) )
| ( omega @ regular @ ( omega @ regular @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1829:[bind(A,$thf( omega @ regular ))]]) ).
thf(126,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[125]) ).
thf(9928,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9840,126]) ).
thf(9929,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[9928]) ).
thf(9930,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[9929:[bind(A,$thf( choice ))]]) ).
thf(141,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[140]) ).
thf(142,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[141]) ).
thf(143,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[142]) ).
thf(1823,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1231,359]) ).
thf(1824,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1823]) ).
thf(1839,plain,
( ( ( y @ complement )
!= omega )
| ( ( omega @ regular )
!= x )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1824]) ).
thf(203,plain,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[202]) ).
thf(478,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( y @ member ) ) )
| ( ( C @ complement )
!= x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[282,370]) ).
thf(479,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( y @ member ) ) )
| ( ( C @ complement )
!= x ) ),
inference(simp,[status(thm)],[478]) ).
thf(490,plain,
! [C: $i,B: $i,A: $i] :
( ( universal_class != C )
| ( ( B @ ( A @ unordered_pair ) )
!= y )
| ( ( C @ complement )
!= x ) ),
inference(simp,[status(thm)],[479]) ).
thf(496,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= y )
| ( ( universal_class @ complement )
!= x ) ),
inference(simp,[status(thm)],[490]) ).
thf(758,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= y )
| ( ( universal_class @ complement )
!= x ) ),
inference(paramod_ordered,[status(thm)],[150,496]) ).
thf(759,plain,
! [A: $i] :
( ( ( A @ singleton )
!= y )
| ( ( universal_class @ complement )
!= x ) ),
inference(pattern_uni,[status(thm)],[758:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(53,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(218,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(69,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(250,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)],[69]) ).
thf(124,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[123]) ).
thf(465,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(rewrite,[status(thm)],[124,118]) ).
thf(100,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(319,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(31,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(171,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(10398,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[232,10338]) ).
thf(10399,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[10398]) ).
thf(10400,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[10399]) ).
thf(157,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[156]) ).
thf(158,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[157]) ).
thf(188,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[187]) ).
thf(189,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[188]) ).
thf(16180,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[189,118]) ).
thf(48,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(207,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)],[48]) ).
thf(1678,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( universal_class @ ( B @ complement @ ( element_relation @ restrict ) ) @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[160,1646]) ).
thf(1679,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[1678:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(1786,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[1679]) ).
thf(8420,plain,
! [A: $i] :
( ( universal_class @ ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(rewrite,[status(thm)],[1786,155]) ).
thf(12294,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( y @ complement @ subclass ) ) )
| ~ ( y @ complement @ ( y @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,3334]) ).
thf(12295,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( y @ complement @ subclass ) ) )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[12294]) ).
thf(12430,plain,
! [A: $i] :
( ( universal_class != y )
| ( A
!= ( y @ complement ) )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[12295]) ).
thf(12463,plain,
( ( universal_class != y )
| ~ ( y @ complement @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[12430]) ).
thf(4341,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ complement @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( x @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[196,3076]) ).
thf(4342,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ complement @ ( x @ subclass ) ) )
| ~ ( x @ ( x @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[4341]) ).
thf(4461,plain,
! [A: $i] :
( ( ( x @ complement )
!= universal_class )
| ( A != x )
| ~ ( x @ ( x @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[4342]) ).
thf(4478,plain,
( ( ( x @ complement )
!= universal_class )
| ~ ( x @ ( x @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[4461]) ).
thf(78,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(272,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)],[78]) ).
thf(112,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(346,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(85,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(287,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)],[85]) ).
thf(375,plain,
( ( ( x @ ( y @ member ) )
!= ( y @ complement @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[116,359]) ).
thf(376,plain,
( ( x @ ( y @ member ) )
!= ( y @ complement @ ( x @ member ) ) ),
inference(simp,[status(thm)],[375]) ).
thf(378,plain,
( ( ( y @ complement )
!= x )
| ( y != x ) ),
inference(simp,[status(thm)],[376]) ).
thf(61,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(235,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(21,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(151,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(98,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(315,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)],[98]) ).
thf(37,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(183,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(6715,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[6669,701]) ).
thf(6716,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[6715:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(984,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[210,701]) ).
thf(985,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[984:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(5596,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[985,701]) ).
thf(5597,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[5596:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1658,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1646,148]) ).
thf(1659,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1658:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(7910,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[1659,155]) ).
thf(11822,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( element_relation @ ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[5597,7910]) ).
thf(13941,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement ) ),
inference(rewrite,[status(thm)],[6716,11822]) ).
thf(153,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[152]) ).
thf(12272,plain,
! [A: $i] :
( ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ complement )
= ( A @ successor ) ),
inference(rewrite,[status(thm)],[153,201]) ).
thf(93,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(305,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)],[93]) ).
thf(64,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(239,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)],[64]) ).
thf(77,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(268,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)],[77]) ).
thf(195,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[194]) ).
thf(16716,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[191,10338]) ).
thf(16717,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16716]) ).
thf(16772,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16717]) ).
thf(16777,plain,
( ( ( universal_class @ complement @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[16772]) ).
thf(18538,plain,
( ( ( successor_relation @ ( universal_class @ ( universal_class @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of )
!= universal_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(rewrite,[status(thm)],[16777,4663]) ).
thf(12615,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ( universal_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[196,12463]) ).
thf(12616,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ complement @ ( y @ subclass ) ) )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[12615]) ).
thf(12701,plain,
! [A: $i] :
( ( ( y @ complement )
!= universal_class )
| ( A != y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[12616]) ).
thf(12707,plain,
( ( ( y @ complement )
!= universal_class )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[12701]) ).
thf(1164,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[351,197]) ).
thf(1165,plain,
( ( null_class @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1164:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(83,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(283,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)],[83]) ).
thf(14,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(136,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(462,plain,
( ( ( choice @ inverse @ function )
!= ( choice @ function ) )
| ( choice @ one_to_one )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[137,436]) ).
thf(463,plain,
( ( ( choice @ inverse @ function )
!= ( choice @ function ) )
| ( choice @ one_to_one ) ),
inference(simp,[status(thm)],[462]) ).
thf(464,plain,
( ( ( choice @ inverse )
!= choice )
| ( choice @ one_to_one ) ),
inference(simp,[status(thm)],[463]) ).
thf(10361,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9840,10338]) ).
thf(10362,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[10361]) ).
thf(10403,plain,
( ( ( universal_class @ complement )
!= identity_relation )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[10362]) ).
thf(10,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(131,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(1579,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ ( element_relation @ inverse @ compose ) @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( B @ domain_of @ intersection ) )
= ( B @ cantor ) ) ),
inference(paramod_ordered,[status(thm)],[210,212]) ).
thf(1580,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[1579:[bind(A,$thf( D @ ( element_relation @ inverse @ compose ) )),bind(B,$thf( D ))]]) ).
thf(1597,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(simp,[status(thm)],[1580]) ).
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(332,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(866,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[552,552]) ).
thf(867,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[866:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(926,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[867]) ).
thf(19279,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ power_class @ ( element_relation @ image ) @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( element_relation @ ( universal_class @ ( A @ complement @ ( element_relation @ image ) @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[926,4663]) ).
thf(95,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(309,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(1821,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( A @ ( y @ member ) ) )
| ( ( A @ complement )
!= x )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1231,370]) ).
thf(1822,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( A @ ( y @ member ) ) )
| ( ( A @ complement )
!= x )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1821]) ).
thf(1842,plain,
! [A: $i] :
( ( omega != A )
| ( ( omega @ regular )
!= y )
| ( ( A @ complement )
!= x )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1822]) ).
thf(1848,plain,
( ( ( omega @ regular )
!= y )
| ( ( omega @ complement )
!= x )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[1842]) ).
thf(484,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( x @ complement @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[282,364]) ).
thf(485,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( x @ complement @ ( y @ member ) ) ),
inference(simp,[status(thm)],[484]) ).
thf(486,plain,
! [B: $i,A: $i] :
( ( ( x @ complement )
!= universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= y ) ),
inference(simp,[status(thm)],[485]) ).
thf(754,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( x @ complement )
!= universal_class )
| ( ( A @ singleton )
!= y ) ),
inference(paramod_ordered,[status(thm)],[150,486]) ).
thf(755,plain,
! [A: $i] :
( ( ( x @ complement )
!= universal_class )
| ( ( A @ singleton )
!= y ) ),
inference(pattern_uni,[status(thm)],[754:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[161]) ).
thf(6668,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(rewrite,[status(thm)],[162,118]) ).
thf(55,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(222,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(1700,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1646,552]) ).
thf(1701,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1700:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(21076,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[1701,4663]) ).
thf(707,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[160,199]) ).
thf(708,plain,
( ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[707:[bind(A,$thf( element_relation ))]]) ).
thf(45416,plain,
$false,
inference(e,[status(thm)],[4560,6544,1840,352,138,6669,408,115,276,308,379,8988,202,120,247,1426,340,174,8816,1046,1601,301,436,257,389,372,751,196,12713,125,344,17097,8819,325,152,289,261,1847,16953,22089,216,8394,164,179,321,211,238,121,147,397,280,132,396,133,312,206,9822,116,243,292,233,117,17257,201,220,1163,16973,349,160,1502,22092,297,192,137,165,4661,229,252,197,493,757,8403,285,224,6656,317,156,1227,169,328,22533,753,193,6541,212,1424,20456,1231,173,129,11610,371,134,244,311,128,237,266,10338,1135,264,161,10420,149,4662,21946,118,204,18884,259,13358,313,6667,144,335,181,159,9808,1090,187,406,172,9840,1048,567,18743,245,303,9890,4477,12464,323,8983,140,213,155,198,330,387,8932,130,399,278,19866,135,299,10424,226,241,8922,167,255,209,749,194,342,123,17393,295,282,150,8163,409,1098,1499,199,3334,370,127,338,359,23138,177,182,18007,1830,154,9930,386,143,497,1839,203,759,218,250,465,319,1101,171,10400,158,351,2944,16180,207,8420,12463,4478,272,346,287,119,378,235,246,151,315,168,146,190,183,13941,4663,12272,305,210,239,268,195,18538,12707,1165,283,126,136,464,1268,10403,131,1597,3076,332,1129,163,200,19279,122,309,1848,755,491,6668,222,496,364,486,21076,708,232]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET518-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.08 % 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
% 0.14/0.39 % Computer : n001.cluster.edu
% 0.14/0.39 % Model : x86_64 x86_64
% 0.14/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.39 % Memory : 8046.5625MB
% 0.14/0.39 % OS : Linux 6.8.0-71-generic
% 0.14/0.39 % CPULimit : 300
% 0.14/0.39 % WCLimit : 300
% 0.14/0.39 % DateTime : Sat Sep 26 07:57:12 UTC 2026
% 0.14/0.40 % CPUTime :
% 0.14/0.40 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
% 1.31/1.28 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.62/1.71 % [INFO] Parsing done (425ms).
% 2.62/1.73 % [INFO] Running in sequential loop mode.
% 3.80/2.34 % [INFO] eprover registered as external prover.
% 3.80/2.35 % [INFO] Scanning for conjecture ...
% 4.22/2.54 % [INFO] Found a conjecture (or negated_conjecture) and 112 axioms. Running axiom selection ...
% 4.69/2.70 % [INFO] Axiom selection finished. Selected 112 axioms (removed 0 axioms).
% 5.03/2.88 % [INFO] Problem is propositional (TPTP CNF).
% 5.32/2.90 % [INFO] Type checking passed.
% 5.32/2.91 % [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 ...
% 188.33/47.32 % External prover 'e' found a proof!
% 188.33/47.32 % [INFO] Killing All external provers ...
% 188.33/47.33 % Time passed: 46710ms (effective reasoning time: 45583ms)
% 188.33/47.33 % Axioms used in derivation (112): operation4, flip2, inverse, range, domain1, inductive3, equal_implies_subclass1, complement1, one_to_one2, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, maps4, range_of, function3, regularity1, singleton_set, identity_relation, union, flip3, element_relation2, unordered_pairs_in_universal, compose2, homomorphism3, compatible3, application_function_defn2, symmetric_difference, cartesian_product1, not_subclass_members1, compose_can_define_singleton, image, definition_of_domain_relation1, replacement, compose_class_definition3, definition_of_composition_function2, successor, ordered_pair, successor_relation1, domain, homomorphism1, one_to_one3, single_valued_term_defn3, single_valued_class1, compose3, omega_is_inductive2, rotate1, choice1, single_valued_term_defn1, power_class2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, maps1, not_subclass_members2, single_valued_term_defn2, successor_relation3, function1, maps3, domain2, compatible4, intersection3, single_valued_class2, compose_class_definition1, unordered_pair_member, apply, element_relation1, operation1, application_function_defn1, restriction1, cartesian_product3, definition_of_domain_relation3, flip1, homomorphism5, maps2, compatible1, unordered_pair2, application_function_defn4, complement2, operation2, rotate3, compose_class_definition2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, intersection2, operation3, application_function_defn3, definition_of_domain_relation2, choice2, definition_of_composition_function1, homomorphism2, power_class_definition, definition_of_composition_function3, unordered_pair3, subset_relation, compose1
% 188.33/47.33 % No. of inferences in proof: 637
% 188.33/47.33 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 46710 ms resp. 45583 ms w/o parsing
% 178.29/47.43 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 178.29/47.44 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------