%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET097-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n012.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 08:29:24 AM UTC 2026
% Result : Unsatisfiable 77.14s 18.39s
% Output : Refutation 63.67s
% Verified :
% SZS Type : Refutation
% Derivation depth : 27
% Number of leaves : 94
% Syntax : Number of formulae : 508 ( 139 unt; 0 typ; 0 def)
% Number of atoms : 1303 ( 585 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 5063 ( 759 ~; 639 |; 0 &;3665 @)
% ( 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 : 51 ( 48 usr; 11 con; 0-3 aty)
% Number of variables : 612 ( 0 ^; 612 !; 0 ?; 612 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(x_decl,type,
x: $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(flip_decl,type,
flip: $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(rotate_decl,type,
rotate: $i > $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(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $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(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(omega_decl,type,
omega: $i ).
thf(apply_decl,type,
apply: $i > $i > $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(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(20,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(137,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(138,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[137]) ).
thf(26,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(147,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(148,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[147]) ).
thf(9,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(112,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(113,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[112]) ).
thf(114,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[113]) ).
thf(3,negated_conjecture,
x != null_class,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_number_of_elements_in_class_3) ).
thf(97,plain,
x != null_class,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(101,plain,
x != null_class,
inference(polarity_switch,[status(thm)],[97]) ).
thf(102,plain,
null_class != x,
inference(lifteq,[status(thm)],[101]) ).
thf(391,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[114,102]) ).
thf(392,plain,
! [A: $i] :
( ( A != x )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[391:[bind(A,$thf( null_class ))]]) ).
thf(1248,plain,
( ~ ( null_class @ ( x @ subclass ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[392]) ).
thf(10241,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,1248]) ).
thf(10242,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10241]) ).
thf(10253,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != x )
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10242]) ).
thf(10267,plain,
( ( omega != x )
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[10253]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(121,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(122,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[121]) ).
thf(6,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(105,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(106,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[105]) ).
thf(3434,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[122,106]) ).
thf(73,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(249,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(35,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(167,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(7686,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,138]) ).
thf(7687,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[7686]) ).
thf(7688,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[7687:[bind(A,$thf( omega ))]]) ).
thf(21,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(139,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(1657,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[139,1248]) ).
thf(1658,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1657]) ).
thf(1671,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( A != x )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1658]) ).
thf(1685,plain,
( ( universal_class != null_class )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1671]) ).
thf(1823,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[139,1685]) ).
thf(1824,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1823]) ).
thf(1832,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1824]) ).
thf(1837,plain,
( ( universal_class != x )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1832]) ).
thf(2170,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != x )
| ( universal_class != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[114,1837]) ).
thf(2171,plain,
! [A: $i] :
( ( A != x )
| ( universal_class != null_class )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2170:[bind(A,$thf( universal_class ))]]) ).
thf(2177,plain,
( ( universal_class != null_class )
| ~ ( universal_class @ ( x @ subclass ) )
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[2171]) ).
thf(5086,plain,
( ( universal_class != null_class )
| ~ $true
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[2177,139]) ).
thf(5087,plain,
( ( universal_class != null_class )
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[5086]) ).
thf(7792,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7688,5087]) ).
thf(7793,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[7792]) ).
thf(7798,plain,
( ( omega != x )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[7793]) ).
thf(28,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(151,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(152,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[151]) ).
thf(2903,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[152,106]) ).
thf(53,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(206,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(207,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[206]) ).
thf(1458,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)],[207,207]) ).
thf(1459,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1458:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1508,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1459]) ).
thf(2925,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ power_class @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( C @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2903,1508]) ).
thf(2926,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2925:[bind(A,$thf( element_relation )),bind(B,$thf( D @ power_class )),bind(C,$thf( D ))]]) ).
thf(2976,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[2926]) ).
thf(10,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(115,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(85,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(276,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(1655,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[139,1248]) ).
thf(1656,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1655]) ).
thf(1677,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != null_class )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1656]) ).
thf(1687,plain,
( ( universal_class != x )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1677]) ).
thf(7676,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[138,1687]) ).
thf(7677,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7676]) ).
thf(7714,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7677]) ).
thf(7726,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7714]) ).
thf(116,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[115]) ).
thf(7,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(107,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(14,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(123,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(124,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[123]) ).
thf(4574,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[107,124]) ).
thf(4575,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4574]) ).
thf(4576,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4575:[bind(A,$thf( choice ))]]) ).
thf(4602,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[116,4576]) ).
thf(4603,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4602:[bind(A,$thf( choice ))]]) ).
thf(10237,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,5087]) ).
thf(10238,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10237]) ).
thf(10247,plain,
! [A: $i] :
( ( A != x )
| ( omega != universal_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10238]) ).
thf(10261,plain,
( ( omega != universal_class )
| ( universal_class != null_class )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[10247]) ).
thf(11252,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,10261]) ).
thf(11253,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[11252]) ).
thf(11266,plain,
( ( omega != x )
| ( omega != universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[11253]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(267,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(268,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[267]) ).
thf(269,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[268]) ).
thf(89,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(284,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(285,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[284]) ).
thf(72,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(247,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(248,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[247]) ).
thf(13952,plain,
! [A: $i] :
( ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ complement )
= ( A @ successor ) ),
inference(rewrite,[status(thm)],[285,248]) ).
thf(14175,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ complement )
!= ( B @ complement ) )
| ( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[13952,207]) ).
thf(14176,plain,
! [A: $i] :
( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[14175:[bind(A,$thf( G )),bind(B,$thf( G @ singleton @ complement @ ( G @ complement @ intersection ) ))]]) ).
thf(14324,plain,
! [A: $i] :
( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ power_class ) ),
inference(simp,[status(thm)],[14176]) ).
thf(12,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(119,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(120,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[119]) ).
thf(71,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(245,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)],[71]) ).
thf(246,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)],[245]) ).
thf(1704,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[116,246]) ).
thf(1705,plain,
( ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[1704:[bind(A,$thf( element_relation ))]]) ).
thf(23,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function3) ).
thf(141,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)],[23]) ).
thf(142,plain,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[141]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(153,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)],[29]) ).
thf(1913,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[207,1508]) ).
thf(1914,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1913:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(8215,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,7726]) ).
thf(8216,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[8215]) ).
thf(8239,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[8216]) ).
thf(76,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(256,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(257,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[256]) ).
thf(1519,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)],[257,207]) ).
thf(1520,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1519:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1581,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1520]) ).
thf(1966,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)],[257,1581]) ).
thf(1967,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1966:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(43,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(184,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(15,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(125,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(22,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(140,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(5106,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,5087]) ).
thf(5107,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[5106]) ).
thf(5130,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( successor_relation != universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[5107]) ).
thf(30,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(156,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(157,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[156]) ).
thf(2402,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,140]) ).
thf(2403,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2402:[bind(A,$thf( universal_class ))]]) ).
thf(7707,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[138,5087]) ).
thf(7708,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7707]) ).
thf(7716,plain,
! [A: $i] :
( ( A != x )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7708]) ).
thf(7728,plain,
( ( ( x @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[7716]) ).
thf(91,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(289,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)],[91]) ).
thf(25,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(146,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(2168,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( universal_class != x )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[114,1837]) ).
thf(2169,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != null_class )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2168:[bind(A,$thf( universal_class ))]]) ).
thf(2176,plain,
( ( universal_class != x )
| ~ ( universal_class @ ( null_class @ subclass ) )
| ~ ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[2169]) ).
thf(5025,plain,
( ( universal_class != x )
| ~ $true
| ~ ( null_class @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[2176,139]) ).
thf(5026,plain,
( ( universal_class != x )
| ~ ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[5025]) ).
thf(5065,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[146,5026]) ).
thf(5066,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[5065]) ).
thf(5077,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[5066]) ).
thf(78,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(260,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(261,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[260]) ).
thf(58,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(216,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)],[58]) ).
thf(41,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(179,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)],[41]) ).
thf(1905,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1508,207]) ).
thf(1906,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1905:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1960,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1906]) ).
thf(154,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[153]) ).
thf(155,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) ) ),
inference(lifteq,[status(thm)],[154]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(265,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(266,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[265]) ).
thf(12041,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ singleton @ ( B @ image ) @ sum_class )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ singleton @ ( C @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[155,266]) ).
thf(1857,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[257,1508]) ).
thf(1858,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1857:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1941,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1858]) ).
thf(3588,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1941,1508]) ).
thf(3589,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3588:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(3702,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3589]) ).
thf(75,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(253,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(87,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(280,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(93,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(293,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(5124,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[146,5087]) ).
thf(5125,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[5124]) ).
thf(5137,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( element_relation != universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[5125]) ).
thf(77,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(258,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(259,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[258]) ).
thf(1,negated_conjecture,
~ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_number_of_elements_in_class_1) ).
thf(95,plain,
~ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(98,plain,
~ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ),
inference(polarity_switch,[status(thm)],[95]) ).
thf(301,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) ) )
| ~ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[259,98]) ).
thf(305,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[301]) ).
thf(307,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[259,305]) ).
thf(310,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[307]) ).
thf(312,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[310]) ).
thf(70,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(243,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)],[70]) ).
thf(65,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(233,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)],[65]) ).
thf(11,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(117,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)],[11]) ).
thf(2471,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ~ ( x @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,1685]) ).
thf(2472,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( x @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2471]) ).
thf(2473,plain,
( ~ ( x @ ( null_class @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2472:[bind(A,$thf( universal_class ))]]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(270,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(50,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(201,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)],[50]) ).
thf(2100,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)],[116,261]) ).
thf(2101,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[2100:[bind(A,$thf( element_relation ))]]) ).
thf(2164,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)],[2101]) ).
thf(60,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(220,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)],[60]) ).
thf(7691,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[138,1685]) ).
thf(7692,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7691]) ).
thf(7718,plain,
! [A: $i] :
( ( A != x )
| ( ( A @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7692]) ).
thf(7730,plain,
( ( ( x @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[7718]) ).
thf(94,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(295,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(2276,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,295]) ).
thf(2277,plain,
( ( null_class @ ( omega @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2276:[bind(A,$thf( universal_class ))]]) ).
thf(32,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(160,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)],[32]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(165,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(63,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(229,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)],[63]) ).
thf(49,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(197,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)],[49]) ).
thf(7766,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( omega @ ( successor_relation @ image ) ) )
| ( omega @ ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[2903,7688]) ).
thf(7767,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[7766:[bind(A,$thf( successor_relation )),bind(B,$thf( omega ))]]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(188,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(2949,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)],[2903,207]) ).
thf(2950,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2949:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2967,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2950]) ).
thf(3077,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)],[2967,207]) ).
thf(3078,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3077:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(47,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(193,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(118,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[117]) ).
thf(1842,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)],[118,106]) ).
thf(56,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(212,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)],[56]) ).
thf(2,negated_conjecture,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_number_of_elements_in_class_2) ).
thf(96,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(1867,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)],[1508,1508]) ).
thf(1868,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1867:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1942,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)],[1868]) ).
thf(38,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(173,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(17,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(129,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(19,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(134,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)],[19]) ).
thf(6408,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2976,1508]) ).
thf(6409,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[6408:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(7768,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7688,1685]) ).
thf(7769,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[7768]) ).
thf(7802,plain,
( ( omega != x )
| ( ( omega @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[7769]) ).
thf(16,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(127,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(128,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[127]) ).
thf(67,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(237,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(296,plain,
( ( ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[295,98]) ).
thf(297,plain,
( ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[296]) ).
thf(298,plain,
( ( ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) )
!= universal_class )
| ( ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= omega ) ),
inference(simp,[status(thm)],[297]) ).
thf(3018,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)],[116,2967]) ).
thf(3019,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3018:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(3130,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)],[3019]) ).
thf(2918,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)],[2903,1581]) ).
thf(2919,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[2918:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(2974,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)],[2919]) ).
thf(6025,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2974,1581]) ).
thf(6026,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[6025:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(2274,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,295]) ).
thf(2275,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2274:[bind(A,$thf( omega ))]]) ).
thf(27,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(149,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(90,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(286,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)],[90]) ).
thf(92,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(291,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)],[92]) ).
thf(5095,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ~ ( x @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,5087]) ).
thf(5096,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( x @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[5095]) ).
thf(5097,plain,
( ~ ( x @ ( universal_class @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[5096:[bind(A,$thf( universal_class ))]]) ).
thf(135,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)],[134]) ).
thf(136,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)],[135]) ).
thf(5752,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)],[136,106]) ).
thf(52,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(204,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)],[52]) ).
thf(2868,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,1687]) ).
thf(2869,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2868]) ).
thf(2891,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2869]) ).
thf(10212,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,1687]) ).
thf(10213,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10212]) ).
thf(10243,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != x )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10213]) ).
thf(10257,plain,
( ( omega != x )
| ( universal_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[10243]) ).
thf(8853,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( ( x @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,7730]) ).
thf(8854,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( ( x @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[8853]) ).
thf(8877,plain,
( ( omega != x )
| ( ( x @ ( successor_relation @ image ) )
!= null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[8854]) ).
thf(2206,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,102]) ).
thf(2207,plain,
! [A: $i] :
( ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2206:[]]) ).
thf(2753,plain,
x @ ( x @ regular @ member ),
inference(simp,[status(thm)],[2207]) ).
thf(7756,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7688,1687]) ).
thf(7757,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[7756]) ).
thf(7800,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[7757]) ).
thf(334,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[116,259]) ).
thf(335,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[334:[bind(A,$thf( subset_relation ))]]) ).
thf(31,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(158,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(159,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[158]) ).
thf(2025,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1581,207]) ).
thf(2026,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2025:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(2084,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[2026]) ).
thf(84,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(274,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(2256,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,259]) ).
thf(2257,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2256:[bind(A,$thf( subset_relation ))]]) ).
thf(54,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(208,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(209,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[208]) ).
thf(51,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(203,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(1588,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)],[209,203]) ).
thf(1589,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1588:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(2886,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[146,1687]) ).
thf(2887,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2886]) ).
thf(2896,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2887]) ).
thf(4,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(103,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(74,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(251,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(8,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(108,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(322,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[203,312]) ).
thf(323,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(simp,[status(thm)],[322]) ).
thf(330,plain,
! [B: $i,A: $i] :
( ( identity_relation != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( identity_relation @ not_subclass_element ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(simp,[status(thm)],[323]) ).
thf(8257,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( ( x @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,7728]) ).
thf(8258,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( ( x @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[8257]) ).
thf(8281,plain,
( ( omega != x )
| ( ( x @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[8258]) ).
thf(18,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(130,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(2400,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,140]) ).
thf(2401,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2400:[bind(A,$thf( universal_class ))]]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(278,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(5246,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)],[4576,128]) ).
thf(5247,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[5246]) ).
thf(5248,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[5247:[bind(A,$thf( choice ))]]) ).
thf(61,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(223,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)],[61]) ).
thf(302,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) ) )
| ~ ( identity_relation @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[259,98]) ).
thf(306,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[302]) ).
thf(10214,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,5026]) ).
thf(10215,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10214]) ).
thf(10250,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != universal_class )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10215]) ).
thf(10264,plain,
( ( omega != universal_class )
| ( universal_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[10250]) ).
thf(69,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(241,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)],[69]) ).
thf(62,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(226,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)],[62]) ).
thf(10225,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,1685]) ).
thf(10226,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10225]) ).
thf(10255,plain,
! [A: $i] :
( ( A != x )
| ( omega != null_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10226]) ).
thf(10269,plain,
( ( omega != null_class )
| ( universal_class != null_class )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[10255]) ).
thf(24,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(143,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)],[24]) ).
thf(144,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[143]) ).
thf(145,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[144]) ).
thf(1908,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1508,207]) ).
thf(1909,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1908:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1961,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1909]) ).
thf(150,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[149]) ).
thf(13303,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ~ ( x @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,10267]) ).
thf(13304,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[13303]) ).
thf(13339,plain,
( ( omega != null_class )
| ( omega != x )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[13304]) ).
thf(14385,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( omega != null_class )
| ( omega != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,13339]) ).
thf(14386,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( omega != null_class )
| ( omega != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14385]) ).
thf(14433,plain,
! [A: $i] :
( ( A != x )
| ( omega != null_class )
| ( omega != null_class )
| ( omega != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14386]) ).
thf(14455,plain,
( ( omega != null_class )
| ( omega != x )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[14433]) ).
thf(14465,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != null_class )
| ( omega != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,14455]) ).
thf(14466,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != null_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[14465]) ).
thf(14479,plain,
( ( omega != x )
| ( omega != null_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[14466]) ).
thf(14485,plain,
( ( omega != null_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[14479]) ).
thf(2587,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,146]) ).
thf(2588,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2587:[bind(A,$thf( universal_class ))]]) ).
thf(7831,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[116,7767]) ).
thf(7832,plain,
omega @ ( universal_class @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[7831:[bind(A,$thf( universal_class @ ( omega @ ( successor_relation @ restrict ) ) ))]]) ).
thf(2396,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,140]) ).
thf(2397,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2396:[bind(A,$thf( successor_relation ))]]) ).
thf(88,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(282,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(2927,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( C @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2903,1508]) ).
thf(2928,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2927:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2977,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[2928]) ).
thf(11360,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[167,10264]) ).
thf(11361,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[11360]) ).
thf(11374,plain,
( ( omega != null_class )
| ( omega != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[11361]) ).
thf(2335,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class != x )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,1837]) ).
thf(2336,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2335]) ).
thf(2337,plain,
( ( universal_class != x )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2336:[bind(A,$thf( universal_class ))]]) ).
thf(1825,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[146,1685]) ).
thf(1826,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1825]) ).
thf(1830,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( element_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1826]) ).
thf(7678,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[138,5026]) ).
thf(7679,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7678]) ).
thf(7724,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7679]) ).
thf(7737,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7724]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(177,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(42,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(182,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)],[42]) ).
thf(3847,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1960,1508]) ).
thf(3848,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3847:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(3970,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3848]) ).
thf(6774,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2977,1508]) ).
thf(6775,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[6774:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(39,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(175,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(2583,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,146]) ).
thf(2584,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( element_relation @ ( element_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2583:[bind(A,$thf( element_relation ))]]) ).
thf(3564,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1581,1941]) ).
thf(3565,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3564:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(59,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(218,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)],[59]) ).
thf(5,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(104,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(2642,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,107]) ).
thf(2643,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2642:[bind(A,$thf( choice ))]]) ).
thf(64,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(231,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)],[64]) ).
thf(44,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(186,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(37,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(171,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(10239,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( x @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[148,1248]) ).
thf(10240,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10239]) ).
thf(10254,plain,
! [A: $i] :
( ( A != x )
| ( omega != null_class )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[10240]) ).
thf(10268,plain,
( ( omega != null_class )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[10254]) ).
thf(3858,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1508,1960]) ).
thf(3859,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3858:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(2089,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)],[257,261]) ).
thf(2090,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[2089:[bind(A,$thf( D @ ( element_relation @ inverse @ compose ) )),bind(B,$thf( D ))]]) ).
thf(2162,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(simp,[status(thm)],[2090]) ).
thf(83,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(272,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)],[83]) ).
thf(5047,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,5026]) ).
thf(5048,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[5047]) ).
thf(5072,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[5048]) ).
thf(57,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(214,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)],[57]) ).
thf(66,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(235,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(313,plain,
( ( ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[295,312]) ).
thf(314,plain,
( ( ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(simp,[status(thm)],[313]) ).
thf(315,plain,
( ( identity_relation != universal_class )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
!= omega )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(simp,[status(thm)],[314]) ).
thf(2254,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,259]) ).
thf(2255,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2254:[bind(A,$thf( subset_relation ))]]) ).
thf(46,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(190,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(36,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(168,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)],[36]) ).
thf(79,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(262,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(55,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(210,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(68,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(239,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)],[68]) ).
thf(126,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[125]) ).
thf(5299,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ single_valued_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,5248]) ).
thf(5300,plain,
( ( null_class @ single_valued_class )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[5299:[bind(A,$thf( choice ))]]) ).
thf(340,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
!= omega )
| ( subset_relation != x )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[116,315]) ).
thf(341,plain,
( ( identity_relation != universal_class )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
!= omega )
| ( subset_relation != x )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) ) ),
inference(pattern_uni,[status(thm)],[340:[bind(A,$thf( subset_relation ))]]) ).
thf(48,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(195,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)],[48]) ).
thf(7758,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7688,5026]) ).
thf(7759,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[7758]) ).
thf(7808,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[7759]) ).
thf(1807,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,1685]) ).
thf(1808,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1807]) ).
thf(1829,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( successor_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[1808]) ).
thf(2466,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[269,167]) ).
thf(2467,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2466:[bind(A,$thf( omega ))]]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(163,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(109,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[108]) ).
thf(110,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[109]) ).
thf(111,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[110]) ).
thf(336,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( subset_relation != x )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[116,312]) ).
thf(337,plain,
( ( subset_relation != x )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ~ ( identity_relation @ ( null_class @ ( identity_relation @ not_subclass_element ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[336:[bind(A,$thf( subset_relation ))]]) ).
thf(99,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(polarity_switch,[status(thm)],[96]) ).
thf(100,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(lifteq,[status(thm)],[99]) ).
thf(27445,plain,
$false,
inference(e,[status(thm)],[138,10267,3434,249,7798,2976,115,276,7726,4603,11266,269,14324,120,247,1705,142,153,1914,8239,1967,257,184,125,5130,157,2403,284,7728,289,5077,261,216,179,106,121,1960,1581,12041,3702,253,147,280,293,5137,265,7688,312,206,116,243,233,248,117,2473,270,201,2164,220,4576,102,7730,2277,260,160,137,165,229,1837,197,97,1687,7767,156,188,3078,141,256,193,1842,212,96,124,1942,173,129,5026,134,6409,105,266,7802,128,237,298,3130,6026,2275,148,149,286,291,5097,5752,204,259,13952,2891,10257,8877,2753,7800,335,159,245,98,2084,274,2257,1589,2896,208,103,140,251,2974,108,330,8281,130,2401,278,5248,223,306,10264,267,241,167,226,2967,10269,209,112,123,145,1961,150,14485,95,2588,1941,7832,2397,295,282,2977,11374,2337,127,1830,2903,7737,177,182,3970,6775,175,143,2584,3565,1685,203,218,104,2643,10261,231,1508,258,158,186,114,171,139,10268,5087,3859,119,207,2162,272,5072,214,235,246,151,315,2255,146,190,1248,168,262,305,210,107,239,126,5300,341,195,7808,1829,2467,163,111,337,100]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : SET097-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.04 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.05/0.33 % Computer : n012.cluster.edu
% 0.05/0.33 % Model : x86_64 x86_64
% 0.05/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.05/0.33 % Memory : 8046.5625MB
% 0.05/0.33 % OS : Linux 6.8.0-71-generic
% 0.05/0.33 % CPULimit : 300
% 0.05/0.33 % WCLimit : 300
% 0.05/0.33 % DateTime : Sat Sep 26 07:06:20 UTC 2026
% 0.05/0.33 % CPUTime :
% 0.05/0.33 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.61/0.77 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.95/1.01 % [INFO] Parsing done (236ms).
% 0.95/1.02 % [INFO] Running in sequential loop mode.
% 1.74/1.28 % [INFO] eprover registered as external prover.
% 1.74/1.28 % [INFO] Scanning for conjecture ...
% 1.96/1.35 % [INFO] Found a conjecture (or negated_conjecture) and 91 axioms. Running axiom selection ...
% 2.19/1.41 % [INFO] Axiom selection finished. Selected 91 axioms (removed 0 axioms).
% 2.19/1.47 % [INFO] Problem is propositional (TPTP CNF).
% 2.19/1.48 % [INFO] Type checking passed.
% 2.19/1.48 % [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 ...
% 77.14/18.38 % External prover 'e' found a proof!
% 77.14/18.38 % [INFO] Killing All external provers ...
% 77.14/18.39 % Time passed: 17948ms (effective reasoning time: 17360ms)
% 77.14/18.39 % Axioms used in derivation (91): operation4, flip2, inverse, range, domain1, inductive3, operation3, equal_implies_subclass1, complement1, one_to_one2, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, function3, regularity1, singleton_set, identity_relation, union, flip3, element_relation2, unordered_pairs_in_universal, compose2, homomorphism3, compatible3, symmetric_difference, cartesian_product1, not_subclass_members1, image, replacement, successor, ordered_pair, successor_relation1, domain, homomorphism1, one_to_one3, single_valued_class1, compose3, omega_is_inductive2, rotate1, choice1, power_class2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, not_subclass_members2, successor_relation3, domain2, function1, compatible4, intersection3, single_valued_class2, unordered_pair_member, apply, element_relation1, operation1, restriction1, cartesian_product3, flip1, homomorphism5, compatible1, unordered_pair2, complement2, operation2, rotate3, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, intersection2, choice2, homomorphism2, power_class_definition, unordered_pair3, subset_relation, compose1
% 77.14/18.39 % No. of inferences in proof: 508
% 77.14/18.39 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 17948 ms resp. 17360 ms w/o parsing
% 63.67/18.54 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 63.67/18.54 % [INFO] Killing All external provers ...
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