%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET076-7 : 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 : n019.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 08:29:16 AM UTC 2026
% Result : Unsatisfiable 142.30s 35.91s
% Output : Refutation 144.43s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 121
% Syntax : Number of formulae : 811 ( 246 unt; 0 typ; 0 def)
% Number of atoms : 1879 ( 933 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 7468 (1078 ~; 830 |; 0 &;5560 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 13 con; 0-3 aty)
% Number of variables : 1105 ( 0 ^;1105 !; 0 ?;1105 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(x_decl,type,
x: $i ).
thf(z_decl,type,
z: $i ).
thf(y_decl,type,
y: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $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(intersection_decl,type,
intersection: $i > $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(image_decl,type,
image: $i > $i > $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(power_class_decl,type,
power_class: $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(flip_decl,type,
flip: $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(121,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(379,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(380,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[379]) ).
thf(381,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[380]) ).
thf(2,negated_conjecture,
z @ ( y @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_unordered_pair_is_subset_2) ).
thf(123,plain,
z @ ( y @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(59,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(245,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(246,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[245]) ).
thf(382,plain,
! [A: $i] :
( ( ( z @ ( y @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[123,246]) ).
thf(383,plain,
! [A: $i] :
( ( z @ ( y @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[382]) ).
thf(386,plain,
! [A: $i] :
( ( null_class != z )
| ( y != A ) ),
inference(simp,[status(thm)],[383]) ).
thf(388,plain,
null_class != z,
inference(simp,[status(thm)],[386]) ).
thf(1251,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,388]) ).
thf(1252,plain,
! [A: $i] :
( ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1251:[]]) ).
thf(1633,plain,
z @ ( z @ regular @ member ),
inference(simp,[status(thm)],[1252]) ).
thf(107,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(347,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(61,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(249,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(28,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(177,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(178,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[177]) ).
thf(17,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(153,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(154,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[153]) ).
thf(1,negated_conjecture,
z @ ( x @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_unordered_pair_is_subset_1) ).
thf(122,plain,
z @ ( x @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(42,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(205,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[205]) ).
thf(11914,plain,
! [C: $i,B: $i,A: $i] :
( ( ( z @ ( x @ member ) )
!= ( C @ ( B @ intersection ) @ ( A @ member ) ) )
| ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[122,206]) ).
thf(11915,plain,
! [C: $i,B: $i,A: $i] :
( ( ( z @ ( x @ member ) )
!= ( C @ ( B @ intersection ) @ ( A @ member ) ) )
| ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[11914]) ).
thf(12036,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ intersection ) )
!= z )
| ( x != A )
| ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[11915]) ).
thf(12047,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ intersection ) )
!= z )
| ( A @ ( x @ member ) ) ),
inference(simp,[status(thm)],[12036]) ).
thf(12372,plain,
! [B: $i,A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( B @ ( A @ intersection ) ) )
| ( identity_relation != z )
| ( A @ ( x @ member ) ) ),
inference(paramod_ordered,[status(thm)],[154,12047]) ).
thf(12373,plain,
( ( identity_relation != z )
| ( subset_relation @ inverse @ ( x @ member ) ) ),
inference(pattern_uni,[status(thm)],[12372:[bind(A,$thf( subset_relation @ inverse )),bind(B,$thf( subset_relation ))]]) ).
thf(12415,plain,
! [A: $i] :
( ( ( subset_relation @ inverse @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[12373,246]) ).
thf(12416,plain,
! [A: $i] :
( ( ( subset_relation @ inverse @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[12415]) ).
thf(12428,plain,
! [A: $i] :
( ( ( subset_relation @ inverse )
!= null_class )
| ( x != A )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[12416]) ).
thf(12437,plain,
( ( ( subset_relation @ inverse )
!= null_class )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[12428]) ).
thf(12730,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= null_class )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[178,12437]) ).
thf(12731,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= null_class )
| ( identity_relation != z ) ),
inference(pattern_uni,[status(thm)],[12730:[bind(A,$thf( subset_relation ))]]) ).
thf(53,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(231,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(232,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[231]) ).
thf(233,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[232]) ).
thf(234,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[233]) ).
thf(60,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(247,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(248,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[247]) ).
thf(68,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(263,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(390,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[263,246]) ).
thf(391,plain,
! [A: $i] :
( ( universal_class @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[390]) ).
thf(392,plain,
! [A: $i] :
( ( null_class != universal_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[391]) ).
thf(393,plain,
null_class != universal_class,
inference(simp,[status(thm)],[392]) ).
thf(1264,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,393]) ).
thf(1265,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1264:[]]) ).
thf(1638,plain,
universal_class @ ( universal_class @ regular @ member ),
inference(simp,[status(thm)],[1265]) ).
thf(48,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(217,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(218,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[217]) ).
thf(21203,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1638,218]) ).
thf(21204,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[21203]) ).
thf(21205,plain,
! [A: $i] : ( A @ ( universal_class @ regular @ unordered_pair ) @ ( universal_class @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[21204:[bind(A,$thf( universal_class @ regular )),bind(B,$thf( B ))]]) ).
thf(21358,plain,
! [A: $i] : ( A @ ( universal_class @ regular @ unordered_pair ) @ ( universal_class @ regular @ member ) ),
inference(simp,[status(thm)],[21205]) ).
thf(23703,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( universal_class @ regular @ unordered_pair ) ) )
| ( A @ singleton @ ( universal_class @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[248,21358]) ).
thf(23704,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ member ),
inference(pattern_uni,[status(thm)],[23703:[bind(A,$thf( universal_class @ regular )),bind(B,$thf( universal_class @ regular ))]]) ).
thf(23870,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[23704,246]) ).
thf(23871,plain,
! [A: $i] :
( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[23870]) ).
thf(23889,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton )
!= null_class )
| ( ( universal_class @ regular )
!= A ) ),
inference(simp,[status(thm)],[23871]) ).
thf(23904,plain,
( ( universal_class @ regular @ singleton )
!= null_class ),
inference(simp,[status(thm)],[23889]) ).
thf(9,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(136,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(137,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[136]) ).
thf(6,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(129,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)],[6]) ).
thf(130,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[129]) ).
thf(481,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[137,130]) ).
thf(10,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(138,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)],[10]) ).
thf(43,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(207,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(208,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[207]) ).
thf(3,negated_conjecture,
~ ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_unordered_pair_is_subset_3) ).
thf(124,plain,
~ ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(125,plain,
~ ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ),
inference(polarity_switch,[status(thm)],[124]) ).
thf(12449,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[208,125]) ).
thf(12450,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[12449]) ).
thf(12451,plain,
~ ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) ),
inference(pattern_uni,[status(thm)],[12450:[bind(A,$thf( y @ ( x @ unordered_pair ) )),bind(B,$thf( z ))]]) ).
thf(12620,plain,
( ( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[123,12451]) ).
thf(12621,plain,
( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( y @ member ) ) ),
inference(simp,[status(thm)],[12620]) ).
thf(12645,plain,
( ( z != z )
| ( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= y ) ),
inference(simp,[status(thm)],[12621]) ).
thf(12677,plain,
( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= y ),
inference(simp,[status(thm)],[12645]) ).
thf(12699,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= y ) ),
inference(paramod_ordered,[status(thm)],[248,12677]) ).
thf(12709,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= y ) ),
inference(simp,[status(thm)],[12699]) ).
thf(12710,plain,
( ( y != x )
| ( ( z @ ( x @ singleton @ not_subclass_element ) )
!= y ) ),
inference(simp,[status(thm)],[12709]) ).
thf(1442,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,154]) ).
thf(1443,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1442:[bind(A,$thf( subset_relation ))]]) ).
thf(21235,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[263,218]) ).
thf(21236,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[21235]) ).
thf(21237,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[21236:[bind(A,$thf( null_class ))]]) ).
thf(21362,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( null_class @ member ) ),
inference(simp,[status(thm)],[21237]) ).
thf(22163,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ unordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21362,246]) ).
thf(22164,plain,
! [B: $i,A: $i] :
( ( A @ ( null_class @ unordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[22163]) ).
thf(22175,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ unordered_pair ) )
!= null_class )
| ( null_class != B ) ),
inference(simp,[status(thm)],[22164]) ).
thf(22193,plain,
! [A: $i] :
( ( A @ ( null_class @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[22175]) ).
thf(41,axiom,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',left_cancellation) ).
thf(202,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(16,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(151,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(152,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[151]) ).
thf(98,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(330,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(90,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(314,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(315,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[314]) ).
thf(401,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[330,315]) ).
thf(402,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[401]) ).
thf(412,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( omega != A ) ),
inference(simp,[status(thm)],[402]) ).
thf(417,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[412]) ).
thf(523,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[152,417]) ).
thf(524,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[523:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(557,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[524]) ).
thf(772,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[154,557]) ).
thf(779,plain,
! [A: $i] :
( ( identity_relation != subset_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[772]) ).
thf(789,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[779]) ).
thf(1070,plain,
( ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != subset_relation )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[154,789]) ).
thf(1078,plain,
( ( ( identity_relation @ domain_of )
!= subset_relation )
| ( ( subset_relation @ inverse @ diagonalise )
!= ( subset_relation @ inverse ) )
| ( identity_relation != subset_relation )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[1070]) ).
thf(54,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(235,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(236,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[235]) ).
thf(237,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[236]) ).
thf(97,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(329,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(426,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[329,125]) ).
thf(427,plain,
! [A: $i] :
( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[426]) ).
thf(428,plain,
! [A: $i] :
( ( universal_class != z )
| ( A
!= ( y @ ( x @ unordered_pair ) ) ) ),
inference(simp,[status(thm)],[427]) ).
thf(429,plain,
universal_class != z,
inference(simp,[status(thm)],[428]) ).
thf(25252,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[237,429]) ).
thf(25253,plain,
! [A: $i] :
( ( A != z )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[25252:[bind(A,$thf( universal_class ))]]) ).
thf(28699,plain,
( ~ ( universal_class @ ( z @ subclass ) )
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[25253]) ).
thf(30199,plain,
( ~ $true
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[28699,329]) ).
thf(30200,plain,
~ ( z @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[30199]) ).
thf(30235,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[208,30200]) ).
thf(30236,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[30235]) ).
thf(30237,plain,
~ ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) ),
inference(pattern_uni,[status(thm)],[30236:[bind(A,$thf( universal_class )),bind(B,$thf( z ))]]) ).
thf(30267,plain,
( ( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( z @ regular @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1633,30237]) ).
thf(30268,plain,
( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[30267]) ).
thf(30452,plain,
( ( z != z )
| ( ( z @ ( universal_class @ not_subclass_element ) )
!= ( z @ regular ) ) ),
inference(simp,[status(thm)],[30268]) ).
thf(30478,plain,
( ( z @ ( universal_class @ not_subclass_element ) )
!= ( z @ regular ) ),
inference(simp,[status(thm)],[30452]) ).
thf(47,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(214,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(215,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[214]) ).
thf(216,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[215]) ).
thf(20,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(160,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(161,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[160]) ).
thf(18316,plain,
! [A: $i] :
( ( A @ ( A @ singleton @ ( choice @ image ) @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ( A = null_class ) ),
inference(rewrite,[status(thm)],[216,161]) ).
thf(729,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ~ ( z @ ( A @ singleton @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,125]) ).
thf(740,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ~ ( z @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[729]) ).
thf(747,plain,
( ( y != x )
| ~ ( z @ ( x @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[740]) ).
thf(12,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(142,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)],[12]) ).
thf(13,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(144,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)],[13]) ).
thf(145,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)],[144]) ).
thf(32,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(184,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)],[32]) ).
thf(185,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ restrict ) ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(lifteq,[status(thm)],[184]) ).
thf(4802,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(rewrite,[status(thm)],[185,130]) ).
thf(4869,plain,
! [C: $i,B: $i,A: $i] :
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( A @ ( C @ ( B @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[145,4802]) ).
thf(4870,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[4869:[bind(A,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(1518,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[381,315]) ).
thf(1519,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1518]) ).
thf(1520,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[1519:[bind(A,$thf( F @ ( F @ complement @ intersection ) )),bind(B,$thf( F @ ( F @ complement @ intersection ) @ regular )),bind(C,$thf( F ))]]) ).
thf(1678,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[1520]) ).
thf(25,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(171,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(172,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[171]) ).
thf(567,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)],[152,172]) ).
thf(568,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[567:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(595,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[568]) ).
thf(1894,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ power_class )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1678,595]) ).
thf(1895,plain,
( ( null_class @ domain_of @ power_class )
= ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1894:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(1868,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise )
= ( null_class @ domain_of @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[1678,152]) ).
thf(1869,plain,
( ( identity_relation @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[1868:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(2228,plain,
( ( null_class @ domain_of @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[1895,1869]) ).
thf(2274,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2228,172]) ).
thf(2275,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2274:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(2843,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2275,172]) ).
thf(2844,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2843:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(104,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(340,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(21217,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[330,218]) ).
thf(21218,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[21217]) ).
thf(21219,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[21218:[bind(A,$thf( omega ))]]) ).
thf(21360,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(simp,[status(thm)],[21219]) ).
thf(21390,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) ) )
| ( A @ singleton @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[248,21360]) ).
thf(21391,plain,
omega @ singleton @ ( omega @ member ),
inference(pattern_uni,[status(thm)],[21390:[bind(A,$thf( omega )),bind(B,$thf( omega ))]]) ).
thf(21534,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21391,315]) ).
thf(21535,plain,
! [B: $i,A: $i] :
( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[21534]) ).
thf(21552,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( omega @ singleton ) )
| ( omega != A ) ),
inference(simp,[status(thm)],[21535]) ).
thf(21563,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[21552]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(228,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(229,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[228]) ).
thf(230,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) ) ),
inference(lifteq,[status(thm)],[229]) ).
thf(117,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(371,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(430,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[371,125]) ).
thf(431,plain,
! [A: $i] :
( ( A @ ( null_class @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[430]) ).
thf(432,plain,
! [A: $i] :
( ( A != z )
| ( ( y @ ( x @ unordered_pair ) )
!= null_class ) ),
inference(simp,[status(thm)],[431]) ).
thf(433,plain,
( ( y @ ( x @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[432]) ).
thf(23643,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( B @ singleton )
!= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[230,433]) ).
thf(23644,plain,
( ( ( x @ singleton )
!= null_class )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[23643:[bind(A,$thf( y )),bind(B,$thf( x ))]]) ).
thf(31466,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( null_class != null_class )
| ( universal_class @ ( y @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,23644]) ).
thf(31467,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( universal_class @ ( y @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[31466]) ).
thf(31468,plain,
( ( universal_class @ ( y @ member ) )
| ( x @ singleton @ ( x @ singleton @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[31467:[bind(A,$thf( x @ singleton ))]]) ).
thf(1440,plain,
! [A: $i] :
( ( A
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,154]) ).
thf(1441,plain,
( ( ( subset_relation @ ( null_class @ intersection ) )
= identity_relation )
| ( subset_relation @ inverse @ ( subset_relation @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1440:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(80,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(288,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(93,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(320,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(65,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(257,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(23,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(166,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(167,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[166]) ).
thf(4,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(126,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(127,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[126]) ).
thf(3711,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[167,127]) ).
thf(3753,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)],[3711,172]) ).
thf(3754,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3753:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3833,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3754]) ).
thf(7235,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(rewrite,[status(thm)],[3833,130]) ).
thf(4865,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[4802,145]) ).
thf(4866,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[4865:[bind(A,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(118,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(372,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(1509,plain,
! [A: $i] :
( ( A
!= ( y @ ( x @ unordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,433]) ).
thf(1510,plain,
! [A: $i] :
( ( A
!= ( y @ ( x @ unordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[1509]) ).
thf(1511,plain,
y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[1510:[bind(A,$thf( y @ ( x @ unordered_pair ) ))]]) ).
thf(1836,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( A @ singleton @ ( y @ ( x @ unordered_pair ) @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[248,1511]) ).
thf(1856,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( A @ singleton @ ( y @ ( x @ unordered_pair ) @ regular @ member ) ) ),
inference(simp,[status(thm)],[1836]) ).
thf(1860,plain,
( ( y != x )
| ( x @ singleton @ ( y @ ( x @ unordered_pair ) @ regular @ member ) ) ),
inference(simp,[status(thm)],[1856]) ).
thf(403,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[123,315]) ).
thf(404,plain,
! [B: $i,A: $i] :
( ( z @ ( y @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[403]) ).
thf(410,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= z )
| ( y != A ) ),
inference(simp,[status(thm)],[404]) ).
thf(415,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= z ),
inference(simp,[status(thm)],[410]) ).
thf(423,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[154,415]) ).
thf(424,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[423]) ).
thf(425,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[424]) ).
thf(1278,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,425]) ).
thf(1279,plain,
( ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1278:[bind(A,$thf( subset_relation ))]]) ).
thf(38,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(196,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(197,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[196]) ).
thf(22833,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( null_class @ unordered_pair ) ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[197,22193]) ).
thf(22834,plain,
! [A: $i] :
( ( null_class @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[22833:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(19,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(157,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(37,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(195,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(398,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,125]) ).
thf(399,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[398]) ).
thf(400,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( y @ ( x @ unordered_pair ) )
!= element_relation ) ),
inference(simp,[status(thm)],[399]) ).
thf(737,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( A @ singleton )
!= element_relation ) ),
inference(paramod_ordered,[status(thm)],[248,400]) ).
thf(744,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( A @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[737]) ).
thf(751,plain,
( ( y != x )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( x @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[744]) ).
thf(579,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[172,415]) ).
thf(580,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= z ),
inference(pattern_uni,[status(thm)],[579:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(592,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= z ),
inference(simp,[status(thm)],[580]) ).
thf(790,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ ( B @ power_class @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[152,592]) ).
thf(791,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= z ),
inference(pattern_uni,[status(thm)],[790:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(802,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= z ),
inference(simp,[status(thm)],[791]) ).
thf(22143,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( null_class @ unordered_pair ) ) )
| ( B @ ( A @ unordered_pair ) @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[197,21362]) ).
thf(22144,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[22143:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(34,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(188,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(189,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[188]) ).
thf(407,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[154,315]) ).
thf(411,plain,
! [B: $i,A: $i] :
( ( B != subset_relation )
| ( ( B @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[407]) ).
thf(416,plain,
! [A: $i] :
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[411]) ).
thf(91,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(316,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(25532,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[237,388]) ).
thf(25533,plain,
! [A: $i] :
( ( A != z )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[25532:[bind(A,$thf( null_class ))]]) ).
thf(28793,plain,
( ~ ( null_class @ ( z @ subclass ) )
| ~ ( z @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[25533]) ).
thf(31350,plain,
( ~ ( null_class @ ( z @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[28793,371]) ).
thf(31351,plain,
~ ( null_class @ ( z @ subclass ) ),
inference(simp,[status(thm)],[31350]) ).
thf(95,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(325,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(22,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(164,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(12346,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( B @ ( A @ intersection ) ) )
| ( subset_relation != z )
| ( A @ ( x @ member ) ) ),
inference(paramod_ordered,[status(thm)],[4866,12047]) ).
thf(12347,plain,
( ( subset_relation != z )
| ( universal_class @ ( universal_class @ cross_product ) @ ( x @ member ) ) ),
inference(pattern_uni,[status(thm)],[12346:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) )),bind(B,$thf( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) ))]]) ).
thf(29,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(179,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(30246,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,30200]) ).
thf(30247,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[30246]) ).
thf(30253,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( universal_class != element_relation ) ),
inference(simp,[status(thm)],[30247]) ).
thf(120,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(376,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(1835,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( y @ ( x @ unordered_pair ) @ ( A @ singleton @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[248,1511]) ).
thf(1855,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( y @ ( x @ unordered_pair ) @ ( A @ singleton @ regular @ member ) ) ),
inference(simp,[status(thm)],[1835]) ).
thf(1864,plain,
( ( y != x )
| ( y @ ( x @ unordered_pair ) @ ( x @ singleton @ regular @ member ) ) ),
inference(simp,[status(thm)],[1855]) ).
thf(55,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(238,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(110,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(353,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(46,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(213,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(478,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[213,125]) ).
thf(479,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[478]) ).
thf(480,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= z )
| ( ( A @ rotate )
!= ( y @ ( x @ unordered_pair ) ) ) ),
inference(simp,[status(thm)],[479]) ).
thf(44,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality3) ).
thf(209,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(210,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[209]) ).
thf(211,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( B = A ) ),
inference(lifteq,[status(thm)],[210]) ).
thf(63,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(253,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)],[63]) ).
thf(14,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(146,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(147,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[146]) ).
thf(1870,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[152,1678]) ).
thf(1871,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[1870:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1928,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[1871]) ).
thf(50,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(221,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(82,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(293,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(450,plain,
! [A: $i] :
( ( ( A @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[234,125]) ).
thf(451,plain,
! [A: $i] :
( ( A @ ( A @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[450]) ).
thf(452,plain,
! [A: $i] :
( ( A != z )
| ( A
!= ( y @ ( x @ unordered_pair ) ) ) ),
inference(simp,[status(thm)],[451]) ).
thf(453,plain,
( ( y @ ( x @ unordered_pair ) )
!= z ),
inference(simp,[status(thm)],[452]) ).
thf(7,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(131,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)],[7]) ).
thf(132,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[131]) ).
thf(133,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[132]) ).
thf(99,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(331,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(1432,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,331]) ).
thf(1433,plain,
( ( null_class @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1432:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(58,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(243,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)],[58]) ).
thf(89,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(312,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(1853,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1511,315]) ).
thf(1854,plain,
! [B: $i,A: $i] :
( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1853]) ).
thf(1859,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( y @ ( x @ unordered_pair ) @ regular )
!= A ) ),
inference(simp,[status(thm)],[1854]) ).
thf(1863,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( y @ ( x @ unordered_pair ) ) ),
inference(simp,[status(thm)],[1859]) ).
thf(40,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(200,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)],[40]) ).
thf(201,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[200]) ).
thf(12444,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( x @ singleton @ subclass ) ) )
| ( y != x )
| ~ $true
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[208,747]) ).
thf(12445,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( z @ ( x @ singleton @ subclass ) ) )
| ( y != x )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[12444]) ).
thf(12446,plain,
( ( y != x )
| ~ ( z @ ( z @ ( x @ singleton @ not_subclass_element ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[12445:[bind(A,$thf( x @ singleton )),bind(B,$thf( z ))]]) ).
thf(30373,plain,
( ( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[123,30237]) ).
thf(30374,plain,
( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( y @ member ) ) ),
inference(simp,[status(thm)],[30373]) ).
thf(30426,plain,
( ( z != z )
| ( ( z @ ( universal_class @ not_subclass_element ) )
!= y ) ),
inference(simp,[status(thm)],[30374]) ).
thf(30466,plain,
( ( z @ ( universal_class @ not_subclass_element ) )
!= y ),
inference(simp,[status(thm)],[30426]) ).
thf(72,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(270,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(49,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(219,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)],[49]) ).
thf(220,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[219]) ).
thf(115,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(366,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(420,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[331,125]) ).
thf(421,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[420]) ).
thf(422,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( y @ ( x @ unordered_pair ) )
!= successor_relation ) ),
inference(simp,[status(thm)],[421]) ).
thf(734,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( A @ singleton )
!= successor_relation ) ),
inference(paramod_ordered,[status(thm)],[248,422]) ).
thf(739,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( A @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[734]) ).
thf(746,plain,
( ( y != x )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( ( x @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[739]) ).
thf(74,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(275,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)],[74]) ).
thf(165,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[164]) ).
thf(3598,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,1443]) ).
thf(3599,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3598:[bind(A,$thf( subset_relation ))]]) ).
thf(3664,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[3599]) ).
thf(1682,plain,
! [A: $i] :
( ( A
!= ( z @ regular ) )
| ( z @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,1633]) ).
thf(1683,plain,
( ( z @ ( null_class @ member ) )
| ( z @ regular @ ( z @ regular @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1682:[bind(A,$thf( z @ regular ))]]) ).
thf(2376,plain,
! [A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true
| ( z @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1683,416]) ).
thf(2377,plain,
! [A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2376]) ).
thf(2391,plain,
! [A: $i] :
( ( ( z @ regular )
!= identity_relation )
| ( ( z @ regular @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2377]) ).
thf(2396,plain,
( ( ( z @ regular )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2391]) ).
thf(21506,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ singleton @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,21391]) ).
thf(21507,plain,
( ( null_class @ singleton @ ( omega @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[21506:[bind(A,$thf( omega ))]]) ).
thf(101,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(334,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)],[101]) ).
thf(67,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(260,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)],[67]) ).
thf(108,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(349,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(113,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(361,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(20635,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[1678,802]) ).
thf(20636,plain,
( ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= z ),
inference(pattern_uni,[status(thm)],[20635:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(20758,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= z ),
inference(rewrite,[status(thm)],[20636,1869]) ).
thf(139,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second ) ),
inference(lifteq,[status(thm)],[138]) ).
thf(522,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( A @ ( C @ ( B @ singleton @ cross_product ) @ intersection ) @ not_subclass_element ) @ second ) ),
inference(rewrite,[status(thm)],[139,130]) ).
thf(541,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[152,415]) ).
thf(542,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ),
inference(pattern_uni,[status(thm)],[541:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(560,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ),
inference(simp,[status(thm)],[542]) ).
thf(694,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[154,560]) ).
thf(703,plain,
! [A: $i] :
( ( identity_relation != subset_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ) ),
inference(simp,[status(thm)],[694]) ).
thf(710,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= z ) ),
inference(simp,[status(thm)],[703]) ).
thf(611,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[178,154]) ).
thf(612,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[611:[bind(A,$thf( subset_relation ))]]) ).
thf(2754,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( ( y @ ( x @ unordered_pair ) )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[612,1863]) ).
thf(2762,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( ( y @ ( x @ unordered_pair ) )
!= identity_relation ) ),
inference(simp,[status(thm)],[2754]) ).
thf(2769,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( ( y @ ( x @ unordered_pair ) )
!= identity_relation ) ),
inference(simp,[status(thm)],[2762]) ).
thf(250,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[249]) ).
thf(33088,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[250,246]) ).
thf(33089,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[33088]) ).
thf(33090,plain,
~ ( null_class @ inductive ),
inference(pattern_uni,[status(thm)],[33089:[bind(A,$thf( null_class )),bind(B,$thf( null_class ))]]) ).
thf(21428,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21360,246]) ).
thf(21429,plain,
! [B: $i,A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( null_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[21428]) ).
thf(21450,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= null_class )
| ( omega != B ) ),
inference(simp,[status(thm)],[21429]) ).
thf(21461,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[21450]) ).
thf(22045,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( omega @ unordered_pair ) ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[197,21461]) ).
thf(22046,plain,
! [A: $i] :
( ( omega @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[22045:[bind(A,$thf( A )),bind(B,$thf( omega )),bind(C,$thf( A ))]]) ).
thf(78,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(285,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(51,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(224,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)],[51]) ).
thf(22123,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( null_class @ unordered_pair ) ) )
| ( A @ singleton @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[248,21362]) ).
thf(22124,plain,
null_class @ singleton @ ( null_class @ member ),
inference(pattern_uni,[status(thm)],[22123:[bind(A,$thf( null_class )),bind(B,$thf( null_class ))]]) ).
thf(22315,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[22124,246]) ).
thf(22316,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[22315]) ).
thf(22319,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[22316]) ).
thf(22343,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[22319]) ).
thf(22675,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,22343]) ).
thf(22676,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[22675]) ).
thf(22677,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[22676:[bind(A,$thf( null_class @ singleton ))]]) ).
thf(18,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(155,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(156,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[155]) ).
thf(3751,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3711,2275]) ).
thf(3752,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3751:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement ))]]) ).
thf(9376,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[3752,130]) ).
thf(10106,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( A @ ( C @ ( B @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[4866,4802]) ).
thf(10107,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[10106:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(36,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(193,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)],[36]) ).
thf(2386,plain,
! [A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( z @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1683,246]) ).
thf(2387,plain,
! [A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2386]) ).
thf(2392,plain,
! [A: $i] :
( ( ( z @ regular )
!= null_class )
| ( ( z @ regular @ regular )
!= A )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2387]) ).
thf(2397,plain,
( ( ( z @ regular )
!= null_class )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2392]) ).
thf(2412,plain,
! [B: $i,A: $i] :
( ( ( z @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( ( z @ regular )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2397,315]) ).
thf(2413,plain,
! [B: $i,A: $i] :
( ( ( z @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2412]) ).
thf(2415,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= z )
| ( null_class != A )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2413]) ).
thf(2419,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= z )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2415]) ).
thf(733,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[248,433]) ).
thf(741,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[733]) ).
thf(748,plain,
( ( y != x )
| ( ( x @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[741]) ).
thf(1482,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( y != x )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,748]) ).
thf(1483,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( y != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[1482]) ).
thf(1484,plain,
( ( y != x )
| ( x @ singleton @ ( x @ singleton @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1483:[bind(A,$thf( x @ singleton ))]]) ).
thf(2101,plain,
! [B: $i,A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( y != x ) ),
inference(paramod_ordered,[status(thm)],[1484,315]) ).
thf(2102,plain,
! [B: $i,A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( y != x ) ),
inference(simp,[status(thm)],[2101]) ).
thf(2105,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( x @ singleton ) )
| ( ( x @ singleton @ regular )
!= A )
| ( y != x ) ),
inference(simp,[status(thm)],[2102]) ).
thf(2108,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( x @ singleton ) )
| ( y != x ) ),
inference(simp,[status(thm)],[2105]) ).
thf(45,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(212,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(563,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)],[172,172]) ).
thf(564,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[563:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(594,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[564]) ).
thf(2287,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2228,594]) ).
thf(2288,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2287:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(7947,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( A @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2288,594]) ).
thf(7948,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[7947:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(26,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(173,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(7912,plain,
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2275,2288]) ).
thf(7913,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[7912:[]]) ).
thf(3742,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3711,2228]) ).
thf(3743,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3742:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement ))]]) ).
thf(3842,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[3743,130]) ).
thf(3966,plain,
! [A: $i] :
( ( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
!= ( A @ complement ) )
| ( ( A @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[3842,1678]) ).
thf(3967,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3966:[bind(A,$thf( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of ))]]) ).
thf(8,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(134,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(244,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(cnf,[status(esa)],[243]) ).
thf(35,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(190,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)],[35]) ).
thf(191,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[190]) ).
thf(192,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) ) ),
inference(lifteq,[status(thm)],[191]) ).
thf(7052,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ singleton @ ( C @ image ) @ sum_class )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ singleton @ ( A @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[192,161]) ).
thf(22317,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[22124,315]) ).
thf(22318,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[22317]) ).
thf(22336,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[22318]) ).
thf(22339,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[22336]) ).
thf(5,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(128,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(84,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(298,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)],[84]) ).
thf(731,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( A @ singleton )
!= z ) ),
inference(paramod_ordered,[status(thm)],[248,453]) ).
thf(743,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( A @ singleton )
!= z ) ),
inference(simp,[status(thm)],[731]) ).
thf(750,plain,
( ( y != x )
| ( ( x @ singleton )
!= z ) ),
inference(simp,[status(thm)],[743]) ).
thf(504,plain,
! [A: $i] :
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( subset_relation != z ) ),
inference(paramod_ordered,[status(thm)],[145,415]) ).
thf(506,plain,
! [A: $i] :
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= A )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( A @ complement ) )
| ( subset_relation != z ) ),
inference(simp,[status(thm)],[504]) ).
thf(513,plain,
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ complement )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( subset_relation != z ) ),
inference(simp,[status(thm)],[506]) ).
thf(4872,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[4802,3842]) ).
thf(4873,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ ( element_relation @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[4872:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement )),bind(C,$thf( universal_class ))]]) ).
thf(5358,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ ( element_relation @ intersection ) @ inverse @ domain_of @ complement )
!= ( A @ complement ) )
| ( ( A @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[4873,1678]) ).
thf(5359,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ ( element_relation @ intersection ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[5358:[bind(A,$thf( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ ( element_relation @ intersection ) @ inverse @ domain_of ))]]) ).
thf(21532,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21391,246]) ).
thf(21533,plain,
! [A: $i] :
( ( omega @ singleton @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[21532]) ).
thf(21544,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= null_class )
| ( omega != A ) ),
inference(simp,[status(thm)],[21533]) ).
thf(21558,plain,
( ( omega @ singleton )
!= null_class ),
inference(simp,[status(thm)],[21544]) ).
thf(21581,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,21558]) ).
thf(21582,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[21581]) ).
thf(21583,plain,
omega @ singleton @ ( omega @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[21582:[bind(A,$thf( omega @ singleton ))]]) ).
thf(69,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(264,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)],[69]) ).
thf(12571,plain,
( ( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( z @ regular @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1633,12451]) ).
thf(12572,plain,
( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[12571]) ).
thf(12667,plain,
( ( z != z )
| ( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= ( z @ regular ) ) ),
inference(simp,[status(thm)],[12572]) ).
thf(12683,plain,
( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= ( z @ regular ) ),
inference(simp,[status(thm)],[12667]) ).
thf(76,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(279,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)],[76]) ).
thf(180,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[179]) ).
thf(21482,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21391,416]) ).
thf(21483,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[21482]) ).
thf(21538,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= identity_relation )
| ( omega != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[21483]) ).
thf(21555,plain,
( ( ( omega @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[21538]) ).
thf(23745,plain,
! [B: $i,A: $i] :
( ( ( A @ ( universal_class @ regular @ unordered_pair ) @ ( universal_class @ regular @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21358,246]) ).
thf(23746,plain,
! [B: $i,A: $i] :
( ( A @ ( universal_class @ regular @ unordered_pair ) @ ( universal_class @ regular @ member ) )
!= ( null_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[23745]) ).
thf(23769,plain,
! [B: $i,A: $i] :
( ( ( A @ ( universal_class @ regular @ unordered_pair ) )
!= null_class )
| ( ( universal_class @ regular )
!= B ) ),
inference(simp,[status(thm)],[23746]) ).
thf(23791,plain,
! [A: $i] :
( ( A @ ( universal_class @ regular @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[23769]) ).
thf(29938,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( universal_class @ regular @ unordered_pair ) ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[197,23791]) ).
thf(29939,plain,
! [A: $i] :
( ( universal_class @ regular @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[29938:[bind(A,$thf( A )),bind(B,$thf( universal_class @ regular )),bind(C,$thf( A ))]]) ).
thf(15,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(148,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)],[15]) ).
thf(27,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(175,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(176,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[175]) ).
thf(79,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(286,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(2388,plain,
! [B: $i,A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( z @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1683,315]) ).
thf(2389,plain,
! [B: $i,A: $i] :
( ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2388]) ).
thf(2395,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( z @ regular ) )
| ( ( z @ regular @ regular )
!= A )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2389]) ).
thf(2398,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( z @ regular ) )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2395]) ).
thf(4678,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ ( null_class @ inverse @ intersection ) ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3664,415]) ).
thf(4713,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[4678]) ).
thf(4726,plain,
( ( ( null_class @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[4713]) ).
thf(6759,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( ( null_class @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[4726,246]) ).
thf(6760,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( ( null_class @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[6759]) ).
thf(6764,plain,
! [A: $i] :
( ( subset_relation != null_class )
| ( ( subset_relation @ regular )
!= A )
| ( ( null_class @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[6760]) ).
thf(6774,plain,
( ( subset_relation != null_class )
| ( ( null_class @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[6764]) ).
thf(203,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(cnf,[status(esa)],[202]) ).
thf(204,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( B = C )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[203]) ).
thf(23721,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( universal_class @ regular @ unordered_pair ) ) )
| ( B @ ( A @ unordered_pair ) @ ( universal_class @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[197,21358]) ).
thf(23722,plain,
! [A: $i] : ( universal_class @ regular @ ( A @ unordered_pair ) @ ( universal_class @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[23721:[bind(A,$thf( A )),bind(B,$thf( universal_class @ regular )),bind(C,$thf( A ))]]) ).
thf(3625,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( null_class @ inverse @ intersection ) ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1443,415]) ).
thf(3660,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[3625]) ).
thf(3663,plain,
( ( ( subset_relation @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[3660]) ).
thf(30,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(181,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(158,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[157]) ).
thf(159,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[158]) ).
thf(774,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[612,557]) ).
thf(785,plain,
! [A: $i] :
( ( identity_relation != subset_relation )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[774]) ).
thf(787,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[785]) ).
thf(57,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(242,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(735,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)],[248,242]) ).
thf(736,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[735:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(19442,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( A @ ( A @ singleton @ ( choice @ image ) @ sum_class @ member ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[330,18316]) ).
thf(19443,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( A @ ( A @ singleton @ ( choice @ image ) @ sum_class @ member ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[19442]) ).
thf(19444,plain,
( ( omega @ ( omega @ singleton @ ( choice @ image ) @ sum_class @ member ) )
| ( omega = null_class ) ),
inference(pattern_uni,[status(thm)],[19443:[bind(A,$thf( omega ))]]) ).
thf(2283,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2228,594]) ).
thf(2284,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2283:[bind(A,$thf( null_class @ domain_of ))]]) ).
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(318,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(39,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(198,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)],[39]) ).
thf(199,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[198]) ).
thf(8943,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ $true
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[199,246]) ).
thf(8944,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[8943]) ).
thf(8945,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[8944:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( G )),bind(D,$thf( null_class )),bind(E,$thf( G @ ( I @ ( H @ ordered_pair ) @ ordered_pair ) ))]]) ).
thf(8986,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[8945]) ).
thf(86,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(303,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)],[86]) ).
thf(12642,plain,
( ( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[122,12451]) ).
thf(12643,plain,
( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( x @ member ) ) ),
inference(simp,[status(thm)],[12642]) ).
thf(12675,plain,
( ( z != z )
| ( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= x ) ),
inference(simp,[status(thm)],[12643]) ).
thf(12676,plain,
( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= x ),
inference(simp,[status(thm)],[12675]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(186,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(187,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[186]) ).
thf(475,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[212,125]) ).
thf(476,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( z @ ( y @ ( x @ unordered_pair ) @ subclass ) ) ),
inference(simp,[status(thm)],[475]) ).
thf(477,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= z )
| ( ( A @ flip )
!= ( y @ ( x @ unordered_pair ) ) ) ),
inference(simp,[status(thm)],[476]) ).
thf(106,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(345,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)],[106]) ).
thf(94,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(323,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(1833,plain,
! [A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1511,416]) ).
thf(1834,plain,
! [A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[1833]) ).
thf(1858,plain,
! [A: $i] :
( ( ( y @ ( x @ unordered_pair ) )
!= identity_relation )
| ( ( y @ ( x @ unordered_pair ) @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[1834]) ).
thf(1862,plain,
( ( ( y @ ( x @ unordered_pair ) )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[1858]) ).
thf(12725,plain,
! [A: $i] :
( ( A
!= ( subset_relation @ inverse ) )
| ( null_class != null_class )
| ( identity_relation != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,12437]) ).
thf(12726,plain,
! [A: $i] :
( ( A
!= ( subset_relation @ inverse ) )
| ( identity_relation != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[12725]) ).
thf(12727,plain,
( ( identity_relation != z )
| ( subset_relation @ inverse @ ( subset_relation @ inverse @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[12726:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(23872,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[23704,315]) ).
thf(23873,plain,
! [B: $i,A: $i] :
( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[23872]) ).
thf(23886,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) )
| ( ( universal_class @ regular )
!= A ) ),
inference(simp,[status(thm)],[23873]) ).
thf(23902,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) ),
inference(simp,[status(thm)],[23886]) ).
thf(24,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(168,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)],[24]) ).
thf(169,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)],[168]) ).
thf(170,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)],[169]) ).
thf(4264,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)],[170,161]) ).
thf(11,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(140,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)],[11]) ).
thf(56,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(240,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)],[56]) ).
thf(62,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(251,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(222,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[221]) ).
thf(223,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(lifteq,[status(thm)],[222]) ).
thf(70,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(267,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(194,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[193]) ).
thf(8258,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ $true
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[194,246]) ).
thf(8259,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[8258]) ).
thf(8260,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[8259:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( I )),bind(D,$thf( null_class )),bind(E,$thf( G @ ( I @ ( H @ ordered_pair ) @ ordered_pair ) ))]]) ).
thf(8291,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[8260]) ).
thf(1065,plain,
! [A: $i] :
( ( ( A @ diagonalise )
!= ( subset_relation @ inverse @ diagonalise ) )
| ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[152,789]) ).
thf(1066,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( subset_relation @ inverse @ intersection ) @ domain_of @ complement @ intersection ) )
!= universal_class ) ),
inference(pattern_uni,[status(thm)],[1065:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(1444,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,154]) ).
thf(1445,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1444:[bind(A,$thf( subset_relation ))]]) ).
thf(135,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[134]) ).
thf(241,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[240]) ).
thf(3749,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3711,2275]) ).
thf(3750,plain,
( ( universal_class @ ( null_class @ domain_of @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3749:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ power_class ))]]) ).
thf(8990,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(rewrite,[status(thm)],[3750,130]) ).
thf(621,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[612,417]) ).
thf(649,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[621]) ).
thf(653,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[649]) ).
thf(2088,plain,
! [A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true
| ( y != x ) ),
inference(paramod_ordered,[status(thm)],[1484,416]) ).
thf(2089,plain,
! [A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( y != x ) ),
inference(simp,[status(thm)],[2088]) ).
thf(2104,plain,
! [A: $i] :
( ( ( x @ singleton )
!= identity_relation )
| ( ( x @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( y != x ) ),
inference(simp,[status(thm)],[2089]) ).
thf(2107,plain,
( ( ( x @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( y != x ) ),
inference(simp,[status(thm)],[2104]) ).
thf(21,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(30242,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[331,30200]) ).
thf(30243,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[30242]) ).
thf(30255,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[30243]) ).
thf(64,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(255,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)],[64]) ).
thf(21404,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( omega @ unordered_pair ) ) )
| ( B @ ( A @ unordered_pair ) @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[197,21360]) ).
thf(21405,plain,
! [A: $i] : ( omega @ ( A @ unordered_pair ) @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[21404:[bind(A,$thf( A )),bind(B,$thf( omega )),bind(C,$thf( A ))]]) ).
thf(111,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality1) ).
thf(355,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(105,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',right_cancellation) ).
thf(342,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(77,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(282,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)],[77]) ).
thf(96,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(327,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(22263,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[22124,416]) ).
thf(22264,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[22263]) ).
thf(22320,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= identity_relation )
| ( null_class != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[22264]) ).
thf(22344,plain,
( ( ( null_class @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[22320]) ).
thf(2881,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( A @ ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[2275,1678]) ).
thf(2882,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[2881:[bind(A,$thf( null_class @ domain_of @ power_class @ ( element_relation @ image ) ))]]) ).
thf(83,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality4) ).
thf(295,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(149,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[148]) ).
thf(150,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[149]) ).
thf(696,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[612,560]) ).
thf(704,plain,
! [A: $i] :
( ( identity_relation != subset_relation )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= z ) ),
inference(simp,[status(thm)],[696]) ).
thf(707,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ diagonalise @ intersection ) )
!= z ) ),
inference(simp,[status(thm)],[704]) ).
thf(103,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(338,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(182,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[181]) ).
thf(1535,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,195]) ).
thf(1536,plain,
( ( null_class @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1535:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(1954,plain,
! [A: $i] :
( ( A
!= ( identity_relation @ complement ) )
| ( ( null_class @ diagonalise )
= ( null_class @ domain_of @ complement ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,1869]) ).
thf(1955,plain,
( ( ( null_class @ diagonalise )
= ( null_class @ domain_of @ complement ) )
| ( identity_relation @ complement @ ( identity_relation @ complement @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1954:[bind(A,$thf( identity_relation @ complement ))]]) ).
thf(720,plain,
( ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != subset_relation )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[154,710]) ).
thf(727,plain,
( ( ( identity_relation @ domain_of )
!= subset_relation )
| ( ( subset_relation @ inverse @ diagonalise )
!= ( subset_relation @ inverse ) )
| ( identity_relation != subset_relation )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[720]) ).
thf(434,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[330,416]) ).
thf(435,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[434]) ).
thf(442,plain,
! [A: $i] :
( ( identity_relation != universal_class )
| ( omega != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[435]) ).
thf(446,plain,
( ( identity_relation != universal_class )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[442]) ).
thf(143,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[142]) ).
thf(102,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(336,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(21504,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ singleton @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,21391]) ).
thf(21505,plain,
( ( omega @ singleton @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[21504:[bind(A,$thf( omega ))]]) ).
thf(23931,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,23904]) ).
thf(23932,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[23931]) ).
thf(23933,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[23932:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(23595,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( B @ singleton )
!= z )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[230,453]) ).
thf(23596,plain,
( ( ( x @ singleton )
!= z )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[23595:[bind(A,$thf( y )),bind(B,$thf( x ))]]) ).
thf(12711,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= ( z @ regular ) ) ),
inference(paramod_ordered,[status(thm)],[248,12683]) ).
thf(12721,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= ( z @ regular ) ) ),
inference(simp,[status(thm)],[12711]) ).
thf(12722,plain,
( ( y != x )
| ( ( z @ ( x @ singleton @ not_subclass_element ) )
!= ( z @ regular ) ) ),
inference(simp,[status(thm)],[12721]) ).
thf(116,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(368,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(3695,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[1279,246]) ).
thf(3696,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[3695]) ).
thf(3703,plain,
! [A: $i] :
( ( subset_relation != null_class )
| ( ( subset_relation @ regular )
!= A )
| ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[3696]) ).
thf(3709,plain,
( ( subset_relation != null_class )
| ( ( null_class @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[3703]) ).
thf(66,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(258,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)],[66]) ).
thf(2390,plain,
( ~ $true
| ( ( z @ regular @ ( z @ regular @ regular @ member ) )
!= ( z @ ( null_class @ member ) ) )
| ( z @ ( null_class @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[1683]) ).
thf(2394,plain,
( ( ( z @ regular )
!= z )
| ( ( z @ regular @ regular )
!= null_class )
| ( z @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2390]) ).
thf(12687,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= x ) ),
inference(paramod_ordered,[status(thm)],[248,12676]) ).
thf(12697,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( z @ ( A @ singleton @ not_subclass_element ) )
!= x ) ),
inference(simp,[status(thm)],[12687]) ).
thf(12698,plain,
( ( y != x )
| ( ( z @ ( x @ singleton @ not_subclass_element ) )
!= x ) ),
inference(simp,[status(thm)],[12697]) ).
thf(4806,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ ( D @ ( universal_class @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ domain_of )
= ( D @ sum_class ) ) ),
inference(paramod_ordered,[status(thm)],[4802,481]) ).
thf(4807,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(pattern_uni,[status(thm)],[4806:[bind(A,$thf( element_relation )),bind(B,$thf( universal_class )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(5008,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(simp,[status(thm)],[4807]) ).
thf(639,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[612,415]) ).
thf(647,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[639]) ).
thf(652,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[647]) ).
thf(109,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(351,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(30405,plain,
( ( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[122,30237]) ).
thf(30406,plain,
( ( z @ ( z @ ( universal_class @ not_subclass_element ) @ member ) )
!= ( z @ ( x @ member ) ) ),
inference(simp,[status(thm)],[30405]) ).
thf(30435,plain,
( ( z != z )
| ( ( z @ ( universal_class @ not_subclass_element ) )
!= x ) ),
inference(simp,[status(thm)],[30406]) ).
thf(30472,plain,
( ( z @ ( universal_class @ not_subclass_element ) )
!= x ),
inference(simp,[status(thm)],[30435]) ).
thf(81,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(290,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(1904,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[1678,560]) ).
thf(1905,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
!= z ),
inference(pattern_uni,[status(thm)],[1904:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(2023,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
!= z ),
inference(rewrite,[status(thm)],[1905,1869]) ).
thf(2050,plain,
( ( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[154,2023]) ).
thf(2066,plain,
( ( ( null_class @ domain_of )
!= subset_relation )
| ( ( null_class @ domain_of @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[2050]) ).
thf(715,plain,
! [A: $i] :
( ( ( A @ diagonalise )
!= ( subset_relation @ inverse @ diagonalise ) )
| ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ intersection ) )
!= z ) ),
inference(paramod_ordered,[status(thm)],[152,710]) ).
thf(716,plain,
( ( identity_relation != subset_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( subset_relation @ inverse @ intersection ) @ domain_of @ complement @ intersection ) )
!= z ) ),
inference(pattern_uni,[status(thm)],[715:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality2) ).
thf(300,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(1956,plain,
! [A: $i] :
( ( A != identity_relation )
| ( ( null_class @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[381,1869]) ).
thf(1957,plain,
( ( ( null_class @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) )
| ( identity_relation @ ( identity_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1956:[bind(A,$thf( identity_relation ))]]) ).
thf(31,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(183,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(73,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(273,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(12624,plain,
( ( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( null_class @ member ) ) )
| ~ $true
| ( ( z @ regular )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2397,12451]) ).
thf(12625,plain,
( ( ( z @ ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) @ member ) )
!= ( z @ ( null_class @ member ) ) )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[12624]) ).
thf(12660,plain,
( ( z != z )
| ( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= null_class )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[12625]) ).
thf(12681,plain,
( ( ( z @ ( y @ ( x @ unordered_pair ) @ not_subclass_element ) )
!= null_class )
| ( ( z @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[12660]) ).
thf(119,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(373,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)],[119]) ).
thf(5911,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( ( subset_relation @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(paramod_ordered,[status(thm)],[3663,246]) ).
thf(5912,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[5911]) ).
thf(5920,plain,
! [A: $i] :
( ( subset_relation != null_class )
| ( ( subset_relation @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[5912]) ).
thf(5926,plain,
( ( subset_relation != null_class )
| ( ( subset_relation @ complement )
!= ( null_class @ inverse ) )
| ( identity_relation != z ) ),
inference(simp,[status(thm)],[5920]) ).
thf(25754,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[237,393]) ).
thf(25755,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[25754:[bind(A,$thf( null_class ))]]) ).
thf(28864,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[25755]) ).
thf(32792,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[28864,329]) ).
thf(32793,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[32792]) ).
thf(87,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(305,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(71,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(268,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)],[71]) ).
thf(112,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(358,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(3187,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( y != x ) ),
inference(paramod_ordered,[status(thm)],[1864,315]) ).
thf(3188,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( y != x ) ),
inference(simp,[status(thm)],[3187]) ).
thf(3192,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( x @ singleton @ regular )
!= A )
| ( y != x ) ),
inference(simp,[status(thm)],[3188]) ).
thf(3196,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( y != x ) ),
inference(simp,[status(thm)],[3192]) ).
thf(1888,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[1678,557]) ).
thf(1889,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[1888:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(1976,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
!= universal_class ),
inference(rewrite,[status(thm)],[1889,1869]) ).
thf(2003,plain,
( ( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[154,1976]) ).
thf(2021,plain,
( ( ( null_class @ domain_of )
!= subset_relation )
| ( ( null_class @ domain_of @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[2003]) ).
thf(239,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[238]) ).
thf(1881,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[172,1678]) ).
thf(1882,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[1881:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1930,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[1882]) ).
thf(100,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(332,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(2997,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[1678,1928]) ).
thf(2998,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[2997:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(3057,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
= null_class ),
inference(rewrite,[status(thm)],[2998,1869]) ).
thf(75,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(277,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(141,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first ) ),
inference(lifteq,[status(thm)],[140]) ).
thf(728,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( A @ ( C @ singleton @ ( B @ cross_product ) @ intersection ) @ not_subclass_element ) @ first ) ),
inference(rewrite,[status(thm)],[141,130]) ).
thf(23810,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[23704,416]) ).
thf(23811,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[23810]) ).
thf(23894,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton )
!= identity_relation )
| ( ( universal_class @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[23811]) ).
thf(23908,plain,
( ( ( universal_class @ regular @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[23894]) ).
thf(12771,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( subset_relation != z ) ),
inference(paramod_ordered,[status(thm)],[12347,246]) ).
thf(12772,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( subset_relation != z ) ),
inference(simp,[status(thm)],[12771]) ).
thf(12780,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( x != A )
| ( subset_relation != z ) ),
inference(simp,[status(thm)],[12772]) ).
thf(12793,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( subset_relation != z ) ),
inference(simp,[status(thm)],[12780]) ).
thf(4208,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( ( x @ singleton )
!= identity_relation )
| ( y != x ) ),
inference(paramod_ordered,[status(thm)],[612,2108]) ).
thf(4216,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( ( x @ singleton )
!= identity_relation )
| ( y != x ) ),
inference(simp,[status(thm)],[4208]) ).
thf(4229,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( ( x @ singleton )
!= identity_relation )
| ( y != x ) ),
inference(simp,[status(thm)],[4216]) ).
thf(163,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[162]) ).
thf(225,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(cnf,[status(esa)],[224]) ).
thf(226,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(lifteq,[status(thm)],[225]) ).
thf(227,plain,
! [A: $i] :
( ( successor_relation @ ( A @ successor @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ successor @ ( A @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[226]) ).
thf(88,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(309,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)],[88]) ).
thf(114,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(364,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(35582,plain,
$false,
inference(e,[status(thm)],[1633,347,249,12731,234,23904,481,138,12710,217,1443,22193,23704,247,379,202,1078,30478,18316,21358,747,153,142,4870,2844,340,21563,31468,1441,288,320,257,417,7235,4866,184,372,1860,1279,196,22834,157,1511,751,802,125,22144,189,416,152,228,316,31351,325,164,12347,789,179,30253,376,1864,238,353,480,211,253,147,1928,221,293,453,133,1433,243,312,206,1863,248,201,12446,381,30466,270,220,366,12437,746,3711,160,429,275,165,3664,2396,21507,334,260,349,197,361,329,20758,522,2228,710,2769,33090,22046,285,224,22677,156,188,388,9376,22124,1683,10107,124,22343,193,2419,2108,212,393,7948,173,129,7913,3967,371,134,244,7052,205,22339,128,237,23644,12677,298,750,425,513,2275,5359,166,21583,264,12683,161,279,180,21555,29939,148,176,286,2398,6774,204,23722,30200,144,3663,12451,21360,181,1638,159,787,21558,172,219,736,19444,2284,230,245,318,8986,303,208,12676,187,477,345,323,213,1862,12727,4726,155,198,330,23902,4264,594,140,240,251,130,223,2288,267,8291,1066,1445,135,12373,241,8990,653,2107,162,30255,255,21405,209,355,342,123,194,145,3842,282,327,22344,1484,263,2882,2397,295,150,707,23791,199,331,1869,127,338,177,182,1536,154,1955,727,175,314,446,595,612,143,21362,4802,336,21505,23933,4873,218,23596,250,231,12722,368,748,3709,258,186,2394,12698,5008,652,351,30472,171,290,433,207,2066,235,246,716,214,151,300,315,1957,1678,168,146,183,422,273,190,12681,373,5926,32793,305,268,358,195,30237,400,3196,2021,415,239,242,126,136,1930,332,3057,178,277,728,21461,23908,12793,131,4229,163,200,227,122,309,21391,364,12047]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET076-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.08 % 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
% 0.14/0.39 % Computer : n019.cluster.edu
% 0.14/0.39 % Model : x86_64 x86_64
% 0.14/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.39 % Memory : 8046.5625MB
% 0.14/0.39 % OS : Linux 6.8.0-71-generic
% 0.14/0.39 % CPULimit : 300
% 0.14/0.39 % WCLimit : 300
% 0.14/0.39 % DateTime : Sat Sep 26 07:03:33 UTC 2026
% 0.14/0.40 % CPUTime :
% 0.14/0.40 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
% 1.30/1.29 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 2.49/1.72 % [INFO] Parsing done (427ms).
% 2.49/1.74 % [INFO] Running in sequential loop mode.
% 3.46/2.37 % [INFO] eprover registered as external prover.
% 3.46/2.38 % [INFO] Scanning for conjecture ...
% 3.89/2.57 % [INFO] Found a conjecture (or negated_conjecture) and 118 axioms. Running axiom selection ...
% 4.28/2.73 % [INFO] Axiom selection finished. Selected 118 axioms (removed 0 axioms).
% 4.91/2.90 % [INFO] Problem is propositional (TPTP CNF).
% 4.91/2.93 % [INFO] Type checking passed.
% 4.91/2.94 % [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 ...
% 142.30/35.90 % External prover 'e' found a proof!
% 142.30/35.90 % [INFO] Killing All external provers ...
% 142.30/35.91 % Time passed: 35284ms (effective reasoning time: 34155ms)
% 142.30/35.91 % Axioms used in derivation (118): operation4, flip2, domain1, operation3, equal_implies_subclass1, complement1, one_to_one2, null_unordered_pair, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, function3, regularity1, singleton_set, identity_relation, commutativity_of_unordered_pair, union, flip3, element_relation2, unordered_pairs_in_universal, corollary_of_null_class_is_subclass, equality1, compose2, homomorphism3, compatible3, symmetric_difference, cartesian_product1, not_subclass_members1, image, replacement, existence_of_null_class, ordered_pair, successor_relation1, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, singleton_in_unordered_pair1, null_class_is_a_set, one_to_one3, single_valued_class1, compose3, omega_is_inductive2, rotate1, left_cancellation, choice1, power_class2, equality2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, inverse, corollary_to_unordered_pair_axiom3, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, not_subclass_members2, successor_relation3, domain2, function1, right_cancellation, compatible4, intersection3, corollary_to_unordered_pair_axiom1, single_valued_class2, unordered_pair_member, apply, element_relation1, transitivity_of_subclass, operation1, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, flip1, homomorphism5, compatible1, unordered_pair2, operation2, rotate3, singleton_in_unordered_pair2, complement2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, corollary_1_to_cartesian_product, null_class_is_unique, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, subclass_is_reflexive, homomorphism2, power_class_definition, unordered_pair_equals_singleton1, corollary_2_to_cartesian_product, choice2, unordered_pair3, special_classes_lemma, subset_relation, compose1
% 142.30/35.91 % No. of inferences in proof: 811
% 142.30/35.91 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 35284 ms resp. 34155 ms w/o parsing
% 144.43/36.26 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 144.43/36.27 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------