TSTP Solution File: SET122-7 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SET122-7 : TPTP v8.1.2. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri May 19 11:52:42 EDT 2023
% Result : Unsatisfiable 5.98s 2.63s
% Output : Refutation 6.15s
% Verified :
% SZS Type : Refutation
% Derivation depth : 2
% Number of leaves : 220
% Syntax : Number of formulae : 390 ( 93 unt; 51 typ; 0 def)
% Number of atoms : 685 ( 224 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 2928 ( 280 ~; 346 |; 0 &;2302 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 67 ( 67 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 51 usr; 11 con; 0-3 aty)
% Number of variables : 626 ( 0 ^; 626 !; 0 ?; 626 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_type,type,
member: $i > $i > $o ).
thf(y_type,type,
y: $i ).
thf(universal_class_type,type,
universal_class: $i ).
thf(second_type,type,
second: $i > $i ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(x_type,type,
x: $i ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(member_of_type,type,
member_of: $i > $i ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(domain_of_type,type,
domain_of: $i > $i ).
thf(inverse_type,type,
inverse: $i > $i ).
thf(range_of_type,type,
range_of: $i > $i ).
thf(function_type,type,
function: $i > $o ).
thf(choice_type,type,
choice: $i ).
thf(subclass_type,type,
subclass: $i > $i > $o ).
thf(not_subclass_element_type,type,
not_subclass_element: $i > $i > $i ).
thf(intersection_type,type,
intersection: $i > $i > $i ).
thf(complement_type,type,
complement: $i > $i ).
thf(null_class_type,type,
null_class: $i ).
thf(image_type,type,
image: $i > $i > $i ).
thf(element_relation_type,type,
element_relation: $i ).
thf(power_class_type,type,
power_class: $i > $i ).
thf(single_valued_class_type,type,
single_valued_class: $i > $o ).
thf(compose_type,type,
compose: $i > $i > $i ).
thf(identity_relation_type,type,
identity_relation: $i ).
thf(one_to_one_type,type,
one_to_one: $i > $o ).
thf(cross_product_type,type,
cross_product: $i > $i > $i ).
thf(restrict_type,type,
restrict: $i > $i > $i > $i ).
thf(compatible_type,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_type,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_type,type,
operation: $i > $o ).
thf(first_type,type,
first: $i > $i ).
thf(domain_type,type,
domain: $i > $i > $i > $i ).
thf(symmetric_difference_type,type,
symmetric_difference: $i > $i > $i ).
thf(inductive_type,type,
inductive: $i > $o ).
thf(successor_relation_type,type,
successor_relation: $i ).
thf(union_type,type,
union: $i > $i > $i ).
thf(diagonalise_type,type,
diagonalise: $i > $i ).
thf(subset_relation_type,type,
subset_relation: $i ).
thf(cantor_type,type,
cantor: $i > $i ).
thf(regular_type,type,
regular: $i > $i ).
thf(sum_class_type,type,
sum_class: $i > $i ).
thf(apply_type,type,
apply: $i > $i > $i ).
thf(omega_type,type,
omega: $i ).
thf(member_of1_type,type,
member_of1: $i > $i ).
thf(successor_type,type,
successor: $i > $i ).
thf(rotate_type,type,
rotate: $i > $i ).
thf(flip_type,type,
flip: $i > $i ).
thf(not_homomorphism1_type,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_type,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(range_type,type,
range: $i > $i > $i > $i ).
thf(104,axiom,
! [A: $i] :
( ( domain_of @ ( flip @ ( cross_product @ A @ universal_class ) ) )
= ( inverse @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(404,plain,
! [A: $i] :
( ( domain_of @ ( flip @ ( cross_product @ A @ universal_class ) ) )
= ( inverse @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(97,axiom,
! [A: $i] :
( ( member @ ( member_of @ A ) @ universal_class )
| ( ( member_of @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists3) ).
thf(388,plain,
! [A: $i] :
( ( member @ ( member_of @ A ) @ universal_class )
| ( ( member_of @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(87,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( intersection @ B @ C ) )
| ( member @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(365,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( intersection @ B @ C ) )
| ( member @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(136,axiom,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_2) ).
thf(479,plain,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(81,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( A = null_class )
| ( member @ ( apply @ choice @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(347,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( A = null_class )
| ( member @ ( apply @ choice @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( unordered_pair @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(518,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( unordered_pair @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(131,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(468,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( singleton @ B ) )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(31,axiom,
! [B: $i,A: $i] :
( ( A != B )
| ( subclass @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(234,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( subclass @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(1,negated_conjecture,
~ ( member @ y @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_corollary_2_to_OP_determines_components2_1) ).
thf(170,plain,
~ ( member @ y @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(155,axiom,
! [A: $i] :
( ( ( singleton @ ( member_of @ A ) )
!= A )
| ( member @ A @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(523,plain,
! [A: $i] :
( ( ( singleton @ ( member_of @ A ) )
!= A )
| ( member @ A @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(23,axiom,
! [A: $i] :
( ~ ( one_to_one @ A )
| ( function @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(217,plain,
! [A: $i] :
( ~ ( one_to_one @ A )
| ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(36,axiom,
! [A: $i] :
( ~ ( operation @ A )
| ( ( cross_product @ ( domain_of @ ( domain_of @ A ) ) @ ( domain_of @ ( domain_of @ A ) ) )
= ( domain_of @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(247,plain,
! [A: $i] :
( ~ ( operation @ A )
| ( ( cross_product @ ( domain_of @ ( domain_of @ A ) ) @ ( domain_of @ ( domain_of @ A ) ) )
= ( domain_of @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ( ( singleton @ A )
!= ( singleton @ B ) )
| ~ ( member @ B @ universal_class )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(440,plain,
! [B: $i,A: $i] :
( ( ( singleton @ A )
!= ( singleton @ B ) )
| ~ ( member @ B @ universal_class )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(46,axiom,
! [A: $i] :
( ( ( singleton @ ( member_of1 @ A ) )
= A )
| ( ( member_of @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(269,plain,
! [A: $i] :
( ( ( singleton @ ( member_of1 @ A ) )
= A )
| ( ( member_of @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(15,axiom,
! [C: $i,B: $i,A: $i] :
( ( intersection @ ( cross_product @ A @ B ) @ C )
= ( restrict @ C @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(202,plain,
! [C: $i,B: $i,A: $i] :
( ( intersection @ ( cross_product @ A @ B ) @ C )
= ( restrict @ C @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(96,axiom,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( first @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_4) ).
thf(385,plain,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( first @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(133,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ ( unordered_pair @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(472,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ ( unordered_pair @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(162,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ( member @ ( not_subclass_element @ A @ B ) @ A )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality1) ).
thf(538,plain,
! [B: $i,A: $i] :
( ( A = B )
| ( member @ ( not_subclass_element @ A @ B ) @ A )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(90,axiom,
! [B: $i,A: $i] : ( member @ ( unordered_pair @ A @ B ) @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(372,plain,
! [B: $i,A: $i] : ( member @ ( unordered_pair @ A @ B ) @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(148,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ successor_relation )
| ( ( successor @ A )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(504,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ successor_relation )
| ( ( successor @ A )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( first @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_ordered_pair_determines_components1) ).
thf(344,plain,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( first @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(53,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ~ ( member @ ( ordered_pair @ D @ E ) @ ( domain_of @ B ) )
| ( ( apply @ C @ ( ordered_pair @ ( apply @ A @ D ) @ ( apply @ A @ E ) ) )
= ( apply @ A @ ( apply @ B @ ( ordered_pair @ D @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(284,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ~ ( member @ ( ordered_pair @ D @ E ) @ ( domain_of @ B ) )
| ( ( apply @ C @ ( ordered_pair @ ( apply @ A @ D ) @ ( apply @ A @ E ) ) )
= ( apply @ A @ ( apply @ B @ ( ordered_pair @ D @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(109,axiom,
! [B: $i,A: $i] :
( ( ( restrict @ A @ ( singleton @ B ) @ universal_class )
!= null_class )
| ~ ( member @ B @ ( domain_of @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(416,plain,
! [B: $i,A: $i] :
( ( ( restrict @ A @ ( singleton @ B ) @ universal_class )
!= null_class )
| ~ ( member @ B @ ( domain_of @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( singleton @ C ) )
| ~ ( member @ C @ universal_class )
| ( C = A )
| ( C = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(325,plain,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( singleton @ C ) )
| ~ ( member @ C @ universal_class )
| ( C = A )
| ( C = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(55,axiom,
subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(289,plain,
subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(123,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( A = B )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality2) ).
thf(448,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( A = B )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(68,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( singleton @ A )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(316,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( singleton @ A )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(42,axiom,
! [B: $i,A: $i] :
( ( sum_class @ ( image @ A @ ( singleton @ B ) ) )
= ( apply @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(261,plain,
! [B: $i,A: $i] :
( ( sum_class @ ( image @ A @ ( singleton @ B ) ) )
= ( apply @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(22,axiom,
! [A: $i] : ( subclass @ null_class @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(216,plain,
! [A: $i] : ( subclass @ null_class @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(121,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ A @ B )
!= ( ordered_pair @ C @ D ) )
| ~ ( member @ A @ universal_class )
| ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components1) ).
thf(443,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ A @ B )
!= ( ordered_pair @ C @ D ) )
| ~ ( member @ A @ universal_class )
| ( A = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(140,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( power_class @ A ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(485,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( power_class @ A ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(32,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( function @ A )
| ( ( domain_of @ ( domain_of @ B ) )
!= ( domain_of @ A ) )
| ~ ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ C ) ) )
| ( compatible @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(238,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( function @ A )
| ( ( domain_of @ ( domain_of @ B ) )
!= ( domain_of @ A ) )
| ~ ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ C ) ) )
| ( compatible @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(83,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(353,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(101,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( cross_product @ B @ C ) )
| ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(397,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( cross_product @ B @ C ) )
| ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(51,axiom,
! [A: $i] :
( ~ ( one_to_one @ A )
| ( function @ ( inverse @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(280,plain,
! [A: $i] :
( ~ ( one_to_one @ A )
| ( function @ ( inverse @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(25,axiom,
! [B: $i,A: $i] : ( subclass @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(221,plain,
! [B: $i,A: $i] : ( subclass @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(58,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ B )
| ( subclass @ ( singleton @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(293,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ B )
| ( subclass @ ( singleton @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(125,axiom,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( first @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets3) ).
thf(453,plain,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( first @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( subclass @ B @ C )
| ( subclass @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(243,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( subclass @ B @ C )
| ( subclass @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(115,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( complement @ B ) )
| ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(428,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( complement @ B ) )
| ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(44,axiom,
! [A: $i] :
( ~ ( inductive @ A )
| ( subclass @ omega @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(265,plain,
! [A: $i] :
( ~ ( inductive @ A )
| ( subclass @ omega @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( ( singleton @ A )
!= ( singleton @ B ) )
| ~ ( member @ A @ universal_class )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(507,plain,
! [B: $i,A: $i] :
( ( ( singleton @ A )
!= ( singleton @ B ) )
| ~ ( member @ A @ universal_class )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(64,axiom,
! [B: $i,A: $i] :
( ( range_of @ ( restrict @ A @ B @ universal_class ) )
= ( image @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(307,plain,
! [B: $i,A: $i] :
( ( range_of @ ( restrict @ A @ B @ universal_class ) )
= ( image @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(57,axiom,
! [B: $i,A: $i] : ( member @ ( singleton @ A ) @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_member_of_ordered_pair) ).
thf(292,plain,
! [B: $i,A: $i] : ( member @ ( singleton @ A ) @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(94,axiom,
! [A: $i] :
( ~ ( inductive @ A )
| ( member @ null_class @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(380,plain,
! [A: $i] :
( ~ ( inductive @ A )
| ( member @ null_class @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(14,axiom,
! [A: $i] : ( subclass @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(201,plain,
! [A: $i] : ( subclass @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(160,axiom,
! [A: $i] :
( ~ ( member @ null_class @ A )
| ~ ( subclass @ ( image @ successor_relation @ A ) @ A )
| ( inductive @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(534,plain,
! [A: $i] :
( ~ ( member @ null_class @ A )
| ~ ( subclass @ ( image @ successor_relation @ A ) @ A )
| ( inductive @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(113,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( unordered_pair @ B @ C ) )
| ( A = B )
| ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(424,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( unordered_pair @ B @ C ) )
| ( A = B )
| ( A = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(49,axiom,
! [C: $i,B: $i,A: $i] :
( ( intersection @ A @ ( cross_product @ B @ C ) )
= ( restrict @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(275,plain,
! [C: $i,B: $i,A: $i] :
( ( intersection @ A @ ( cross_product @ B @ C ) )
= ( restrict @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(38,axiom,
! [A: $i] :
( ( complement @ ( domain_of @ ( intersection @ A @ identity_relation ) ) )
= ( diagonalise @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(252,plain,
! [A: $i] :
( ( complement @ ( domain_of @ ( intersection @ A @ identity_relation ) ) )
= ( diagonalise @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(12,axiom,
! [A: $i] :
( ~ ( single_valued_class @ A )
| ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(197,plain,
! [A: $i] :
( ~ ( single_valued_class @ A )
| ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(118,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ~ ( member @ ( not_subclass_element @ B @ A ) @ A )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality4) ).
thf(435,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ~ ( member @ ( not_subclass_element @ B @ A ) @ A )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(167,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( member_of @ ( singleton @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(551,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( member_of @ ( singleton @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(126,axiom,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_3) ).
thf(456,plain,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(29,axiom,
! [A: $i] :
( ~ ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation )
| ( single_valued_class @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(229,plain,
! [A: $i] :
( ~ ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation )
| ( single_valued_class @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(139,axiom,
subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(484,plain,
subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(27,axiom,
! [C: $i,B: $i,A: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ A @ B @ ( singleton @ C ) ) @ null_class ) )
= ( domain @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(225,plain,
! [C: $i,B: $i,A: $i] :
( ( first @ ( not_subclass_element @ ( restrict @ A @ B @ ( singleton @ C ) ) @ null_class ) )
= ( domain @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(168,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ null_class @ ( singleton @ ( singleton @ A ) ) )
= ( ordered_pair @ B @ A ) )
| ( member @ B @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_2_of_ordered_pair) ).
thf(554,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ null_class @ ( singleton @ ( singleton @ A ) ) )
= ( ordered_pair @ B @ A ) )
| ( member @ B @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(10,axiom,
! [A: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ A ) ) )
= ( power_class @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(193,plain,
! [A: $i] :
( ( complement @ ( image @ element_relation @ ( complement @ A ) ) )
= ( power_class @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(156,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( member_of @ ( singleton @ A ) ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists1) ).
thf(526,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( member_of @ ( singleton @ A ) ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(142,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( complement @ B ) )
| ~ ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(489,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ ( complement @ B ) )
| ~ ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(20,axiom,
! [A: $i] :
( ~ ( operation @ A )
| ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(212,plain,
! [A: $i] :
( ~ ( operation @ A )
| ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(72,axiom,
! [B: $i,A: $i] :
( ( ( unordered_pair @ null_class @ ( singleton @ null_class ) )
= ( ordered_pair @ A @ B ) )
| ( member @ A @ universal_class )
| ( member @ B @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_3_of_ordered_pair) ).
thf(328,plain,
! [B: $i,A: $i] :
( ( ( unordered_pair @ null_class @ ( singleton @ null_class ) )
= ( ordered_pair @ A @ B ) )
| ( member @ A @ universal_class )
| ( member @ B @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(132,axiom,
! [B: $i,A: $i] : ( member @ ( unordered_pair @ A @ ( singleton @ B ) ) @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member_of_ordered_pair) ).
thf(471,plain,
! [B: $i,A: $i] : ( member @ ( unordered_pair @ A @ ( singleton @ B ) ) @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(40,axiom,
! [A: $i] :
( ( intersection @ ( domain_of @ A ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ A ) ) )
= ( cantor @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(256,plain,
! [A: $i] :
( ( intersection @ ( domain_of @ A ) @ ( diagonalise @ ( compose @ ( inverse @ element_relation ) @ A ) ) )
= ( cantor @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(99,axiom,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ ( singleton @ B ) ) )
= ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(393,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ ( singleton @ B ) ) )
= ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(152,axiom,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( second @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_ordered_pair_determines_components1) ).
thf(515,plain,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( second @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ ( first @ ( ordered_pair @ A @ B ) ) @ ( second @ ( ordered_pair @ A @ B ) ) ) @ ( cross_product @ universal_class @ universal_class ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_1) ).
thf(311,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ ( first @ ( ordered_pair @ A @ B ) ) @ ( second @ ( ordered_pair @ A @ B ) ) ) @ ( cross_product @ universal_class @ universal_class ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(163,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ A @ B )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(541,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ A @ B )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(158,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(530,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(60,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( compose @ C @ D ) )
| ( member @ B @ ( image @ C @ ( image @ D @ ( singleton @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(298,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( compose @ C @ D ) )
| ( member @ B @ ( image @ C @ ( image @ D @ ( singleton @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(107,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ D )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ ( flip @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(412,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ D )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ ( flip @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(5,axiom,
! [A: $i] :
( ( domain_of @ ( inverse @ A ) )
= ( range_of @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(180,plain,
! [A: $i] :
( ( domain_of @ ( inverse @ A ) )
= ( range_of @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(92,axiom,
! [A: $i] :
~ ( member @ A @ null_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(375,plain,
! [A: $i] :
~ ( member @ A @ null_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(151,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ C @ B )
| ( subclass @ ( unordered_pair @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(513,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ C @ B )
| ( subclass @ ( unordered_pair @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(116,axiom,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( unordered_pair @ A @ B )
= ( singleton @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(430,plain,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( unordered_pair @ A @ B )
= ( singleton @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(112,axiom,
! [B: $i,A: $i] : ( member @ ( singleton @ A ) @ ( unordered_pair @ B @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(423,plain,
! [B: $i,A: $i] : ( member @ ( singleton @ A ) @ ( unordered_pair @ B @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(16,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( ( domain_of @ ( domain_of @ B ) )
= ( domain_of @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(204,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( ( domain_of @ ( domain_of @ B ) )
= ( domain_of @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(165,axiom,
! [B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
= null_class )
| ( member @ A @ universal_class )
| ( member @ B @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(545,plain,
! [B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
= null_class )
| ( member @ A @ universal_class )
| ( member @ B @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(98,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ A @ B )
| ( member @ ( ordered_pair @ A @ B ) @ element_relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(391,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( member @ A @ B )
| ( member @ ( ordered_pair @ A @ B ) @ element_relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(95,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ A ) ) )
= ( singleton @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists2) ).
thf(382,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( singleton @ ( member_of @ ( singleton @ A ) ) )
= ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(67,axiom,
! [A: $i] :
( ( member @ ( member_of1 @ A ) @ universal_class )
| ( ( member_of @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(313,plain,
! [A: $i] :
( ( member @ ( member_of1 @ A ) @ universal_class )
| ( ( member_of @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(146,axiom,
! [A: $i] :
( ( member @ ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class ) @ A )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_number_of_elements_in_class) ).
thf(498,plain,
! [A: $i] :
( ( member @ ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class ) @ A )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(130,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ C @ D )
| ( member @ ( ordered_pair @ A @ C ) @ ( cross_product @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(466,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ C @ D )
| ( member @ ( ordered_pair @ A @ C ) @ ( cross_product @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(75,axiom,
! [A: $i] : ( subclass @ ( rotate @ A ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(335,plain,
! [A: $i] : ( subclass @ ( rotate @ A ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(82,axiom,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( second @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets4) ).
thf(350,plain,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( second @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(8,axiom,
! [B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( subclass @ B @ A )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(187,plain,
! [B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( subclass @ B @ A )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(105,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ B @ A )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(406,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( unordered_pair @ B @ A )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(24,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( function @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(48,axiom,
inductive @ omega,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(274,plain,
inductive @ omega,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(110,axiom,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
!= A )
| ( member @ A @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_ordered_pairs_are_sets) ).
thf(419,plain,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
!= A )
| ( member @ A @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(93,axiom,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets1) ).
thf(377,plain,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(35,axiom,
! [A: $i] :
( ~ ( operation @ A )
| ( function @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(245,plain,
! [A: $i] :
( ~ ( operation @ A )
| ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(62,axiom,
! [A: $i] :
( ~ ( member @ A @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ A @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_ordered_pairs_are_sets) ).
thf(303,plain,
! [A: $i] :
( ~ ( member @ A @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ A @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(18,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(208,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( compatible @ A @ B @ C )
| ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( unordered_pair @ B @ A )
= ( singleton @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(362,plain,
! [B: $i,A: $i] :
( ( member @ A @ universal_class )
| ( ( unordered_pair @ B @ A )
= ( singleton @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(124,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ A @ C )
| ( member @ A @ ( intersection @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(451,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ B )
| ~ ( member @ A @ C )
| ( member @ A @ ( intersection @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(135,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( sum_class @ A ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(477,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ ( sum_class @ A ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(144,axiom,
( ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( complement @ ( compose @ ( complement @ element_relation ) @ ( inverse @ element_relation ) ) ) ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(494,plain,
( ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( intersection @ ( cross_product @ universal_class @ universal_class ) @ ( complement @ ( compose @ ( complement @ element_relation ) @ ( inverse @ element_relation ) ) ) ) )
= subset_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(88,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ ( ordered_pair @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets2) ).
thf(367,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ ( ordered_pair @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(138,axiom,
member @ omega @ universal_class,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(483,plain,
member @ omega @ universal_class,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(154,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ B ) )
| ~ ( member @ ( ordered_pair @ A @ C ) @ ( cross_product @ universal_class @ universal_class ) )
| ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',right_cancellation) ).
thf(520,plain,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ B ) )
| ~ ( member @ ( ordered_pair @ A @ C ) @ ( cross_product @ universal_class @ universal_class ) )
| ( A = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(128,axiom,
! [B: $i,A: $i] :
~ ( member @ A @ ( intersection @ ( complement @ B ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(462,plain,
! [B: $i,A: $i] :
~ ( member @ A @ ( intersection @ ( complement @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( operation @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(267,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( operation @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(119,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( operation @ A )
| ~ ( operation @ B )
| ~ ( compatible @ C @ A @ B )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ C @ A @ B ) @ ( not_homomorphism2 @ C @ A @ B ) ) @ ( domain_of @ A ) )
| ( homomorphism @ C @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(438,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( operation @ A )
| ~ ( operation @ B )
| ~ ( compatible @ C @ A @ B )
| ( member @ ( ordered_pair @ ( not_homomorphism1 @ C @ A @ B ) @ ( not_homomorphism2 @ C @ A @ B ) ) @ ( domain_of @ A ) )
| ( homomorphism @ C @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(33,axiom,
! [A: $i] :
( ~ ( inductive @ A )
| ( subclass @ ( image @ successor_relation @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(241,plain,
! [A: $i] :
( ~ ( inductive @ A )
| ( subclass @ ( image @ successor_relation @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(166,axiom,
! [A: $i] :
( ( A = null_class )
| ( member @ ( not_subclass_element @ A @ null_class ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(548,plain,
! [A: $i] :
( ( A = null_class )
| ( member @ ( not_subclass_element @ A @ null_class ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(141,axiom,
! [A: $i] :
( ~ ( subclass @ A @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation )
| ( function @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function3) ).
thf(487,plain,
! [A: $i] :
( ~ ( subclass @ A @ ( cross_product @ universal_class @ universal_class ) )
| ~ ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation )
| ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(79,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( unordered_pair @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(342,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( unordered_pair @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(84,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( ( unordered_pair @ A @ B )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(355,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( ( unordered_pair @ A @ B )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(169,axiom,
! [C: $i,B: $i,A: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ A @ ( singleton @ B ) @ C ) @ null_class ) )
= ( range @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(557,plain,
! [C: $i,B: $i,A: $i] :
( ( second @ ( not_subclass_element @ ( restrict @ A @ ( singleton @ B ) @ C ) @ null_class ) )
= ( range @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(59,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ ( ordered_pair @ A @ B ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets1) ).
thf(295,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( first @ ( ordered_pair @ A @ B ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( union @ ( singleton @ A ) @ ( singleton @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_and_singletons) ).
thf(282,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( union @ ( singleton @ A ) @ ( singleton @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(108,axiom,
! [A: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ A ) )
= ( sum_class @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(414,plain,
! [A: $i] :
( ( domain_of @ ( restrict @ element_relation @ universal_class @ A ) )
= ( sum_class @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(106,axiom,
! [A: $i] :
( ( member @ A @ universal_class )
| ( ( singleton @ A )
= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(409,plain,
! [A: $i] :
( ( member @ A @ universal_class )
| ( ( singleton @ A )
= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( compatible @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(263,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( compatible @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(13,axiom,
! [A: $i] :
( ~ ( function @ ( inverse @ A ) )
| ~ ( function @ A )
| ( one_to_one @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(199,plain,
! [A: $i] :
( ~ ( function @ ( inverse @ A ) )
| ~ ( function @ A )
| ( one_to_one @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(6,axiom,
function @ choice,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(182,plain,
function @ choice,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(73,axiom,
! [A: $i] : ( member @ ( singleton @ A ) @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(331,plain,
! [A: $i] : ( member @ ( singleton @ A ) @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(89,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(370,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( B = A )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality3) ).
thf(338,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( B = A )
| ( member @ ( not_subclass_element @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(164,axiom,
member @ null_class @ ( singleton @ null_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(544,plain,
member @ null_class @ ( singleton @ null_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(76,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( subclass @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(336,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( not_subclass_element @ A @ B ) @ B )
| ( subclass @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(3,axiom,
! [A: $i] :
( ( ( singleton @ ( member_of @ A ) )
= A )
| ( ( member_of @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists4) ).
thf(175,plain,
! [A: $i] :
( ( ( singleton @ ( member_of @ A ) )
= A )
| ( ( member_of @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(122,axiom,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(446,plain,
! [A: $i] :
( ~ ( member @ A @ universal_class )
| ( member @ A @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(137,axiom,
! [A: $i] : ( subclass @ A @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(482,plain,
! [A: $i] : ( subclass @ A @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(37,axiom,
! [B: $i,A: $i] :
( ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) )
= ( union @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(250,plain,
! [B: $i,A: $i] :
( ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) )
= ( union @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(30,axiom,
! [B: $i,A: $i] :
( ~ ( subclass @ A @ ( singleton @ B ) )
| ( A = null_class )
| ( ( singleton @ B )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(231,plain,
! [B: $i,A: $i] :
( ~ ( subclass @ A @ ( singleton @ B ) )
| ( A = null_class )
| ( ( singleton @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(69,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( operation @ A )
| ~ ( operation @ B )
| ~ ( compatible @ C @ A @ B )
| ( ( apply @ B @ ( ordered_pair @ ( apply @ C @ ( not_homomorphism1 @ C @ A @ B ) ) @ ( apply @ C @ ( not_homomorphism2 @ C @ A @ B ) ) ) )
!= ( apply @ C @ ( apply @ A @ ( ordered_pair @ ( not_homomorphism1 @ C @ A @ B ) @ ( not_homomorphism2 @ C @ A @ B ) ) ) ) )
| ( homomorphism @ C @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(319,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( operation @ A )
| ~ ( operation @ B )
| ~ ( compatible @ C @ A @ B )
| ( ( apply @ B @ ( ordered_pair @ ( apply @ C @ ( not_homomorphism1 @ C @ A @ B ) ) @ ( apply @ C @ ( not_homomorphism2 @ C @ A @ B ) ) ) )
!= ( apply @ C @ ( apply @ A @ ( ordered_pair @ ( not_homomorphism1 @ C @ A @ B ) @ ( not_homomorphism2 @ C @ A @ B ) ) ) ) )
| ( homomorphism @ C @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(41,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( intersection @ A @ ( regular @ A ) )
= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(258,plain,
! [A: $i] :
( ( A = null_class )
| ( ( intersection @ A @ ( regular @ A ) )
= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(103,axiom,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( restrict @ B @ ( singleton @ A ) @ universal_class )
= null_class )
| ( member @ A @ ( domain_of @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(401,plain,
! [B: $i,A: $i] :
( ~ ( member @ A @ universal_class )
| ( ( restrict @ B @ ( singleton @ A ) @ universal_class )
= null_class )
| ( member @ A @ ( domain_of @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(2,negated_conjecture,
( ( second @ ( ordered_pair @ x @ y ) )
!= ( ordered_pair @ x @ y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_corollary_2_to_OP_determines_components2_2) ).
thf(171,plain,
( ( second @ ( ordered_pair @ x @ y ) )
!= ( ordered_pair @ x @ y ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(56,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ ( flip @ D ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(290,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ ( flip @ D ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ C ) @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(17,axiom,
! [B: $i,A: $i] : ( subclass @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(207,plain,
! [B: $i,A: $i] : ( subclass @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(47,axiom,
! [A: $i] :
( ( union @ A @ ( singleton @ A ) )
= ( successor @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(272,plain,
! [A: $i] :
( ( union @ A @ ( singleton @ A ) )
= ( successor @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( member @ ( not_subclass_element @ A @ B ) @ A )
| ( subclass @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(536,plain,
! [B: $i,A: $i] :
( ( member @ ( not_subclass_element @ A @ B ) @ A )
| ( subclass @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(117,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( member @ C @ A )
| ( member @ C @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(433,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( subclass @ A @ B )
| ~ ( member @ C @ A )
| ( member @ C @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(150,axiom,
! [A: $i] :
( ( A = null_class )
| ( member @ ( regular @ A ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(510,plain,
! [A: $i] :
( ( A = null_class )
| ( member @ ( regular @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(21,axiom,
! [A: $i] :
( ~ ( function @ A )
| ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(214,plain,
! [A: $i] :
( ~ ( function @ A )
| ( subclass @ ( compose @ A @ ( inverse @ A ) ) @ identity_relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(54,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ ( rotate @ D ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ C ) @ A ) @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(287,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ ( rotate @ D ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ B @ C ) @ A ) @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(61,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ A @ C ) )
| ~ ( member @ ( ordered_pair @ B @ C ) @ ( cross_product @ universal_class @ universal_class ) )
| ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',left_cancellation) ).
thf(300,plain,
! [C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ A @ C ) )
| ~ ( member @ ( ordered_pair @ B @ C ) @ ( cross_product @ universal_class @ universal_class ) )
| ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(9,axiom,
! [A: $i] :
( ( ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class )
!= ( not_subclass_element @ A @ null_class ) )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_number_of_elements_in_class) ).
thf(190,plain,
! [A: $i] :
( ( ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class )
!= ( not_subclass_element @ A @ null_class ) )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(7,axiom,
! [B: $i,A: $i] :
( ( A != B )
| ( subclass @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(183,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( subclass @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(111,axiom,
! [B: $i,A: $i] : ( member @ ( ordered_pair @ A @ B ) @ universal_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_is_set) ).
thf(422,plain,
! [B: $i,A: $i] : ( member @ ( ordered_pair @ A @ B ) @ universal_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(91,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( image @ B @ ( image @ C @ ( singleton @ D ) ) ) )
| ~ ( member @ ( ordered_pair @ D @ A ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ D @ A ) @ ( compose @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(373,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( image @ B @ ( image @ C @ ( singleton @ D ) ) ) )
| ~ ( member @ ( ordered_pair @ D @ A ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ D @ A ) @ ( compose @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(63,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( intersection @ B @ C ) )
| ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(305,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( member @ A @ ( intersection @ B @ C ) )
| ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(19,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( operation @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( homomorphism @ A @ B @ C )
| ( operation @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(78,axiom,
! [A: $i] : ( subclass @ ( flip @ A ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(341,plain,
! [A: $i] : ( subclass @ ( flip @ A ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( ( successor @ A )
!= B )
| ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ A @ B ) @ successor_relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(358,plain,
! [B: $i,A: $i] :
( ( ( successor @ A )
!= B )
| ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) )
| ( member @ ( ordered_pair @ A @ B ) @ successor_relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(11,axiom,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(195,plain,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(134,axiom,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets2) ).
thf(474,plain,
! [A: $i] :
( ( member @ ( ordered_pair @ ( first @ A ) @ ( second @ A ) ) @ ( cross_product @ universal_class @ universal_class ) )
| ( ( second @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(102,axiom,
member @ null_class @ universal_class,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(400,plain,
member @ null_class @ universal_class,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(159,axiom,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ element_relation )
| ( member @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(532,plain,
! [B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ element_relation )
| ( member @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(100,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ D )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ C @ A ) @ B ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ C @ A ) @ B ) @ ( rotate @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(395,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ C ) @ D )
| ~ ( member @ ( ordered_pair @ ( ordered_pair @ C @ A ) @ B ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) )
| ( member @ ( ordered_pair @ ( ordered_pair @ C @ A ) @ B ) @ ( rotate @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(129,axiom,
! [A: $i] :
( ~ ( function @ A )
| ( subclass @ A @ ( cross_product @ universal_class @ universal_class ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(464,plain,
! [A: $i] :
( ~ ( function @ A )
| ( subclass @ A @ ( cross_product @ universal_class @ universal_class ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(74,axiom,
! [A: $i] :
( ( member @ ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class ) @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(332,plain,
! [A: $i] :
( ( member @ ( not_subclass_element @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) @ null_class ) @ ( intersection @ ( complement @ ( singleton @ ( not_subclass_element @ A @ null_class ) ) ) @ A ) )
| ( ( singleton @ ( not_subclass_element @ A @ null_class ) )
= A )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(114,axiom,
! [B: $i,A: $i] : ( subclass @ ( compose @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(427,plain,
! [B: $i,A: $i] : ( subclass @ ( compose @ A @ B ) @ ( cross_product @ universal_class @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ~ ( function @ A )
| ~ ( member @ B @ universal_class )
| ( member @ ( image @ A @ B ) @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(528,plain,
! [B: $i,A: $i] :
( ~ ( function @ A )
| ~ ( member @ B @ universal_class )
| ( member @ ( image @ A @ B ) @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(127,axiom,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( second @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_5) ).
thf(459,plain,
! [A: $i] :
( ( ( ordered_pair @ ( first @ A ) @ ( second @ A ) )
= A )
| ( ( second @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(4,axiom,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(178,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(50,axiom,
! [A: $i] :
( ~ ( subclass @ A @ null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(277,plain,
! [A: $i] :
( ~ ( subclass @ A @ null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(147,axiom,
! [B: $i,A: $i] :
( ( ( singleton @ ( member_of @ A ) )
!= A )
| ~ ( member @ B @ A )
| ( ( member_of @ A )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(501,plain,
! [B: $i,A: $i] :
( ( ( singleton @ ( member_of @ A ) )
!= A )
| ~ ( member @ B @ A )
| ( ( member_of @ A )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(70,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ A @ B )
!= ( ordered_pair @ C @ D ) )
| ~ ( member @ B @ universal_class )
| ( B = D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components2) ).
thf(322,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ A @ B )
!= ( ordered_pair @ C @ D ) )
| ~ ( member @ B @ universal_class )
| ( B = D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(65,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ ( unordered_pair @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(309,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ A @ ( unordered_pair @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(143,axiom,
! [B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ null_class ) )
= ( ordered_pair @ A @ B ) )
| ( member @ B @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(491,plain,
! [B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ null_class ) )
= ( ordered_pair @ A @ B ) )
| ( member @ B @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(26,axiom,
! [A: $i] :
( ~ ( function @ A )
| ( ( cross_product @ ( domain_of @ ( domain_of @ A ) ) @ ( domain_of @ ( domain_of @ A ) ) )
!= ( domain_of @ A ) )
| ~ ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ A ) ) )
| ( operation @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(222,plain,
! [A: $i] :
( ~ ( function @ A )
| ( ( cross_product @ ( domain_of @ ( domain_of @ A ) ) @ ( domain_of @ ( domain_of @ A ) ) )
!= ( domain_of @ A ) )
| ~ ( subclass @ ( range_of @ A ) @ ( domain_of @ ( domain_of @ A ) ) )
| ( operation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(145,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ universal_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(496,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( member @ ( ordered_pair @ A @ B ) @ ( cross_product @ C @ D ) )
| ( member @ B @ universal_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(39,axiom,
( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(254,plain,
( ( intersection @ ( inverse @ subset_relation ) @ subset_relation )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(28,axiom,
! [B: $i,A: $i] :
( ( intersection @ ( complement @ ( intersection @ A @ B ) ) @ ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) ) )
= ( symmetric_difference @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(227,plain,
! [B: $i,A: $i] :
( ( intersection @ ( complement @ ( intersection @ A @ B ) ) @ ( complement @ ( intersection @ ( complement @ A ) @ ( complement @ B ) ) ) )
= ( symmetric_difference @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(570,plain,
$false,
inference(cvc4,[status(thm)],[404,388,365,479,347,518,468,234,170,523,217,247,440,269,202,385,472,538,372,504,344,284,416,325,289,448,316,261,216,443,485,238,353,397,280,221,293,453,243,428,265,507,307,292,380,201,534,424,275,252,197,435,551,456,229,484,225,554,193,526,489,212,328,471,256,393,515,311,541,530,298,412,180,375,513,430,423,204,545,391,382,313,498,466,335,350,187,406,219,274,419,377,245,303,208,362,451,477,494,367,483,520,462,267,438,241,548,487,342,355,557,295,282,414,409,263,199,182,331,370,338,544,336,175,446,482,250,231,319,258,401,171,290,207,272,536,433,510,214,287,300,190,183,422,373,305,210,341,358,195,474,400,532,395,464,332,427,528,459,178,277,501,322,309,491,222,496,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : SET122-7 : TPTP v8.1.2. Bugfixed v7.3.0.
% 0.10/0.16 % Command : run_Leo-III %s %d
% 0.15/0.37 % Computer : n002.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Thu May 18 19:04:20 EDT 2023
% 0.15/0.37 % CPUTime :
% 0.92/0.91 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.59/1.14 % [INFO] Parsing done (227ms).
% 1.59/1.15 % [INFO] Running in sequential loop mode.
% 1.89/1.38 % [INFO] eprover registered as external prover.
% 1.89/1.38 % [INFO] cvc4 registered as external prover.
% 1.89/1.38 % [INFO] Scanning for conjecture ...
% 2.15/1.49 % [INFO] Found a conjecture and 167 axioms. Running axiom selection ...
% 2.32/1.57 % [INFO] Axiom selection finished. Selected 167 axioms (removed 0 axioms).
% 2.84/1.67 % [INFO] Problem is propositional (TPTP CNF).
% 2.84/1.68 % [INFO] Type checking passed.
% 2.84/1.69 % [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 ...
% 5.98/2.62 % External prover 'cvc4' found a proof!
% 5.98/2.62 % [INFO] Killing All external provers ...
% 5.98/2.63 % Time passed: 2073ms (effective reasoning time: 1472ms)
% 5.98/2.63 % Solved by 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)>
% 5.98/2.63 % Axioms used in derivation (167): operation4, equal_implies_subclass1, complement1, one_to_one2, null_unordered_pair, property_3_of_ordered_pair, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, singleton_in_unordered_pair3, function3, corollary_to_set_in_its_singleton, 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, image, unique_1st_and_2nd_in_pair_of_non_sets1, replacement, singletons_are_sets, not_subclass_members1, existence_of_null_class, ordered_pair, successor_relation1, property_2_of_ordered_pair, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, corollary_2_to_ordered_pair_determines_components1, singleton_in_unordered_pair1, singleton_identified_by_element2, member_of_non_singleton_unique1, null_class_is_a_set, member_exists2, one_to_one3, ordered_pair_determines_components1, single_valued_class1, existence_of_1st_and_2nd_5, compose3, omega_is_inductive2, rotate1, corollary_1_to_ordered_pairs_are_sets, left_cancellation, choice1, property_1_of_ordered_pair, power_class2, equality2, regularity2, singleton_member_of_ordered_pair, equal_implies_subclass2, element_relation3, homomorphism6, existence_of_1st_and_2nd_1, intersection1, existence_of_1st_and_2nd_4, flip2, two_subsets_of_singleton, ordered_pair_is_set, corollary_1_to_singletons_are_sets, inverse, corollary_1_to_number_of_elements_in_class, unique_1st_and_2nd_in_pair_of_non_sets4, corollary_to_unordered_pair_axiom3, sum_class_definition, corollary_1_to_ordered_pair_determines_components1, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, only_member_in_singleton, not_subclass_members2, ordered_pair_determines_components2, successor_relation3, member_exists3, domain2, function1, right_cancellation, compatible4, property_of_singletons2, intersection3, singleton_identified_by_element1, corollary_to_unordered_pair_axiom1, member_of_non_singleton_unique2, corollary_2_to_singletons_are_sets, single_valued_class2, unordered_pair_member, apply, corollary_2_to_number_of_elements_in_class, element_relation1, transitivity_of_subclass, operation1, member_of_singleton_is_unique, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, existence_of_1st_and_2nd_2, flip1, homomorphism5, unique_1st_and_2nd_in_pair_of_sets1, compatible1, unordered_pair2, unique_1st_and_2nd_in_pair_of_non_sets3, corollary_2_to_ordered_pairs_are_sets, number_of_elements_in_class, unordered_pair_member_of_ordered_pair, operation2, member_exists4, rotate3, property_of_singletons1, singleton_in_unordered_pair2, complement2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, unique_1st_and_2nd_in_pair_of_sets2, set_in_its_singleton, domain1, unordered_pair_is_subset, null_class_in_its_singleton, corollary_1_to_cartesian_product, null_class_is_unique, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, unordered_pairs_and_singletons, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, singleton_is_null_class, existence_of_1st_and_2nd_3, operation3, subclass_is_reflexive, homomorphism2, power_class_definition, unordered_pair_equals_singleton1, corollary_2_to_cartesian_product, choice2, unordered_pair3, special_classes_lemma, subset_relation, unique_1st_and_2nd_in_pair_of_non_sets2, compose1, member_exists1
% 5.98/2.63 % No. of inferences in proof: 339
% 5.98/2.63 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 2073 ms resp. 1472 ms w/o parsing
% 6.15/2.75 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 6.15/2.76 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------