%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET099-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n011.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 08:29:25 AM UTC 2026
% Result : Unsatisfiable 175.52s 51.24s
% Output : Refutation 164.52s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 141
% Syntax : Number of formulae : 490 ( 158 unt; 0 typ; 0 def)
% Number of atoms : 1051 ( 487 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 4502 ( 567 ~; 497 |; 0 &;3438 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 11 con; 0-3 aty)
% Number of variables : 842 ( 0 ^; 842 !; 0 ?; 842 :)
% Comments :
%------------------------------------------------------------------------------
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(x_decl,type,
x: $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
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(function_decl,type,
function: $i > $o ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(member_of1_decl,type,
member_of1: $i > $i ).
thf(member_of_decl,type,
member_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(choice_decl,type,
choice: $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(102,axiom,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(365,plain,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(140,axiom,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(455,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(456,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[455]) ).
thf(457,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
= null_class ) ),
inference(lifteq,[status(thm)],[456]) ).
thf(1,negated_conjecture,
( ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_1) ).
thf(142,plain,
( ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(145,plain,
( ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ),
inference(lifteq,[status(thm)],[142]) ).
thf(682,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton ) )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[457,145]) ).
thf(683,plain,
( ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) )
| ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[682:[bind(A,$thf( null_class @ ( x @ not_subclass_element ) ))]]) ).
thf(75,axiom,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(301,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(302,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[301]) ).
thf(303,plain,
! [A: $i] :
( ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(lifteq,[status(thm)],[302]) ).
thf(758,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[683,303]) ).
thf(759,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[758]) ).
thf(760,plain,
( ( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= null_class )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(pattern_uni,[status(thm)],[759:[bind(A,$thf( null_class @ ( x @ not_subclass_element ) ))]]) ).
thf(88,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(334,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(335,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[334]) ).
thf(1831,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ singleton @ ( C @ unordered_pair ) ) )
| ( A @ singleton @ ( B @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[335,365]) ).
thf(1832,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1831:[bind(A,$thf( D @ singleton )),bind(B,$thf( D )),bind(C,$thf( D @ singleton ))]]) ).
thf(1850,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[1832]) ).
thf(87,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(332,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(333,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[332]) ).
thf(50,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product3) ).
thf(249,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)],[50]) ).
thf(12,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(168,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)],[12]) ).
thf(169,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)],[168]) ).
thf(170,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)],[169]) ).
thf(16,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip2) ).
thf(176,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)],[16]) ).
thf(177,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)],[176]) ).
thf(86,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(330,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(331,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[330]) ).
thf(2211,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)],[177,331]) ).
thf(2212,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)],[2211]) ).
thf(2213,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[2212:[bind(A,$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(2231,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[2213]) ).
thf(22,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(187,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(188,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(lifteq,[status(thm)],[187]) ).
thf(17,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(178,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(179,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(2398,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ complement )
= ( C @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[179,333]) ).
thf(2399,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2398:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2421,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2399]) ).
thf(2448,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ complement @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[188,2421]) ).
thf(2449,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2448:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(2495,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2449]) ).
thf(78,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(308,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(49,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(247,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(95,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(350,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(115,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(395,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(396,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[395]) ).
thf(499,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[350,396]) ).
thf(500,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[499]) ).
thf(506,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[500]) ).
thf(511,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[506]) ).
thf(61,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair3) ).
thf(269,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(35,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible2) ).
thf(217,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)],[35]) ).
thf(64,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/sandbox2/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(276,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)],[64]) ).
thf(1866,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ singleton ) )
| ( null_class @ singleton @ ( B @ singleton @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[457,1850]) ).
thf(1867,plain,
! [A: $i] :
( ( null_class @ singleton @ ( A @ singleton @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[1866:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(15,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(174,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(175,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[174]) ).
thf(2007,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( A @ ( B @ unordered_pair ) @ ( C @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[175,365]) ).
thf(2008,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[2007:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(2042,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[2008]) ).
thf(20,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip1) ).
thf(185,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(124,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(417,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(132,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(436,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(54,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(257,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(118,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset_relation) ).
thf(403,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)],[118]) ).
thf(404,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)],[403]) ).
thf(106,axiom,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(372,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(26,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(196,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(7,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(157,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(1837,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[365,331]) ).
thf(1838,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[1837]) ).
thf(1841,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[1838]) ).
thf(1846,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[1841]) ).
thf(2015,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[175,1846]) ).
thf(2016,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[2015:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(2039,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2016]) ).
thf(19,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(182,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(183,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(cnf,[status(esa)],[182]) ).
thf(184,plain,
! [A: $i] :
( ( universal_class @ ( A @ member_of1 @ member ) )
| ( ( A @ member_of )
= A ) ),
inference(lifteq,[status(thm)],[183]) ).
thf(29,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(205,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(418,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[417]) ).
thf(505,plain,
! [B: $i,A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( B @ ( B @ complement @ intersection ) ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[418,396]) ).
thf(509,plain,
! [B: $i,A: $i] :
( ( subset_relation != B )
| ( ( subset_relation @ inverse )
!= ( B @ complement ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[505]) ).
thf(514,plain,
! [A: $i] :
( ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[509]) ).
thf(545,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( identity_relation @ ( C @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[205,514]) ).
thf(546,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( identity_relation @ ( C @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[545]) ).
thf(554,plain,
! [C: $i,B: $i,A: $i] :
( ( identity_relation != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= C )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[546]) ).
thf(559,plain,
( ( identity_relation != universal_class )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[554]) ).
thf(23,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(189,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(126,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(421,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(84,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(325,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(5,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(152,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)],[5]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(228,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(70,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation4) ).
thf(289,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)],[70]) ).
thf(137,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists4) ).
thf(448,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(316,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(10,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(164,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_members) ).
thf(238,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(94,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction2) ).
thf(348,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)],[94]) ).
thf(55,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(258,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(478,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[258,303]) ).
thf(479,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[478]) ).
thf(480,plain,
( ( omega @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[479:[bind(A,$thf( omega ))]]) ).
thf(32,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(211,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(52,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(253,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(2,negated_conjecture,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_2) ).
thf(143,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(146,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(polarity_switch,[status(thm)],[143]) ).
thf(147,plain,
( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= x ),
inference(lifteq,[status(thm)],[146]) ).
thf(116,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(397,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(66,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(280,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(36,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(220,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(221,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[220]) ).
thf(135,axiom,
null_class @ singleton @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(444,plain,
null_class @ singleton @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(501,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[444,396]) ).
thf(502,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[501]) ).
thf(510,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[502]) ).
thf(515,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[510]) ).
thf(942,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[333,515]) ).
thf(943,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= ( null_class @ singleton ) ),
inference(pattern_uni,[status(thm)],[942:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(956,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[943]) ).
thf(139,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(453,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(648,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[221,418]) ).
thf(649,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[648:[bind(A,$thf( subset_relation ))]]) ).
thf(122,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(412,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(413,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[412]) ).
thf(970,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[413,515]) ).
thf(971,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ),
inference(pattern_uni,[status(thm)],[970:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(993,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[971]) ).
thf(1418,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 ) )
!= ( null_class @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[649,993]) ).
thf(1429,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[1418]) ).
thf(1433,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[1429]) ).
thf(128,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(426,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(427,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[426]) ).
thf(428,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[427]) ).
thf(30,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists3) ).
thf(206,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(77,axiom,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(307,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(71,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation2) ).
thf(292,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(233,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)],[43]) ).
thf(109,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(381,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(103,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range) ).
thf(366,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)],[103]) ).
thf(57,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(260,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(8,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(159,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)],[8]) ).
thf(160,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(cnf,[status(esa)],[159]) ).
thf(73,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(297,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)],[73]) ).
thf(127,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(424,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(129,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(429,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(165,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[164]) ).
thf(1862,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1850,396]) ).
thf(1863,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1862]) ).
thf(1878,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( ( A @ singleton )
!= B ) ),
inference(simp,[status(thm)],[1863]) ).
thf(1883,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ singleton )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[1878]) ).
thf(25,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation3) ).
thf(192,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)],[25]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(407,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(100,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(361,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(68,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair2) ).
thf(285,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(38,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(224,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(6,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(155,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(156,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[155]) ).
thf(112,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality2) ).
thf(388,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)],[112]) ).
thf(749,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[683,396]) ).
thf(750,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[749]) ).
thf(774,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( ( null_class @ ( x @ not_subclass_element ) )
!= A )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[750]) ).
thf(783,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= universal_class )
| ( ( null_class @ ( x @ ( null_class @ complement @ intersection ) @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[774]) ).
thf(85,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(328,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)],[85]) ).
thf(487,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[418,145]) ).
thf(488,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[487]) ).
thf(964,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)],[413,511]) ).
thf(965,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[964:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(991,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[965]) ).
thf(1143,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)],[418,991]) ).
thf(1149,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[1143]) ).
thf(1155,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[1149]) ).
thf(33,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate3) ).
thf(212,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)],[33]) ).
thf(114,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction1) ).
thf(393,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)],[114]) ).
thf(14,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(173,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(92,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists2) ).
thf(343,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(549,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[444,514]) ).
thf(550,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[549]) ).
thf(553,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= identity_relation )
| ( null_class != A )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[550]) ).
thf(558,plain,
( ( ( null_class @ singleton )
!= identity_relation )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[553]) ).
thf(933,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)],[333,333]) ).
thf(934,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[933:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(953,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[934]) ).
thf(11,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(166,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(59,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation2) ).
thf(264,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)],[59]) ).
thf(9,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible4) ).
thf(161,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)],[9]) ).
thf(531,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[205,303]) ).
thf(532,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[531]) ).
thf(533,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[532:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ unordered_pair ) ))]]) ).
thf(540,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[533]) ).
thf(2027,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[175,540]) ).
thf(2028,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[2027:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(523,plain,
( ( ( null_class @ ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( x @ not_subclass_element ) ) )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[145,488]) ).
thf(525,plain,
( ( null_class != null_class )
| ( ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) )
!= x )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[523]) ).
thf(528,plain,
( ( ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) )
!= x )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[525]) ).
thf(569,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement @ intersection ) ) )
| ( identity_relation != x )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[418,528]) ).
thf(570,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ( identity_relation != x )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[569]) ).
thf(573,plain,
( ( identity_relation != x )
| ( subset_relation != x )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[570]) ).
thf(18,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/sandbox2/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(180,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)],[18]) ).
thf(3,negated_conjecture,
x != null_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_3) ).
thf(144,plain,
x != null_class,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(148,plain,
x != null_class,
inference(polarity_switch,[status(thm)],[144]) ).
thf(149,plain,
null_class != x,
inference(lifteq,[status(thm)],[148]) ).
thf(107,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(375,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(1140,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)],[649,991]) ).
thf(1153,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[1140]) ).
thf(1154,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[1153]) ).
thf(56,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(259,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(113,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(391,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)],[113]) ).
thf(136,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(445,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)],[136]) ).
thf(80,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(313,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(1864,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ singleton ) )
| ( B @ singleton @ singleton @ ( null_class @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[457,1850]) ).
thf(1865,plain,
! [A: $i] :
( ( A @ singleton @ singleton @ ( null_class @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[1864:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(44,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class2) ).
thf(236,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(181,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(cnf,[status(esa)],[180]) ).
thf(661,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[649,515]) ).
thf(668,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[661]) ).
thf(672,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[668]) ).
thf(13,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate2) ).
thf(171,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)],[13]) ).
thf(172,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)],[171]) ).
thf(680,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton ) )
| ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ complement ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[457,488]) ).
thf(681,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ complement ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) )
| ( universal_class @ ( null_class @ ( x @ not_subclass_element ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[680:[bind(A,$thf( null_class @ ( x @ not_subclass_element ) ))]]) ).
thf(63,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(274,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)],[63]) ).
thf(125,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cantor_class) ).
thf(419,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(957,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)],[413,333]) ).
thf(958,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[957:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(996,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[958]) ).
thf(41,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_definition) ).
thf(230,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(48,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(245,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(62,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/sandbox2/benchmark/theBenchmark.p',right_cancellation) ).
thf(271,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)],[62]) ).
thf(101,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(362,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)],[101]) ).
thf(138,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(451,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(131,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(434,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(459,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[258,331]) ).
thf(460,plain,
! [A: $i] :
( ( universal_class @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[459]) ).
thf(461,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( omega != A ) ),
inference(simp,[status(thm)],[460]) ).
thf(462,plain,
universal_class != null_class,
inference(simp,[status(thm)],[461]) ).
thf(46,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(240,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(51,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/sandbox2/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(251,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)],[51]) ).
thf(65,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(278,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(2059,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[175,2039]) ).
thf(2060,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[2059:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(2064,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2060]) ).
thf(74,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(299,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(39,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(226,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)],[39]) ).
thf(60,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/sandbox2/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(267,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)],[60]) ).
thf(133,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality1) ).
thf(438,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)],[133]) ).
thf(47,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(241,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(53,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(255,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(31,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(209,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)],[31]) ).
thf(643,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( B @ range_of ) ) ),
inference(paramod_ordered,[status(thm)],[221,188]) ).
thf(644,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ range_of ) ),
inference(pattern_uni,[status(thm)],[643:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1080,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( B @ range_of ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( B @ inverse @ domain_of ) ) ),
inference(paramod_ordered,[status(thm)],[644,188]) ).
thf(1081,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[1080:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(91,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(341,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(342,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[341]) ).
thf(98,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(355,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(141,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(458,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(1164,plain,
( ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( subset_relation != identity_relation )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[418,1155]) ).
thf(1166,plain,
( ( ( identity_relation @ domain_of )
!= subset_relation )
| ( ( subset_relation @ inverse @ diagonalise )
!= ( subset_relation @ inverse ) )
| ( subset_relation != identity_relation )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[1164]) ).
thf(72,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(295,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(67,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/sandbox2/benchmark/theBenchmark.p',left_cancellation) ).
thf(282,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)],[67]) ).
thf(4,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(150,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(123,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(414,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(938,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[333,511]) ).
thf(939,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[938:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(954,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[939]) ).
thf(121,axiom,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(409,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(79,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality3) ).
thf(310,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)],[79]) ).
thf(197,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[196]) ).
thf(198,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[197]) ).
thf(199,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[198]) ).
thf(1421,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[418,993]) ).
thf(1427,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[1421]) ).
thf(1432,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[1427]) ).
thf(134,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(441,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(105,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetric_difference) ).
thf(370,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)],[105]) ).
thf(90,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(338,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(89,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(336,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(153,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[152]) ).
thf(154,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)],[153]) ).
thf(111,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(386,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(231,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[230]) ).
thf(28,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product2) ).
thf(203,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)],[28]) ).
thf(489,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ~ $true
| ( ( B @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[307,303]) ).
thf(490,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( ( B @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[489]) ).
thf(491,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[490:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(495,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(simp,[status(thm)],[491]) ).
thf(104,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain) ).
thf(368,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)],[104]) ).
thf(82,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(319,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(76,axiom,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(304,plain,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(21,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate1) ).
thf(186,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(158,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[157]) ).
thf(640,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ~ ( identity_relation @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[221,514]) ).
thf(641,plain,
! [A: $i] :
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[640:[bind(A,$thf( subset_relation ))]]) ).
thf(652,plain,
! [A: $i] :
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[641]) ).
thf(96,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(351,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)],[96]) ).
thf(108,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality4) ).
thf(378,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)],[108]) ).
thf(110,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(383,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(93,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(346,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)],[93]) ).
thf(1040,plain,
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( x @ ( null_class @ complement @ intersection ) ) )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= null_class )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[418,760]) ).
thf(1045,plain,
( ( subset_relation != x )
| ( ( subset_relation @ inverse )
!= ( null_class @ complement ) )
| ( ( null_class @ ( x @ not_subclass_element ) @ singleton )
!= null_class )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(simp,[status(thm)],[1040]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product4) ).
thf(214,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)],[34]) ).
thf(1160,plain,
! [A: $i] :
( ( ( A @ diagonalise )
!= ( subset_relation @ inverse @ diagonalise ) )
| ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[413,1155]) ).
thf(1161,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( subset_relation @ inverse @ intersection ) @ domain_of @ complement @ intersection ) )
!= universal_class ) ),
inference(pattern_uni,[status(thm)],[1160:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(69,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class1) ).
thf(287,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(151,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[150]) ).
thf(1570,plain,
( ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( subset_relation != identity_relation )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[418,1432]) ).
thf(1578,plain,
( ( ( identity_relation @ domain_of )
!= subset_relation )
| ( ( subset_relation @ inverse @ diagonalise )
!= ( subset_relation @ inverse ) )
| ( subset_relation != identity_relation )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[1570]) ).
thf(130,axiom,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists1) ).
thf(432,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(119,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(405,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(24,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product1) ).
thf(190,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)],[24]) ).
thf(58,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(262,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)],[58]) ).
thf(99,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(358,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(117,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(400,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(634,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( subset_relation != x )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(paramod_ordered,[status(thm)],[221,488]) ).
thf(635,plain,
( ( subset_relation != x )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( null_class @ ( x @ not_subclass_element ) @ singleton @ complement ) )
| ( ( null_class @ ( identity_relation @ not_subclass_element ) )
= ( null_class @ ( x @ not_subclass_element ) ) ) ),
inference(pattern_uni,[status(thm)],[634:[bind(A,$thf( subset_relation ))]]) ).
thf(1800,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)],[172,331]) ).
thf(1801,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)],[1800]) ).
thf(1802,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[1801:[bind(A,$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(1816,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[1802]) ).
thf(471,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[444,331]) ).
thf(472,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[471]) ).
thf(473,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[472]) ).
thf(474,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[473]) ).
thf(2406,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ ( C @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[179,954]) ).
thf(2407,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[2406:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2417,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2407]) ).
thf(1860,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( identity_relation @ ( B @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1850,514]) ).
thf(1861,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( identity_relation @ ( B @ member ) ) )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[1860]) ).
thf(1880,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= identity_relation )
| ( ( A @ singleton )
!= B )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[1861]) ).
thf(1885,plain,
! [A: $i] :
( ( ( A @ singleton @ singleton )
!= identity_relation )
| ( ( subset_relation @ inverse )
!= ( subset_relation @ complement ) ) ),
inference(simp,[status(thm)],[1880]) ).
thf(3852,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[428,149]) ).
thf(3853,plain,
! [A: $i] :
( ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3852:[]]) ).
thf(4674,plain,
x @ ( x @ regular @ member ),
inference(simp,[status(thm)],[3853]) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[161]) ).
thf(163,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[162]) ).
thf(27,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(200,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(83,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/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(322,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)],[83]) ).
thf(97,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(354,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(1564,plain,
! [A: $i] :
( ( ( A @ diagonalise )
!= ( subset_relation @ inverse @ diagonalise ) )
| ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[413,1432]) ).
thf(1565,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( identity_relation @ ( subset_relation @ inverse @ intersection ) @ domain_of @ complement @ intersection ) )
!= ( null_class @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[1564:[bind(A,$thf( subset_relation @ inverse ))]]) ).
thf(638,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[221,559]) ).
thf(639,plain,
( ( identity_relation != universal_class )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) ) ),
inference(pattern_uni,[status(thm)],[638:[bind(A,$thf( subset_relation ))]]) ).
thf(37,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible3) ).
thf(222,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)],[37]) ).
thf(42,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(232,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(74966,plain,
$false,
inference(e,[status(thm)],[365,760,1850,333,249,170,683,2231,2495,308,247,511,269,217,276,142,1867,2042,185,417,301,436,257,174,404,372,196,157,2039,184,559,189,421,325,152,228,289,448,316,164,179,238,514,348,480,211,253,147,397,280,221,956,453,396,1433,428,206,307,292,233,381,220,366,334,260,160,297,424,429,165,1883,192,407,361,285,224,156,403,188,388,783,328,488,1155,212,393,515,457,173,343,205,1846,558,953,412,166,264,161,2028,573,180,444,149,375,1154,176,259,413,391,445,313,144,1865,335,236,181,350,540,159,672,187,172,681,274,419,996,230,245,303,271,362,451,649,434,155,330,462,240,251,278,455,2064,299,226,267,438,241,167,255,209,1081,342,355,458,145,426,1166,295,282,150,414,954,409,310,993,199,1432,331,441,370,338,177,182,336,154,143,991,175,386,231,203,495,368,319,304,258,186,171,418,158,652,351,378,383,346,1045,214,1161,2421,287,151,1578,432,405,190,168,262,358,400,635,341,1816,474,2417,395,332,528,644,178,1885,4674,163,200,322,354,1565,639,222,232]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET099-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.13/5.56 % Computer : n011.cluster.edu
% 0.13/5.56 % Model : x86_64 x86_64
% 0.13/5.56 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/5.56 % Memory : 8046.5625MB
% 0.13/5.56 % OS : Linux 6.8.0-71-generic
% 0.13/5.56 % CPULimit : 300
% 0.13/5.56 % WCLimit : 300
% 0.13/5.56 % DateTime : Sat Sep 26 07:07:04 UTC 2026
% 0.13/5.56 % CPUTime :
% 0.13/5.56 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.00/6.15 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.81/6.43 % [INFO] Parsing done (280ms).
% 1.81/6.45 % [INFO] Running in sequential loop mode.
% 2.49/6.85 % [INFO] eprover registered as external prover.
% 2.49/6.86 % [INFO] Scanning for conjecture ...
% 2.73/6.99 % [INFO] Found a conjecture (or negated_conjecture) and 141 axioms. Running axiom selection ...
% 2.99/7.11 % [INFO] Axiom selection finished. Selected 138 axioms (removed 3 axioms).
% 3.54/7.23 % [INFO] Problem is propositional (TPTP CNF).
% 3.54/7.24 % [INFO] Type checking passed.
% 3.54/7.25 % [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 ...
% 175.52/51.23 % External prover 'e' found a proof!
% 175.52/51.23 % [INFO] Killing All external provers ...
% 175.52/51.24 % Time passed: 45529ms (effective reasoning time: 44785ms)
% 175.52/51.24 % Axioms used in derivation (138): operation4, number_of_elements_in_class, equal_implies_subclass1, complement1, null_unordered_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, replacement, singletons_are_sets, not_subclass_members1, existence_of_null_class, ordered_pair, successor_relation1, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, singleton_in_unordered_pair1, singleton_identified_by_element2, member_of_non_singleton_unique1, null_class_is_a_set, member_exists2, single_valued_class1, compose3, omega_is_inductive2, rotate1, left_cancellation, choice1, power_class2, equality2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, flip2, two_subsets_of_singleton, corollary_1_to_singletons_are_sets, inverse, corollary_to_unordered_pair_axiom3, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, only_member_in_singleton, not_subclass_members2, 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, element_relation1, transitivity_of_subclass, operation1, member_of_singleton_is_unique, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, flip1, homomorphism5, compatible1, unordered_pair2, operation2, member_exists4, rotate3, property_of_singletons1, singleton_in_unordered_pair2, complement2, subclass_implies_equal, function2, cantor_class, successor_relation2, 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, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, singleton_is_null_class, 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, compose1, member_exists1
% 175.52/51.24 % No. of inferences in proof: 490
% 175.52/51.24 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 45529 ms resp. 44785 ms w/o parsing
% 164.09/51.50 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 164.52/51.51 % [INFO] Killing All external provers ...
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