%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET108-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n002.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 08:29:27 AM UTC 2026
% Result : Unsatisfiable 119.96s 30.63s
% Output : Refutation 112.33s
% Verified :
% SZS Type : Refutation
% Derivation depth : 36
% Number of leaves : 154
% Syntax : Number of formulae : 1002 ( 265 unt; 0 typ; 0 def)
% Number of atoms : 2479 (1093 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 9259 (1422 ~;1101 |; 0 &;6736 @)
% ( 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 : 54 ( 51 usr; 12 con; 0-3 aty)
% Number of variables : 1379 ( 0 ^;1379 !; 0 ?;1379 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(y_decl,type,
y: $i ).
thf(z_decl,type,
z: $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(member_of_decl,type,
member_of: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $i ).
thf(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(null_class_decl,type,
null_class: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(member_of1_decl,type,
member_of1: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(95,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(364,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(365,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[364]) ).
thf(88,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(347,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)],[88]) ).
thf(28,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(212,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(213,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[212]) ).
thf(9,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(173,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(174,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[173]) ).
thf(175,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[174]) ).
thf(147,axiom,
null_class @ singleton @ ( null_class @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(480,plain,
null_class @ singleton @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(81,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(331,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(332,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[331]) ).
thf(507,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[480,332]) ).
thf(508,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[507]) ).
thf(509,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[508]) ).
thf(510,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[509]) ).
thf(881,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[175,510]) ).
thf(882,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( null_class @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[881:[bind(A,$thf( null_class @ singleton )),bind(B,$thf( B ))]]) ).
thf(1604,plain,
( ~ ( null_class @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[882]) ).
thf(20,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(194,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(5023,plain,
( ~ $true
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1604,194]) ).
thf(5024,plain,
~ ( null_class @ ( null_class @ singleton @ subclass ) ),
inference(simp,[status(thm)],[5023]) ).
thf(7873,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[213,5024]) ).
thf(7874,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7873]) ).
thf(7913,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7874]) ).
thf(7922,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7913]) ).
thf(142,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(468,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(38,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(233,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(234,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[233]) ).
thf(44,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(246,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(247,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[246]) ).
thf(248,plain,
! [A: $i] :
( ~ ( null_class @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[247]) ).
thf(134,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(449,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(450,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[449]) ).
thf(451,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[450]) ).
thf(89,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(350,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(499,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[350,332]) ).
thf(500,plain,
! [A: $i] :
( ( universal_class @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[499]) ).
thf(503,plain,
! [A: $i] :
( ( null_class != universal_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[500]) ).
thf(505,plain,
null_class != universal_class,
inference(simp,[status(thm)],[503]) ).
thf(3373,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,505]) ).
thf(3374,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3373:[]]) ).
thf(3646,plain,
universal_class @ ( universal_class @ regular @ member ),
inference(simp,[status(thm)],[3374]) ).
thf(61,axiom,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(283,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(284,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[283]) ).
thf(285,plain,
! [A: $i] :
( ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(lifteq,[status(thm)],[284]) ).
thf(3783,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3646,285]) ).
thf(3784,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3783]) ).
thf(3785,plain,
( ( universal_class @ regular @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[3784:[bind(A,$thf( universal_class @ regular ))]]) ).
thf(15842,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,3785]) ).
thf(15843,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15842]) ).
thf(15844,plain,
~ ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[15843:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(122,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(422,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(553,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[422,285]) ).
thf(554,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[553]) ).
thf(555,plain,
( ( omega @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[554:[bind(A,$thf( omega ))]]) ).
thf(6,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(165,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(19,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(192,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(193,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[192]) ).
thf(4659,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[165,193]) ).
thf(4660,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4659]) ).
thf(4661,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4660:[bind(A,$thf( choice ))]]) ).
thf(1,negated_conjecture,
universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_existence_of_1st_and_2nd_1_1) ).
thf(155,plain,
universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(511,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,332]) ).
thf(512,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[511]) ).
thf(513,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( z @ ( y @ ordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[512]) ).
thf(514,plain,
( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class ),
inference(simp,[status(thm)],[513]) ).
thf(1063,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[175,514]) ).
thf(1064,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1063:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) )),bind(B,$thf( B ))]]) ).
thf(1435,plain,
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ~ ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[1064]) ).
thf(1774,plain,
( ~ $true
| ~ ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(rewrite,[status(thm)],[1435,194]) ).
thf(1775,plain,
~ ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ),
inference(simp,[status(thm)],[1774]) ).
thf(4716,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4661,1775]) ).
thf(4717,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[4716]) ).
thf(4750,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[4717]) ).
thf(4875,plain,
! [A: $i] :
( ( A != identity_relation )
| ( null_class != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,4750]) ).
thf(4876,plain,
! [A: $i] :
( ( A != identity_relation )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[4875]) ).
thf(4877,plain,
( ( ( choice @ inverse @ ( choice @ compose ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( identity_relation @ ( identity_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4876:[bind(A,$thf( identity_relation ))]]) ).
thf(148,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(481,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)],[148]) ).
thf(66,axiom,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(298,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(545,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ~ $true
| ( ( B @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[298,285]) ).
thf(546,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( ( B @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[545]) ).
thf(547,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[546:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(564,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(simp,[status(thm)],[547]) ).
thf(3525,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,564]) ).
thf(3526,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3525]) ).
thf(3527,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[3526:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(3690,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(simp,[status(thm)],[3527]) ).
thf(115,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(409,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(410,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[409]) ).
thf(5248,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3690,410]) ).
thf(5249,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[5248]) ).
thf(5255,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( ( A @ singleton @ singleton @ regular )
!= B ) ),
inference(simp,[status(thm)],[5249]) ).
thf(5260,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ singleton )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[5255]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(231,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(232,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[231]) ).
thf(711,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[175,555]) ).
thf(712,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( omega @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( omega @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[711:[bind(A,$thf( omega @ singleton )),bind(B,$thf( B ))]]) ).
thf(1488,plain,
( ~ ( omega @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[712]) ).
thf(4013,plain,
( ~ $true
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1488,194]) ).
thf(4014,plain,
~ ( null_class @ ( omega @ singleton @ subclass ) ),
inference(simp,[status(thm)],[4013]) ).
thf(11337,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,4014]) ).
thf(11338,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11337]) ).
thf(11355,plain,
! [A: $i] :
( ( A != null_class )
| ( ( omega @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11338]) ).
thf(11372,plain,
( ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11355]) ).
thf(82,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists2) ).
thf(333,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(45,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(249,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(3499,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[451,410]) ).
thf(3500,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[3499]) ).
thf(3501,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3500:[bind(A,$thf( F @ ( F @ complement @ intersection ) )),bind(B,$thf( F @ ( F @ complement @ intersection ) @ regular )),bind(C,$thf( F ))]]) ).
thf(3681,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3501]) ).
thf(4,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(161,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(162,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[161]) ).
thf(98,axiom,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(370,plain,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(7018,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)],[370,332]) ).
thf(7019,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[7018]) ).
thf(7022,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[7019]) ).
thf(7045,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7022]) ).
thf(7584,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( ( B @ ( A @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[162,7045]) ).
thf(7585,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[7584:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( E )),bind(D,$thf( B ))]]) ).
thf(7594,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7585]) ).
thf(7625,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)],[162,7594]) ).
thf(7626,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[7625:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(7641,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7626]) ).
thf(71,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(308,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(3088,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,510]) ).
thf(3089,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3088]) ).
thf(3090,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3089:[bind(A,$thf( null_class @ singleton ))]]) ).
thf(3854,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3090,285]) ).
thf(3855,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3854]) ).
thf(3899,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= universal_class )
| ( ( null_class @ singleton @ regular )
!= A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3855]) ).
thf(3910,plain,
( ( ( null_class @ singleton )
!= universal_class )
| ( ( null_class @ singleton @ regular @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3899]) ).
thf(821,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[175,505]) ).
thf(822,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[821:[bind(A,$thf( null_class ))]]) ).
thf(1558,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[822]) ).
thf(2193,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[1558,194]) ).
thf(2194,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[2193]) ).
thf(11333,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,2194]) ).
thf(11334,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11333]) ).
thf(11357,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11334]) ).
thf(11374,plain,
( ( omega != universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11357]) ).
thf(11393,plain,
! [B: $i,A: $i] :
( ( A != omega )
| ( B != universal_class )
| ~ ( null_class @ inductive )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[175,11374]) ).
thf(11394,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ inductive )
| ~ ( omega @ ( A @ subclass ) )
| ~ ( A @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11393:[bind(A,$thf( omega ))]]) ).
thf(11401,plain,
( ~ ( null_class @ inductive )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[11394]) ).
thf(121,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(421,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(13706,plain,
( ~ ( null_class @ inductive )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[11401,421]) ).
thf(13707,plain,
( ~ ( null_class @ inductive )
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[13706]) ).
thf(102,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(379,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)],[102]) ).
thf(655,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[480,410]) ).
thf(656,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[655]) ).
thf(669,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[656]) ).
thf(677,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[669]) ).
thf(30,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(216,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(217,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[216]) ).
thf(58,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(276,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(93,axiom,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(359,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(360,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[359]) ).
thf(361,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
= null_class ) ),
inference(lifteq,[status(thm)],[360]) ).
thf(42,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(243,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(214,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(215,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[214]) ).
thf(8385,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[215,1775]) ).
thf(8386,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[8385]) ).
thf(8387,plain,
! [A: $i] :
( ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[8386:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) )),bind(B,$thf( B )),bind(C,$thf( null_class ))]]) ).
thf(8437,plain,
! [A: $i] :
( ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[8387]) ).
thf(11341,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( B @ subclass ) ) )
| ~ $true
| ~ ( B @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,8437]) ).
thf(11342,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( B @ subclass ) ) )
| ~ ( B @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11341]) ).
thf(11343,plain,
( ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[11342:[bind(A,$thf( null_class )),bind(B,$thf( omega ))]]) ).
thf(12406,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,11343]) ).
thf(12407,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[12406]) ).
thf(12487,plain,
( ( omega != null_class )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[12407]) ).
thf(13377,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,12487]) ).
thf(13378,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[13377]) ).
thf(13379,plain,
( ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[13378:[bind(A,$thf( omega ))]]) ).
thf(22505,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true
| ( omega @ ( omega @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[421,13379]) ).
thf(22506,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[22505]) ).
thf(22589,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[22506]) ).
thf(22621,plain,
( ( omega != universal_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[22589]) ).
thf(23518,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(paramod_ordered,[status(thm)],[22621,285]) ).
thf(23519,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[23518]) ).
thf(23567,plain,
! [A: $i] :
( ( omega != universal_class )
| ( ( omega @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[23519]) ).
thf(23579,plain,
( ( ( omega @ regular @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[23567]) ).
thf(23599,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( omega @ regular @ singleton ) )
| ( null_class != null_class )
| ( omega != universal_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[361,23579]) ).
thf(23600,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( omega @ regular @ singleton ) )
| ( omega != universal_class )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[23599]) ).
thf(23601,plain,
( ( omega != universal_class )
| ( universal_class @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[23600:[bind(A,$thf( omega @ regular ))]]) ).
thf(131,axiom,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(440,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)],[131]) ).
thf(25,axiom,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(202,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(79,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(328,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(582,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)],[328,285]) ).
thf(583,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)],[582]) ).
thf(584,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[583:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ unordered_pair ) ))]]) ).
thf(593,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[584]) ).
thf(596,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)],[162,593]) ).
thf(597,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[596:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(105,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(384,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(47,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_and_singletons) ).
thf(253,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(254,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ singleton @ union ) )
= ( B @ ( A @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[253]) ).
thf(250,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[249]) ).
thf(17801,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
= ( B @ ( A @ unordered_pair ) ) ),
inference(rewrite,[status(thm)],[254,250]) ).
thf(17870,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( universal_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ member ) ) ),
inference(paramod_ordered,[status(thm)],[17801,328]) ).
thf(17871,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ member ) ),
inference(pattern_uni,[status(thm)],[17870:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(11381,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,11374]) ).
thf(11382,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[11381]) ).
thf(11399,plain,
( ( omega != null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[11382]) ).
thf(144,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(472,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(7906,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[213,2194]) ).
thf(7907,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7906]) ).
thf(7914,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7907]) ).
thf(7924,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7914]) ).
thf(8175,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,7924]) ).
thf(8176,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[8175]) ).
thf(8210,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[8176]) ).
thf(59,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(278,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)],[59]) ).
thf(279,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)],[278]) ).
thf(27701,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,279]) ).
thf(27702,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[27701]) ).
thf(27703,plain,
z @ ( y @ unordered_pair ) @ ( y @ member ),
inference(pattern_uni,[status(thm)],[27702:[bind(A,$thf( y )),bind(B,$thf( z )),bind(C,$thf( universal_class )),bind(D,$thf( universal_class ))]]) ).
thf(27905,plain,
! [A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[27703,332]) ).
thf(27906,plain,
! [A: $i] :
( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[27905]) ).
thf(27921,plain,
! [A: $i] :
( ( ( z @ ( y @ unordered_pair ) )
!= null_class )
| ( y != A ) ),
inference(simp,[status(thm)],[27906]) ).
thf(27931,plain,
( ( z @ ( y @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[27921]) ).
thf(15,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(185,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(27940,plain,
! [A: $i] :
( ( A
!= ( z @ ( y @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,27931]) ).
thf(27941,plain,
! [A: $i] :
( ( A
!= ( z @ ( y @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[27940]) ).
thf(27942,plain,
~ ( null_class @ ( z @ ( y @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[27941:[bind(A,$thf( z @ ( y @ unordered_pair ) ))]]) ).
thf(28086,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ ( y @ unordered_pair ) @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,27942]) ).
thf(28087,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ ( y @ unordered_pair ) @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[28086]) ).
thf(28114,plain,
! [A: $i] :
( ( A != null_class )
| ( ( z @ ( y @ unordered_pair ) )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[28087]) ).
thf(28133,plain,
( ( ( z @ ( y @ unordered_pair ) )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[28114]) ).
thf(129,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(436,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)],[129]) ).
thf(53,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(264,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(265,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[264]) ).
thf(22084,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3090,265]) ).
thf(22085,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22084]) ).
thf(22086,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[22085:[bind(A,$thf( null_class @ singleton @ regular )),bind(B,$thf( null_class @ singleton ))]]) ).
thf(7875,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,213]) ).
thf(7876,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[7875]) ).
thf(7877,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[7876:[bind(A,$thf( omega ))]]) ).
thf(7938,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,5024]) ).
thf(7939,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7938]) ).
thf(7993,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[7939]) ).
thf(3181,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,514]) ).
thf(3182,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3181]) ).
thf(3183,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[3182:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(3716,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3183,285]) ).
thf(3717,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3716]) ).
thf(3766,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( universal_class @ ( universal_class @ cross_product ) @ regular )
!= A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3717]) ).
thf(3773,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( universal_class @ ( universal_class @ cross_product ) @ regular @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3766]) ).
thf(4119,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) @ regular @ singleton ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( null_class != null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[361,3773]) ).
thf(4120,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) @ regular @ singleton ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4119]) ).
thf(4121,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4120:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) @ regular ))]]) ).
thf(4229,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[4121,410]) ).
thf(4230,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4229]) ).
thf(4233,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( ( universal_class @ ( universal_class @ cross_product ) @ regular )
!= A )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4230]) ).
thf(4238,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= universal_class )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4233]) ).
thf(3,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists4) ).
thf(158,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(159,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(cnf,[status(esa)],[158]) ).
thf(160,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[159]) ).
thf(551,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[155,285]) ).
thf(552,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[551]) ).
thf(560,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( z @ ( y @ ordered_pair ) )
!= A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[552]) ).
thf(562,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( z @ ( y @ ordered_pair ) @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[560]) ).
thf(569,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( z @ ( y @ ordered_pair ) @ singleton ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[160,562]) ).
thf(570,plain,
( ( ( z @ ( y @ ordered_pair ) @ singleton )
!= ( null_class @ member_of @ singleton ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[569]) ).
thf(14,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(184,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(3114,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,555]) ).
thf(3115,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3114]) ).
thf(3116,plain,
omega @ singleton @ ( omega @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3115:[bind(A,$thf( omega @ singleton ))]]) ).
thf(32,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(220,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(221,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[220]) ).
thf(657,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[221,410]) ).
thf(667,plain,
! [B: $i,A: $i] :
( ( B != subset_relation )
| ( ( B @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[657]) ).
thf(675,plain,
! [A: $i] :
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[667]) ).
thf(686,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[480,675]) ).
thf(687,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[686]) ).
thf(695,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= identity_relation )
| ( null_class != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[687]) ).
thf(702,plain,
( ( ( null_class @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[695]) ).
thf(28027,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ~ ( null_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[17801,27942]) ).
thf(28028,plain,
~ ( null_class @ ( z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement @ subclass ) ),
inference(pattern_uni,[status(thm)],[28027:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(91,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(354,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(355,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[354]) ).
thf(277,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[276]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(163,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(164,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[163]) ).
thf(26077,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[277,164]) ).
thf(26086,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( omega @ ( successor_relation @ image ) ) )
| ( omega @ ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[26077,7877]) ).
thf(26087,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[26086:[bind(A,$thf( successor_relation )),bind(B,$thf( omega ))]]) ).
thf(26488,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[355,26087]) ).
thf(26489,plain,
omega @ ( universal_class @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[26488:[bind(A,$thf( universal_class @ ( omega @ ( successor_relation @ restrict ) ) ))]]) ).
thf(21,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(195,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(196,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[195]) ).
thf(2,negated_conjecture,
~ ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_existence_of_1st_and_2nd_1_2) ).
thf(156,plain,
~ ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(157,plain,
~ ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) ),
inference(polarity_switch,[status(thm)],[156]) ).
thf(12221,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,11372]) ).
thf(12222,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[12221]) ).
thf(12254,plain,
( ( omega != null_class )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[12222]) ).
thf(107,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality4) ).
thf(389,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)],[107]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(188,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(189,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[188]) ).
thf(16359,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,15844]) ).
thf(16360,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16359]) ).
thf(16371,plain,
! [A: $i] :
( ( A != null_class )
| ( ( universal_class @ regular @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16360]) ).
thf(16392,plain,
( ( ( universal_class @ regular @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[16371]) ).
thf(25198,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( universal_class @ regular @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,16392]) ).
thf(25199,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( universal_class @ regular @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[25198]) ).
thf(25247,plain,
( ( omega != null_class )
| ( ( universal_class @ regular @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[25199]) ).
thf(63,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components2) ).
thf(289,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(10,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(176,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(177,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[176]) ).
thf(4423,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[177,3681]) ).
thf(4424,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[4423:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(4460,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[4424]) ).
thf(75,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(316,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)],[75]) ).
thf(51,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(261,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)],[51]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(178,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(179,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(2436,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[185,1775]) ).
thf(2437,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[2436]) ).
thf(2456,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[2437]) ).
thf(5786,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) )
| ( ( A @ singleton )
!= null_class )
| ( ( B @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(paramod_ordered,[status(thm)],[179,2456]) ).
thf(5787,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ( ( A @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(pattern_uni,[status(thm)],[5786:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(7224,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ singleton ) )
| ( null_class != null_class )
| ( ( B @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[361,5787]) ).
thf(7225,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ singleton ) )
| ( ( B @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[7224]) ).
thf(7226,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[7225:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(11321,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,1775]) ).
thf(11322,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11321]) ).
thf(11359,plain,
! [A: $i] :
( ( A != null_class )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11322]) ).
thf(11376,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11359]) ).
thf(11514,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,11376]) ).
thf(11515,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega ) ),
inference(simp,[status(thm)],[11514]) ).
thf(11545,plain,
( ( omega != null_class )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega ) ),
inference(simp,[status(thm)],[11515]) ).
thf(11562,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,11545]) ).
thf(11563,plain,
! [A: $i] :
( ( A != omega )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[11562]) ).
thf(11564,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[11563:[bind(A,$thf( omega ))]]) ).
thf(6970,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)],[179,370]) ).
thf(6971,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6970:[bind(A,$thf( D @ singleton )),bind(B,$thf( D )),bind(C,$thf( D @ singleton ))]]) ).
thf(7047,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[6971]) ).
thf(35,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(227,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(228,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[227]) ).
thf(9384,plain,
! [A: $i] :
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ ( null_class @ ( A @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,8437]) ).
thf(9385,plain,
! [A: $i] :
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[9384]) ).
thf(9489,plain,
! [A: $i] :
( ( omega != A )
| ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[9385]) ).
thf(9521,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[9489]) ).
thf(11312,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,9521]) ).
thf(11313,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11312]) ).
thf(11314,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[11313:[bind(A,$thf( null_class ))]]) ).
thf(31,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(218,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(219,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[218]) ).
thf(1895,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)],[219,177]) ).
thf(1896,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1895:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1946,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1896]) ).
thf(8897,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ power_class )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3681,1946]) ).
thf(8898,plain,
( ( null_class @ domain_of @ power_class )
= ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[8897:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(4411,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise )
= ( null_class @ domain_of @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3681,219]) ).
thf(4412,plain,
( ( identity_relation @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[4411:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(9104,plain,
( ( null_class @ domain_of @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[8898,4412]) ).
thf(26142,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[26077,9104]) ).
thf(26143,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[26142:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement ))]]) ).
thf(7910,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[213,4014]) ).
thf(7911,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7910]) ).
thf(7919,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7911]) ).
thf(7931,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7919]) ).
thf(9295,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,7931]) ).
thf(9296,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9295]) ).
thf(9330,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9296]) ).
thf(132,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(443,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(1823,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)],[177,177]) ).
thf(1824,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1823:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1858,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1824]) ).
thf(3932,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3116,675]) ).
thf(3933,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3932]) ).
thf(3961,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= identity_relation )
| ( ( omega @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3933]) ).
thf(3973,plain,
( ( ( omega @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3961]) ).
thf(1956,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[355,221]) ).
thf(1957,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[1956:[bind(A,$thf( subset_relation ))]]) ).
thf(660,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[422,410]) ).
thf(661,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( omega @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[660]) ).
thf(664,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( omega != A ) ),
inference(simp,[status(thm)],[661]) ).
thf(672,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[664]) ).
thf(2021,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[1957,672]) ).
thf(2062,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[2021]) ).
thf(2073,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[2062]) ).
thf(11301,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,5024]) ).
thf(11302,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11301]) ).
thf(11358,plain,
! [A: $i] :
( ( A != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11302]) ).
thf(11375,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11358]) ).
thf(12261,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,11375]) ).
thf(12262,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[12261]) ).
thf(12296,plain,
( ( omega != null_class )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[12262]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(238,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(60,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(280,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(7953,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,7877]) ).
thf(7954,plain,
( ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7953:[bind(A,$thf( omega ))]]) ).
thf(116,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(411,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(126,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(428,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(7955,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,7877]) ).
thf(7956,plain,
( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7955:[bind(A,$thf( omega ))]]) ).
thf(15563,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ $true
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7956,248]) ).
thf(15564,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[15563]) ).
thf(15565,plain,
( ( ( omega @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[15564:[bind(A,$thf( omega @ ( successor_relation @ image ) ))]]) ).
thf(16936,plain,
! [A: $i] :
( ( A != omega )
| ( ( null_class @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,15565]) ).
thf(16937,plain,
( ( ( null_class @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16936:[bind(A,$thf( omega ))]]) ).
thf(17008,plain,
( ( ( null_class @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[16937]) ).
thf(4413,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[219,3681]) ).
thf(4414,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[4413:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(4483,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[4414]) ).
thf(4748,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4661,4014]) ).
thf(4749,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[4748]) ).
thf(4751,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[4749]) ).
thf(110,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components1) ).
thf(397,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(517,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
!= ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,157]) ).
thf(518,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
!= ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[517]) ).
thf(524,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) )
!= ( z @ ( y @ ordered_pair ) ) ) ),
inference(simp,[status(thm)],[518]) ).
thf(527,plain,
( ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) )
!= ( z @ ( y @ ordered_pair ) ) ),
inference(simp,[status(thm)],[524]) ).
thf(73,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(312,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(7951,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,7877]) ).
thf(7952,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7951:[bind(A,$thf( successor_relation ))]]) ).
thf(64,axiom,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(292,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)],[64]) ).
thf(55,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_number_of_elements_in_class) ).
thf(268,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(269,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[268]) ).
thf(270,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) )
| ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[269]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(200,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(201,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[200]) ).
thf(135,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(452,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)],[135]) ).
thf(13718,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ $true
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[243,13707]) ).
thf(13719,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[13718]) ).
thf(13787,plain,
( ( omega != null_class )
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[13719]) ).
thf(15977,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,13787]) ).
thf(15978,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15977]) ).
thf(15979,plain,
( ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[15978:[bind(A,$thf( omega ))]]) ).
thf(103,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(381,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(96,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(366,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)],[96]) ).
thf(3920,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3116,285]) ).
thf(3921,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3920]) ).
thf(3960,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= universal_class )
| ( ( omega @ singleton @ regular )
!= A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3921]) ).
thf(3972,plain,
( ( ( omega @ singleton )
!= universal_class )
| ( ( omega @ singleton @ regular @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3960]) ).
thf(68,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(302,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(5453,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4661,201]) ).
thf(5454,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[5453]) ).
thf(5455,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[5454:[bind(A,$thf( choice ))]]) ).
thf(27899,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[27703,410]) ).
thf(27900,plain,
! [B: $i,A: $i] :
( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[27899]) ).
thf(27909,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ( y != A ) ),
inference(simp,[status(thm)],[27900]) ).
thf(27926,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( z @ ( y @ unordered_pair ) ) ),
inference(simp,[status(thm)],[27909]) ).
thf(108,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(392,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)],[108]) ).
thf(29834,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( z @ ( y @ unordered_pair ) )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,28133]) ).
thf(29835,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( z @ ( y @ unordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[29834]) ).
thf(29878,plain,
( ( omega != null_class )
| ( ( z @ ( y @ unordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[29835]) ).
thf(124,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(424,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(57,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(274,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(275,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[274]) ).
thf(8588,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,7922]) ).
thf(8589,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[8588]) ).
thf(8623,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[8589]) ).
thf(50,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(260,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(27989,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ singleton @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[179,27942]) ).
thf(28097,plain,
! [A: $i] :
( ( A != z )
| ( A != y )
| ~ ( null_class @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[27989]) ).
thf(28127,plain,
( ( z != y )
| ~ ( null_class @ ( y @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[28097]) ).
thf(28294,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( y @ singleton @ subclass ) ) )
| ( z != y )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[232,28127]) ).
thf(28295,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( y @ singleton @ subclass ) ) )
| ( z != y )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[28294]) ).
thf(28313,plain,
! [A: $i] :
( ( A != null_class )
| ( ( y @ singleton )
!= omega )
| ( z != y )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[28295]) ).
thf(28335,plain,
( ( ( y @ singleton )
!= omega )
| ( z != y )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[28313]) ).
thf(114,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(407,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(46,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(251,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(252,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[251]) ).
thf(22,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(197,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(27866,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ ( y @ member ) ) ),
inference(paramod_ordered,[status(thm)],[17801,27703]) ).
thf(27867,plain,
z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement @ ( y @ member ),
inference(pattern_uni,[status(thm)],[27866:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(29126,plain,
! [B: $i,A: $i] :
( ( ( z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[27867,265]) ).
thf(29127,plain,
! [B: $i,A: $i] :
( ( ( z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement @ ( y @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[29126]) ).
thf(29128,plain,
z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement @ ( y @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[29127:[bind(A,$thf( y )),bind(B,$thf( z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement ))]]) ).
thf(80,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(329,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)],[80]) ).
thf(22058,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3116,265]) ).
thf(22059,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22058]) ).
thf(22060,plain,
omega @ singleton @ ( omega @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[22059:[bind(A,$thf( omega @ singleton @ regular )),bind(B,$thf( omega @ singleton ))]]) ).
thf(137,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',right_cancellation) ).
thf(456,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)],[137]) ).
thf(36,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(229,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(149,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(484,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(653,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,410]) ).
thf(654,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[653]) ).
thf(666,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( z @ ( y @ ordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[654]) ).
thf(674,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ),
inference(simp,[status(thm)],[666]) ).
thf(2015,plain,
! [A: $i] :
( ( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
!= ( A @ ( A @ complement @ intersection ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[1957,674]) ).
thf(2059,plain,
! [A: $i] :
( ( subset_relation != A )
| ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( A @ complement ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= identity_relation ) ),
inference(simp,[status(thm)],[2015]) ).
thf(2068,plain,
( ( ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of )
!= ( subset_relation @ complement ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= identity_relation ) ),
inference(simp,[status(thm)],[2059]) ).
thf(34,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(224,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(27855,plain,
! [A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[27703,285]) ).
thf(27856,plain,
! [A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( y @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[27855]) ).
thf(27924,plain,
! [A: $i] :
( ( ( z @ ( y @ unordered_pair ) )
!= universal_class )
| ( y != A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[27856]) ).
thf(27932,plain,
( ( ( z @ ( y @ unordered_pair ) )
!= universal_class )
| ( ( y @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[27924]) ).
thf(9442,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( B @ subclass ) ) )
| ~ $true
| ~ ( B @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[213,8437]) ).
thf(9443,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( B @ subclass ) ) )
| ~ ( B @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9442]) ).
thf(9444,plain,
( ~ ( null_class @ ( successor_relation @ image ) @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[9443:[bind(A,$thf( null_class )),bind(B,$thf( null_class @ ( successor_relation @ image ) ))]]) ).
thf(92,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(356,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(8,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(169,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(52,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_member_of_ordered_pair) ).
thf(263,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(2655,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[263,332]) ).
thf(2656,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2655]) ).
thf(2660,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[2656]) ).
thf(2666,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2660]) ).
thf(15049,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,2666]) ).
thf(15050,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15049]) ).
thf(15051,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[15050:[bind(A,$thf( E @ ( D @ ordered_pair ) )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(16242,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[15051]) ).
thf(14503,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,11399]) ).
thf(14504,plain,
! [A: $i] :
( ( A != omega )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14503]) ).
thf(14505,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[14504:[bind(A,$thf( omega ))]]) ).
thf(151,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(488,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(84,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(339,plain,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(3214,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,243]) ).
thf(3215,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3214:[bind(A,$thf( omega ))]]) ).
thf(22098,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[3215,265]) ).
thf(22099,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[22098]) ).
thf(22100,plain,
( ( omega @ ( omega @ regular @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[22099:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(3822,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,3785]) ).
thf(3823,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3822]) ).
thf(3824,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3823:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(4342,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3824,675]) ).
thf(4343,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[4342]) ).
thf(4381,plain,
! [A: $i] :
( ( ( universal_class @ regular @ singleton )
!= identity_relation )
| ( ( universal_class @ regular @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[4343]) ).
thf(4389,plain,
( ( ( universal_class @ regular @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[4381]) ).
thf(27939,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,27931]) ).
thf(27970,plain,
! [A: $i] :
( ( A != z )
| ( A != y )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[27939]) ).
thf(27971,plain,
( ( z != y )
| ( ( y @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[27970]) ).
thf(28145,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ singleton ) )
| ( z != y )
| ( null_class != null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[361,27971]) ).
thf(28146,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ singleton ) )
| ( z != y )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[28145]) ).
thf(28147,plain,
( ( z != y )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[28146:[bind(A,$thf( y ))]]) ).
thf(5534,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[3681,4483]) ).
thf(5535,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[5534:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(5662,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
= null_class ),
inference(rewrite,[status(thm)],[5535,4412]) ).
thf(9386,plain,
! [A: $i] :
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ $true
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[7877,8437]) ).
thf(9387,plain,
! [A: $i] :
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[9386]) ).
thf(9498,plain,
! [A: $i] :
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= A )
| ~ ( A @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[9387]) ).
thf(9530,plain,
( ( omega != null_class )
| ~ ( omega @ ( successor_relation @ image ) @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[9498]) ).
thf(684,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,675]) ).
thf(685,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[684]) ).
thf(699,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= identity_relation )
| ( ( z @ ( y @ ordered_pair ) )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[685]) ).
thf(706,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[699]) ).
thf(682,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[350,675]) ).
thf(683,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[682]) ).
thf(694,plain,
! [A: $i] :
( ( identity_relation != universal_class )
| ( null_class != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[683]) ).
thf(701,plain,
( ( identity_relation != universal_class )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[694]) ).
thf(1893,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(paramod_ordered,[status(thm)],[219,674]) ).
thf(1894,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ),
inference(pattern_uni,[status(thm)],[1893:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1945,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ),
inference(simp,[status(thm)],[1894]) ).
thf(2806,plain,
! [A: $i] :
( ( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
!= ( identity_relation @ ( A @ intersection ) ) )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(paramod_ordered,[status(thm)],[221,1945]) ).
thf(2836,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[2806]) ).
thf(2851,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[2836]) ).
thf(17917,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[17801,7594]) ).
thf(17918,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ complement @ ( B @ singleton @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[17917:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(18176,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ complement @ ( B @ singleton @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(simp,[status(thm)],[17918]) ).
thf(3200,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,221]) ).
thf(3201,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3200:[bind(A,$thf( subset_relation ))]]) ).
thf(9124,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[9104,1858]) ).
thf(9125,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[9124:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(11915,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( A @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[9125,1858]) ).
thf(11916,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[11915:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(99,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(371,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)],[99]) ).
thf(43,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(244,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)],[43]) ).
thf(14452,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ singleton @ ( B @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,7641]) ).
thf(14453,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ singleton @ ( B @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14452]) ).
thf(14454,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[14453:[bind(A,$thf( F @ singleton @ ( D @ unordered_pair ) )),bind(B,$thf( D )),bind(C,$thf( F ))]]) ).
thf(16080,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[14454]) ).
thf(54,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(266,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)],[54]) ).
thf(26,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(205,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(17811,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[17801,597]) ).
thf(17812,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[17811:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(86,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(343,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)],[86]) ).
thf(39,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(235,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(236,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(cnf,[status(esa)],[235]) ).
thf(237,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[236]) ).
thf(15550,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,12487]) ).
thf(15551,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15550]) ).
thf(15552,plain,
( ~ ( omega @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[15551:[bind(A,$thf( omega ))]]) ).
thf(127,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(430,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)],[127]) ).
thf(7,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_number_of_elements_in_class) ).
thf(166,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(12,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(180,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(123,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(423,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
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/sandbox/benchmark/theBenchmark.p',equality2) ).
thf(402,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(62,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(286,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)],[62]) ).
thf(14712,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,12296]) ).
thf(14713,plain,
! [A: $i] :
( ( A != omega )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14712]) ).
thf(14714,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[14713:[bind(A,$thf( omega ))]]) ).
thf(290,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(cnf,[status(esa)],[289]) ).
thf(291,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( universal_class @ ( B @ member ) )
| ( B = D )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[290]) ).
thf(203,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[202]) ).
thf(204,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ subclass ) )
| ( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[203]) ).
thf(49,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(258,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)],[49]) ).
thf(259,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)],[258]) ).
thf(117,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(413,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)],[117]) ).
thf(23507,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( omega != universal_class ) ),
inference(paramod_ordered,[status(thm)],[22621,265]) ).
thf(23508,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[23507]) ).
thf(23509,plain,
( ( omega @ ( omega @ regular @ singleton @ subclass ) )
| ( omega != universal_class ) ),
inference(pattern_uni,[status(thm)],[23508:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(104,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(382,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(23591,plain,
! [A: $i] :
( ( A
!= ( omega @ regular @ singleton ) )
| ( null_class != null_class )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,23579]) ).
thf(23592,plain,
! [A: $i] :
( ( A
!= ( omega @ regular @ singleton ) )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[23591]) ).
thf(23593,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ regular @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[23592:[bind(A,$thf( omega @ regular @ singleton ))]]) ).
thf(141,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(466,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)],[141]) ).
thf(181,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[180]) ).
thf(14547,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,12254]) ).
thf(14548,plain,
! [A: $i] :
( ( A != omega )
| ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14547]) ).
thf(14549,plain,
( ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[14548:[bind(A,$thf( omega ))]]) ).
thf(22101,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3824,265]) ).
thf(22102,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22101]) ).
thf(22103,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[22102:[bind(A,$thf( universal_class @ regular @ singleton @ regular )),bind(B,$thf( universal_class @ regular @ singleton ))]]) ).
thf(22192,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3690,265]) ).
thf(22193,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22192]) ).
thf(22194,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[22193:[bind(A,$thf( H )),bind(B,$thf( H @ singleton @ singleton @ regular )),bind(C,$thf( H @ singleton @ singleton ))]]) ).
thf(22259,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ singleton @ subclass ) ),
inference(simp,[status(thm)],[22194]) ).
thf(170,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[169]) ).
thf(171,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[170]) ).
thf(172,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[171]) ).
thf(101,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(377,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)],[101]) ).
thf(230,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[229]) ).
thf(245,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[244]) ).
thf(69,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(303,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(56,axiom,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',left_cancellation) ).
thf(271,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)],[56]) ).
thf(4372,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3824,410]) ).
thf(4373,plain,
! [B: $i,A: $i] :
( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4372]) ).
thf(4379,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) )
| ( ( universal_class @ regular @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[4373]) ).
thf(4388,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) ),
inference(simp,[status(thm)],[4379]) ).
thf(14876,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,11545]) ).
thf(14877,plain,
! [A: $i] :
( ( A != omega )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14876]) ).
thf(14878,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[14877:[bind(A,$thf( omega ))]]) ).
thf(1782,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[308,1775]) ).
thf(1783,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[1782]) ).
thf(1806,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= null_class )
| ( ( A @ flip )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[1783]) ).
thf(120,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(419,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)],[120]) ).
thf(287,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)],[286]) ).
thf(288,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)],[287]) ).
thf(29886,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ singleton @ ( B @ image ) @ sum_class )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ singleton @ ( C @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[288,228]) ).
thf(15329,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,7594]) ).
thf(15330,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15329]) ).
thf(15331,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[15330:[bind(A,$thf( E @ ( F @ singleton @ unordered_pair ) )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(16258,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[15331]) ).
thf(94,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(362,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)],[94]) ).
thf(146,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(477,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_2_of_ordered_pair) ).
thf(494,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(87,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(345,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)],[87]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(323,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(14496,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ unordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,597]) ).
thf(14497,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ unordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14496]) ).
thf(14498,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[14497:[bind(A,$thf( F @ ( E @ unordered_pair ) @ singleton )),bind(B,$thf( F )),bind(C,$thf( E ))]]) ).
thf(16101,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[14498]) ).
thf(198,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[197]) ).
thf(139,axiom,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists1) ).
thf(462,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(1826,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)],[177,672]) ).
thf(1827,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[1826:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1859,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[1827]) ).
thf(2151,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ ( B @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[177,1859]) ).
thf(2152,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ ( A @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[2151:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(2182,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ ( A @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2152]) ).
thf(106,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(387,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(41,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(240,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)],[41]) ).
thf(262,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)],[261]) ).
thf(21244,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)],[262,332]) ).
thf(21245,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)],[21244]) ).
thf(21246,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21245:[bind(A,$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(21314,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[21246]) ).
thf(17964,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[17801,7045]) ).
thf(17965,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[17964:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(18121,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(simp,[status(thm)],[17965]) ).
thf(9193,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[9104,177]) ).
thf(9194,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[9193:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(10274,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[9194,177]) ).
thf(10275,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[10274:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(222,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(223,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[222]) ).
thf(109,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(394,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(67,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/sandbox/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(299,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)],[67]) ).
thf(225,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[224]) ).
thf(226,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[225]) ).
thf(267,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[266]) ).
thf(27951,plain,
! [A: $i] :
( ( A
!= ( z @ ( y @ unordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,27931]) ).
thf(27952,plain,
! [A: $i] :
( ( A
!= ( z @ ( y @ unordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[27951]) ).
thf(27953,plain,
z @ ( y @ unordered_pair ) @ ( z @ ( y @ unordered_pair ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[27952:[bind(A,$thf( z @ ( y @ unordered_pair ) ))]]) ).
thf(130,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(438,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)],[130]) ).
thf(7890,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[213,1775]) ).
thf(7891,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7890]) ).
thf(7912,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7891]) ).
thf(7921,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7912]) ).
thf(8256,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,7921]) ).
thf(8257,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[8256]) ).
thf(8291,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[8257]) ).
thf(1784,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[302,1775]) ).
thf(1785,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[1784]) ).
thf(1809,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= null_class )
| ( ( A @ rotate )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[1785]) ).
thf(125,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function3) ).
thf(426,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)],[125]) ).
thf(48,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(255,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)],[48]) ).
thf(27,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(209,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)],[27]) ).
thf(7981,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,2194]) ).
thf(7982,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[7981]) ).
thf(7992,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[7982]) ).
thf(27844,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ( A @ singleton @ ( y @ member ) ) ),
inference(paramod_ordered,[status(thm)],[179,27703]) ).
thf(27925,plain,
! [A: $i] :
( ( A != z )
| ( A != y )
| ( A @ singleton @ ( y @ member ) ) ),
inference(simp,[status(thm)],[27844]) ).
thf(27933,plain,
( ( z != y )
| ( y @ singleton @ ( y @ member ) ) ),
inference(simp,[status(thm)],[27925]) ).
thf(12542,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of1 @ singleton )
!= ( B @ singleton ) )
| ( A != null_class )
| ( ( B @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[237,5787]) ).
thf(12543,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ member_of1 @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( A @ member_of )
= A ) ),
inference(pattern_uni,[status(thm)],[12542:[bind(A,$thf( C )),bind(B,$thf( C @ member_of1 ))]]) ).
thf(12740,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( null_class @ member_of1 @ singleton ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[12543]) ).
thf(256,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)],[255]) ).
thf(257,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)],[256]) ).
thf(19080,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ singleton @ ( C @ image ) @ sum_class )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ singleton @ ( A @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[257,228]) ).
thf(22108,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3183,265]) ).
thf(22109,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22108]) ).
thf(22110,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[22109:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) @ regular )),bind(B,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(100,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(374,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)],[100]) ).
thf(519,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[422,157]) ).
thf(520,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[519]) ).
thf(525,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[520]) ).
thf(65,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_3_of_ordered_pair) ).
thf(295,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(281,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(cnf,[status(esa)],[280]) ).
thf(282,plain,
! [A: $i] :
( ( universal_class @ ( A @ member_of1 @ member ) )
| ( ( A @ member_of )
= A ) ),
inference(lifteq,[status(thm)],[281]) ).
thf(28349,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( z @ ( y @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[27953,265]) ).
thf(28350,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ unordered_pair ) @ ( z @ ( y @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[28349]) ).
thf(28351,plain,
z @ ( y @ unordered_pair ) @ ( z @ ( y @ unordered_pair ) @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[28350:[bind(A,$thf( z @ ( y @ unordered_pair ) @ regular )),bind(B,$thf( z @ ( y @ unordered_pair ) ))]]) ).
thf(23,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(199,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(7985,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,4014]) ).
thf(7986,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7985]) ).
thf(7988,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[7986]) ).
thf(143,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(470,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(22154,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,265]) ).
thf(22155,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[22154]) ).
thf(22156,plain,
universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[22155:[bind(A,$thf( z @ ( y @ ordered_pair ) )),bind(B,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(3714,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3183,157]) ).
thf(3715,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ ( universal_class @ cross_product ) @ ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[3714]) ).
thf(3763,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) @ regular )
!= ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) ) ) ),
inference(simp,[status(thm)],[3715]) ).
thf(3769,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ regular )
!= ( z @ ( y @ ordered_pair ) @ second @ ( z @ ( y @ ordered_pair ) @ first @ ordered_pair ) ) ),
inference(simp,[status(thm)],[3763]) ).
thf(27949,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( z @ ( y @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[17801,27931]) ).
thf(27950,plain,
( ( z @ singleton @ complement @ ( y @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[27949:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(9120,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[9104,1858]) ).
thf(9121,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[9120:[bind(A,$thf( null_class @ domain_of ))]]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(182,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(9830,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,3201]) ).
thf(9831,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9830:[bind(A,$thf( subset_relation ))]]) ).
thf(9911,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[9831]) ).
thf(83,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists3) ).
thf(336,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(3954,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3116,410]) ).
thf(3955,plain,
! [B: $i,A: $i] :
( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3954]) ).
thf(3958,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( omega @ singleton ) )
| ( ( omega @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3955]) ).
thf(3970,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[3958]) ).
thf(2156,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ ( B @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[219,1859]) ).
thf(2157,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[2156:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(2183,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2157]) ).
thf(21458,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[3681,2183]) ).
thf(21459,plain,
( ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[21458:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(21676,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(rewrite,[status(thm)],[21459,4412]) ).
thf(74,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(314,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)],[74]) ).
thf(154,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(497,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(133,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(446,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(571,plain,
( ( ( z @ ( y @ ordered_pair ) )
!= ( null_class @ member_of ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= universal_class )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[569]) ).
thf(9118,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[9104,4460]) ).
thf(9119,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[9118:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(16,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(186,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)],[16]) ).
thf(187,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[186]) ).
thf(2907,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(rewrite,[status(thm)],[187,164]) ).
thf(9924,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ ( null_class @ ( successor_relation @ image ) @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[243,9444]) ).
thf(9925,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ~ ( null_class @ ( successor_relation @ image ) @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[9924]) ).
thf(10018,plain,
( ( omega != null_class )
| ~ ( null_class @ ( successor_relation @ image ) @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[9925]) ).
thf(76,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(319,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)],[76]) ).
thf(97,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_is_set) ).
thf(369,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(623,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[369,285]) ).
thf(624,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[623]) ).
thf(625,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[624:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ ordered_pair ) ))]]) ).
thf(632,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[625]) ).
thf(5038,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4661,5024]) ).
thf(5039,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5038]) ).
thf(5087,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[5039]) ).
thf(119,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member_of_ordered_pair) ).
thf(418,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(18,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(190,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(191,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[190]) ).
thf(4250,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[191,164]) ).
thf(90,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(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)],[90]) ).
thf(210,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)],[209]) ).
thf(211,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)],[210]) ).
thf(7312,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[211,164]) ).
thf(128,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(433,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(1903,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)],[219,672]) ).
thf(1904,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[1903:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1948,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[1904]) ).
thf(2383,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)],[221,1948]) ).
thf(2414,plain,
! [A: $i] :
( ( subset_relation != identity_relation )
| ( ( subset_relation @ inverse )
!= A )
| ( ( identity_relation @ domain_of @ ( A @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[2383]) ).
thf(2421,plain,
( ( subset_relation != identity_relation )
| ( ( identity_relation @ domain_of @ ( subset_relation @ inverse @ diagonalise @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[2414]) ).
thf(12322,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,7954]) ).
thf(12323,plain,
( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[12322:[bind(A,$thf( omega ))]]) ).
thf(12391,plain,
( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[12323]) ).
thf(15793,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,632]) ).
thf(15794,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15793]) ).
thf(15795,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[15794:[bind(A,$thf( F @ ( E @ ordered_pair ) @ singleton )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(16165,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[15795]) ).
thf(4718,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[355,4661]) ).
thf(4719,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4718:[bind(A,$thf( choice ))]]) ).
thf(113,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(405,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(183,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[182]) ).
thf(272,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(cnf,[status(esa)],[271]) ).
thf(273,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( B = C )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[272]) ).
thf(3202,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[451,221]) ).
thf(3203,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3202:[bind(A,$thf( subset_relation ))]]) ).
thf(167,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(168,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(lifteq,[status(thm)],[167]) ).
thf(70,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality3) ).
thf(305,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)],[70]) ).
thf(78,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(326,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)],[78]) ).
thf(241,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[240]) ).
thf(242,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[241]) ).
thf(293,plain,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
inference(cnf,[status(esa)],[292]) ).
thf(294,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( universal_class @ ( C @ member ) )
| ( C = B )
| ( C = A )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
inference(lifteq,[status(thm)],[293]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(341,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)],[85]) ).
thf(145,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality1) ).
thf(474,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)],[145]) ).
thf(111,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(400,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(7220,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of @ singleton )
!= ( B @ singleton ) )
| ( A != null_class )
| ( ( B @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[160,5787]) ).
thf(7221,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ member_of @ singleton )
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( ( A @ member_of )
= A ) ),
inference(pattern_uni,[status(thm)],[7220:[bind(A,$thf( C )),bind(B,$thf( C @ member_of ))]]) ).
thf(7280,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( null_class @ member_of @ singleton ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[7221]) ).
thf(118,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(415,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(437,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)],[436]) ).
thf(140,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(464,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)],[140]) ).
thf(13712,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( universal_class @ subclass ) ) )
| ~ ( null_class @ inductive )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,13707]) ).
thf(13713,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( universal_class @ subclass ) ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[13712]) ).
thf(13793,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[13713]) ).
thf(13807,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[13793]) ).
thf(528,plain,
( ( ( z @ ( y @ ordered_pair ) @ second )
!= z )
| ( ( z @ ( y @ ordered_pair ) @ first )
!= y ) ),
inference(simp,[status(thm)],[527]) ).
thf(138,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(459,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(7000,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)],[162,370]) ).
thf(7001,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[7000:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(7035,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[7001]) ).
thf(26887,plain,
! [A: $i] :
( ( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
!= ( A @ complement ) )
| ( ( A @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[26143,3681]) ).
thf(26888,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[26887:[bind(A,$thf( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of ))]]) ).
thf(11924,plain,
! [A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1858,9125]) ).
thf(11925,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[11924:[bind(A,$thf( null_class @ domain_of ))]]) ).
thf(14512,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,8210]) ).
thf(14513,plain,
! [A: $i] :
( ( A != omega )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[14512]) ).
thf(14514,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[14513:[bind(A,$thf( omega ))]]) ).
thf(7957,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7877,1775]) ).
thf(7958,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ ( universal_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[7957]) ).
thf(7987,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( universal_class @ ( universal_class @ cross_product ) ) ) ),
inference(simp,[status(thm)],[7958]) ).
thf(150,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(486,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(72,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(309,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(15931,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[248,564]) ).
thf(15932,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[15931]) ).
thf(15933,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[15932:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(16203,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(simp,[status(thm)],[15933]) ).
thf(4746,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4661,2194]) ).
thf(4747,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[4746]) ).
thf(4752,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4747]) ).
thf(152,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(491,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(136,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(454,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(20231,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)],[259,332]) ).
thf(20232,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)],[20231]) ).
thf(20233,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20232:[bind(A,$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(20263,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20233]) ).
thf(57898,plain,
$false,
inference(e,[status(thm)],[365,347,7922,468,234,15844,555,4877,481,5260,11372,333,249,3681,7641,308,3910,13707,449,379,677,217,276,23601,440,202,597,384,17871,11399,472,8210,27931,185,27942,28133,436,22086,7993,5024,4238,570,184,22621,3116,702,7877,28028,26489,196,157,12254,389,189,25247,289,4460,316,261,7226,421,11564,7047,228,2456,11314,26143,179,9330,443,253,1858,3973,514,2073,480,5787,216,1946,164,12296,238,280,7931,221,7954,411,243,428,17008,265,4483,4751,397,527,312,7952,292,248,270,201,13379,11545,233,452,15979,381,220,366,3972,302,160,5455,27926,392,29878,424,275,165,4661,8623,260,28335,192,407,252,197,361,29128,3090,329,22060,456,229,484,11376,2068,285,224,27932,156,9444,356,169,593,16242,188,14505,488,339,22100,193,4389,4750,28147,5662,2666,328,9530,212,706,701,2851,18176,173,3201,3215,11916,371,3824,244,16080,266,205,3183,17812,28127,343,237,9104,15552,674,298,430,166,4412,264,161,279,11343,180,176,423,402,286,14714,291,9521,204,259,413,23509,3773,382,9125,17801,3646,23593,7924,466,350,181,14549,22103,22259,672,27703,172,7594,219,377,230,245,303,271,4388,14878,274,1806,419,29886,16258,362,451,477,494,27971,345,323,16101,155,198,462,2182,387,26087,213,240,251,11375,21314,18121,10275,278,223,394,1775,299,226,267,27953,438,505,8291,1809,426,162,255,209,7992,355,27933,194,12740,675,19080,22110,374,525,12487,295,282,7956,7045,409,263,28351,199,331,370,11374,7988,470,22156,3769,27950,9121,177,182,9911,359,9194,336,3970,7921,21676,175,314,497,446,571,9119,2194,218,2907,250,231,23579,10018,4121,319,186,632,5087,418,4250,258,158,351,27867,7312,2183,235,246,433,510,214,2421,12391,15565,3785,410,16165,16392,4719,405,190,183,422,273,3690,3203,4014,1957,168,262,564,305,326,242,294,341,268,195,474,400,7280,283,415,437,13787,464,332,13807,528,459,7035,8437,178,26888,11925,369,14514,7987,163,200,227,486,26077,354,309,16203,215,4752,491,222,454,364,232,20263]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET108-7 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.12/0.51 % Computer : n002.cluster.edu
% 0.12/0.51 % Model : x86_64 x86_64
% 0.12/0.51 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.51 % Memory : 8046.5625MB
% 0.12/0.51 % OS : Linux 6.8.0-71-generic
% 0.12/0.51 % CPULimit : 300
% 0.12/0.51 % WCLimit : 300
% 0.12/0.51 % DateTime : Sat Sep 26 07:11:09 UTC 2026
% 0.12/0.51 % CPUTime :
% 0.12/0.51 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.89/1.10 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.34/1.39 % [INFO] Parsing done (282ms).
% 1.79/1.40 % [INFO] Running in sequential loop mode.
% 2.45/1.80 % [INFO] eprover registered as external prover.
% 2.45/1.81 % [INFO] Scanning for conjecture ...
% 2.65/1.94 % [INFO] Found a conjecture (or negated_conjecture) and 152 axioms. Running axiom selection ...
% 2.89/2.06 % [INFO] Axiom selection finished. Selected 152 axioms (removed 0 axioms).
% 3.16/2.18 % [INFO] Problem is propositional (TPTP CNF).
% 3.32/2.20 % [INFO] Type checking passed.
% 3.32/2.20 % [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 ...
% 119.96/30.61 % External prover 'e' found a proof!
% 119.96/30.62 % [INFO] Killing All external provers ...
% 119.96/30.62 % Time passed: 29960ms (effective reasoning time: 29209ms)
% 119.96/30.63 % Axioms used in derivation (152): operation4, number_of_elements_in_class, equal_implies_subclass1, complement1, one_to_one2, null_unordered_pair, property_3_of_ordered_pair, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, singleton_in_unordered_pair3, function3, corollary_to_set_in_its_singleton, regularity1, singleton_set, identity_relation, commutativity_of_unordered_pair, union, flip3, element_relation2, unordered_pairs_in_universal, corollary_of_null_class_is_subclass, equality1, compose2, homomorphism3, compatible3, symmetric_difference, cartesian_product1, image, replacement, singletons_are_sets, not_subclass_members1, existence_of_null_class, ordered_pair, successor_relation1, property_2_of_ordered_pair, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, singleton_in_unordered_pair1, singleton_identified_by_element2, member_of_non_singleton_unique1, null_class_is_a_set, member_exists2, one_to_one3, ordered_pair_determines_components1, single_valued_class1, compose3, omega_is_inductive2, rotate1, left_cancellation, choice1, property_1_of_ordered_pair, power_class2, equality2, regularity2, singleton_member_of_ordered_pair, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, flip2, two_subsets_of_singleton, ordered_pair_is_set, corollary_1_to_singletons_are_sets, inverse, corollary_1_to_number_of_elements_in_class, corollary_to_unordered_pair_axiom3, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, only_member_in_singleton, not_subclass_members2, ordered_pair_determines_components2, successor_relation3, member_exists3, domain2, function1, right_cancellation, compatible4, property_of_singletons2, intersection3, singleton_identified_by_element1, corollary_to_unordered_pair_axiom1, member_of_non_singleton_unique2, corollary_2_to_singletons_are_sets, single_valued_class2, unordered_pair_member, apply, corollary_2_to_number_of_elements_in_class, element_relation1, transitivity_of_subclass, operation1, member_of_singleton_is_unique, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, flip1, homomorphism5, compatible1, unordered_pair2, unordered_pair_member_of_ordered_pair, operation2, member_exists4, rotate3, property_of_singletons1, singleton_in_unordered_pair2, complement2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, set_in_its_singleton, domain1, unordered_pair_is_subset, null_class_in_its_singleton, corollary_1_to_cartesian_product, null_class_is_unique, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, unordered_pairs_and_singletons, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, singleton_is_null_class, 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
% 119.96/30.63 % No. of inferences in proof: 1002
% 119.96/30.63 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 29960 ms resp. 29209 ms w/o parsing
% 112.33/31.16 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 112.33/31.19 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------