%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUM009-1 : TPTP v9.3.1. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n017.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 08:08:05 AM UTC 2026
% Result : Unsatisfiable 15.12s 10.99s
% Output : Refutation 17.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 2
% Number of leaves : 220
% Syntax : Number of formulae : 441 ( 25 unt; 0 typ; 0 def)
% Number of atoms : 1275 ( 150 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 5376 ( 698 ~; 834 |; 0 &;3844 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 112 ( 110 usr; 13 con; 0-5 aty)
% Number of variables : 968 ( 0 ^; 968 !; 0 ?; 968 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(natural_numbers_decl,type,
natural_numbers: $i ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(function_decl,type,
function: $i > $o ).
thf(group_decl,type,
group: $i > $i > $o ).
thf(identity_decl,type,
identity: $i > $i > $i > $o ).
thf(f39_decl,type,
f39: $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(singleton_set_decl,type,
singleton_set: $i > $i ).
thf(proper_subset_decl,type,
proper_subset: $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $i > $i > $o ).
thf(closed_decl,type,
closed: $i > $i > $o ).
thf(one_to_one_function_decl,type,
one_to_one_function: $i > $o ).
thf(little_set_decl,type,
little_set: $i > $o ).
thf(powerset_decl,type,
powerset: $i > $i ).
thf(relation_decl,type,
relation: $i > $o ).
thf(single_valued_set_decl,type,
single_valued_set: $i > $o ).
thf(associative_decl,type,
associative: $i > $i > $o ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(sigma_decl,type,
sigma: $i > $i ).
thf(f20_decl,type,
f20: $i > $i ).
thf(f21_decl,type,
f21: $i > $i ).
thf(f25_decl,type,
f25: $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(ordered_pair_predicate_decl,type,
ordered_pair_predicate: $i > $o ).
thf(f3_decl,type,
f3: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(infinity_decl,type,
infinity: $i ).
thf(inverse_decl,type,
inverse: $i > $i > $i > $i > $o ).
thf(non_ordered_pair_decl,type,
non_ordered_pair: $i > $i > $i ).
thf(finite_decl,type,
finite: $i > $o ).
thf(f58_decl,type,
f58: $i > $i ).
thf(f57_decl,type,
f57: $i > $i ).
thf(converse_decl,type,
converse: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(apply_to_two_arguments_decl,type,
apply_to_two_arguments: $i > $i > $i > $i ).
thf(f32_decl,type,
f32: $i > $i > $i > $i > $i > $i ).
thf(f33_decl,type,
f33: $i > $i > $i > $i > $i > $i ).
thf(f1_decl,type,
f1: $i > $i > $i ).
thf(f40_decl,type,
f40: $i > $i > $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(f2_decl,type,
f2: $i > $i ).
thf(f38_decl,type,
f38: $i > $i > $i > $i > $i ).
thf(commutes_decl,type,
commutes: $i > $i > $o ).
thf(f41_decl,type,
f41: $i > $i > $i ).
thf(f42_decl,type,
f42: $i > $i > $i ).
thf(f19_decl,type,
f19: $i > $i ).
thf(f34_decl,type,
f34: $i > $i > $i ).
thf(f35_decl,type,
f35: $i > $i > $i ).
thf(f36_decl,type,
f36: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(f18_decl,type,
f18: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i ).
thf(universal_set_decl,type,
universal_set: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(f22_decl,type,
f22: $i > $i > $i > $i ).
thf(even_numbers_decl,type,
even_numbers: $i ).
thf(plus_decl,type,
plus: $i ).
thf(f59_decl,type,
f59: $i > $i ).
thf(flip_range_of_decl,type,
flip_range_of: $i > $i ).
thf(f13_decl,type,
f13: $i > $i > $i ).
thf(times_decl,type,
times: $i ).
thf(f50_decl,type,
f50: $i > $i > $i ).
thf(f53_decl,type,
f53: $i > $i > $i ).
thf(f49_decl,type,
f49: $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(f30_decl,type,
f30: $i > $i > $i > $i ).
thf(f16_decl,type,
f16: $i > $i > $i ).
thf(prime_numbers_decl,type,
prime_numbers: $i ).
thf(f55_decl,type,
f55: $i > $i ).
thf(rotate_right_decl,type,
rotate_right: $i > $i ).
thf(f9_decl,type,
f9: $i > $i > $i ).
thf(f10_decl,type,
f10: $i > $i > $i ).
thf(f11_decl,type,
f11: $i > $i > $i ).
thf(disjoint_decl,type,
disjoint: $i > $i > $o ).
thf(f23_decl,type,
f23: $i > $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(f5_decl,type,
f5: $i > $i > $i ).
thf(f6_decl,type,
f6: $i > $i > $i ).
thf(f7_decl,type,
f7: $i > $i > $i ).
thf(twin_prime_numbers_decl,type,
twin_prime_numbers: $i ).
thf(f31_decl,type,
f31: $i > $i > $i > $i ).
thf(f27_decl,type,
f27: $i > $i > $i ).
thf(f26_decl,type,
f26: $i > $i ).
thf(f12_decl,type,
f12: $i > $i > $i ).
thf(f14_decl,type,
f14: $i > $i > $i ).
thf(estin_decl,type,
estin: $i ).
thf(f4_decl,type,
f4: $i > $i > $i ).
thf(f51_decl,type,
f51: $i > $i > $i ).
thf(f52_decl,type,
f52: $i > $i > $i ).
thf(f28_decl,type,
f28: $i > $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(f54_decl,type,
f54: $i > $i ).
thf(f17_decl,type,
f17: $i > $i > $i ).
thf(f44_decl,type,
f44: $i > $i ).
thf(f45_decl,type,
f45: $i > $i > $i ).
thf(f46_decl,type,
f46: $i > $i > $i ).
thf(f47_decl,type,
f47: $i > $i > $i ).
thf(f48_decl,type,
f48: $i > $i > $i ).
thf(f37_decl,type,
f37: $i > $i > $i > $i ).
thf(f56_decl,type,
f56: $i > $i ).
thf(f29_decl,type,
f29: $i > $i > $i > $i ).
thf(f43_decl,type,
f43: $i > $i > $i ).
thf(f8_decl,type,
f8: $i > $i > $i ).
thf(f24_decl,type,
f24: $i > $i ).
thf(176,axiom,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f8 ) @ first ) )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain3) ).
thf(629,plain,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f8 ) @ first ) )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(190,axiom,
! [A: $i] :
( ( A @ ( A @ f24 @ disjoint ) )
| ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(666,plain,
! [A: $i] :
( ( A @ ( A @ f24 @ disjoint ) )
| ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(197,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( B @ ( A @ f30 ) ) @ ( C @ ( B @ ( A @ f31 ) ) @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose6) ).
thf(683,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( B @ ( A @ f30 ) ) @ ( C @ ( B @ ( A @ f31 ) ) @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(26,axiom,
! [A: $i] :
( ( A @ converse @ function )
| ~ ( A @ one_to_one_function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function2) ).
thf(276,plain,
! [A: $i] :
( ( A @ converse @ function )
| ~ ( A @ one_to_one_function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(39,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subset ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps3) ).
thf(308,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subset ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(99,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) )
| ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(449,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) )
| ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(13,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ ( A @ subset ) )
| ( B @ ( A @ proper_subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset3) ).
thf(247,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ ( A @ subset ) )
| ( B @ ( A @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(194,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f16 ) @ ( A @ member ) )
| ~ ( B @ sigma @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma2) ).
thf(677,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f16 ) @ ( A @ member ) )
| ~ ( B @ sigma @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(164,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f48 ) @ successor @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus10) ).
thf(597,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f48 ) @ successor @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(132,axiom,
! [A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( estin @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(523,plain,
! [A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( estin @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(157,axiom,
! [C: $i,B: $i,A: $i] :
( ( A
= ( C @ ( B @ ( A @ f30 ) ) @ ( C @ ( B @ ( A @ f29 ) ) @ ordered_pair ) ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose4) ).
thf(582,plain,
! [C: $i,B: $i,A: $i] :
( ( A
= ( C @ ( B @ ( A @ f30 ) ) @ ( C @ ( B @ ( A @ f29 ) ) @ ordered_pair ) ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(185,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(655,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(127,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( C @ ( B @ ( A @ f37 ) ) @ ( B @ apply_to_two_arguments ) ) )
!= ( C @ ( B @ ( A @ f37 ) ) ) )
| ( ( C @ ( B @ ( A @ f37 ) ) @ ( C @ ( B @ apply_to_two_arguments ) ) )
!= ( C @ ( B @ ( A @ f37 ) ) ) )
| ~ ( A @ ( C @ member ) )
| ( C @ ( B @ ( A @ identity ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity5) ).
thf(511,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( C @ ( B @ ( A @ f37 ) ) @ ( B @ apply_to_two_arguments ) ) )
!= ( C @ ( B @ ( A @ f37 ) ) ) )
| ( ( C @ ( B @ ( A @ f37 ) ) @ ( C @ ( B @ apply_to_two_arguments ) ) )
!= ( C @ ( B @ ( A @ f37 ) ) ) )
| ~ ( A @ ( C @ member ) )
| ( C @ ( B @ ( A @ identity ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(73,axiom,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f11 ) @ ( B @ ( A @ f10 ) @ ordered_pair ) @ ( B @ ( A @ f9 ) @ ordered_pair ) ) )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right4) ).
thf(385,plain,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f11 ) @ ( B @ ( A @ f10 ) @ ordered_pair ) @ ( B @ ( A @ f9 ) @ ordered_pair ) ) )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(144,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( B @ ( A @ f22 ) ) @ member ) )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution2) ).
thf(550,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( B @ ( A @ f22 ) ) @ member ) )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(212,axiom,
! [A: $i] :
( ( A @ f54 @ little_set )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times11) ).
thf(715,plain,
! [A: $i] :
( ( A @ f54 @ little_set )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(204,axiom,
! [A: $i] :
( ( A @ f44 @ little_set )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers3) ).
thf(698,plain,
! [A: $i] :
( ( A @ f44 @ little_set )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(54,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ apply_to_two_arguments ) ) )
= ( C @ ( B @ ordered_pair ) @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply_to_two_arguments) ).
thf(340,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ apply_to_two_arguments ) ) )
= ( C @ ( B @ ordered_pair ) @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(138,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ subset ) )
| ~ ( A @ little_set )
| ( B @ powerset @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset2) ).
thf(538,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ subset ) )
| ~ ( A @ little_set )
| ( B @ powerset @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ( B
= ( B @ ( A @ f7 ) @ ( B @ ( A @ f6 ) @ ordered_pair ) ) )
| ~ ( B @ second @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second3) ).
thf(404,plain,
! [B: $i,A: $i] :
( ( B
= ( B @ ( A @ f7 ) @ ( B @ ( A @ f6 ) @ ordered_pair ) ) )
| ~ ( B @ second @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f27 ) @ ordered_pair_predicate )
| ~ ( B @ range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of1) ).
thf(417,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f27 ) @ ordered_pair_predicate )
| ~ ( B @ range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(44,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( E @ ( D @ closed ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(320,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( E @ ( D @ closed ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(183,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ first @ ( A @ second @ ordered_pair ) @ member ) )
| ~ ( B @ converse @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse2) ).
thf(651,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ first @ ( A @ second @ ordered_pair ) @ member ) )
| ~ ( B @ converse @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(152,axiom,
! [C: $i,B: $i,A: $i] :
( ( A != B )
| ~ ( A @ little_set )
| ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair2) ).
thf(570,plain,
! [C: $i,B: $i,A: $i] :
( ( A != B )
| ~ ( A @ little_set )
| ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(214,axiom,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ~ ( A @ little_set )
| ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair3) ).
thf(719,plain,
! [C: $i,B: $i,A: $i] :
( ( A != C )
| ~ ( A @ little_set )
| ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(124,axiom,
! [B: $i,A: $i] :
( ( A @ f54 @ ( empty_set @ ( B @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times12) ).
thf(504,plain,
! [B: $i,A: $i] :
( ( A @ f54 @ ( empty_set @ ( B @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(159,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f28 ) ) @ second @ ( A @ member ) )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply4) ).
thf(587,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f28 ) ) @ second @ ( A @ member ) )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(206,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ member ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(702,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ member ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(56,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ restrict ) )
= ( universal_set @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restrict) ).
thf(344,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ restrict ) )
= ( universal_set @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(62,axiom,
! [A: $i] :
( ( A @ f49 @ little_set )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus11) ).
thf(357,plain,
! [A: $i] :
( ( A @ f49 @ little_set )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(140,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f43 ) @ successor @ member ) )
| ~ ( B @ ( empty_set @ member ) )
| ~ ( B @ little_set )
| ~ ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers2) ).
thf(542,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f43 ) @ successor @ member ) )
| ~ ( B @ ( empty_set @ member ) )
| ~ ( B @ little_set )
| ~ ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(105,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ apply ) @ ( E @ ( B @ apply_to_two_arguments ) ) )
= C )
| ~ ( A @ ( E @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse3) ).
thf(460,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ apply ) @ ( E @ ( B @ apply_to_two_arguments ) ) )
= C )
| ~ ( A @ ( E @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(179,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ( B
!= ( D @ ( C @ ordered_pair ) ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ( B @ first @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first5) ).
thf(638,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ( B
!= ( D @ ( C @ ordered_pair ) ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ( B @ first @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(88,axiom,
! [A: $i] :
( ( A @ ( A @ f26 @ member ) )
| ( A = empty_set )
| ~ ( A @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(421,plain,
! [A: $i] :
( ( A @ ( A @ f26 @ member ) )
| ( A = empty_set )
| ~ ( A @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(168,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f6 ) @ little_set )
| ~ ( B @ second @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second1) ).
thf(606,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f6 ) @ little_set )
| ~ ( B @ second @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(29,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f40 ) @ ( B @ ( A @ f39 ) @ ( B @ ( A @ inverse ) ) ) )
| ~ ( B @ ( A @ group ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',group4) ).
thf(284,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f40 ) @ ( B @ ( A @ f39 ) @ ( B @ ( A @ inverse ) ) ) )
| ~ ( B @ ( A @ group ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ f3 @ little_set )
| ~ ( A @ ordered_pair_predicate ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate2) ).
thf(261,plain,
! [A: $i] :
( ( A @ f3 @ little_set )
| ~ ( A @ ordered_pair_predicate ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(112,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f28 ) ) @ ordered_pair_predicate )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply1) ).
thf(475,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f28 ) ) @ ordered_pair_predicate )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f48 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus8) ).
thf(591,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f48 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(181,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ f57 @ member ) )
| ~ ( A @ finite ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finite1) ).
thf(646,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ f57 @ member ) )
| ~ ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(116,axiom,
! [A: $i] :
( ~ ( A @ f54 @ ( A @ member ) )
| ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times14) ).
thf(485,plain,
! [A: $i] :
( ~ ( A @ f54 @ ( A @ member ) )
| ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(128,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f36 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',associative_system4) ).
thf(514,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f36 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(60,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f53 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times8) ).
thf(353,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f53 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(114,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ f22 ) ) @ second )
= A )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution4) ).
thf(480,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ f22 ) ) @ second )
= A )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(58,axiom,
! [A: $i] :
( ( ( A @ f59 @ ( A @ f59 @ ( plus @ apply_to_two_arguments ) ) )
= A )
| ~ ( even_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',even_numbers3) ).
thf(348,plain,
! [A: $i] :
( ( ( A @ f59 @ ( A @ f59 @ ( plus @ apply_to_two_arguments ) ) )
= A )
| ~ ( even_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(153,axiom,
! [A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( identity_relation @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation1) ).
thf(574,plain,
! [A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(166,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f48 ) @ successor @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus5) ).
thf(602,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f48 ) @ successor @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(199,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset1) ).
thf(687,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(201,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ member ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(692,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ member ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(1,negated_conjecture,
~ ( natural_numbers @ ( empty_set @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_zero_is_a_natural) ).
thf(221,plain,
~ ( natural_numbers @ ( empty_set @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(101,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f53 ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times4) ).
thf(453,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f53 ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(83,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f53 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times3) ).
thf(411,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f53 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(11,axiom,
! [A: $i] :
( ~ ( A @ single_valued_set )
| ~ ( A @ relation )
| ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(243,plain,
! [A: $i] :
( ~ ( A @ single_valued_set )
| ~ ( A @ relation )
| ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(134,axiom,
! [A: $i] :
( ( ( A @ f56 @ ( A @ f55 @ ( times @ apply_to_two_arguments ) ) )
= A )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers7) ).
thf(527,plain,
! [A: $i] :
( ( ( A @ f56 @ ( A @ f55 @ ( times @ apply_to_two_arguments ) ) )
= A )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(41,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( A @ ( D @ maps ) ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse1) ).
thf(312,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( A @ ( D @ maps ) ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(32,axiom,
! [A: $i] :
( ~ ( A @ converse @ function )
| ~ ( A @ function )
| ( A @ one_to_one_function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function3) ).
thf(292,plain,
! [A: $i] :
( ~ ( A @ converse @ function )
| ~ ( A @ function )
| ( A @ one_to_one_function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(6,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(233,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(71,axiom,
! [A: $i] :
( ( A @ ( A @ f20 @ ( A @ f19 @ ordered_pair ) @ member ) )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set5) ).
thf(380,plain,
! [A: $i] :
( ( A @ ( A @ f20 @ ( A @ f19 @ ordered_pair ) @ member ) )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(95,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f4 ) @ ( A @ member ) )
| ~ ( B @ first @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first4) ).
thf(439,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f4 ) @ ( A @ member ) )
| ~ ( B @ first @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(23,axiom,
! [A: $i] :
( ( A @ single_valued_set )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(270,plain,
! [A: $i] :
( ( A @ single_valued_set )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(188,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ apply_to_two_arguments ) ) )
= D )
| ~ ( A @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity2) ).
thf(661,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ apply_to_two_arguments ) ) )
= D )
| ~ ( A @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(142,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f53 ) @ ( plus @ apply_to_two_arguments ) ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times10) ).
thf(546,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f53 ) @ ( plus @ apply_to_two_arguments ) ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ ( B @ ( A @ f1 ) @ member ) )
| ~ ( A @ ( B @ ( A @ f1 ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality3) ).
thf(366,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ ( B @ ( A @ f1 ) @ member ) )
| ~ ( A @ ( B @ ( A @ f1 ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(137,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( D @ ( E @ ordered_pair ) @ ( C @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) @ ( C @ ordered_pair ) ) )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( A @ little_set )
| ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range6) ).
thf(534,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( D @ ( E @ ordered_pair ) @ ( C @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) @ ( C @ ordered_pair ) ) )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( A @ little_set )
| ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(51,axiom,
! [A: $i] :
( ( A @ f21 @ little_set )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set4) ).
thf(334,plain,
! [A: $i] :
( ( A @ f21 @ little_set )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(89,axiom,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f14 ) @ ( B @ ( A @ f13 ) @ ordered_pair ) @ ( B @ ( A @ f12 ) @ ordered_pair ) ) )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range4) ).
thf(424,plain,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f14 ) @ ( B @ ( A @ f13 ) @ ordered_pair ) @ ( B @ ( A @ f12 ) @ ordered_pair ) ) )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(81,axiom,
! [A: $i] :
( ~ ( prime_numbers @ ( A @ successor @ successor @ member ) )
| ~ ( prime_numbers @ ( A @ member ) )
| ( twin_prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',twin_primes3) ).
thf(407,plain,
! [A: $i] :
( ~ ( prime_numbers @ ( A @ successor @ successor @ member ) )
| ~ ( prime_numbers @ ( A @ member ) )
| ( twin_prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(91,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( C @ ( E @ ordered_pair ) @ ( D @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) @ ( C @ ordered_pair ) ) )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( A @ little_set )
| ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right6) ).
thf(429,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( C @ ( E @ ordered_pair ) @ ( D @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) @ ( C @ ordered_pair ) ) )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( A @ little_set )
| ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(218,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ member ) )
| ~ ( A @ ( A @ ( D @ maps ) ) )
| ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse4) ).
thf(729,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ member ) )
| ~ ( A @ ( A @ ( D @ maps ) ) )
| ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(108,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ second @ member ) )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product3) ).
thf(467,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ second @ member ) )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(5,axiom,
! [B: $i,A: $i] :
( ( A != B )
| ~ ( B @ ( A @ proper_subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset2) ).
thf(229,plain,
! [B: $i,A: $i] :
( ( A != B )
| ~ ( B @ ( A @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(15,axiom,
! [A: $i] :
( ( ( A @ f20 )
!= ( A @ f21 ) )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set7) ).
thf(252,plain,
! [A: $i] :
( ( ( A @ f20 )
!= ( A @ f21 ) )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(64,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f30 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(361,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f30 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(93,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(435,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(48,axiom,
! [B: $i,A: $i] : ( B @ ( A @ non_ordered_pair ) @ little_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair4) ).
thf(329,plain,
! [B: $i,A: $i] : ( B @ ( A @ non_ordered_pair ) @ little_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(120,axiom,
! [A: $i] :
( ~ ( A @ f44 @ ( A @ member ) )
| ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers6) ).
thf(493,plain,
! [A: $i] :
( ~ ( A @ f44 @ ( A @ member ) )
| ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(210,axiom,
! [B: $i,A: $i] :
( ( A @ f49 @ ( B @ ( B @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus12) ).
thf(710,plain,
! [B: $i,A: $i] :
( ( A @ f49 @ ( B @ ( B @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(46,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ f36 ) @ ( B @ ( A @ f35 ) @ ( B @ ( A @ f34 ) @ ( B @ apply_to_two_arguments ) ) @ ( B @ apply_to_two_arguments ) ) )
!= ( B @ ( A @ f36 ) @ ( B @ ( A @ f35 ) @ ( B @ apply_to_two_arguments ) ) @ ( B @ ( A @ f34 ) @ ( B @ apply_to_two_arguments ) ) ) )
| ( B @ ( A @ associative ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',associative_system5) ).
thf(324,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ f36 ) @ ( B @ ( A @ f35 ) @ ( B @ ( A @ f34 ) @ ( B @ apply_to_two_arguments ) ) @ ( B @ apply_to_two_arguments ) ) )
!= ( B @ ( A @ f36 ) @ ( B @ ( A @ f35 ) @ ( B @ apply_to_two_arguments ) ) @ ( B @ ( A @ f34 ) @ ( B @ apply_to_two_arguments ) ) ) )
| ( B @ ( A @ associative ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(43,axiom,
! [A: $i] :
( ( A
= ( A @ f3 @ ( A @ f2 @ ordered_pair ) ) )
| ~ ( A @ ordered_pair_predicate ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate3) ).
thf(317,plain,
! [A: $i] :
( ( A
= ( A @ f3 @ ( A @ f2 @ ordered_pair ) ) )
| ~ ( A @ ordered_pair_predicate ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(216,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f13 ) @ ( B @ ( A @ f14 ) @ ordered_pair ) @ ( B @ ( A @ f12 ) @ ordered_pair ) @ member ) )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range5) ).
thf(725,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f13 ) @ ( B @ ( A @ f14 ) @ ordered_pair ) @ ( B @ ( A @ f12 ) @ ordered_pair ) @ member ) )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(74,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f23 ) @ member ) )
| ( B @ ( A @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint2) ).
thf(388,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f23 ) @ member ) )
| ( B @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(162,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f53 ) @ ( plus @ apply_to_two_arguments ) ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times5) ).
thf(593,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( B @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f53 ) @ ( plus @ apply_to_two_arguments ) ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(130,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f22 ) ) @ ordered_pair_predicate )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution1) ).
thf(519,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f22 ) ) @ ordered_pair_predicate )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(155,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ first @ ( A @ second @ ordered_pair ) @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ converse @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse3) ).
thf(578,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ first @ ( A @ second @ ordered_pair ) @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ converse @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(3,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f39 ) @ ( B @ ( A @ identity ) ) )
| ~ ( B @ ( A @ group ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',group3) ).
thf(225,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f39 ) @ ( B @ ( A @ identity ) ) )
| ~ ( B @ ( A @ group ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(191,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f8 ) @ member ) )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(669,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f8 ) @ member ) )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(180,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ ( empty_set @ successor @ non_ordered_pair ) @ ( B @ member ) )
| ( ( C @ ( B @ ( times @ apply_to_two_arguments ) ) )
!= A )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers4) ).
thf(642,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( empty_set @ successor @ non_ordered_pair ) @ ( B @ member ) )
| ( ( C @ ( B @ ( times @ apply_to_two_arguments ) ) )
!= A )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(118,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f10 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right2) ).
thf(489,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f10 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(195,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f47 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus2) ).
thf(679,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f47 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(208,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f34 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',associative_system2) ).
thf(706,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f34 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(110,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint1) ).
thf(471,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).
thf(256,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(173,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f7 ) @ little_set )
| ~ ( B @ second @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second2) ).
thf(622,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f7 ) @ little_set )
| ~ ( B @ second @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(103,axiom,
! [B: $i,A: $i] :
( ( B @ ordered_pair_predicate )
| ~ ( A @ ( B @ member ) )
| ~ ( A @ relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation1) ).
thf(457,plain,
! [B: $i,A: $i] :
( ( B @ ordered_pair_predicate )
| ~ ( A @ ( B @ member ) )
| ~ ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(68,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(371,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(8,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one_function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one_function1) ).
thf(237,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one_function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(186,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( B @ ( A @ f17 ) @ member ) )
| ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset3) ).
thf(657,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( B @ ( A @ f17 ) @ member ) )
| ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f23 ) @ member ) )
| ( B @ ( A @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint3) ).
thf(398,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f23 ) @ member ) )
| ( B @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(147,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f28 ) ) @ member ) )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply2) ).
thf(558,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f28 ) ) @ member ) )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(193,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ f56 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers6) ).
thf(674,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ f56 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(135,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ f49 @ ( D @ successor @ ( C @ ( B @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( A @ f49 @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( D @ member ) )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus13) ).
thf(530,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ f49 @ ( D @ successor @ ( C @ ( B @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( A @ f49 @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( D @ member ) )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(35,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ proper_subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',proper_subset1) ).
thf(298,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(172,axiom,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ second ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of4) ).
thf(618,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ second ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(97,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ member ) )
| ~ ( even_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',even_numbers1) ).
thf(444,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ member ) )
| ~ ( even_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(200,axiom,
! [A: $i] :
( ( f25 @ ( A @ f26 @ ( A @ ordered_pair ) @ member ) )
| ( A = empty_set )
| ~ ( A @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice3) ).
thf(689,plain,
! [A: $i] :
( ( f25 @ ( A @ f26 @ ( A @ ordered_pair ) @ member ) )
| ( A = empty_set )
| ~ ( A @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(34,axiom,
! [A: $i] :
( ( A @ f2 @ little_set )
| ~ ( A @ ordered_pair_predicate ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate1) ).
thf(296,plain,
! [A: $i] :
( ( A @ f2 @ little_set )
| ~ ( A @ ordered_pair_predicate ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f43 ) @ member ) )
| ~ ( B @ ( empty_set @ member ) )
| ~ ( B @ little_set )
| ~ ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers1) ).
thf(562,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f43 ) @ member ) )
| ~ ( B @ ( empty_set @ member ) )
| ~ ( B @ little_set )
| ~ ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(202,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f29 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(694,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f29 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(79,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f5 ) @ little_set )
| ~ ( B @ first @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first2) ).
thf(402,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f5 ) @ little_set )
| ~ ( B @ first @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(30,axiom,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ~ ( C @ little_set )
| ~ ( B @ little_set )
| ( A @ ordered_pair_predicate ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_predicate4) ).
thf(286,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ~ ( C @ little_set )
| ~ ( B @ little_set )
| ( A @ ordered_pair_predicate ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(28,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ( B @ ( A @ f1 ) @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality1) ).
thf(281,plain,
! [B: $i,A: $i] :
( ( A = B )
| ( B @ ( A @ f1 ) @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(18,axiom,
! [B: $i,A: $i] :
( ( A @ ( A @ ( A @ cross_product ) @ ( B @ maps ) ) )
| ~ ( B @ ( A @ closed ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed3) ).
thf(259,plain,
! [B: $i,A: $i] :
( ( A @ ( A @ ( A @ cross_product ) @ ( B @ maps ) ) )
| ~ ( B @ ( A @ closed ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(84,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f31 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(413,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ f31 ) ) @ little_set )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(72,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',non_ordered_pair1) ).
thf(382,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ non_ordered_pair ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(207,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f4 ) @ little_set )
| ~ ( B @ first @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first1) ).
thf(704,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f4 ) @ little_set )
| ~ ( B @ first @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(139,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f31 ) ) @ ( C @ ( B @ ( A @ f29 ) ) @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose5) ).
thf(540,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f31 ) ) @ ( C @ ( B @ ( A @ f29 ) ) @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(167,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f48 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus3) ).
thf(604,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f48 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(196,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f46 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus1) ).
thf(681,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f46 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(25,axiom,
! [A: $i] :
( ( A @ ( A @ f57 @ ( A @ f58 @ maps ) ) )
| ~ ( A @ finite ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finite2) ).
thf(274,plain,
! [A: $i] :
( ( A @ ( A @ f57 @ ( A @ f58 @ maps ) ) )
| ~ ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(87,axiom,
! [B: $i,A: $i] :
( ( A @ little_set )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a2) ).
thf(419,plain,
! [B: $i,A: $i] :
( ( A @ little_set )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(70,axiom,
! [A: $i] :
( ~ ( A @ ( empty_set @ successor @ non_ordered_pair ) @ ( A @ f55 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers8) ).
thf(377,plain,
! [A: $i] :
( ~ ( A @ ( empty_set @ successor @ non_ordered_pair ) @ ( A @ f55 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(100,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity1) ).
thf(451,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(12,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ associative ) )
| ~ ( B @ ( A @ group ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',group2) ).
thf(245,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ associative ) )
| ~ ( B @ ( A @ group ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(126,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f48 ) @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus4) ).
thf(509,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f48 ) @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f45 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(165,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ f28 ) ) @ first )
= C )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply3) ).
thf(599,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ f28 ) ) @ first )
= C )
| ~ ( C @ ( B @ apply ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(171,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ second @ member ) )
| ~ ( B @ ( A @ first @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product4) ).
thf(616,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ second @ member ) )
| ~ ( B @ ( A @ first @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(37,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ f42 ) @ ( B @ ( A @ f41 ) @ ( B @ apply_to_two_arguments ) ) )
!= ( B @ ( A @ f41 ) @ ( B @ ( A @ f42 ) @ ( B @ apply_to_two_arguments ) ) ) )
| ( B @ ( A @ commutes ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutes4) ).
thf(303,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ f42 ) @ ( B @ ( A @ f41 ) @ ( B @ apply_to_two_arguments ) ) )
!= ( B @ ( A @ f41 ) @ ( B @ ( A @ f42 ) @ ( B @ apply_to_two_arguments ) ) ) )
| ( B @ ( A @ commutes ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(113,axiom,
! [A: $i] :
( ( ( A @ first )
!= ( A @ second ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( identity_relation @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation3) ).
thf(477,plain,
! [A: $i] :
( ( ( A @ first )
!= ( A @ second ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( identity_relation @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(145,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ second )
!= A )
| ~ ( B @ ( D @ first @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( D @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( C @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution5) ).
thf(552,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ second )
!= A )
| ~ ( B @ ( D @ first @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( D @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( C @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(115,axiom,
! [A: $i] :
( ~ ( A @ little_set )
| ( universal_set @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',universal_set) ).
thf(483,plain,
! [A: $i] :
( ~ ( A @ little_set )
| ( universal_set @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(205,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f12 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range1) ).
thf(700,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f12 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(49,axiom,
! [A: $i] :
( ( A @ relation )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(330,plain,
! [A: $i] :
( ( A @ relation )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(27,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ ( C @ apply_to_two_arguments ) ) @ ( A @ apply ) )
!= ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ ( A @ apply ) @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ ( A @ apply ) @ ( E @ apply_to_two_arguments ) ) ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism7) ).
thf(278,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ ( C @ apply_to_two_arguments ) ) @ ( A @ apply ) )
!= ( E @ ( D @ ( C @ ( B @ ( A @ f33 ) ) ) ) @ ( A @ apply ) @ ( E @ ( D @ ( C @ ( B @ ( A @ f32 ) ) ) ) @ ( A @ apply ) @ ( E @ apply_to_two_arguments ) ) ) )
| ~ ( D @ ( B @ ( A @ maps ) ) )
| ~ ( E @ ( D @ closed ) )
| ~ ( C @ ( B @ closed ) )
| ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(102,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f27 ) @ member ) )
| ~ ( B @ range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of2) ).
thf(455,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f27 ) @ member ) )
| ~ ( B @ range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(21,axiom,
infinity @ little_set,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity1) ).
thf(267,plain,
infinity @ little_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(198,axiom,
! [A: $i] :
( ( A @ second @ ( A @ first @ member ) )
| ~ ( estin @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(685,plain,
! [A: $i] :
( ( A @ second @ ( A @ first @ member ) )
| ~ ( estin @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(131,axiom,
! [B: $i,A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( B @ converse @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',converse1) ).
thf(521,plain,
! [B: $i,A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( B @ converse @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(2,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).
thf(223,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(169,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ one_to_one_function )
| ( ( C @ range_of )
!= A )
| ~ ( A @ ( B @ ( C @ maps ) ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ( A @ finite ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finite5) ).
thf(608,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ one_to_one_function )
| ( ( C @ range_of )
!= A )
| ~ ( A @ ( B @ ( C @ maps ) ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(76,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ second @ ( A @ member ) )
| ( ( D @ first )
!= C )
| ~ ( B @ ( D @ member ) )
| ~ ( D @ ordered_pair_predicate )
| ( C @ ( B @ apply ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply5) ).
thf(394,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ second @ ( A @ member ) )
| ( ( D @ first )
!= C )
| ~ ( B @ ( D @ member ) )
| ~ ( D @ ordered_pair_predicate )
| ( C @ ( B @ apply ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(38,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ closed ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(306,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ closed ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(10,axiom,
! [B: $i,A: $i] :
( ( A @ little_set )
| ~ ( B @ ( A @ closed ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed1) ).
thf(241,plain,
! [B: $i,A: $i] :
( ( A @ little_set )
| ~ ( B @ ( A @ closed ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(184,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f52 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times7) ).
thf(653,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f52 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(111,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f51 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times1) ).
thf(473,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f51 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(160,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f51 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times6) ).
thf(589,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f51 ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(143,axiom,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).
thf(548,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(16,axiom,
f25 @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(255,plain,
f25 @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(129,axiom,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( G @ ( F @ ( C @ apply_to_two_arguments ) ) @ ( A @ apply ) )
= ( G @ ( A @ apply ) @ ( F @ ( A @ apply ) @ ( E @ apply_to_two_arguments ) ) ) )
| ~ ( B @ ( G @ member ) )
| ~ ( B @ ( F @ member ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(516,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( G @ ( F @ ( C @ apply_to_two_arguments ) ) @ ( A @ apply ) )
= ( G @ ( A @ apply ) @ ( F @ ( A @ apply ) @ ( E @ apply_to_two_arguments ) ) ) )
| ~ ( B @ ( G @ member ) )
| ~ ( B @ ( F @ member ) )
| ~ ( E @ ( D @ ( C @ ( B @ ( A @ homomorphism ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(55,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ image ) @ little_set )
| ~ ( B @ function )
| ~ ( A @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution6) ).
thf(342,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ image ) @ little_set )
| ~ ( B @ function )
| ~ ( A @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(61,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ f59 @ member ) )
| ~ ( even_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',even_numbers2) ).
thf(355,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ f59 @ member ) )
| ~ ( even_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(182,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ f55 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers5) ).
thf(648,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ f55 @ member ) )
| ( A
= ( empty_set @ successor ) )
| ( A = empty_set )
| ~ ( natural_numbers @ ( A @ member ) )
| ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(117,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ( B @ sigma @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma3) ).
thf(487,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ( B @ sigma @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(133,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product1) ).
thf(525,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ordered_pair_predicate )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(211,axiom,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f27 ) @ second ) )
| ~ ( B @ range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of3) ).
thf(712,plain,
! [B: $i,A: $i] :
( ( A
= ( B @ ( A @ f27 ) @ second ) )
| ~ ( B @ range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(156,axiom,
! [A: $i] :
( ( A @ ( A @ f18 @ member ) )
| ( A @ relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation2) ).
thf(580,plain,
! [A: $i] :
( ( A @ ( A @ f18 @ member ) )
| ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(47,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(327,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f42 ) @ member ) )
| ( B @ ( A @ commutes ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutes3) ).
thf(409,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f42 ) @ member ) )
| ( B @ ( A @ commutes ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(20,axiom,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ range_of @ subset ) )
| ( ( A @ domain_of )
!= B )
| ~ ( A @ function )
| ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps4) ).
thf(263,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ range_of @ subset ) )
| ( ( A @ domain_of )
!= B )
| ~ ( A @ function )
| ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ f20 @ little_set )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set3) ).
thf(310,plain,
! [A: $i] :
( ( A @ f20 @ little_set )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(96,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ apply_to_two_arguments ) ) )
= ( C @ ( D @ ( B @ apply_to_two_arguments ) ) ) )
| ~ ( A @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ commutes ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutes1) ).
thf(441,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ apply_to_two_arguments ) ) )
= ( C @ ( D @ ( B @ apply_to_two_arguments ) ) ) )
| ~ ( A @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ commutes ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(158,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f47 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus7) ).
thf(585,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f47 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(213,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f48 ) @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus9) ).
thf(717,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f48 ) @ ( B @ ( A @ f47 ) @ ( B @ ( A @ f46 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(106,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f52 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times2) ).
thf(463,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f52 ) @ member ) )
| ( natural_numbers @ ( B @ ( A @ f50 ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(53,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ ( A @ non_ordered_pair ) @ ( A @ singleton_set @ non_ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(338,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ ( A @ non_ordered_pair ) @ ( A @ singleton_set @ non_ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(154,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ first @ member ) )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product2) ).
thf(576,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ first @ member ) )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(63,axiom,
! [A: $i] :
( ( A @ ( A @ f21 @ ( A @ f19 @ ordered_pair ) @ member ) )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set6) ).
thf(359,plain,
! [A: $i] :
( ( A @ ( A @ f21 @ ( A @ f19 @ ordered_pair ) @ member ) )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(141,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f46 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus6) ).
thf(544,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( natural_numbers @ ( B @ ( A @ f46 ) @ member ) )
| ~ ( B @ ( B @ ( A @ f45 ) @ ( B @ ( A @ f45 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(52,axiom,
! [A: $i] :
( ~ ( A @ f18 @ ordered_pair_predicate )
| ( A @ relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',relation3) ).
thf(336,plain,
! [A: $i] :
( ~ ( A @ f18 @ ordered_pair_predicate )
| ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(217,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f7 ) @ ( A @ member ) )
| ~ ( B @ second @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second4) ).
thf(727,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f7 ) @ ( A @ member ) )
| ~ ( B @ second @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(42,axiom,
! [A: $i] :
( ( ( A @ f58 @ range_of )
= A )
| ~ ( A @ finite ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finite3) ).
thf(314,plain,
! [A: $i] :
( ( ( A @ f58 @ range_of )
= A )
| ~ ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(98,axiom,
! [B: $i,A: $i] :
( ( B
= ( B @ ( A @ f5 ) @ ( B @ ( A @ f4 ) @ ordered_pair ) ) )
| ~ ( B @ first @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first3) ).
thf(446,plain,
! [B: $i,A: $i] :
( ( B
= ( B @ ( A @ f5 ) @ ( B @ ( A @ f4 ) @ ordered_pair ) ) )
| ~ ( B @ first @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(163,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f8 ) @ ordered_pair_predicate )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(595,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f8 ) @ ordered_pair_predicate )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(170,axiom,
! [A: $i] :
( ( A
!= ( empty_set @ successor ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers3) ).
thf(612,plain,
! [A: $i] :
( ( A
!= ( empty_set @ successor ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(122,axiom,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( C @ ( E @ ( F @ ordered_pair ) @ member ) )
| ~ ( B @ ( F @ ( D @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) ) )
| ~ ( F @ little_set )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( A @ little_set )
| ( C @ ( B @ compose ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose7) ).
thf(497,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( C @ ( E @ ( F @ ordered_pair ) @ member ) )
| ~ ( B @ ( F @ ( D @ ordered_pair ) @ member ) )
| ( A
!= ( E @ ( D @ ordered_pair ) ) )
| ~ ( F @ little_set )
| ~ ( E @ little_set )
| ~ ( D @ little_set )
| ~ ( A @ little_set )
| ( C @ ( B @ compose ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(175,axiom,
! [A: $i] :
( ( infinity @ ( A @ successor @ member ) )
| ~ ( infinity @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity3) ).
thf(627,plain,
! [A: $i] :
( ( infinity @ ( A @ successor @ member ) )
| ~ ( infinity @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(192,axiom,
! [A: $i] :
( ( A @ ( A @ f24 @ member ) )
| ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(671,plain,
! [A: $i] :
( ( A @ ( A @ f24 @ member ) )
| ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(121,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( A @ little_set )
| ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(495,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( A @ little_set )
| ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(14,axiom,
! [A: $i] :
( ( A @ sigma @ little_set )
| ~ ( A @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma4) ).
thf(250,plain,
! [A: $i] :
( ( A @ sigma @ little_set )
| ~ ( A @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(107,axiom,
! [A: $i] :
( ( prime_numbers @ ( A @ member ) )
| ~ ( twin_prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',twin_primes1) ).
thf(465,plain,
! [A: $i] :
( ( prime_numbers @ ( A @ member ) )
| ~ ( twin_prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(65,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C = D )
| ~ ( A @ ( D @ ( B @ ordered_pair ) @ member ) )
| ~ ( A @ ( C @ ( B @ ordered_pair ) @ member ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( B @ little_set )
| ~ ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set1) ).
thf(363,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C = D )
| ~ ( A @ ( D @ ( B @ ordered_pair ) @ member ) )
| ~ ( A @ ( C @ ( B @ ordered_pair ) @ member ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ~ ( B @ little_set )
| ~ ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(146,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f14 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range3) ).
thf(556,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f14 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(59,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f13 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip_range2) ).
thf(351,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f13 ) @ little_set )
| ~ ( B @ flip_range_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(75,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( A @ member ) )
| ( B
!= ( D @ ( C @ ordered_pair ) ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ( B @ second @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second5) ).
thf(390,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( A @ member ) )
| ( B
!= ( D @ ( C @ ordered_pair ) ) )
| ~ ( D @ little_set )
| ~ ( C @ little_set )
| ( B @ second @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(177,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ ( D @ ( B @ apply_to_two_arguments ) ) )
= D )
| ~ ( A @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity3) ).
thf(632,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ ( D @ ( B @ apply_to_two_arguments ) ) )
= D )
| ~ ( A @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ identity ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(219,axiom,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ first ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ domain_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain4) ).
thf(731,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ first ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( B @ domain_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(151,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f11 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right3) ).
thf(568,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f11 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(31,axiom,
! [B: $i,A: $i] :
( ~ ( A @ ( A @ ( A @ cross_product ) @ ( B @ maps ) ) )
| ~ ( B @ little_set )
| ~ ( A @ little_set )
| ( B @ ( A @ closed ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed4) ).
thf(290,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( A @ ( A @ cross_product ) @ ( B @ maps ) ) )
| ~ ( B @ little_set )
| ~ ( A @ little_set )
| ( B @ ( A @ closed ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(7,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ closed ) )
| ~ ( B @ ( A @ group ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',group1) ).
thf(235,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ closed ) )
| ~ ( B @ ( A @ group ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(92,axiom,
! [A: $i] :
( ( natural_numbers @ ( A @ member ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers1) ).
thf(433,plain,
! [A: $i] :
( ( natural_numbers @ ( A @ member ) )
| ~ ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ singleton_set )
= ( A @ ( A @ non_ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(272,plain,
! [A: $i] :
( ( A @ singleton_set )
= ( A @ ( A @ non_ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(57,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f22 ) ) @ first @ member ) )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_and_substitution3) ).
thf(346,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( B @ ( A @ f22 ) ) @ first @ member ) )
| ~ ( C @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(36,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ apply ) @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( B @ apply_to_two_arguments ) ) )
!= C )
| ( ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ apply ) @ ( B @ apply_to_two_arguments ) ) )
!= C )
| ~ ( A @ ( A @ ( D @ maps ) ) )
| ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse5) ).
thf(300,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ apply ) @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( B @ apply_to_two_arguments ) ) )
!= C )
| ( ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ ( C @ ( B @ ( A @ f38 ) ) ) @ ( D @ apply ) @ ( B @ apply_to_two_arguments ) ) )
!= C )
| ~ ( A @ ( A @ ( D @ maps ) ) )
| ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(69,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( B @ ( plus @ apply_to_two_arguments ) ) )
!= A )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( natural_numbers @ ( A @ member ) )
| ( even_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',even_numbers4) ).
thf(373,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ ( plus @ apply_to_two_arguments ) ) )
!= A )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( natural_numbers @ ( A @ member ) )
| ( even_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(220,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f41 ) @ member ) )
| ( B @ ( A @ commutes ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutes2) ).
thf(735,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f41 ) @ member ) )
| ( B @ ( A @ commutes ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(150,axiom,
! [A: $i] :
( ( A != empty_set )
| ~ ( prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prime_numbers2) ).
thf(564,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(9,axiom,
! [A: $i] :
( ( A @ powerset @ little_set )
| ~ ( A @ little_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset3) ).
thf(239,plain,
! [A: $i] :
( ( A @ powerset @ little_set )
| ~ ( A @ little_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ f58 @ one_to_one_function )
| ~ ( A @ finite ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finite4) ).
thf(294,plain,
! [A: $i] :
( ( A @ f58 @ one_to_one_function )
| ~ ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(94,axiom,
! [A: $i] :
( ( prime_numbers @ ( A @ successor @ successor @ member ) )
| ~ ( twin_prime_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',twin_primes2) ).
thf(437,plain,
! [A: $i] :
( ( prime_numbers @ ( A @ successor @ successor @ member ) )
| ~ ( twin_prime_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(189,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ ( A @ f37 ) ) @ member ) )
| ~ ( A @ ( C @ member ) )
| ( C @ ( B @ ( A @ identity ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity4) ).
thf(664,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ ( A @ f37 ) ) @ member ) )
| ~ ( A @ ( C @ member ) )
| ( C @ ( B @ ( A @ identity ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(78,axiom,
! [A: $i] :
( ~ ( A @ f49 @ ( A @ member ) )
| ( plus @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',plus14) ).
thf(400,plain,
! [A: $i] :
( ~ ( A @ f49 @ ( A @ member ) )
| ( plus @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(178,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ( B @ ( B @ ( A @ f1 ) @ member ) )
| ( A @ ( B @ ( A @ f1 ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality2) ).
thf(635,plain,
! [B: $i,A: $i] :
( ( A = B )
| ( B @ ( B @ ( A @ f1 ) @ member ) )
| ( A @ ( B @ ( A @ f1 ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(136,axiom,
! [B: $i,A: $i] :
( ( A @ f44 @ ( B @ successor @ member ) )
| ~ ( A @ f44 @ ( B @ member ) )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers5) ).
thf(532,plain,
! [B: $i,A: $i] :
( ( A @ f44 @ ( B @ successor @ member ) )
| ~ ( A @ f44 @ ( B @ member ) )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(203,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f53 ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times9) ).
thf(696,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ ( B @ ( A @ f53 ) @ ( B @ ( A @ f52 ) @ ( B @ ( A @ f51 ) @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( empty_set @ ( B @ ( A @ f50 ) @ ( empty_set @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( B @ little_set )
| ~ ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ powerset @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',powerset1) ).
thf(415,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ powerset @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(125,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( E @ ( D @ apply ) @ ( B @ apply_to_two_arguments ) ) )
= C )
| ~ ( A @ ( E @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse2) ).
thf(506,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( E @ ( D @ apply ) @ ( B @ apply_to_two_arguments ) ) )
= C )
| ~ ( A @ ( E @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(22,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) )
| ~ ( C @ ( B @ ( A @ identity ) ) )
| ~ ( B @ ( A @ associative ) )
| ~ ( B @ ( A @ closed ) )
| ( B @ ( A @ group ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',group5) ).
thf(268,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( C @ ( B @ ( A @ inverse ) ) ) )
| ~ ( C @ ( B @ ( A @ identity ) ) )
| ~ ( B @ ( A @ associative ) )
| ~ ( B @ ( A @ closed ) )
| ( B @ ( A @ group ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(174,axiom,
! [A: $i] :
( ( ( A @ first )
= ( A @ second ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation2) ).
thf(624,plain,
! [A: $i] :
( ( ( A @ first )
= ( A @ second ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(50,axiom,
! [B: $i,A: $i] :
( ( B @ little_set )
| ~ ( B @ ( A @ closed ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed2) ).
thf(332,plain,
! [B: $i,A: $i] :
( ( B @ little_set )
| ~ ( B @ ( A @ closed ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(90,axiom,
! [A: $i] :
( ~ ( A @ second @ ( A @ first @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( estin @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(427,plain,
! [A: $i] :
( ~ ( A @ second @ ( A @ first @ member ) )
| ~ ( A @ ordered_pair_predicate )
| ~ ( A @ little_set )
| ( estin @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(104,axiom,
infinity @ ( empty_set @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity2) ).
thf(459,plain,
infinity @ ( empty_set @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(215,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f35 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',associative_system3) ).
thf(723,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f35 ) @ member ) )
| ( B @ ( A @ associative ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(123,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ ( C @ ( B @ apply_to_two_arguments ) ) @ ( B @ apply_to_two_arguments ) ) )
= ( E @ ( D @ ( B @ apply_to_two_arguments ) ) @ ( C @ ( B @ apply_to_two_arguments ) ) ) )
| ~ ( A @ ( E @ member ) )
| ~ ( A @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ associative ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',associative_system1) ).
thf(501,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ ( C @ ( B @ apply_to_two_arguments ) ) @ ( B @ apply_to_two_arguments ) ) )
= ( E @ ( D @ ( B @ apply_to_two_arguments ) ) @ ( C @ ( B @ apply_to_two_arguments ) ) ) )
| ~ ( A @ ( E @ member ) )
| ~ ( A @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ associative ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(67,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f16 ) @ member ) )
| ~ ( B @ sigma @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sigma1) ).
thf(369,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f16 ) @ member ) )
| ~ ( B @ sigma @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(4,axiom,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton_set @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(227,plain,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton_set @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(45,axiom,
! [A: $i] :
( ( A @ f19 @ little_set )
| ( A @ single_valued_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_set2) ).
thf(322,plain,
! [A: $i] :
( ( A @ f19 @ little_set )
| ( A @ single_valued_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(109,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ f9 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right1) ).
thf(469,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ f9 ) @ little_set )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(187,axiom,
! [A: $i] :
( ( A @ f44 @ ( empty_set @ member ) )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',natural_numbers4) ).
thf(659,plain,
! [A: $i] :
( ( A @ f44 @ ( empty_set @ member ) )
| ~ ( A @ little_set )
| ( natural_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(148,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f9 ) @ ( B @ ( A @ f11 ) @ ordered_pair ) @ ( B @ ( A @ f10 ) @ ordered_pair ) @ member ) )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate_right5) ).
thf(560,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ f9 ) @ ( B @ ( A @ f11 ) @ ordered_pair ) @ ( B @ ( A @ f10 ) @ ordered_pair ) @ member ) )
| ~ ( B @ rotate_right @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(119,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f17 ) @ member ) )
| ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset2) ).
thf(491,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ( A @ f17 ) @ member ) )
| ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(209,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ f54 @ ( C @ ( D @ ( plus @ apply_to_two_arguments ) ) @ ( C @ ( B @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( A @ f54 @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( D @ member ) )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',times13) ).
thf(708,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ f54 @ ( C @ ( D @ ( plus @ apply_to_two_arguments ) ) @ ( C @ ( B @ successor @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( A @ f54 @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( natural_numbers @ ( D @ member ) )
| ~ ( natural_numbers @ ( C @ member ) )
| ~ ( natural_numbers @ ( B @ member ) )
| ~ ( A @ little_set )
| ( times @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(754,plain,
$false,
inference(e,[status(thm)],[629,666,683,276,308,449,247,677,597,523,582,655,511,385,550,715,698,340,538,404,417,320,651,570,719,504,587,702,344,357,542,460,638,421,606,284,261,475,591,646,485,514,353,480,348,574,602,687,692,221,453,411,243,527,312,292,233,380,439,270,661,546,366,534,334,424,407,429,729,467,229,252,361,435,329,493,710,324,317,725,388,593,519,578,225,669,642,489,679,706,471,256,622,457,371,237,657,398,558,674,530,298,618,444,689,296,562,694,402,286,281,259,413,382,704,540,604,681,274,419,377,451,245,509,599,616,303,477,552,483,700,330,278,455,267,685,521,223,608,394,306,241,653,473,589,548,255,516,342,355,648,487,525,712,580,327,409,263,310,441,585,717,463,338,576,359,544,336,727,314,446,595,612,497,627,671,495,250,465,363,556,351,390,632,731,568,290,235,433,272,346,300,373,735,564,239,294,437,664,400,635,532,696,415,506,268,624,332,427,459,723,501,369,227,322,469,659,560,491,708]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : NUM009-1 : TPTP v9.3.1. Bugfixed v1.2.1.
% 0.00/0.12 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.23/5.52 % Computer : n017.cluster.edu
% 0.23/5.52 % Model : x86_64 x86_64
% 0.23/5.52 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.23/5.52 % Memory : 8046.5625MB
% 0.23/5.52 % OS : Linux 6.8.0-71-generic
% 0.23/5.52 % CPULimit : 300
% 0.23/5.52 % WCLimit : 300
% 0.23/5.52 % DateTime : Sat Sep 26 01:48:10 UTC 2026
% 0.23/5.53 % CPUTime :
% 0.23/5.53 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.37/6.44 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.83/6.92 % [INFO] Parsing done (483ms).
% 2.83/6.95 % [INFO] Running in sequential loop mode.
% 4.27/7.58 % [INFO] eprover registered as external prover.
% 4.27/7.60 % [INFO] Scanning for conjecture ...
% 4.94/7.81 % [INFO] Found a conjecture (or negated_conjecture) and 219 axioms. Running axiom selection ...
% 5.15/8.03 % [INFO] Axiom selection finished. Selected 219 axioms (removed 0 axioms).
% 6.61/8.31 % [INFO] Problem is propositional (TPTP CNF).
% 6.61/8.34 % [INFO] Type checking passed.
% 6.61/8.35 % [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 ...
% 15.12/10.98 % External prover 'e' found a proof!
% 15.12/10.98 % [INFO] Killing All external provers ...
% 15.12/10.98 % Time passed: 5234ms (effective reasoning time: 4025ms)
% 15.12/10.99 % Axioms used in derivation (219): twin_primes3, cross_product4, complement1, plus10, single_valued_set4, image_and_substitution2, second2, maps4, first2, plus12, first1, function3, apply5, homomorphism4, flip_range5, closed3, times12, associative_system3, infinity3, range_of3, apply1, regularity1, sigma2, domain4, singleton_set, compose4, sigma1, proper_subset3, prime_numbers5, relation1, ordered_pair_predicate2, single_valued_set7, one_to_one_function3, natural_numbers3, powerset1, union, rotate_right1, plus8, element_relation2, identity_relation2, apply_to_two_arguments, times3, compose2, homomorphism3, group2, image_and_substitution5, finite1, second5, inverse4, rotate_right3, prime_numbers8, commutes2, successor, non_ordered_pair2, empty_set, even_numbers4, homomorphism7, choice3, ordered_pair, associative_system2, prime_numbers2, inverse3, image_and_substitution3, identity3, homomorphism1, flip_range4, identity2, closed4, rotate_right6, disjoint2, times2, infinity2, times11, prime_numbers6, subset3, compose3, apply4, plus4, image_and_substitution6, range_of4, choice1, second3, single_valued_set3, converse3, natural_numbers5, domain3, regularity2, first3, cross_product3, rotate_right2, element_relation3, plus7, homomorphism6, extensionality2, intersection1, subset2, compose7, flip_range1, group3, commutes3, finite2, relation2, non_ordered_pair1, ordered_pair_predicate1, even_numbers3, times7, finite4, proper_subset1, maps1, sigma4, times6, natural_numbers2, one_to_one_function2, times1, prime_numbers1, first4, prime_numbers3, function1, compose6, commutes1, maps3, domain2, intersection3, natural_numbers1, identity4, second4, rotate_right5, associative_system5, identity_relation3, plus3, closed1, converse2, disjoint1, disjoint3, element_relation1, twin_primes1, infinity1, times10, plus1, image_and_substitution4, plus6, second1, inverse2, cross_product2, one_to_one_function1, finite3, extensionality1, non_ordered_pair4, sigma3, ordered_pair_predicate4, plus14, single_valued_set2, powerset3, plus2, times8, finite5, times5, homomorphism5, commutes4, flip_range2, maps2, group4, even_numbers2, apply3, relation3, image_and_substitution1, associative_system1, rotate_right4, complement2, flip_range6, twin_primes2, cross_product1, plus5, function2, compose5, apply2, range_of2, inverse5, prime_numbers4, natural_numbers6, associative_system4, plus13, universal_set, domain1, closed2, identity_relation1, subset1, times13, proper_subset2, converse1, single_valued_set5, identity5, intersection2, powerset2, identity1, times4, first5, natural_numbers4, non_ordered_pair3, single_valued_set1, times14, even_numbers1, ordered_pair_predicate3, group1, choice2, restrict, homomorphism2, a2, plus11, times9, group5, plus9, prime_numbers7, single_valued_set6, extensionality3, inverse1, flip_range3, compose1, range_of1
% 15.12/10.99 % No. of inferences in proof: 441
% 15.12/10.99 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 5234 ms resp. 4025 ms w/o parsing
% 17.09/11.30 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 17.09/11.31 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------