TSTP Solution File: SEU208+2 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SEU208+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n023.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri May 19 11:57:33 EDT 2023
% Result : Theorem 111.36s 21.42s
% Output : Refutation 111.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 223
% Syntax : Number of formulae : 413 ( 69 unt; 35 typ; 0 def)
% Number of atoms : 1282 ( 284 equ; 0 cnn)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 4039 ( 140 ~; 22 |; 322 &;2995 @)
% ( 71 <=>; 489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 57 ( 57 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 35 usr; 2 con; 0-3 aty)
% Number of variables : 948 ( 0 ^; 879 !; 69 ?; 948 :)
% Comments :
%------------------------------------------------------------------------------
thf(relation_type,type,
relation: $i > $o ).
thf(in_type,type,
in: $i > $i > $o ).
thf(relation_inverse_image_type,type,
relation_inverse_image: $i > $i > $i ).
thf(relation_rng_type,type,
relation_rng: $i > $i ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(proper_subset_type,type,
proper_subset: $i > $i > $o ).
thf(cast_to_subset_type,type,
cast_to_subset: $i > $i ).
thf(empty_type,type,
empty: $i > $o ).
thf(set_difference_type,type,
set_difference: $i > $i > $i ).
thf(empty_set_type,type,
empty_set: $i ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(cartesian_product2_type,type,
cartesian_product2: $i > $i > $i ).
thf(set_union2_type,type,
set_union2: $i > $i > $i ).
thf(element_type,type,
element: $i > $i > $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(union_of_subsets_type,type,
union_of_subsets: $i > $i > $i ).
thf(disjoint_type,type,
disjoint: $i > $i > $o ).
thf(set_intersection2_type,type,
set_intersection2: $i > $i > $i ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(subset_difference_type,type,
subset_difference: $i > $i > $i > $i ).
thf(subset_complement_type,type,
subset_complement: $i > $i > $i ).
thf(union_type,type,
union: $i > $i ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(meet_of_subsets_type,type,
meet_of_subsets: $i > $i > $i ).
thf(complements_of_subsets_type,type,
complements_of_subsets: $i > $i > $i ).
thf(set_meet_type,type,
set_meet: $i > $i ).
thf(relation_field_type,type,
relation_field: $i > $i ).
thf(relation_dom_type,type,
relation_dom: $i > $i ).
thf(relation_dom_restriction_type,type,
relation_dom_restriction: $i > $i > $i ).
thf(relation_composition_type,type,
relation_composition: $i > $i > $i ).
thf(identity_relation_type,type,
identity_relation: $i > $i ).
thf(are_equipotent_type,type,
are_equipotent: $i > $i > $o ).
thf(relation_image_type,type,
relation_image: $i > $i > $i ).
thf(relation_inverse_type,type,
relation_inverse: $i > $i ).
thf(relation_rng_restriction_type,type,
relation_rng_restriction: $i > $i > $i ).
thf(162,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_intersection2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_xboole_0) ).
thf(1059,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_intersection2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(157,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
<=> ( ( in @ A @ C )
& ( in @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l55_zfmisc_1) ).
thf(1036,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
=> ( ( in @ A @ C )
& ( in @ B @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(46,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t33_xboole_1) ).
thf(352,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(51,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k7_setfam_1) ).
thf(365,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(180,axiom,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ~ ( ( in @ B @ ( subset_complement @ A @ C ) )
& ( in @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t54_subset_1) ).
thf(1190,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ~ ( ( in @ B @ ( subset_complement @ A @ C ) )
& ( in @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(143,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_composition @ A @ B ) ) @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t44_relat_1) ).
thf(909,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_composition @ A @ B ) ) @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(41,axiom,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(333,plain,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(113,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( cartesian_product2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_zfmisc_1) ).
thf(683,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( cartesian_product2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) )
& ( ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) )
& ( ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( cartesian_product2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(153,axiom,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(994,plain,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(170,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_composition @ A @ B ) ) @ ( relation_rng @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t45_relat_1) ).
thf(1142,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_composition @ A @ B ) ) @ ( relation_rng @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(118,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ B @ ( relation_rng @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t20_relat_1) ).
thf(730,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ B @ ( relation_rng @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ C ) )
& ( subset @ ( cartesian_product2 @ C @ A ) @ ( cartesian_product2 @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t118_zfmisc_1) ).
thf(217,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ C ) )
& ( subset @ ( cartesian_product2 @ C @ A ) @ ( cartesian_product2 @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(114,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
| ( ( relation_rng @ A )
= empty_set ) )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t64_relat_1) ).
thf(715,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
| ( ( relation_rng @ A )
= empty_set ) )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(85,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ B @ ( relation_field @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t30_relat_1) ).
thf(472,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ B @ ( relation_field @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(187,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_relat_1) ).
thf(1233,plain,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(61,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t39_xboole_1) ).
thf(404,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(66,axiom,
! [A: $i] : ( subset @ empty_set @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(417,plain,
! [A: $i] : ( subset @ empty_set @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(37,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( ( unordered_pair @ A @ B )
= ( unordered_pair @ C @ D ) )
& ( A != C )
& ( A != D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t10_zfmisc_1) ).
thf(320,plain,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( ( unordered_pair @ A @ B )
= ( unordered_pair @ C @ D ) )
& ( A != C )
& ( A != D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(73,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( meet_of_subsets @ A @ B )
= ( set_meet @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k6_setfam_1) ).
thf(436,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( meet_of_subsets @ A @ B )
= ( set_meet @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(19,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(257,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) )
& ( ( ( subset @ A @ B )
& ( subset @ B @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(140,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_image @ B @ A )
= ( relation_image @ B @ ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t145_relat_1) ).
thf(902,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_image @ B @ A )
= ( relation_image @ B @ ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(98,axiom,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ B @ A ) )
& ( relation @ ( relation_composition @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc10_relat_1) ).
thf(559,plain,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ B @ A ) )
& ( relation @ ( relation_composition @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(27,axiom,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(284,plain,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(166,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_relat_1) ).
thf(1115,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(86,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc8_relat_1) ).
thf(475,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(141,axiom,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k4_relat_1) ).
thf(905,plain,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(174,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_difference @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_xboole_0) ).
thf(1151,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_difference @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ~ ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ~ ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(146,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_relat_1) ).
thf(937,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(156,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( B
= ( relation_inverse @ A ) )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( in @ ( ordered_pair @ D @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_relat_1) ).
thf(1020,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( B
= ( relation_inverse @ A ) )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
=> ( B
= ( relation_inverse @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k6_setfam_1) ).
thf(348,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(107,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t146_relat_1) ).
thf(634,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(78,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(453,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(58,axiom,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(396,plain,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(189,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(1242,plain,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ A @ C ) )
=> ( subset @ A @ ( set_intersection2 @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t19_xboole_1) ).
thf(312,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ A @ C ) )
=> ( subset @ A @ ( set_intersection2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(6,axiom,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_boole) ).
thf(206,plain,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(70,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).
thf(428,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(182,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d6_relat_1) ).
thf(1201,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(307,plain,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(74,axiom,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_boole) ).
thf(439,plain,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(112,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_dom @ A )
= empty_set )
<=> ( ( relation_rng @ A )
= empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t65_relat_1) ).
thf(678,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
=> ( ( relation_rng @ A )
= empty_set ) )
& ( ( ( relation_rng @ A )
= empty_set )
=> ( ( relation_dom @ A )
= empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(95,axiom,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(546,plain,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(4,axiom,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',antisymmetry_r2_xboole_0) ).
thf(201,plain,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(91,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_subset) ).
thf(534,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(160,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
=> ( ( element @ B @ A )
<=> ( in @ B @ A ) ) )
& ( ( empty @ A )
=> ( ( element @ B @ A )
<=> ( empty @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_subset_1) ).
thf(1047,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
=> ( ( ( element @ B @ A )
=> ( in @ B @ A ) )
& ( ( in @ B @ A )
=> ( element @ B @ A ) ) ) )
& ( ( empty @ A )
=> ( ( ( element @ B @ A )
=> ( empty @ B ) )
& ( ( empty @ B )
=> ( element @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(184,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l71_subset_1) ).
thf(1206,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(13,axiom,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(233,plain,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(164,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3_subset_1) ).
thf(1091,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(158,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_rng @ B ) )
=> ( ( relation_rng @ ( relation_composition @ B @ A ) )
= ( relation_rng @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t47_relat_1) ).
thf(1042,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_rng @ B ) )
=> ( ( relation_rng @ ( relation_composition @ B @ A ) )
= ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(62,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( meet_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t48_setfam_1) ).
thf(407,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( meet_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(128,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc7_relat_1) ).
thf(830,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(324,plain,
! [A: $i] :
( ( set_intersection2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(10,axiom,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(224,plain,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(145,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_rng_restriction @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d12_relat_1) ).
thf(916,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_rng_restriction @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) )
& ( ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) )
& ( ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_rng_restriction @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(83,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(467,plain,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(79,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( union_of_subsets @ A @ B )
= ( union @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k5_setfam_1) ).
thf(456,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( union_of_subsets @ A @ B )
= ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(132,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t116_relat_1) ).
thf(842,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(36,axiom,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t48_xboole_1) ).
thf(317,plain,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(110,axiom,
! [A: $i,B: $i] :
( ( B
= ( union @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_tarski) ).
thf(650,plain,
! [A: $i,B: $i] :
( ( ( B
= ( union @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) )
& ( ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) )
& ( ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) )
=> ( in @ C @ B ) ) )
=> ( B
= ( union @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(117,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(725,plain,
! [A: $i,B: $i] :
( ( ( subset @ ( singleton @ A ) @ B )
=> ( in @ A @ B ) )
& ( ( in @ A @ B )
=> ( subset @ ( singleton @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(149,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( relation_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( relation_inverse @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_relat_1) ).
thf(980,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( relation_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( relation_inverse @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(167,axiom,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(1118,plain,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(43,axiom,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
<=> ( ( subset @ A @ B )
& ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_xboole_0) ).
thf(339,plain,
! [A: $i,B: $i] :
( ( ( proper_subset @ A @ B )
=> ( ( subset @ A @ B )
& ( A != B ) ) )
& ( ( ( subset @ A @ B )
& ( A != B ) )
=> ( proper_subset @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(176,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t118_relat_1) ).
thf(1182,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(92,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_boole) ).
thf(537,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(108,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
<=> ( in @ C @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_tarski) ).
thf(637,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( ( in @ C @ A )
=> ( in @ C @ B ) )
& ( ( in @ C @ B )
=> ( in @ C @ A ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(101,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t88_relat_1) ).
thf(590,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(97,axiom,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( in @ C @ ( set_intersection2 @ A @ B ) ) )
& ~ ( ? [C: $i] : ( in @ C @ ( set_intersection2 @ A @ B ) )
& ( disjoint @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_xboole_0) ).
thf(554,plain,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( in @ C @ ( set_intersection2 @ A @ B ) ) )
& ~ ( ? [C: $i] : ( in @ C @ ( set_intersection2 @ A @ B ) )
& ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(121,axiom,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
( ( subset @ E @ C )
=> ( in @ E @ D ) ) ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t9_tarski) ).
thf(753,plain,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
( ( subset @ E @ C )
=> ( in @ E @ D ) ) ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(150,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc6_relat_1) ).
thf(985,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(181,axiom,
! [A: $i] :
( ( relation @ A )
<=> ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_relat_1) ).
thf(1192,plain,
! [A: $i] :
( ( ( relation @ A )
=> ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) ) )
& ( ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) )
=> ( relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(144,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(911,plain,
! [A: $i,B: $i] :
( ( ( subset @ ( singleton @ A ) @ B )
=> ( in @ A @ B ) )
& ( ( in @ A @ B )
=> ( subset @ ( singleton @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(109,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom @ ( relation_dom_restriction @ B @ A ) )
= ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t90_relat_1) ).
thf(647,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom @ ( relation_dom_restriction @ B @ A ) )
= ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(154,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_rng @ A ) @ ( relation_dom @ B ) )
=> ( ( relation_dom @ ( relation_composition @ A @ B ) )
= ( relation_dom @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_relat_1) ).
thf(997,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_rng @ A ) @ ( relation_dom @ B ) )
=> ( ( relation_dom @ ( relation_composition @ A @ B ) )
= ( relation_dom @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(16,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_intersection2 @ A @ B )
= empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_xboole_0) ).
thf(244,plain,
! [A: $i,B: $i] :
( ( ( disjoint @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= empty_set ) )
& ( ( ( set_intersection2 @ A @ B )
= empty_set )
=> ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(49,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_xboole_1) ).
thf(360,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(123,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( unordered_pair @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( D = A )
| ( D = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_tarski) ).
thf(765,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( unordered_pair @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D = A )
| ( D = B ) ) )
& ( ( ( D = A )
| ( D = B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D = A )
| ( D = B ) ) )
& ( ( ( D = A )
| ( D = B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( unordered_pair @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(59,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(398,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(125,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_relat_1) ).
thf(794,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(138,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ ( singleton @ B ) )
= A )
<=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t65_zfmisc_1) ).
thf(888,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ ( singleton @ B ) )
= A )
=> ~ ( in @ B @ A ) )
& ( ~ ( in @ B @ A )
=> ( ( set_difference @ A @ ( singleton @ B ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(64,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t63_xboole_1) ).
thf(412,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(69,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(425,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(172,axiom,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k6_relat_1) ).
thf(1146,plain,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(25,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k3_subset_1) ).
thf(279,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(119,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
| ( subset @ A @ ( set_difference @ B @ ( singleton @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3_zfmisc_1) ).
thf(733,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
| ( subset @ A @ ( set_difference @ B @ ( singleton @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(99,axiom,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_subset) ).
thf(562,plain,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(26,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(281,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(127,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_union2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
| ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_xboole_0) ).
thf(804,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_union2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
| ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
| ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
| ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
| ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_union2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(45,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_subset_1) ).
thf(350,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(93,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) )
<=> ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t86_relat_1) ).
thf(540,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) )
=> ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) ) )
& ( ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) )
=> ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(185,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( relation_dom_restriction @ ( relation_rng_restriction @ A @ C ) @ B )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t140_relat_1) ).
thf(1209,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( relation_dom_restriction @ ( relation_rng_restriction @ A @ C ) @ B )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(23,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_difference @ A @ B @ C )
= ( set_difference @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k6_subset_1) ).
thf(274,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_difference @ A @ B @ C )
= ( set_difference @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(67,axiom,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_boole) ).
thf(419,plain,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(120,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ A ) ) )
=> ( ( C
= ( complements_of_subsets @ A @ B ) )
<=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( in @ D @ C )
<=> ( in @ ( subset_complement @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_setfam_1) ).
thf(736,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ A ) ) )
=> ( ( ( C
= ( complements_of_subsets @ A @ B ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( in @ D @ C )
=> ( in @ ( subset_complement @ A @ D ) @ B ) )
& ( ( in @ ( subset_complement @ A @ D ) @ B )
=> ( in @ D @ C ) ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( in @ D @ C )
=> ( in @ ( subset_complement @ A @ D ) @ B ) )
& ( ( in @ ( subset_complement @ A @ D ) @ B )
=> ( in @ D @ C ) ) ) )
=> ( C
= ( complements_of_subsets @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(152,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) )
<=> ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t115_relat_1) ).
thf(990,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) )
=> ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) ) )
& ( ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) )
=> ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(12,axiom,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(230,plain,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(50,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_setfam_1) ).
thf(362,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(171,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(1144,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(169,axiom,
! [A: $i,B: $i] :
( ( B
= ( powerset @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ( subset @ C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_zfmisc_1) ).
thf(1122,plain,
! [A: $i,B: $i] :
( ( ( B
= ( powerset @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ( subset @ C @ A ) )
& ( ( subset @ C @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ( subset @ C @ A ) )
& ( ( subset @ C @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( powerset @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(21,axiom,
empty @ empty_set,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(271,plain,
empty @ empty_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(103,axiom,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( powerset @ C ) @ B ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t136_zfmisc_1) ).
thf(599,plain,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( powerset @ C ) @ B ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(136,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_subset) ).
thf(880,plain,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(72,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k7_setfam_1) ).
thf(434,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(56,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_difference @ A @ B )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(387,plain,
! [A: $i,B: $i] :
( ( ( disjoint @ A @ B )
=> ( ( set_difference @ A @ B )
= A ) )
& ( ( ( set_difference @ A @ B )
= A )
=> ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(106,axiom,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_tarski) ).
thf(631,plain,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(17,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(251,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(139,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ordered_pair @ A @ B )
= ( ordered_pair @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t33_zfmisc_1) ).
thf(895,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ordered_pair @ A @ B )
= ( ordered_pair @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(40,axiom,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_boole) ).
thf(330,plain,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(81,axiom,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(462,plain,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(3,axiom,
! [A: $i,B: $i] :
~ ( proper_subset @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(198,plain,
! [A: $i] :
~ ( proper_subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(175,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_image @ C @ B ) )
<=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t143_relat_1) ).
thf(1177,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_image @ C @ B ) )
=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) )
=> ( in @ A @ ( relation_image @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(148,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_dom_restriction @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d11_relat_1) ).
thf(959,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_dom_restriction @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) )
& ( ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) )
& ( ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_dom_restriction @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(57,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ D ) )
=> ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t119_zfmisc_1) ).
thf(394,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ D ) )
=> ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(155,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( B
= ( relation_rng @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_relat_1) ).
thf(1000,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( B
= ( relation_rng @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(134,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_zfmisc_1) ).
thf(851,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(71,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( union_of_subsets @ A @ B ) )
= ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t47_setfam_1) ).
thf(431,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( union_of_subsets @ A @ B ) )
= ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(147,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( B
= ( relation_dom @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_relat_1) ).
thf(939,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( B
= ( relation_dom @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( relation_dom @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(47,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(355,plain,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(130,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_dom @ B ) )
& ( subset @ ( relation_rng @ A ) @ ( relation_rng @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t25_relat_1) ).
thf(836,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_dom @ B ) )
& ( subset @ ( relation_rng @ A ) @ ( relation_rng @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(65,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ B )
= ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_subset_1) ).
thf(414,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ B )
= ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(39,axiom,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_boole) ).
thf(327,plain,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(124,axiom,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k7_relat_1) ).
thf(791,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(53,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k3_subset_1) ).
thf(370,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( ( A != empty_set )
=> ( ( B
= ( set_meet @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) ) ) )
& ( ( A = empty_set )
=> ( ( B
= ( set_meet @ A ) )
<=> ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_setfam_1) ).
thf(502,plain,
! [A: $i,B: $i] :
( ( ( A != empty_set )
=> ( ( ( B
= ( set_meet @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) )
& ( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) )
& ( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) )
=> ( in @ C @ B ) ) )
=> ( B
= ( set_meet @ A ) ) ) ) )
& ( ( A = empty_set )
=> ( ( ( B
= ( set_meet @ A ) )
=> ( B = empty_set ) )
& ( ( B = empty_set )
=> ( B
= ( set_meet @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(163,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
<=> ( ( in @ A @ C )
& ( in @ B @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t106_zfmisc_1) ).
thf(1085,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
=> ( ( in @ A @ C )
& ( in @ B @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(179,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_dom_restriction @ B @ A ) ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t99_relat_1) ).
thf(1188,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_dom_restriction @ B @ A ) ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(178,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k5_relat_1) ).
thf(1186,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(94,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_subset) ).
thf(544,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(142,axiom,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_relat_1) ).
thf(907,plain,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(42,axiom,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t40_xboole_1) ).
thf(336,plain,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(35,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(314,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(105,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation @ D )
=> ( ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) )
<=> ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t74_relat_1) ).
thf(627,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation @ D )
=> ( ( ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) )
=> ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(77,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_intersection2 @ A @ C ) @ ( set_intersection2 @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t26_xboole_1) ).
thf(450,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_intersection2 @ A @ C ) @ ( set_intersection2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(88,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ! [B: $i,C: $i] :
~ ( in @ ( ordered_pair @ B @ C ) @ A )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t56_relat_1) ).
thf(482,plain,
! [A: $i] :
( ( relation @ A )
=> ( ! [B: $i,C: $i] :
~ ( in @ ( ordered_pair @ B @ C ) @ A )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(5,axiom,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_subset_1) ).
thf(203,plain,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(52,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ B ) )
=> ( subset @ ( set_union2 @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_xboole_1) ).
thf(368,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ B ) )
=> ( subset @ ( set_union2 @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(133,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
<=> ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_tarski) ).
thf(844,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
=> ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) ) )
& ( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(31,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k3_xboole_0) ).
thf(304,plain,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(60,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(401,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(29,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ ( singleton @ B ) )
<=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t39_zfmisc_1) ).
thf(290,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ ( singleton @ B ) )
=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) )
& ( ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) )
=> ( subset @ A @ ( singleton @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ B )
= empty_set )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l32_xboole_1) ).
thf(235,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ B )
= empty_set )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( ( set_difference @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(87,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_relat_1) ).
thf(478,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( subset @ A @ B )
=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
& ( ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
=> ( subset @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(186,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( B
= ( identity_relation @ A ) )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( ( in @ C @ A )
& ( C = D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d10_relat_1) ).
thf(1212,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( ( B
= ( identity_relation @ A ) )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( ( in @ C @ A )
& ( C = D ) ) )
& ( ( ( in @ C @ A )
& ( C = D ) )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( ( in @ C @ A )
& ( C = D ) ) )
& ( ( ( in @ C @ A )
& ( C = D ) )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
=> ( B
= ( identity_relation @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(55,axiom,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(383,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(96,axiom,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t60_relat_1) ).
thf(549,plain,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(159,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc5_relat_1) ).
thf(1045,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(28,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(287,plain,
! [A: $i] :
( ( set_union2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(122,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t117_relat_1) ).
thf(763,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(22,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k6_subset_1) ).
thf(272,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(30,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_zfmisc_1) ).
thf(300,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(76,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ( ( disjoint @ B @ C )
<=> ( subset @ B @ ( subset_complement @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t43_subset_1) ).
thf(447,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ( ( ( disjoint @ B @ C )
=> ( subset @ B @ ( subset_complement @ A @ C ) ) )
& ( ( subset @ B @ ( subset_complement @ A @ C ) )
=> ( disjoint @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(173,axiom,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( subset_complement @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t50_subset_1) ).
thf(1148,plain,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( subset_complement @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(68,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t9_zfmisc_1) ).
thf(422,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(54,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ ( singleton @ B ) )
<=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l4_zfmisc_1) ).
thf(373,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ ( singleton @ B ) )
=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) )
& ( ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) )
=> ( subset @ A @ ( singleton @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(129,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom_restriction @ B @ A )
= ( relation_composition @ ( identity_relation @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t94_relat_1) ).
thf(833,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom_restriction @ B @ A )
= ( relation_composition @ ( identity_relation @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(63,axiom,
! [A: $i,B: $i] : ( subset @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(410,plain,
! [A: $i] : ( subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(135,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d13_relat_1) ).
thf(854,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(1,conjecture,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_inverse_image @ C @ B ) )
<=> ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t166_relat_1) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_inverse_image @ C @ B ) )
<=> ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(190,plain,
~ ! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_inverse_image @ C @ B ) )
=> ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) )
=> ( in @ A @ ( relation_inverse_image @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(100,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_inverse_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d14_relat_1) ).
thf(564,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_inverse_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_inverse_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ B )
= empty_set )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_xboole_1) ).
thf(210,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ B )
= empty_set )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( ( set_difference @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(15,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k5_setfam_1) ).
thf(242,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(20,axiom,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(268,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(48,axiom,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(358,plain,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(161,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_rng_restriction @ A @ B ) )
= ( set_intersection2 @ ( relation_rng @ B ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t119_relat_1) ).
thf(1056,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_rng_restriction @ A @ B ) )
= ( set_intersection2 @ ( relation_rng @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(115,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k4_relat_1) ).
thf(720,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(126,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( in @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(796,plain,
! [A: $i] :
( ( ( A = empty_set )
=> ! [B: $i] :
~ ( in @ B @ A ) )
& ( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(82,axiom,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t69_enumset1) ).
thf(464,plain,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(80,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t45_xboole_1) ).
thf(459,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(116,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k8_relat_1) ).
thf(723,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(24,axiom,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(277,plain,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(75,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_subset) ).
thf(442,plain,
! [A: $i,B: $i] :
( ( ( element @ A @ ( powerset @ B ) )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( element @ A @ ( powerset @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(168,axiom,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_relat_1) ).
thf(1120,plain,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(104,axiom,
! [A: $i,B: $i] :
( ( B
= ( singleton @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_tarski) ).
thf(607,plain,
! [A: $i,B: $i] :
( ( ( B
= ( singleton @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ( C = A ) )
& ( ( C = A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ( C = A ) )
& ( ( C = A )
=> ( in @ C @ B ) ) )
=> ( B
= ( singleton @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(165,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_composition @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_relat_1) ).
thf(1093,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_composition @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) )
& ( ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) )
& ( ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_composition @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(131,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_composition @ A @ B ) )
= ( relation_image @ B @ ( relation_rng @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t160_relat_1) ).
thf(839,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_composition @ A @ B ) )
= ( relation_image @ B @ ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(188,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ ( unordered_pair @ A @ B ) @ C )
<=> ( ( in @ A @ C )
& ( in @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t38_zfmisc_1) ).
thf(1236,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ ( unordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ C )
& ( in @ B @ C ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ C ) )
=> ( subset @ ( unordered_pair @ A @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(151,axiom,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ A @ B ) )
& ( relation @ ( relation_composition @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc9_relat_1) ).
thf(987,plain,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ A @ B ) )
& ( relation @ ( relation_composition @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(89,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( A = B )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
<=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_relat_1) ).
thf(486,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( A = B )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
& ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ C @ D ) @ A ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
& ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ C @ D ) @ A ) ) )
=> ( A = B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(84,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l23_zfmisc_1) ).
thf(469,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(177,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(1184,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(33,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(309,plain,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(111,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(676,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(183,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t144_relat_1) ).
thf(1204,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(9,axiom,
? [A: $i] : ( empty @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(222,plain,
? [A: $i] : ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(102,axiom,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t71_relat_1) ).
thf(592,plain,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(137,axiom,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ( in @ C @ B ) ) )
& ~ ( ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
& ( disjoint @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_xboole_0) ).
thf(882,plain,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ( in @ C @ B ) ) )
& ~ ( ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
& ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(18,axiom,
? [A: $i] :
~ ( empty @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(254,plain,
? [A: $i] :
~ ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(11,axiom,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_boole) ).
thf(227,plain,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(74286,plain,
$false,
inference(cvc4,[status(thm)],[1059,1036,352,365,1190,909,333,683,994,1142,730,217,715,472,1233,404,417,320,436,257,902,559,284,1115,475,905,1151,937,1020,348,634,453,396,1242,312,206,428,1201,307,439,678,546,201,534,1047,1206,233,1091,1042,407,830,324,224,916,467,456,842,317,650,725,980,1118,339,1182,537,637,590,554,753,985,1192,911,647,997,244,360,765,398,794,888,412,425,1146,279,733,562,281,804,350,540,1209,274,419,736,990,230,362,1144,1122,271,599,880,434,387,631,251,895,330,462,198,1177,959,394,1000,851,431,939,355,836,414,327,791,370,502,1085,1188,1186,544,907,336,314,627,450,482,203,368,844,304,401,290,235,478,1212,383,549,1045,287,763,272,300,447,1148,422,373,833,410,854,190,564,210,242,268,358,1056,720,796,464,459,723,277,442,1120,607,1093,839,1236,987,486,469,1184,309,676,1204,222,592,882,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU208+2 : TPTP v8.1.2. Released v3.3.0.
% 0.12/0.15 % Command : run_Leo-III %s %d
% 0.15/0.35 % Computer : n023.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu May 18 13:06:00 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.98/0.87 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.39/1.08 % [INFO] Parsing done (210ms).
% 1.72/1.09 % [INFO] Running in sequential loop mode.
% 2.07/1.29 % [INFO] eprover registered as external prover.
% 2.07/1.29 % [INFO] cvc4 registered as external prover.
% 2.07/1.29 % [INFO] Scanning for conjecture ...
% 2.48/1.37 % [INFO] Found a conjecture and 204 axioms. Running axiom selection ...
% 2.48/1.45 % [INFO] Axiom selection finished. Selected 187 axioms (removed 17 axioms).
% 2.97/1.57 % [INFO] Problem is first-order (TPTP FOF).
% 2.97/1.59 % [INFO] Type checking passed.
% 2.97/1.59 % [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 ...
% 111.36/21.41 % External prover 'cvc4' found a proof!
% 111.36/21.42 % [INFO] Killing All external provers ...
% 111.36/21.42 % Time passed: 20912ms (effective reasoning time: 20324ms)
% 111.36/21.42 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 111.36/21.42 % Axioms used in derivation (187): involutiveness_k7_setfam_1, t43_subset_1, fc1_zfmisc_1, d1_tarski, t30_relat_1, t19_xboole_1, t99_relat_1, t17_xboole_1, t90_relat_1, d5_relat_1, idempotence_k3_xboole_0, t1_boole, t3_xboole_1, fc1_xboole_0, dt_k2_subset_1, d5_tarski, fc4_relat_1, d12_relat_1, t44_relat_1, t1_subset, t144_relat_1, fc4_subset_1, t117_relat_1, t2_boole, t47_relat_1, fc9_relat_1, l3_subset_1, t99_zfmisc_1, t47_setfam_1, t56_relat_1, t37_xboole_1, cc1_relat_1, redefinition_k6_setfam_1, commutativity_k3_xboole_0, d2_tarski, d4_subset_1, involutiveness_k3_subset_1, fc2_xboole_0, d1_xboole_0, t64_relat_1, l50_zfmisc_1, t118_zfmisc_1, t6_boole, rc2_subset_1, dt_k6_subset_1, d1_relat_1, d13_relat_1, t46_zfmisc_1, t1_xboole_1, t116_relat_1, commutativity_k2_xboole_0, fc3_subset_1, t5_subset, d4_relat_1, t4_xboole_0, l25_zfmisc_1, t1_zfmisc_1, fc2_subset_1, d8_xboole_0, t2_tarski, t33_xboole_1, l32_xboole_1, t8_zfmisc_1, t4_boole, dt_k4_relat_1, t4_subset, fc3_xboole_0, d2_xboole_0, d4_tarski, t9_zfmisc_1, redefinition_k5_setfam_1, involutiveness_k4_relat_1, t7_boole, t118_relat_1, d10_relat_1, t143_relat_1, d2_subset_1, t48_xboole_1, fc1_subset_1, t33_zfmisc_1, t160_relat_1, d11_relat_1, t74_relat_1, fc10_relat_1, rc1_subset_1, t83_xboole_1, t45_relat_1, l28_zfmisc_1, t140_relat_1, d14_relat_1, d7_xboole_0, fc5_relat_1, t46_setfam_1, l4_zfmisc_1, t3_subset, t115_relat_1, d2_zfmisc_1, existence_m1_subset_1, d7_relat_1, antisymmetry_r2_xboole_0, dt_k6_setfam_1, l3_zfmisc_1, commutativity_k2_tarski, t106_zfmisc_1, d8_relat_1, l71_subset_1, d5_subset_1, dt_k7_relat_1, l2_zfmisc_1, t94_relat_1, t48_setfam_1, t92_zfmisc_1, t146_relat_1, d1_zfmisc_1, t69_enumset1, rc2_relat_1, t38_zfmisc_1, l55_zfmisc_1, dt_k8_relat_1, d8_setfam_1, t26_xboole_1, t63_xboole_1, t8_xboole_1, t37_relat_1, t28_xboole_1, d3_tarski, t39_zfmisc_1, t119_zfmisc_1, rc1_relat_1, t60_relat_1, dt_k5_relat_1, reflexivity_r1_tarski, t6_zfmisc_1, dt_k6_relat_1, t2_xboole_1, fc6_relat_1, t46_relat_1, d4_xboole_0, dt_k5_setfam_1, t36_xboole_1, t20_relat_1, t54_subset_1, dt_k7_setfam_1, t136_zfmisc_1, fc1_relat_1, d6_relat_1, l23_zfmisc_1, t8_boole, rc1_xboole_0, t12_xboole_1, d3_relat_1, fc7_relat_1, t86_relat_1, fc8_relat_1, t10_zfmisc_1, t2_subset, t37_zfmisc_1, fc2_relat_1, t65_relat_1, t21_relat_1, d1_setfam_1, symmetry_r1_xboole_0, idempotence_k2_xboole_0, t9_tarski, t145_relat_1, t50_subset_1, d10_xboole_0, t7_xboole_1, t60_xboole_1, t3_xboole_0, t3_boole, redefinition_k6_subset_1, t25_relat_1, rc2_xboole_0, t45_xboole_1, t40_xboole_1, irreflexivity_r2_xboole_0, t65_zfmisc_1, d3_xboole_0, d2_relat_1, t39_xboole_1, t71_relat_1, t88_relat_1, dt_k3_subset_1, l1_zfmisc_1, t119_relat_1, antisymmetry_r2_hidden
% 111.36/21.42 % No. of inferences in proof: 378
% 111.36/21.42 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 20912 ms resp. 20324 ms w/o parsing
% 111.63/21.52 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 111.63/21.52 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------