%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SEU180+2 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:37:55 AM UTC 2026
% Result : Theorem 216.02s 56.57s
% Output : Refutation 213.08s
% Verified :
% SZS Type : Refutation
% Derivation depth : 26
% Number of leaves : 155
% Syntax : Number of formulae : 695 ( 260 unt; 0 typ; 20 def)
% Number of atoms : 2053 ( 724 equ; 0 cnn)
% Maximal formula atoms : 40 ( 2 avg)
% Number of connectives : 5747 ( 769 ~; 397 |; 305 &;3868 @)
% ( 47 <=>; 361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 9 con; 0-3 aty)
% Number of variables : 1395 ( 26 ^;1278 !; 71 ?;1395 :)
% ( 0 !>; 0 ?*; 0 @-; 20 @+)
% Comments :
%------------------------------------------------------------------------------
thf(relation_decl,type,
relation: $i > $o ).
thf(in_decl,type,
in: $i > $i > $o ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(relation_field_decl,type,
relation_field: $i > $i ).
thf(proper_subset_decl,type,
proper_subset: $i > $i > $o ).
thf(cast_to_subset_decl,type,
cast_to_subset: $i > $i ).
thf(empty_decl,type,
empty: $i > $o ).
thf(set_difference_decl,type,
set_difference: $i > $i > $i ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(cartesian_product2_decl,type,
cartesian_product2: $i > $i > $i ).
thf(set_union2_decl,type,
set_union2: $i > $i > $i ).
thf(element_decl,type,
element: $i > $i > $o ).
thf(powerset_decl,type,
powerset: $i > $i ).
thf(union_of_subsets_decl,type,
union_of_subsets: $i > $i > $i ).
thf(disjoint_decl,type,
disjoint: $i > $i > $o ).
thf(set_intersection2_decl,type,
set_intersection2: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(subset_difference_decl,type,
subset_difference: $i > $i > $i > $i ).
thf(subset_complement_decl,type,
subset_complement: $i > $i > $i ).
thf(union_decl,type,
union: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(meet_of_subsets_decl,type,
meet_of_subsets: $i > $i > $i ).
thf(complements_of_subsets_decl,type,
complements_of_subsets: $i > $i > $i ).
thf(set_meet_decl,type,
set_meet: $i > $i ).
thf(are_equipotent_decl,type,
are_equipotent: $i > $i > $o ).
thf(relation_dom_decl,type,
relation_dom: $i > $i ).
thf(relation_rng_decl,type,
relation_rng: $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i ).
thf(sk3_decl,type,
sk3: $i ).
thf(sk4_decl,type,
sk4: $i ).
thf(sk5_decl,type,
sk5: $i ).
thf(sk6_decl,type,
sk6: $i > $i ).
thf(sk8_decl,type,
sk8: $i > $i ).
thf(sk9_decl,type,
sk9: $i ).
thf(sk11_decl,type,
sk11: $i > $i ).
thf(sk12_decl,type,
sk12: $i > $i > $i ).
thf(sk13_decl,type,
sk13: $i > $i > $i ).
thf(sk22_decl,type,
sk22: $i > $i ).
thf(sk23_decl,type,
sk23: $i > $i > $i ).
thf(sk24_decl,type,
sk24: $i > $i > $i > $o ).
thf(sk25_decl,type,
sk25: $i > $i > $i > $i ).
thf(sk26_decl,type,
sk26: $i > $i > $i > $i ).
thf(sk27_decl,type,
sk27: $i > $i ).
thf(sk55_decl,type,
sk55: $i > $i > $i > $i ).
thf(sk56_decl,type,
sk56: $i > $i > $o ).
thf(sk57_decl,type,
sk57: $i > $i > $i ).
thf(sk58_decl,type,
sk58: $i > $i > $i ).
thf(sk59_decl,type,
sk59: $i > $i > $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ! [B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_field @ ( B @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: $i] :
~ ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ ( sk1 @ ordered_pair ) @ in ) )
=> ( ( B @ relation_field @ ( A @ in ) )
& ( B @ relation_field @ ( sk1 @ in ) ) ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( @+[A: $i] :
~ ( ( A @ relation )
=> ( ( A @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
=> ( ( A @ relation_field @ ( sk2 @ in ) )
& ( A @ relation_field @ ( sk1 @ in ) ) ) ) ) ) ) ).
thf(sk4_def,definition,
( sk4
= ( @+[A: $i] : ( A @ empty ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( @+[A: $i] :
~ ( A @ empty ) ) ) ).
thf(sk6_def,definition,
( sk6
= ( ^ [A: $i] :
@+[B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ) ) ).
thf(sk8_def,definition,
( sk8
= ( ^ [A: $i] :
@+[B: $i] : ( A @ ( B @ element ) ) ) ) ).
thf(sk9_def,definition,
( sk9
= ( @+[A: $i] :
( ( A @ relation )
& ( A @ empty ) ) ) ) ).
thf(sk11_def,definition,
( sk11
= ( ^ [A: $i] :
@+[B: $i] :
( ~ ? [C: $i] :
( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( B @ ( C @ powerset @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ) ) ) ).
thf(sk12_def,definition,
( sk12
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ in ) )
=> ( C = B ) ) ) ) ).
thf(sk13_def,definition,
( sk13
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( C = B )
=> ( A @ ( C @ in ) ) ) ) ) ).
thf(sk22_def,definition,
( sk22
= ( ^ [A: $i] :
@+[B: $i] :
( ~ ? [C: $i] :
( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ~ ? [C: $i] :
( ~ ? [D: $i] :
( ! [E: $i] :
( ( C @ ( E @ subset ) )
=> ( D @ ( E @ in ) ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ) ) ) ).
thf(sk23_def,definition,
( sk23
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ! [D: $i] :
( ( A @ ( D @ subset ) )
=> ( C @ ( D @ in ) ) )
& ( B @ sk22 @ ( C @ in ) ) ) ) ) ).
thf(sk25_def,definition,
( sk25
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
~ ( ( A @ ( D @ in ) )
=> ( ( D = B )
| ( D = C ) ) ) ) ) ).
thf(sk26_def,definition,
( sk26
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
~ ( ( ( D = B )
| ( D = C ) )
=> ( A @ ( D @ in ) ) ) ) ) ).
thf(sk27_def,definition,
( sk27
= ( ^ [A: $i] :
@+[B: $i] : ( A @ ( B @ in ) ) ) ) ).
thf(sk55_def,definition,
( sk55
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
~ ( ( C @ ( D @ in ) )
=> ( D @ ( A @ in ) ) ) ) ) ).
thf(sk57_def,definition,
( sk57
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ in ) )
=> ! [D: $i] :
( ( B @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) ) ) ).
thf(sk58_def,definition,
( sk58
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( B @ ( C @ in ) )
=> ( C @ ( B @ ( A @ sk57 ) @ in ) ) ) ) ) ).
thf(sk59_def,definition,
( sk59
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ! [D: $i] :
( ( B @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( A @ ( C @ in ) ) ) ) ) ).
thf(27,axiom,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(233,plain,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(234,plain,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
inference(cnf,[status(esa)],[233]) ).
thf(235,plain,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
inference(lifteq,[status(thm)],[234]) ).
thf(13,axiom,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(179,plain,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(180,plain,
! [B: $i,A: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
inference(cnf,[status(esa)],[179]) ).
thf(12,axiom,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(176,plain,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(177,plain,
! [A: $i] :
( ( A = empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(cnf,[status(esa)],[176]) ).
thf(178,plain,
! [A: $i] :
( ~ ( empty_set @ ( A @ subset ) )
| ( A = empty_set ) ),
inference(lifteq,[status(thm)],[177]) ).
thf(99,axiom,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( C @ ( E @ subset ) )
=> ( D @ ( E @ in ) ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t9_tarski) ).
thf(548,plain,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( C @ ( E @ subset ) )
=> ( D @ ( E @ in ) ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(549,plain,
! [A: $i] :
? [B: $i] :
( ~ ? [C: $i] :
( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ~ ? [C: $i] :
( ~ ? [D: $i] :
( ! [E: $i] :
( ( C @ ( E @ subset ) )
=> ( D @ ( E @ in ) ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
inference(miniscope,[status(thm)],[548]) ).
thf(550,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ sk22 @ ( F @ in ) )
| ( A @ sk22 @ ( F @ are_equipotent ) )
| ~ ( A @ sk22 @ ( F @ subset ) ) )
& ( ( A @ ( D @ sk23 ) @ ( E @ in ) )
| ~ ( D @ ( E @ subset ) )
| ~ ( A @ sk22 @ ( D @ in ) ) )
& ( ( A @ sk22 @ ( A @ ( D @ sk23 ) @ in ) )
| ~ ( A @ sk22 @ ( D @ in ) ) )
& ( ( A @ sk22 @ ( C @ in ) )
| ~ ( B @ ( C @ subset ) )
| ~ ( A @ sk22 @ ( B @ in ) ) )
& ( A @ sk22 @ ( A @ in ) ) ),
inference(cnf,[status(esa)],[549]) ).
thf(553,plain,
! [A: $i] : ( A @ sk22 @ ( A @ in ) ),
inference(cnfConj,[status(thm)],[550]) ).
thf(101,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(586,plain,
! [A: $i] :
( ( ! [B: $i] :
~ ( A @ ( B @ in ) )
=> ( A = empty_set ) )
& ( ( A = empty_set )
=> ! [B: $i] :
~ ( A @ ( B @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(587,plain,
( ! [A: $i] :
( ~ ? [B: $i] : ( A @ ( B @ in ) )
=> ( A = empty_set ) )
& ! [A: $i] :
( ( A = empty_set )
=> ~ ? [B: $i] : ( A @ ( B @ in ) ) ) ),
inference(miniscope,[status(thm)],[586]) ).
thf(588,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C = empty_set )
| ( C @ ( C @ sk27 @ in ) ) )
& ( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ) ),
inference(cnf,[status(esa)],[587]) ).
thf(589,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ),
inference(cnfConj,[status(thm)],[588]) ).
thf(591,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ),
inference(lifteq,[status(thm)],[589]) ).
thf(592,plain,
! [A: $i] :
~ ( empty_set @ ( A @ in ) ),
inference(simp,[status(thm)],[591]) ).
thf(1931,plain,
! [B: $i,A: $i] :
( ( ( A @ sk22 @ ( A @ in ) )
!= ( empty_set @ ( B @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[553,592]) ).
thf(1932,plain,
! [B: $i,A: $i] :
( ( A @ sk22 @ ( A @ in ) )
!= ( empty_set @ ( B @ in ) ) ),
inference(simp,[status(thm)],[1931]) ).
thf(1939,plain,
! [B: $i,A: $i] :
( ( ( A @ sk22 )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[1932]) ).
thf(1950,plain,
! [A: $i] :
( ( A @ sk22 )
!= empty_set ),
inference(simp,[status(thm)],[1939]) ).
thf(2071,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ sk22 ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,1950]) ).
thf(2072,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ sk22 ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2071]) ).
thf(2073,plain,
! [A: $i] :
~ ( empty_set @ ( A @ sk22 @ subset ) ),
inference(pattern_uni,[status(thm)],[2072:[bind(A,$thf( C @ sk22 )),bind(B,$thf( C ))]]) ).
thf(2387,plain,
! [A: $i] :
~ ( empty_set @ ( A @ sk22 @ subset ) ),
inference(simp,[status(thm)],[2073]) ).
thf(2610,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ sk22 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2387]) ).
thf(2611,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ sk22 @ subset ) ) ),
inference(simp,[status(thm)],[2610]) ).
thf(2627,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A
!= ( C @ sk22 ) ) ),
inference(simp,[status(thm)],[2611]) ).
thf(2630,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk22 @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2627]) ).
thf(21,axiom,
empty_set @ empty,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(220,plain,
empty_set @ empty,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(136,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(926,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(927,plain,
~ ? [A: $i,B: $i] : ( B @ ( A @ ordered_pair ) @ empty ),
inference(miniscope,[status(thm)],[926]) ).
thf(928,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
inference(cnf,[status(esa)],[927]) ).
thf(2019,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,928]) ).
thf(2020,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[2019]) ).
thf(2031,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[2020]) ).
thf(57,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( D @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t119_zfmisc_1) ).
thf(347,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( D @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(81,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(419,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(420,plain,
! [B: $i,A: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
inference(cnf,[status(esa)],[419]) ).
thf(3472,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ~ $true
| ( C = empty_set ) ),
inference(paramod_ordered,[status(thm)],[420,178]) ).
thf(3473,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ( C = empty_set ) ),
inference(simp,[status(thm)],[3472]) ).
thf(3474,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_intersection2 ) )
= empty_set ),
inference(pattern_uni,[status(thm)],[3473:[bind(A,$thf( empty_set )),bind(B,$thf( E )),bind(C,$thf( E @ ( empty_set @ set_intersection2 ) ))]]) ).
thf(3490,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_intersection2 ) )
= empty_set ),
inference(simp,[status(thm)],[3474]) ).
thf(16,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ( ( B @ ( A @ set_intersection2 ) )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d7_xboole_0) ).
thf(191,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( A @ set_intersection2 ) )
= empty_set )
=> ( B @ ( A @ disjoint ) ) )
& ( ( B @ ( A @ disjoint ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(192,plain,
( ! [A: $i,B: $i] :
( ( ( B @ ( A @ set_intersection2 ) )
= empty_set )
=> ( B @ ( A @ disjoint ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= empty_set ) ) ),
inference(miniscope,[status(thm)],[191]) ).
thf(193,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ disjoint ) )
| ( ( D @ ( C @ set_intersection2 ) )
!= empty_set ) )
& ( ( ( B @ ( A @ set_intersection2 ) )
= empty_set )
| ~ ( B @ ( A @ disjoint ) ) ) ),
inference(cnf,[status(esa)],[192]) ).
thf(195,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ disjoint ) )
| ( ( B @ ( A @ set_intersection2 ) )
!= empty_set ) ),
inference(cnfConj,[status(thm)],[193]) ).
thf(197,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ disjoint ) )
| ( ( B @ ( A @ set_intersection2 ) )
!= empty_set ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(198,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ disjoint ) )
| ( ( B @ ( A @ set_intersection2 ) )
!= empty_set ) ),
inference(simp,[status(thm)],[197]) ).
thf(3579,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ set_intersection2 ) )
!= ( C @ ( B @ set_intersection2 ) ) )
| ( C @ ( B @ disjoint ) )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[3490,198]) ).
thf(3580,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ set_intersection2 ) )
!= ( C @ ( B @ set_intersection2 ) ) )
| ( C @ ( B @ disjoint ) ) ),
inference(simp,[status(thm)],[3579]) ).
thf(3581,plain,
! [A: $i] : ( A @ ( empty_set @ disjoint ) ),
inference(pattern_uni,[status(thm)],[3580:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(96,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_rng @ ( B @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t20_relat_1) ).
thf(523,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_rng @ ( B @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(69,axiom,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(379,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(131,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ cartesian_product2 ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i,F: $i] :
( ( D
= ( F @ ( E @ ordered_pair ) ) )
& ( B @ ( F @ in ) )
& ( A @ ( E @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_zfmisc_1) ).
thf(878,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ? [E: $i,F: $i] :
( ( D
= ( F @ ( E @ ordered_pair ) ) )
& ( B @ ( F @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i,F: $i] :
( ( D
= ( F @ ( E @ ordered_pair ) ) )
& ( B @ ( F @ in ) )
& ( A @ ( E @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ cartesian_product2 ) ) ) )
& ( ( C
= ( B @ ( A @ cartesian_product2 ) ) )
=> ! [D: $i] :
( ( ? [E: $i,F: $i] :
( ( D
= ( F @ ( E @ ordered_pair ) ) )
& ( B @ ( F @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i,F: $i] :
( ( D
= ( F @ ( E @ ordered_pair ) ) )
& ( B @ ( F @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(18,axiom,
? [A: $i] :
~ ( A @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(202,plain,
? [A: $i] :
~ ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(87,axiom,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(434,plain,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(435,plain,
( ( sk9 @ relation )
& ( sk9 @ empty ) ),
inference(cnf,[status(esa)],[434]) ).
thf(436,plain,
sk9 @ empty,
inference(cnfConj,[status(thm)],[435]) ).
thf(58,axiom,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(349,plain,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(350,plain,
( ( empty_set @ singleton )
= ( empty_set @ powerset ) ),
inference(lifteq,[status(thm)],[349]) ).
thf(20,axiom,
! [A: $i] :
~ ( A @ singleton @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(217,plain,
! [A: $i] :
~ ( A @ singleton @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(218,plain,
~ ? [A: $i] : ( A @ singleton @ empty ),
inference(miniscope,[status(thm)],[217]) ).
thf(219,plain,
! [A: $i] :
~ ( A @ singleton @ empty ),
inference(cnf,[status(esa)],[218]) ).
thf(962,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= ( A @ singleton ) )
| ~ ( empty_set @ powerset @ empty ) ),
inference(paramod_ordered,[status(thm)],[350,219]) ).
thf(963,plain,
~ ( empty_set @ powerset @ empty ),
inference(pattern_uni,[status(thm)],[962:[bind(A,$thf( empty_set ))]]) ).
thf(971,plain,
( ( ( empty_set @ powerset @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,963]) ).
thf(972,plain,
( ( empty_set @ powerset @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[971]) ).
thf(977,plain,
( ( empty_set @ powerset )
!= sk9 ),
inference(simp,[status(thm)],[972]) ).
thf(10,axiom,
! [A: $i,B: $i] :
~ ( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(170,plain,
! [A: $i,B: $i] :
~ ( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(274,plain,
! [A: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(275,plain,
! [A: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
inference(cnf,[status(esa)],[274]) ).
thf(276,plain,
! [A: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
inference(lifteq,[status(thm)],[275]) ).
thf(71,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B != empty_set )
=> ( ( B @ ( A @ union_of_subsets ) @ ( A @ cast_to_subset @ ( A @ subset_difference ) ) )
= ( B @ ( A @ complements_of_subsets ) @ ( A @ meet_of_subsets ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t47_setfam_1) ).
thf(385,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B != empty_set )
=> ( ( B @ ( A @ union_of_subsets ) @ ( A @ cast_to_subset @ ( A @ subset_difference ) ) )
= ( B @ ( A @ complements_of_subsets ) @ ( A @ meet_of_subsets ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(3,axiom,
! [A: $i,B: $i] :
~ ( A @ ( A @ proper_subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(142,plain,
! [A: $i] :
~ ( A @ ( A @ proper_subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(115,axiom,
! [A: $i,B: $i] :
( ( ( A @ empty )
=> ( ( A @ ( B @ element ) )
<=> ( B @ empty ) ) )
& ( ~ ( A @ empty )
=> ( ( A @ ( B @ element ) )
<=> ( A @ ( B @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_subset_1) ).
thf(715,plain,
! [A: $i,B: $i] :
( ( ( A @ empty )
=> ( ( ( B @ empty )
=> ( A @ ( B @ element ) ) )
& ( ( A @ ( B @ element ) )
=> ( B @ empty ) ) ) )
& ( ~ ( A @ empty )
=> ( ( ( A @ ( B @ in ) )
=> ( A @ ( B @ element ) ) )
& ( ( A @ ( B @ element ) )
=> ( A @ ( B @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(1,conjecture,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_field @ ( B @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t30_relat_1) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_field @ ( B @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(137,plain,
~ ! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_field @ ( B @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(138,plain,
( ( ~ ( sk3 @ relation_field @ ( sk2 @ in ) )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) )
& ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
& ( sk3 @ relation ) ),
inference(cnf,[status(esa)],[137]) ).
thf(141,plain,
( ~ ( sk3 @ relation_field @ ( sk2 @ in ) )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(cnfConj,[status(thm)],[138]) ).
thf(942,plain,
( ~ $true
| ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(eqfactor_ordered,[status(thm)],[141]) ).
thf(945,plain,
( ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[942]) ).
thf(6,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(150,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(151,plain,
~ ? [A: $i] :
( ? [B: $i] :
( ( B @ empty )
& ( A != B ) )
& ( A @ empty ) ),
inference(miniscope,[status(thm)],[150]) ).
thf(152,plain,
! [B: $i,A: $i] :
( ~ ( B @ empty )
| ( A = B )
| ~ ( A @ empty ) ),
inference(cnf,[status(esa)],[151]) ).
thf(153,plain,
! [B: $i,A: $i] :
( ~ ( B @ empty )
| ~ ( A @ empty )
| ( A = B ) ),
inference(lifteq,[status(thm)],[152]) ).
thf(112,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B
= ( A @ relation_dom ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] : ( A @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_relat_1) ).
thf(670,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i] :
( ( ? [D: $i] : ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] : ( A @ ( D @ ( C @ ordered_pair ) @ in ) ) ) )
=> ( B
= ( A @ relation_dom ) ) )
& ( ( B
= ( A @ relation_dom ) )
=> ! [C: $i] :
( ( ? [D: $i] : ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] : ( A @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(60,axiom,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(354,plain,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(355,plain,
! [A: $i] :
( ( A @ sk6 @ empty )
& ( A @ powerset @ ( A @ sk6 @ element ) ) ),
inference(cnf,[status(esa)],[354]) ).
thf(357,plain,
! [A: $i] : ( A @ sk6 @ empty ),
inference(cnfConj,[status(thm)],[355]) ).
thf(996,plain,
! [B: $i,A: $i] :
( ( ( A @ sk6 @ empty )
!= ( B @ singleton @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[357,219]) ).
thf(997,plain,
! [B: $i,A: $i] :
( ( A @ sk6 @ empty )
!= ( B @ singleton @ empty ) ),
inference(simp,[status(thm)],[996]) ).
thf(1000,plain,
! [B: $i,A: $i] :
( ( A @ sk6 )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[997]) ).
thf(4802,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ singleton )
!= ( B @ singleton ) )
| ( ( A @ sk6 )
!= ( empty_set @ powerset ) ) ),
inference(paramod_ordered,[status(thm)],[350,1000]) ).
thf(4803,plain,
! [A: $i] :
( ( A @ sk6 )
!= ( empty_set @ powerset ) ),
inference(pattern_uni,[status(thm)],[4802:[bind(A,$thf( A )),bind(B,$thf( empty_set ))]]) ).
thf(45,axiom,
! [A: $i,B: $i] :
( ( ~ ( B @ empty )
& ~ ( A @ empty ) )
=> ~ ( B @ ( A @ cartesian_product2 ) @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_subset_1) ).
thf(301,plain,
! [A: $i,B: $i] :
( ( ~ ( B @ empty )
& ~ ( A @ empty ) )
=> ~ ( B @ ( A @ cartesian_product2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( A @ ( B @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(257,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( A @ ( B @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(19,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(205,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(206,plain,
( ! [A: $i,B: $i] :
( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( A = B ) )
& ! [A: $i,B: $i] :
( ( A = B )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(miniscope,[status(thm)],[205]) ).
thf(207,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C = D )
| ~ ( C @ ( D @ subset ) )
| ~ ( D @ ( C @ subset ) ) )
& ( ( A @ ( B @ subset ) )
| ( A != B ) )
& ( ( B @ ( A @ subset ) )
| ( A != B ) ) ),
inference(cnf,[status(esa)],[206]) ).
thf(210,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[207]) ).
thf(215,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[210]) ).
thf(216,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) )
| ( A = B ) ),
inference(simp,[status(thm)],[215]) ).
thf(140,plain,
sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ),
inference(cnfConj,[status(thm)],[138]) ).
thf(1005,plain,
! [A: $i] :
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( empty_set @ ( A @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,592]) ).
thf(1006,plain,
! [A: $i] :
( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( empty_set @ ( A @ in ) ) ),
inference(simp,[status(thm)],[1005]) ).
thf(1007,plain,
! [A: $i] :
( ( sk3 != empty_set )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[1006]) ).
thf(1008,plain,
sk3 != empty_set,
inference(simp,[status(thm)],[1007]) ).
thf(1321,plain,
! [B: $i,A: $i] :
( ( A != sk3 )
| ( B != empty_set )
| ~ ( B @ empty )
| ~ ( A @ empty ) ),
inference(paramod_ordered,[status(thm)],[153,1008]) ).
thf(1322,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ empty )
| ~ ( sk3 @ empty ) ),
inference(pattern_uni,[status(thm)],[1321:[bind(A,$thf( sk3 ))]]) ).
thf(1400,plain,
( ~ ( empty_set @ empty )
| ~ ( sk3 @ empty ) ),
inference(simp,[status(thm)],[1322]) ).
thf(1513,plain,
( ~ $true
| ~ ( sk3 @ empty ) ),
inference(rewrite,[status(thm)],[1400,220]) ).
thf(1514,plain,
~ ( sk3 @ empty ),
inference(simp,[status(thm)],[1513]) ).
thf(1515,plain,
( ( ( sk9 @ empty )
!= ( sk3 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,1514]) ).
thf(1516,plain,
( ( sk9 @ empty )
!= ( sk3 @ empty ) ),
inference(simp,[status(thm)],[1515]) ).
thf(1531,plain,
sk9 != sk3,
inference(simp,[status(thm)],[1516]) ).
thf(4251,plain,
! [B: $i,A: $i] :
( ( A != sk9 )
| ( B != sk3 )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[216,1531]) ).
thf(4252,plain,
! [A: $i] :
( ( A != sk3 )
| ~ ( sk9 @ ( A @ subset ) )
| ~ ( A @ ( sk9 @ subset ) ) ),
inference(pattern_uni,[status(thm)],[4251:[bind(A,$thf( sk9 ))]]) ).
thf(4524,plain,
( ~ ( sk9 @ ( sk3 @ subset ) )
| ~ ( sk3 @ ( sk9 @ subset ) ) ),
inference(simp,[status(thm)],[4252]) ).
thf(66,axiom,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(371,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(372,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(cnf,[status(esa)],[371]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ singleton ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_tarski) ).
thf(453,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( ( C = A )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( C = A ) ) )
=> ( B
= ( A @ singleton ) ) )
& ( ( B
= ( A @ singleton ) )
=> ! [C: $i] :
( ( ( C = A )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( C = A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(454,plain,
( ! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( C = A )
=> ( B @ ( C @ in ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( C = A ) ) )
=> ( B
= ( A @ singleton ) ) )
& ! [A: $i,B: $i] :
( ( B
= ( A @ singleton ) )
=> ( ! [C: $i] :
( ( C = A )
=> ( B @ ( C @ in ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( C = A ) ) ) ) ),
inference(miniscope,[status(thm)],[453]) ).
thf(455,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F
= ( E @ singleton ) )
| ~ ( F @ ( E @ ( F @ sk13 ) @ in ) )
| ( ( E @ ( F @ sk12 ) )
!= E ) )
& ( ( F
= ( E @ singleton ) )
| ( ( E @ ( F @ sk13 ) )
= E )
| ( ( E @ ( F @ sk12 ) )
!= E ) )
& ( ( F
= ( E @ singleton ) )
| ~ ( F @ ( E @ ( F @ sk13 ) @ in ) )
| ( F @ ( E @ ( F @ sk12 ) @ in ) ) )
& ( ( F
= ( E @ singleton ) )
| ( ( E @ ( F @ sk13 ) )
= E )
| ( F @ ( E @ ( F @ sk12 ) @ in ) ) )
& ( ( B @ ( D @ in ) )
| ( D != A )
| ( B
!= ( A @ singleton ) ) )
& ( ( C = A )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ singleton ) ) ) ),
inference(cnf,[status(esa)],[454]) ).
thf(459,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ in ) )
| ( C != A )
| ( B
!= ( A @ singleton ) ) ),
inference(cnfConj,[status(thm)],[455]) ).
thf(468,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ in ) )
| ( C != A )
| ( B
!= ( A @ singleton ) ) ),
inference(lifteq,[status(thm)],[459]) ).
thf(469,plain,
! [A: $i] : ( A @ singleton @ ( A @ in ) ),
inference(simp,[status(thm)],[468]) ).
thf(946,plain,
( ( ( sk3 @ relation_field )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[942]) ).
thf(947,plain,
( ( sk2 != sk1 )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[946]) ).
thf(1839,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[469,947]) ).
thf(1840,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1839]) ).
thf(1847,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk3 @ relation_field ) )
| ( A != sk1 )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1840]) ).
thf(1860,plain,
( ( ( sk1 @ singleton )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1847]) ).
thf(194,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_intersection2 ) )
= empty_set )
| ~ ( B @ ( A @ disjoint ) ) ),
inference(cnfConj,[status(thm)],[193]) ).
thf(196,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ disjoint ) )
| ( ( B @ ( A @ set_intersection2 ) )
= empty_set ) ),
inference(lifteq,[status(thm)],[194]) ).
thf(203,plain,
~ ! [A: $i] : ( A @ empty ),
inference(miniscope,[status(thm)],[202]) ).
thf(204,plain,
~ ( sk5 @ empty ),
inference(cnf,[status(esa)],[203]) ).
thf(932,plain,
( ( ( sk5 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,204]) ).
thf(933,plain,
( ( sk5 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[932]) ).
thf(934,plain,
sk5 != empty_set,
inference(simp,[status(thm)],[933]) ).
thf(100,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ unordered_pair ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( D = B )
| ( D = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_tarski) ).
thf(559,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( D = B )
| ( D = A ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( D = B )
| ( D = A ) ) ) )
=> ( C
= ( B @ ( A @ unordered_pair ) ) ) )
& ( ( C
= ( B @ ( A @ unordered_pair ) ) )
=> ! [D: $i] :
( ( ( ( D = B )
| ( D = A ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( D = B )
| ( D = A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(82,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_enumset1) ).
thf(421,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(9,axiom,
? [A: $i] : ( A @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(168,plain,
? [A: $i] : ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(169,plain,
sk4 @ empty,
inference(cnf,[status(esa)],[168]) ).
thf(2306,plain,
! [A: $i] :
( ( A != sk3 )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,1008]) ).
thf(2307,plain,
! [A: $i] :
( ( A != sk3 )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2306]) ).
thf(2308,plain,
~ ( empty_set @ ( sk3 @ subset ) ),
inference(pattern_uni,[status(thm)],[2307:[bind(A,$thf( sk3 ))]]) ).
thf(2564,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( sk3 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2308]) ).
thf(2565,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[2564]) ).
thf(2581,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A != sk3 ) ),
inference(simp,[status(thm)],[2565]) ).
thf(2589,plain,
! [A: $i] :
( ( A @ ( sk3 @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2581]) ).
thf(2771,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( sk3 @ set_union2 ) ) )
| ( B != empty_set )
| ~ ( B @ empty )
| ~ ( A @ empty ) ),
inference(paramod_ordered,[status(thm)],[153,2589]) ).
thf(2772,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ~ ( A @ empty )
| ~ ( B @ ( sk3 @ set_union2 ) @ empty ) ),
inference(pattern_uni,[status(thm)],[2771:[bind(A,$thf( E @ ( sk3 @ set_union2 ) )),bind(B,$thf( B )),bind(C,$thf( E ))]]) ).
thf(2787,plain,
! [A: $i] :
( ~ ( empty_set @ empty )
| ~ ( A @ ( sk3 @ set_union2 ) @ empty ) ),
inference(simp,[status(thm)],[2772]) ).
thf(3187,plain,
! [A: $i] :
( ~ $true
| ~ ( A @ ( sk3 @ set_union2 ) @ empty ) ),
inference(rewrite,[status(thm)],[2787,220]) ).
thf(3188,plain,
! [A: $i] :
~ ( A @ ( sk3 @ set_union2 ) @ empty ),
inference(simp,[status(thm)],[3187]) ).
thf(3207,plain,
! [A: $i] :
( ( ( A @ ( sk3 @ set_union2 ) @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,3188]) ).
thf(3208,plain,
! [A: $i] :
( ( A @ ( sk3 @ set_union2 ) @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[3207]) ).
thf(3216,plain,
! [A: $i] :
( ( A @ ( sk3 @ set_union2 ) )
!= sk4 ),
inference(simp,[status(thm)],[3208]) ).
thf(80,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B
= ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t45_xboole_1) ).
thf(416,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B
= ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(25,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ subset_complement ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_subset_1) ).
thf(228,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ subset_complement ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(43,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
<=> ( ( A != B )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_xboole_0) ).
thf(289,plain,
! [A: $i,B: $i] :
( ( ( ( A != B )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ proper_subset ) ) )
& ( ( B @ ( A @ proper_subset ) )
=> ( ( A != B )
& ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(51,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ complements_of_subsets ) @ ( A @ complements_of_subsets ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k7_setfam_1) ).
thf(316,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ complements_of_subsets ) @ ( A @ complements_of_subsets ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(33,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(259,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(260,plain,
~ ? [A: $i,B: $i] : ( B @ ( A @ unordered_pair ) @ empty ),
inference(miniscope,[status(thm)],[259]) ).
thf(261,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
inference(cnf,[status(esa)],[260]) ).
thf(15,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ union_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_setfam_1) ).
thf(189,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ union_of_subsets ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(135,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(924,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(121,axiom,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ ( C @ element ) )
=> ( ~ ( B @ ( C @ in ) )
=> ( B @ ( A @ subset_complement ) @ ( C @ in ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t50_subset_1) ).
thf(789,plain,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ ( C @ element ) )
=> ( ~ ( B @ ( C @ in ) )
=> ( B @ ( A @ subset_complement ) @ ( C @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(53,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) @ ( A @ subset_complement ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k3_subset_1) ).
thf(321,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) @ ( A @ subset_complement ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(68,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( B = C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t9_zfmisc_1) ).
thf(376,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(30,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_zfmisc_1) ).
thf(250,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(251,plain,
! [A: $i,B: $i] :
( ? [C: $i] :
( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( A = B ) ),
inference(miniscope,[status(thm)],[250]) ).
thf(252,plain,
! [C: $i,B: $i,A: $i] :
( ( A = B )
| ( ( A @ singleton )
!= ( C @ ( B @ unordered_pair ) ) ) ),
inference(cnf,[status(esa)],[251]) ).
thf(253,plain,
! [C: $i,B: $i,A: $i] :
( ( A = B )
| ( ( A @ singleton )
!= ( C @ ( B @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[252]) ).
thf(28,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(236,plain,
! [A: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(237,plain,
! [A: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
inference(cnf,[status(esa)],[236]) ).
thf(238,plain,
! [A: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
inference(lifteq,[status(thm)],[237]) ).
thf(935,plain,
( ( ( sk9 @ empty )
!= ( sk5 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,204]) ).
thf(936,plain,
( ( sk9 @ empty )
!= ( sk5 @ empty ) ),
inference(simp,[status(thm)],[935]) ).
thf(937,plain,
sk9 != sk5,
inference(simp,[status(thm)],[936]) ).
thf(124,axiom,
! [A: $i,B: $i,C: $i] :
( ( A @ powerset @ ( C @ element ) )
=> ~ ( ( C @ ( B @ in ) )
& ( C @ ( A @ subset_complement ) @ ( B @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t54_subset_1) ).
thf(821,plain,
! [A: $i,B: $i,C: $i] :
( ( A @ powerset @ ( C @ element ) )
=> ~ ( ( C @ ( B @ in ) )
& ( C @ ( A @ subset_complement ) @ ( B @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(48,axiom,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(309,plain,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(310,plain,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
inference(cnf,[status(esa)],[309]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_subset_1) ).
thf(147,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(148,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(cnf,[status(esa)],[147]) ).
thf(149,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(lifteq,[status(thm)],[148]) ).
thf(2540,plain,
! [A: $i] : ( A @ powerset @ ( A @ element ) ),
inference(rewrite,[status(thm)],[310,149]) ).
thf(2017,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,928]) ).
thf(2018,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[2017]) ).
thf(2034,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= sk9 ),
inference(simp,[status(thm)],[2018]) ).
thf(119,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(766,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(40,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_boole) ).
thf(280,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(22,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ powerset @ ( C @ element ) )
& ( A @ powerset @ ( B @ element ) ) )
=> ( A @ powerset @ ( C @ ( B @ ( A @ subset_difference ) ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_subset_1) ).
thf(221,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ powerset @ ( C @ element ) )
& ( A @ powerset @ ( B @ element ) ) )
=> ( A @ powerset @ ( C @ ( B @ ( A @ subset_difference ) ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(75,axiom,
! [A: $i,B: $i] :
( ( B @ powerset @ ( A @ element ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(396,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( B @ powerset @ ( A @ element ) ) )
& ( ( B @ powerset @ ( A @ element ) )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(109,axiom,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_rng @ ( A @ relation_dom @ cartesian_product2 ) @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_relat_1) ).
thf(660,plain,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_rng @ ( A @ relation_dom @ cartesian_product2 ) @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(992,plain,
! [A: $i] :
( ( ( A @ sk6 @ empty )
!= ( sk5 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[357,204]) ).
thf(993,plain,
! [A: $i] :
( ( A @ sk6 @ empty )
!= ( sk5 @ empty ) ),
inference(simp,[status(thm)],[992]) ).
thf(998,plain,
! [A: $i] :
( ( A @ sk6 )
!= sk5 ),
inference(simp,[status(thm)],[993]) ).
thf(26,axiom,
! [A: $i] :
~ ( A @ powerset @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(230,plain,
! [A: $i] :
~ ( A @ powerset @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(231,plain,
~ ? [A: $i] : ( A @ powerset @ empty ),
inference(miniscope,[status(thm)],[230]) ).
thf(232,plain,
! [A: $i] :
~ ( A @ powerset @ empty ),
inference(cnf,[status(esa)],[231]) ).
thf(983,plain,
! [A: $i] :
( ( ( A @ powerset @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,232]) ).
thf(984,plain,
! [A: $i] :
( ( A @ powerset @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[983]) ).
thf(988,plain,
! [A: $i] :
( ( A @ powerset )
!= empty_set ),
inference(simp,[status(thm)],[984]) ).
thf(1693,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,261]) ).
thf(1694,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[1693]) ).
thf(1697,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= sk4 ),
inference(simp,[status(thm)],[1694]) ).
thf(97,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( ( C @ singleton @ ( B @ set_difference ) @ ( A @ subset ) )
| ( A @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_zfmisc_1) ).
thf(527,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( ( C @ singleton @ ( B @ set_difference ) @ ( A @ subset ) )
| ( A @ ( C @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(1681,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,261]) ).
thf(1682,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[1681]) ).
thf(1696,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= sk9 ),
inference(simp,[status(thm)],[1682]) ).
thf(89,axiom,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( B @ ( C @ powerset @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t136_zfmisc_1) ).
thf(444,plain,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( B @ ( C @ powerset @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(445,plain,
! [A: $i] :
? [B: $i] :
( ~ ? [C: $i] :
( ~ ( B @ ( C @ in ) )
& ~ ( B @ ( C @ are_equipotent ) )
& ( B @ ( C @ subset ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( B @ ( C @ powerset @ in ) ) )
& ! [C: $i,D: $i] :
( ( ( C @ ( D @ subset ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( D @ in ) ) )
& ( B @ ( A @ in ) ) ),
inference(miniscope,[status(thm)],[444]) ).
thf(446,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ sk11 @ ( E @ in ) )
| ( A @ sk11 @ ( E @ are_equipotent ) )
| ~ ( A @ sk11 @ ( E @ subset ) ) )
& ( ( A @ sk11 @ ( D @ powerset @ in ) )
| ~ ( A @ sk11 @ ( D @ in ) ) )
& ( ( A @ sk11 @ ( C @ in ) )
| ~ ( B @ ( C @ subset ) )
| ~ ( A @ sk11 @ ( B @ in ) ) )
& ( A @ sk11 @ ( A @ in ) ) ),
inference(cnf,[status(esa)],[445]) ).
thf(447,plain,
! [A: $i] : ( A @ sk11 @ ( A @ in ) ),
inference(cnfConj,[status(thm)],[446]) ).
thf(1753,plain,
! [B: $i,A: $i] :
( ( ( A @ sk11 @ ( A @ in ) )
!= ( empty_set @ ( B @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[447,592]) ).
thf(1754,plain,
! [B: $i,A: $i] :
( ( A @ sk11 @ ( A @ in ) )
!= ( empty_set @ ( B @ in ) ) ),
inference(simp,[status(thm)],[1753]) ).
thf(1759,plain,
! [B: $i,A: $i] :
( ( ( A @ sk11 )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[1754]) ).
thf(1776,plain,
! [A: $i] :
( ( A @ sk11 )
!= empty_set ),
inference(simp,[status(thm)],[1759]) ).
thf(2194,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ sk11 ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,1776]) ).
thf(2195,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ sk11 ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2194]) ).
thf(2196,plain,
! [A: $i] :
~ ( empty_set @ ( A @ sk11 @ subset ) ),
inference(pattern_uni,[status(thm)],[2195:[bind(A,$thf( C @ sk11 )),bind(B,$thf( C ))]]) ).
thf(2415,plain,
! [A: $i] :
~ ( empty_set @ ( A @ sk11 @ subset ) ),
inference(simp,[status(thm)],[2196]) ).
thf(2743,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ sk11 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2415]) ).
thf(2744,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ sk11 @ subset ) ) ),
inference(simp,[status(thm)],[2743]) ).
thf(2756,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A
!= ( C @ sk11 ) ) ),
inference(simp,[status(thm)],[2744]) ).
thf(2762,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk11 @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2756]) ).
thf(37,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( A != D )
& ( A != C )
& ( ( B @ ( A @ unordered_pair ) )
= ( D @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_zfmisc_1) ).
thf(270,plain,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( A != D )
& ( A != C )
& ( ( B @ ( A @ unordered_pair ) )
= ( D @ ( C @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(17,axiom,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( A @ ( B @ set_union2 ) @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(199,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( A @ ( B @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(200,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ~ ? [B: $i] : ( A @ ( B @ set_union2 ) @ empty ) ),
inference(miniscope,[status(thm)],[199]) ).
thf(201,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ set_union2 ) @ empty )
| ( A @ empty ) ),
inference(cnf,[status(esa)],[200]) ).
thf(64,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ disjoint ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t63_xboole_1) ).
thf(366,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ disjoint ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(950,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,947]) ).
thf(951,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[950]) ).
thf(952,plain,
( ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[951]) ).
thf(83,axiom,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(424,plain,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(104,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_zfmisc_1) ).
thf(630,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(85,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(429,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( C @ cartesian_product2 ) @ ( A @ ( C @ cartesian_product2 ) @ subset ) )
& ( C @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t118_zfmisc_1) ).
thf(162,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( C @ cartesian_product2 ) @ ( A @ ( C @ cartesian_product2 ) @ subset ) )
& ( C @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(163,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ! [C: $i] : ( B @ ( C @ cartesian_product2 ) @ ( A @ ( C @ cartesian_product2 ) @ subset ) )
& ! [C: $i] : ( C @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) ) ) ),
inference(miniscope,[status(thm)],[162]) ).
thf(164,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( D @ cartesian_product2 ) @ ( A @ ( D @ cartesian_product2 ) @ subset ) )
| ~ ( B @ ( A @ subset ) ) )
& ( ( C @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) )
| ~ ( B @ ( A @ subset ) ) ) ),
inference(cnf,[status(esa)],[163]) ).
thf(165,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cartesian_product2 ) @ ( C @ ( A @ cartesian_product2 ) @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[164]) ).
thf(229,plain,
! [B: $i,A: $i] :
( ( A @ powerset @ ( B @ ( A @ subset_complement ) @ element ) )
| ~ ( A @ powerset @ ( B @ element ) ) ),
inference(cnf,[status(esa)],[228]) ).
thf(62,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B != empty_set )
=> ( ( B @ ( A @ complements_of_subsets ) @ ( A @ union_of_subsets ) )
= ( B @ ( A @ meet_of_subsets ) @ ( A @ cast_to_subset @ ( A @ subset_difference ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t48_setfam_1) ).
thf(361,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B != empty_set )
=> ( ( B @ ( A @ complements_of_subsets ) @ ( A @ union_of_subsets ) )
= ( B @ ( A @ meet_of_subsets ) @ ( A @ cast_to_subset @ ( A @ subset_difference ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(560,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( D = B )
| ( D = A ) )
=> ( C @ ( D @ in ) ) )
& ! [D: $i] :
( ( C @ ( D @ in ) )
=> ( ( D = B )
| ( D = A ) ) ) )
=> ( C
= ( B @ ( A @ unordered_pair ) ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ unordered_pair ) ) )
=> ( ! [D: $i] :
( ( ( D = B )
| ( D = A ) )
=> ( C @ ( D @ in ) ) )
& ! [D: $i] :
( ( C @ ( D @ in ) )
=> ( ( D = B )
| ( D = A ) ) ) ) ) ),
inference(miniscope,[status(thm)],[559]) ).
thf(561,plain,
! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( H
= ( G @ ( F @ unordered_pair ) ) )
| ~ ( H @ ( F @ ( G @ ( H @ sk26 ) ) @ in ) )
| ( H @ ( G @ ( F @ sk24 ) ) ) )
& ( ( H
= ( G @ ( F @ unordered_pair ) ) )
| ( ( F @ ( G @ ( H @ sk26 ) ) )
= G )
| ( ( F @ ( G @ ( H @ sk26 ) ) )
= F )
| ( H @ ( G @ ( F @ sk24 ) ) ) )
& ( ( H
= ( G @ ( F @ unordered_pair ) ) )
| ~ ( H @ ( G @ ( F @ sk24 ) ) )
| ( ( F @ ( G @ ( H @ sk25 ) ) )
!= G ) )
& ( ( H
= ( G @ ( F @ unordered_pair ) ) )
| ~ ( H @ ( G @ ( F @ sk24 ) ) )
| ( ( F @ ( G @ ( H @ sk25 ) ) )
!= F ) )
& ( ( H
= ( G @ ( F @ unordered_pair ) ) )
| ~ ( H @ ( G @ ( F @ sk24 ) ) )
| ( H @ ( F @ ( G @ ( H @ sk25 ) ) @ in ) ) )
& ( ( C @ ( E @ in ) )
| ( E != B )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) )
& ( ( C @ ( E @ in ) )
| ( E != A )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) )
& ( ( D = B )
| ( D = A )
| ~ ( C @ ( D @ in ) )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) ) ),
inference(cnf,[status(esa)],[560]) ).
thf(568,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( D @ in ) )
| ( D != A )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) ),
inference(cnfConj,[status(thm)],[561]) ).
thf(582,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( D @ in ) )
| ( D != A )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[568]) ).
thf(583,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ in ) ),
inference(simp,[status(thm)],[582]) ).
thf(54,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ subset ) )
<=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l4_zfmisc_1) ).
thf(324,plain,
! [A: $i,B: $i] :
( ( ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) )
=> ( B @ singleton @ ( A @ subset ) ) )
& ( ( B @ singleton @ ( A @ subset ) )
=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(108,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ordered_pair ) )
= ( D @ ( C @ ordered_pair ) ) )
=> ( ( B = D )
& ( A = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t33_zfmisc_1) ).
thf(650,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ordered_pair ) )
= ( D @ ( C @ ordered_pair ) ) )
=> ( ( B = D )
& ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(74,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_boole) ).
thf(393,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(394,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
inference(cnf,[status(esa)],[393]) ).
thf(395,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
inference(lifteq,[status(thm)],[394]) ).
thf(3379,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ set_intersection2 ) )
!= ( C @ ( B @ set_intersection2 ) ) )
| ( C @ ( B @ disjoint ) )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[395,198]) ).
thf(3380,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ set_intersection2 ) )
!= ( C @ ( B @ set_intersection2 ) ) )
| ( C @ ( B @ disjoint ) ) ),
inference(simp,[status(thm)],[3379]) ).
thf(3381,plain,
! [A: $i] : ( empty_set @ ( A @ disjoint ) ),
inference(pattern_uni,[status(thm)],[3380:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set ))]]) ).
thf(72,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ powerset @ ( B @ ( A @ complements_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k7_setfam_1) ).
thf(388,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ powerset @ ( B @ ( A @ complements_of_subsets ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(356,plain,
! [A: $i] : ( A @ powerset @ ( A @ sk6 @ element ) ),
inference(cnfConj,[status(thm)],[355]) ).
thf(985,plain,
! [A: $i] :
( ( ( A @ powerset @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,232]) ).
thf(986,plain,
! [A: $i] :
( ( A @ powerset @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[985]) ).
thf(989,plain,
! [A: $i] :
( ( A @ powerset )
!= sk4 ),
inference(simp,[status(thm)],[986]) ).
thf(129,axiom,
! [A: $i,B: $i] :
( ( ( A = empty_set )
=> ( ( B
= ( A @ set_meet ) )
<=> ( B = empty_set ) ) )
& ( ( A != empty_set )
=> ( ( B
= ( A @ set_meet ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_setfam_1) ).
thf(843,plain,
! [A: $i,B: $i] :
( ( ( A = empty_set )
=> ( ( ( B = empty_set )
=> ( B
= ( A @ set_meet ) ) )
& ( ( B
= ( A @ set_meet ) )
=> ( B = empty_set ) ) ) )
& ( ( A != empty_set )
=> ( ( ! [C: $i] :
( ( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) )
=> ( B
= ( A @ set_meet ) ) )
& ( ( B
= ( A @ set_meet ) )
=> ! [C: $i] :
( ( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(844,plain,
( ! [A: $i] :
( ( A = empty_set )
=> ( ! [B: $i] :
( ( B = empty_set )
=> ( B
= ( A @ set_meet ) ) )
& ! [B: $i] :
( ( B
= ( A @ set_meet ) )
=> ( B = empty_set ) ) ) )
& ! [A: $i] :
( ( A != empty_set )
=> ( ! [B: $i] :
( ( ! [C: $i] :
( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) )
=> ( B
= ( A @ set_meet ) ) )
& ! [B: $i] :
( ( B
= ( A @ set_meet ) )
=> ( ! [C: $i] :
( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ! [C: $i] :
( ( B @ ( C @ in ) )
=> ! [D: $i] :
( ( A @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) ) ) ) ) ),
inference(miniscope,[status(thm)],[843]) ).
thf(845,plain,
! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( J
= ( H @ set_meet ) )
| ( J != empty_set )
| ( H != empty_set ) )
& ( ( I = empty_set )
| ( I
!= ( H @ set_meet ) )
| ( H != empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ ( F @ sk59 ) @ in ) )
| ( F @ ( A @ sk56 ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ( G @ ( A @ ( F @ sk59 ) @ in ) )
| ~ ( A @ ( G @ in ) )
| ( F @ ( A @ sk56 ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk56 ) )
| ~ ( A @ ( F @ sk58 ) @ ( A @ ( F @ sk57 ) @ in ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk56 ) )
| ( A @ ( A @ ( F @ sk58 ) @ in ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk56 ) )
| ( F @ ( A @ ( F @ sk57 ) @ in ) )
| ( A = empty_set ) )
& ( ( B @ ( E @ in ) )
| ~ ( A @ ( B @ ( E @ sk55 ) ) @ ( E @ in ) )
| ( B
!= ( A @ set_meet ) )
| ( A = empty_set ) )
& ( ( B @ ( E @ in ) )
| ( A @ ( A @ ( B @ ( E @ sk55 ) ) @ in ) )
| ( B
!= ( A @ set_meet ) )
| ( A = empty_set ) )
& ( ( D @ ( C @ in ) )
| ~ ( A @ ( D @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ set_meet ) )
| ( A = empty_set ) ) ),
inference(cnf,[status(esa)],[844]) ).
thf(846,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ set_meet ) )
| ( B != empty_set )
| ( A != empty_set ) ),
inference(cnfConj,[status(thm)],[845]) ).
thf(856,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ set_meet ) )
| ( B != empty_set )
| ( A != empty_set ) ),
inference(lifteq,[status(thm)],[846]) ).
thf(857,plain,
( ( empty_set @ set_meet )
= empty_set ),
inference(simp,[status(thm)],[856]) ).
thf(3203,plain,
! [B: $i,A: $i] :
( ( ( A @ sk6 @ empty )
!= ( B @ ( sk3 @ set_union2 ) @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[357,3188]) ).
thf(3204,plain,
! [B: $i,A: $i] :
( ( A @ sk6 @ empty )
!= ( B @ ( sk3 @ set_union2 ) @ empty ) ),
inference(simp,[status(thm)],[3203]) ).
thf(3218,plain,
! [B: $i,A: $i] :
( ( A @ sk6 )
!= ( B @ ( sk3 @ set_union2 ) ) ),
inference(simp,[status(thm)],[3204]) ).
thf(116,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) )
<=> ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l55_zfmisc_1) ).
thf(725,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) )
=> ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) ) )
& ( ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(93,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ union ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_tarski) ).
thf(488,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) ) ) )
=> ( B
= ( A @ union ) ) )
& ( ( B
= ( A @ union ) )
=> ! [C: $i] :
( ( ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(23,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ powerset @ ( C @ element ) )
& ( A @ powerset @ ( B @ element ) ) )
=> ( ( C @ ( B @ ( A @ subset_difference ) ) )
= ( C @ ( B @ set_difference ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k6_subset_1) ).
thf(223,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ powerset @ ( C @ element ) )
& ( A @ powerset @ ( B @ element ) ) )
=> ( ( C @ ( B @ ( A @ subset_difference ) ) )
= ( C @ ( B @ set_difference ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(224,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ subset_difference ) ) )
= ( C @ ( B @ set_difference ) ) )
| ~ ( A @ powerset @ ( C @ element ) )
| ~ ( A @ powerset @ ( B @ element ) ) ),
inference(cnf,[status(esa)],[223]) ).
thf(225,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( A @ powerset @ ( C @ element ) )
| ~ ( A @ powerset @ ( B @ element ) )
| ( ( C @ ( B @ ( A @ subset_difference ) ) )
= ( C @ ( B @ set_difference ) ) ) ),
inference(lifteq,[status(thm)],[224]) ).
thf(56,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ( ( B @ ( A @ set_difference ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(339,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( A @ set_difference ) )
= A )
=> ( B @ ( A @ disjoint ) ) )
& ( ( B @ ( A @ disjoint ) )
=> ( ( B @ ( A @ set_difference ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(107,axiom,
! [A: $i,B: $i] :
( ( ( B @ singleton @ ( A @ set_difference ) )
= A )
<=> ~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t65_zfmisc_1) ).
thf(642,plain,
! [A: $i,B: $i] :
( ( ~ ( A @ ( B @ in ) )
=> ( ( B @ singleton @ ( A @ set_difference ) )
= A ) )
& ( ( ( B @ singleton @ ( A @ set_difference ) )
= A )
=> ~ ( A @ ( B @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(1841,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= ( A @ singleton ) )
| ( empty_set @ powerset @ ( A @ in ) ) ),
inference(paramod_ordered,[status(thm)],[350,469]) ).
thf(1842,plain,
empty_set @ powerset @ ( empty_set @ in ),
inference(pattern_uni,[status(thm)],[1841:[bind(A,$thf( empty_set ))]]) ).
thf(208,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[207]) ).
thf(211,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[208]) ).
thf(212,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(simp,[status(thm)],[211]) ).
thf(103,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_tarski) ).
thf(622,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) )
=> ( B @ ( A @ subset ) ) )
& ( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(94,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(515,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(1527,plain,
( ( ( sk4 @ empty )
!= ( sk3 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,1514]) ).
thf(1528,plain,
( ( sk4 @ empty )
!= ( sk3 @ empty ) ),
inference(simp,[status(thm)],[1527]) ).
thf(1530,plain,
sk4 != sk3,
inference(simp,[status(thm)],[1528]) ).
thf(1913,plain,
! [A: $i] :
( ( ( A @ sk22 @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[553,945]) ).
thf(1914,plain,
! [A: $i] :
( ( ( A @ sk22 @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) ) ),
inference(simp,[status(thm)],[1913]) ).
thf(1936,plain,
! [A: $i] :
( ( ( A @ sk22 )
!= ( sk3 @ relation_field ) )
| ( A != sk1 )
| ( ( sk3 @ relation_field )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1914]) ).
thf(1947,plain,
( ( ( sk1 @ sk22 )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1936]) ).
thf(3189,plain,
! [A: $i] :
( ( ( A @ ( sk3 @ set_union2 ) @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,3188]) ).
thf(3190,plain,
! [A: $i] :
( ( A @ ( sk3 @ set_union2 ) @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[3189]) ).
thf(3215,plain,
! [A: $i] :
( ( A @ ( sk3 @ set_union2 ) )
!= sk9 ),
inference(simp,[status(thm)],[3190]) ).
thf(590,plain,
! [A: $i] :
( ( A = empty_set )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(cnfConj,[status(thm)],[588]) ).
thf(593,plain,
! [A: $i] :
( ( A @ ( A @ sk27 @ in ) )
| ( A = empty_set ) ),
inference(lifteq,[status(thm)],[590]) ).
thf(594,plain,
! [A: $i] :
( ( A @ ( A @ sk27 @ in ) )
| ( A = empty_set ) ),
inference(simp,[status(thm)],[593]) ).
thf(5848,plain,
! [A: $i] :
( ( A != sk3 )
| ( empty_set != empty_set )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(paramod_ordered,[status(thm)],[594,1008]) ).
thf(5849,plain,
! [A: $i] :
( ( A != sk3 )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(simp,[status(thm)],[5848]) ).
thf(5850,plain,
sk3 @ ( sk3 @ sk27 @ in ),
inference(pattern_uni,[status(thm)],[5849:[bind(A,$thf( sk3 ))]]) ).
thf(11,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_boole) ).
thf(173,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(132,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ subset ) )
=> ( ( B @ relation_rng @ ( A @ relation_rng @ subset ) )
& ( B @ relation_dom @ ( A @ relation_dom @ subset ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_relat_1) ).
thf(911,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ subset ) )
=> ( ( B @ relation_rng @ ( A @ relation_rng @ subset ) )
& ( B @ relation_dom @ ( A @ relation_dom @ subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(122,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ set_difference ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ~ ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_xboole_0) ).
thf(792,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ~ ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ~ ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ set_difference ) ) ) )
& ( ( C
= ( B @ ( A @ set_difference ) ) )
=> ! [D: $i] :
( ( ( ~ ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ~ ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(105,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) )
=> ( C @ ( A @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_subset) ).
thf(633,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) )
=> ( C @ ( A @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(49,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_xboole_1) ).
thf(311,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(110,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(662,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ subset ) ) )
& ( ( B @ ( A @ singleton @ subset ) )
=> ( B @ ( A @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(98,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ powerset @ powerset @ ( C @ element ) )
=> ( ( C
= ( B @ ( A @ complements_of_subsets ) ) )
<=> ! [D: $i] :
( ( A @ powerset @ ( D @ element ) )
=> ( ( C @ ( D @ in ) )
<=> ( B @ ( D @ ( A @ subset_complement ) @ in ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_setfam_1) ).
thf(530,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ powerset @ powerset @ ( C @ element ) )
=> ( ( ! [D: $i] :
( ( A @ powerset @ ( D @ element ) )
=> ( ( ( B @ ( D @ ( A @ subset_complement ) @ in ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( B @ ( D @ ( A @ subset_complement ) @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ complements_of_subsets ) ) ) )
& ( ( C
= ( B @ ( A @ complements_of_subsets ) ) )
=> ! [D: $i] :
( ( A @ powerset @ ( D @ element ) )
=> ( ( ( B @ ( D @ ( A @ subset_complement ) @ in ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( B @ ( D @ ( A @ subset_complement ) @ in ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(5277,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ in ) )
!= ( empty_set @ ( C @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[583,592]) ).
thf(5278,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ in ) )
!= ( empty_set @ ( C @ in ) ) ),
inference(simp,[status(thm)],[5277]) ).
thf(5286,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[5278]) ).
thf(5295,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[5286]) ).
thf(425,plain,
! [A: $i] : ( A @ ( A @ sk8 @ element ) ),
inference(cnf,[status(esa)],[424]) ).
thf(1683,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,261]) ).
thf(1684,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[1683]) ).
thf(1698,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[1684]) ).
thf(2029,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,928]) ).
thf(2030,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[2029]) ).
thf(2033,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= sk4 ),
inference(simp,[status(thm)],[2030]) ).
thf(35,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(264,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_xboole_1) ).
thf(154,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_difference ) )
= empty_set ) )
& ( ( ( B @ ( A @ set_difference ) )
= empty_set )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(155,plain,
( ! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_difference ) )
= empty_set ) )
& ! [A: $i,B: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
=> ( B @ ( A @ subset ) ) ) ),
inference(miniscope,[status(thm)],[154]) ).
thf(156,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ( D @ ( C @ set_difference ) )
= empty_set )
| ~ ( D @ ( C @ subset ) ) )
& ( ( B @ ( A @ subset ) )
| ( ( B @ ( A @ set_difference ) )
!= empty_set ) ) ),
inference(cnf,[status(esa)],[155]) ).
thf(158,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[156]) ).
thf(160,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ subset ) )
| ( ( B @ ( A @ set_difference ) )
= empty_set ) ),
inference(lifteq,[status(thm)],[158]) ).
thf(161,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ subset ) )
| ( ( B @ ( A @ set_difference ) )
= empty_set ) ),
inference(simp,[status(thm)],[160]) ).
thf(1593,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ~ $true
| ( ( C @ ( B @ set_difference ) )
= empty_set ) ),
inference(paramod_ordered,[status(thm)],[212,161]) ).
thf(1594,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( ( C @ ( B @ set_difference ) )
= empty_set ) ),
inference(simp,[status(thm)],[1593]) ).
thf(1595,plain,
! [A: $i] :
( ( A @ ( A @ set_difference ) )
= empty_set ),
inference(pattern_uni,[status(thm)],[1594:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(566,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( D @ in ) )
| ( D != B )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) ),
inference(cnfConj,[status(thm)],[561]) ).
thf(572,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( D @ in ) )
| ( D != B )
| ( C
!= ( B @ ( A @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[566]) ).
thf(573,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( B @ in ) ),
inference(simp,[status(thm)],[572]) ).
thf(67,axiom,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_boole) ).
thf(373,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(374,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
inference(cnf,[status(esa)],[373]) ).
thf(375,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
inference(lifteq,[status(thm)],[374]) ).
thf(123,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(819,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(422,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(cnf,[status(esa)],[421]) ).
thf(423,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[422]) ).
thf(114,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B
= ( A @ relation_rng ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] : ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_relat_1) ).
thf(694,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i] :
( ( ? [D: $i] : ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] : ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) )
=> ( B
= ( A @ relation_rng ) ) )
& ( ( B
= ( A @ relation_rng ) )
=> ! [C: $i] :
( ( ? [D: $i] : ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] : ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(117,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ set_intersection2 ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_xboole_0) ).
thf(732,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ set_intersection2 ) ) ) )
& ( ( C
= ( B @ ( A @ set_intersection2 ) ) )
=> ! [D: $i] :
( ( ( ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(2781,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( sk3 @ set_union2 ) ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,2589]) ).
thf(2782,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( sk3 @ set_union2 ) ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2781]) ).
thf(2783,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( sk3 @ set_union2 ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2782:[bind(A,$thf( D @ ( sk3 @ set_union2 ) )),bind(B,$thf( D ))]]) ).
thf(2791,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( sk3 @ set_union2 ) @ subset ) ),
inference(simp,[status(thm)],[2783]) ).
thf(42,axiom,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ set_union2 ) @ set_difference ) )
= ( B @ ( A @ set_difference ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t40_xboole_1) ).
thf(286,plain,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ set_union2 ) @ set_difference ) )
= ( B @ ( A @ set_difference ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(79,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ union_of_subsets ) )
= ( B @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k5_setfam_1) ).
thf(413,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ union_of_subsets ) )
= ( B @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(76,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ powerset @ ( C @ element ) )
=> ( ( C @ ( B @ disjoint ) )
<=> ( C @ ( A @ subset_complement ) @ ( B @ subset ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t43_subset_1) ).
thf(402,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( A @ powerset @ ( C @ element ) )
=> ( ( ( C @ ( A @ subset_complement ) @ ( B @ subset ) )
=> ( C @ ( B @ disjoint ) ) )
& ( ( C @ ( B @ disjoint ) )
=> ( C @ ( A @ subset_complement ) @ ( B @ subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(70,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_union2 ) )
= ( A @ ( B @ set_union2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).
thf(382,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_union2 ) )
= ( A @ ( B @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(50,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ( ( B @ ( A @ complements_of_subsets ) )
= empty_set )
& ( B != empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_setfam_1) ).
thf(313,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ( ( B @ ( A @ complements_of_subsets ) )
= empty_set )
& ( B != empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(24,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(226,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(227,plain,
! [B: $i,A: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
inference(cnf,[status(esa)],[226]) ).
thf(2137,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ powerset ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,988]) ).
thf(2138,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ powerset ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2137]) ).
thf(2139,plain,
! [A: $i] :
~ ( empty_set @ ( A @ powerset @ subset ) ),
inference(pattern_uni,[status(thm)],[2138:[bind(A,$thf( C @ powerset )),bind(B,$thf( C ))]]) ).
thf(2411,plain,
! [A: $i] :
~ ( empty_set @ ( A @ powerset @ subset ) ),
inference(simp,[status(thm)],[2139]) ).
thf(3332,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ powerset @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[227,2411]) ).
thf(3333,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ powerset @ subset ) ) ),
inference(simp,[status(thm)],[3332]) ).
thf(3342,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( B @ ( A @ set_difference ) )
!= ( C @ powerset ) ) ),
inference(simp,[status(thm)],[3333]) ).
thf(3356,plain,
! [B: $i,A: $i] :
( ( A @ ( empty_set @ set_difference ) )
!= ( B @ powerset ) ),
inference(simp,[status(thm)],[3342]) ).
thf(7534,plain,
! [A: $i] :
( ( A @ powerset )
!= empty_set ),
inference(rewrite,[status(thm)],[3356,375]) ).
thf(143,plain,
~ ? [A: $i] : ( A @ ( A @ proper_subset ) ),
inference(miniscope,[status(thm)],[142]) ).
thf(144,plain,
! [A: $i] :
~ ( A @ ( A @ proper_subset ) ),
inference(cnf,[status(esa)],[143]) ).
thf(55,axiom,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(335,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l32_xboole_1) ).
thf(181,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_difference ) )
= empty_set ) )
& ( ( ( B @ ( A @ set_difference ) )
= empty_set )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(157,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( ( B @ ( A @ set_difference ) )
!= empty_set ) ),
inference(cnfConj,[status(thm)],[156]) ).
thf(159,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( ( B @ ( A @ set_difference ) )
!= empty_set ) ),
inference(lifteq,[status(thm)],[157]) ).
thf(975,plain,
( ( ( empty_set @ powerset @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,963]) ).
thf(976,plain,
( ( empty_set @ powerset @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[975]) ).
thf(979,plain,
( ( empty_set @ powerset )
!= sk4 ),
inference(simp,[status(thm)],[976]) ).
thf(2697,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ powerset @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2411]) ).
thf(2698,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ powerset @ subset ) ) ),
inference(simp,[status(thm)],[2697]) ).
thf(2711,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A
!= ( C @ powerset ) ) ),
inference(simp,[status(thm)],[2698]) ).
thf(2721,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ powerset @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2711]) ).
thf(77,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_intersection2 ) @ ( C @ ( A @ set_intersection2 ) @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_xboole_1) ).
thf(406,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_intersection2 ) @ ( C @ ( A @ set_intersection2 ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(171,plain,
~ ? [A: $i,B: $i] :
( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
inference(miniscope,[status(thm)],[170]) ).
thf(172,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ proper_subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnf,[status(esa)],[171]) ).
thf(29,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ subset ) )
<=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t39_zfmisc_1) ).
thf(239,plain,
! [A: $i,B: $i] :
( ( ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) )
=> ( B @ singleton @ ( A @ subset ) ) )
& ( ( B @ singleton @ ( A @ subset ) )
=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(240,plain,
( ! [A: $i,B: $i] :
( ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) )
=> ( B @ singleton @ ( A @ subset ) ) )
& ! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ subset ) )
=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ) ),
inference(miniscope,[status(thm)],[239]) ).
thf(241,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ singleton @ ( C @ subset ) )
| ( C
!= ( D @ singleton ) ) )
& ( ( D @ singleton @ ( C @ subset ) )
| ( C != empty_set ) )
& ( ( A
= ( B @ singleton ) )
| ( A = empty_set )
| ~ ( B @ singleton @ ( A @ subset ) ) ) ),
inference(cnf,[status(esa)],[240]) ).
thf(242,plain,
! [B: $i,A: $i] :
( ( A
= ( B @ singleton ) )
| ( A = empty_set )
| ~ ( B @ singleton @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[241]) ).
thf(245,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ subset ) )
| ( A
= ( B @ singleton ) )
| ( A = empty_set ) ),
inference(lifteq,[status(thm)],[242]) ).
thf(46,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_difference ) @ ( C @ ( A @ set_difference ) @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t33_xboole_1) ).
thf(303,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_difference ) @ ( C @ ( A @ set_difference ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(5504,plain,
! [A: $i] :
( ( A != sk5 )
| ( empty_set != empty_set )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(paramod_ordered,[status(thm)],[594,934]) ).
thf(5505,plain,
! [A: $i] :
( ( A != sk5 )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(simp,[status(thm)],[5504]) ).
thf(5506,plain,
sk5 @ ( sk5 @ sk27 @ in ),
inference(pattern_uni,[status(thm)],[5505:[bind(A,$thf( sk5 ))]]) ).
thf(92,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
<=> ( B @ ( C @ in ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_tarski) ).
thf(477,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( ( B @ ( C @ in ) )
=> ( A @ ( C @ in ) ) )
& ( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(133,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ unordered_pair ) @ subset ) )
<=> ( ( C @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t38_zfmisc_1) ).
thf(915,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( B @ in ) )
& ( C @ ( A @ in ) ) )
=> ( C @ ( B @ ( A @ unordered_pair ) @ subset ) ) )
& ( ( C @ ( B @ ( A @ unordered_pair ) @ subset ) )
=> ( ( C @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(938,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_field @ ( sk2 @ in ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,141]) ).
thf(939,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_field @ ( sk2 @ in ) ) ),
inference(simp,[status(thm)],[938]) ).
thf(943,plain,
( ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 )
| ~ ( sk3 @ relation_field @ ( sk2 @ in ) ) ),
inference(simp,[status(thm)],[939]) ).
thf(111,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_union2 ) @ relation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_relat_1) ).
thf(668,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_union2 ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(973,plain,
( ( ( empty_set @ powerset @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,963]) ).
thf(974,plain,
( ( empty_set @ powerset @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[973]) ).
thf(978,plain,
( ( empty_set @ powerset )
!= empty_set ),
inference(simp,[status(thm)],[974]) ).
thf(2315,plain,
! [A: $i] :
( ( A
!= ( empty_set @ powerset ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,978]) ).
thf(2316,plain,
! [A: $i] :
( ( A
!= ( empty_set @ powerset ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2315]) ).
thf(2317,plain,
~ ( empty_set @ ( empty_set @ powerset @ subset ) ),
inference(pattern_uni,[status(thm)],[2316:[bind(A,$thf( empty_set @ powerset ))]]) ).
thf(2559,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ powerset @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2317]) ).
thf(2560,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ powerset @ subset ) ) ),
inference(simp,[status(thm)],[2559]) ).
thf(2580,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A
!= ( empty_set @ powerset ) ) ),
inference(simp,[status(thm)],[2560]) ).
thf(2588,plain,
! [A: $i] :
( ( A @ ( empty_set @ powerset @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2580]) ).
thf(2827,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( empty_set @ powerset @ set_union2 ) ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,2588]) ).
thf(2828,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( empty_set @ powerset @ set_union2 ) ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2827]) ).
thf(2829,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( empty_set @ powerset @ set_union2 ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2828:[bind(A,$thf( D @ ( empty_set @ powerset @ set_union2 ) )),bind(B,$thf( D ))]]) ).
thf(2842,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( empty_set @ powerset @ set_union2 ) @ subset ) ),
inference(simp,[status(thm)],[2829]) ).
thf(953,plain,
! [A: $i] :
( ( ( A @ singleton @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,219]) ).
thf(954,plain,
! [A: $i] :
( ( A @ singleton @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[953]) ).
thf(959,plain,
! [A: $i] :
( ( A @ singleton )
!= sk9 ),
inference(simp,[status(thm)],[954]) ).
thf(120,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ powerset ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( A @ ( C @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_zfmisc_1) ).
thf(768,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( ( A @ ( C @ subset ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( A @ ( C @ subset ) ) ) )
=> ( B
= ( A @ powerset ) ) )
& ( ( B
= ( A @ powerset ) )
=> ! [C: $i] :
( ( ( A @ ( C @ subset ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( A @ ( C @ subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(47,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(306,plain,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ meet_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_setfam_1) ).
thf(299,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ meet_of_subsets ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(36,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_difference ) @ ( A @ set_difference ) )
= ( B @ ( A @ set_intersection2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t48_xboole_1) ).
thf(267,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_difference ) @ ( A @ set_difference ) )
= ( B @ ( A @ set_intersection2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(88,axiom,
! [A: $i,B: $i] :
( ~ ( ( B @ ( A @ disjoint ) )
& ? [C: $i] : ( B @ ( A @ set_intersection2 ) @ ( C @ in ) ) )
& ~ ( ! [C: $i] :
~ ( B @ ( A @ set_intersection2 ) @ ( C @ in ) )
& ~ ( B @ ( A @ disjoint ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_xboole_0) ).
thf(438,plain,
! [A: $i,B: $i] :
( ~ ( ( B @ ( A @ disjoint ) )
& ? [C: $i] : ( B @ ( A @ set_intersection2 ) @ ( C @ in ) ) )
& ~ ( ! [C: $i] :
~ ( B @ ( A @ set_intersection2 ) @ ( C @ in ) )
& ~ ( B @ ( A @ disjoint ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(3319,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ sk11 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[227,2415]) ).
thf(3320,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ sk11 @ subset ) ) ),
inference(simp,[status(thm)],[3319]) ).
thf(3344,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( B @ ( A @ set_difference ) )
!= ( C @ sk11 ) ) ),
inference(simp,[status(thm)],[3320]) ).
thf(3358,plain,
! [B: $i,A: $i] :
( ( A @ ( empty_set @ set_difference ) )
!= ( B @ sk11 ) ),
inference(simp,[status(thm)],[3344]) ).
thf(7558,plain,
! [A: $i] :
( ( A @ sk11 )
!= empty_set ),
inference(rewrite,[status(thm)],[3358,375]) ).
thf(336,plain,
~ ? [A: $i] :
( ( A @ singleton )
= empty_set ),
inference(miniscope,[status(thm)],[335]) ).
thf(337,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(cnf,[status(esa)],[336]) ).
thf(338,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(lifteq,[status(thm)],[337]) ).
thf(2090,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,338]) ).
thf(2091,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2090]) ).
thf(2092,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[2091:[bind(A,$thf( C @ singleton )),bind(B,$thf( C ))]]) ).
thf(2390,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ subset ) ),
inference(simp,[status(thm)],[2092]) ).
thf(134,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_subset_1) ).
thf(922,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(166,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ cartesian_product2 ) @ ( A @ ( C @ cartesian_product2 ) @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[164]) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ cartesian_product2 ) @ ( A @ ( C @ cartesian_product2 ) @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[166]) ).
thf(84,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( C @ empty )
& ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_subset) ).
thf(426,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( C @ empty )
& ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(1988,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ~ ( B @ ( C @ proper_subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[372,172]) ).
thf(1989,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ~ ( B @ ( C @ proper_subset ) ) ),
inference(simp,[status(thm)],[1988]) ).
thf(1990,plain,
! [A: $i] :
~ ( empty_set @ ( A @ proper_subset ) ),
inference(pattern_uni,[status(thm)],[1989:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(127,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) )
=> ( B @ powerset @ ( A @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l71_subset_1) ).
thf(836,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) )
=> ( B @ powerset @ ( A @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(4,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
=> ~ ( A @ ( B @ proper_subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_xboole_0) ).
thf(145,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
=> ~ ( A @ ( B @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(940,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk2 @ in ) ) )
| ~ $true
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(paramod_ordered,[status(thm)],[140,141]) ).
thf(941,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk2 @ in ) ) )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[940]) ).
thf(944,plain,
( ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk2 )
| ~ ( sk3 @ relation_field @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[941]) ).
thf(3941,plain,
! [B: $i,A: $i] :
( ( A != sk4 )
| ( B != sk3 )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[216,1530]) ).
thf(3942,plain,
! [A: $i] :
( ( A != sk3 )
| ~ ( sk4 @ ( A @ subset ) )
| ~ ( A @ ( sk4 @ subset ) ) ),
inference(pattern_uni,[status(thm)],[3941:[bind(A,$thf( sk4 ))]]) ).
thf(4722,plain,
( ~ ( sk4 @ ( sk3 @ subset ) )
| ~ ( sk3 @ ( sk4 @ subset ) ) ),
inference(simp,[status(thm)],[3942]) ).
thf(125,axiom,
! [A: $i] :
( ( A @ relation )
<=> ! [B: $i] :
~ ( ! [C: $i,D: $i] :
( B
!= ( D @ ( C @ ordered_pair ) ) )
& ( A @ ( B @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_relat_1) ).
thf(823,plain,
! [A: $i] :
( ( ! [B: $i] :
~ ( ! [C: $i,D: $i] :
( B
!= ( D @ ( C @ ordered_pair ) ) )
& ( A @ ( B @ in ) ) )
=> ( A @ relation ) )
& ( ( A @ relation )
=> ! [B: $i] :
~ ( ! [C: $i,D: $i] :
( B
!= ( D @ ( C @ ordered_pair ) ) )
& ( A @ ( B @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(281,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
inference(cnf,[status(esa)],[280]) ).
thf(282,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
inference(lifteq,[status(thm)],[281]) ).
thf(78,axiom,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(409,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(929,plain,
( ( ( sk5 @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,204]) ).
thf(930,plain,
( ( sk5 @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[929]) ).
thf(931,plain,
sk5 != sk4,
inference(simp,[status(thm)],[930]) ).
thf(4307,plain,
! [B: $i,A: $i] :
( ( A != sk5 )
| ( B != sk4 )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[216,931]) ).
thf(4308,plain,
! [A: $i] :
( ( A != sk4 )
| ~ ( sk5 @ ( A @ subset ) )
| ~ ( A @ ( sk5 @ subset ) ) ),
inference(pattern_uni,[status(thm)],[4307:[bind(A,$thf( sk5 ))]]) ).
thf(4549,plain,
( ~ ( sk5 @ ( sk4 @ subset ) )
| ~ ( sk4 @ ( sk5 @ subset ) ) ),
inference(simp,[status(thm)],[4308]) ).
thf(128,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l23_zfmisc_1) ).
thf(840,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(3301,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ sk22 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[227,2387]) ).
thf(3302,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ sk22 @ subset ) ) ),
inference(simp,[status(thm)],[3301]) ).
thf(3349,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( B @ ( A @ set_difference ) )
!= ( C @ sk22 ) ) ),
inference(simp,[status(thm)],[3302]) ).
thf(3363,plain,
! [B: $i,A: $i] :
( ( A @ ( empty_set @ set_difference ) )
!= ( B @ sk22 ) ),
inference(simp,[status(thm)],[3349]) ).
thf(7662,plain,
! [A: $i] :
( ( A @ sk22 )
!= empty_set ),
inference(rewrite,[status(thm)],[3363,375]) ).
thf(2126,plain,
! [A: $i] :
( ( A != sk5 )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,934]) ).
thf(2127,plain,
! [A: $i] :
( ( A != sk5 )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2126]) ).
thf(2128,plain,
~ ( empty_set @ ( sk5 @ subset ) ),
inference(pattern_uni,[status(thm)],[2127:[bind(A,$thf( sk5 ))]]) ).
thf(2652,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2390]) ).
thf(2653,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[2652]) ).
thf(2667,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A
!= ( C @ singleton ) ) ),
inference(simp,[status(thm)],[2653]) ).
thf(2675,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2667]) ).
thf(102,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ set_union2 ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( B @ ( D @ in ) )
| ( A @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_xboole_0) ).
thf(595,plain,
! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( B @ ( D @ in ) )
| ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ in ) )
| ( A @ ( D @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ set_union2 ) ) ) )
& ( ( C
= ( B @ ( A @ set_union2 ) ) )
=> ! [D: $i] :
( ( ( ( B @ ( D @ in ) )
| ( A @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ in ) )
| ( A @ ( D @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(2572,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( sk5 @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[180,2128]) ).
thf(2573,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ set_union2 ) @ ( A @ subset ) )
!= ( empty_set @ ( sk5 @ subset ) ) ),
inference(simp,[status(thm)],[2572]) ).
thf(2583,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ set_union2 ) )
!= empty_set )
| ( A != sk5 ) ),
inference(simp,[status(thm)],[2573]) ).
thf(2591,plain,
! [A: $i] :
( ( A @ ( sk5 @ set_union2 ) )
!= empty_set ),
inference(simp,[status(thm)],[2583]) ).
thf(2802,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( sk5 @ set_union2 ) ) )
| ( empty_set != empty_set )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[178,2591]) ).
thf(2803,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( sk5 @ set_union2 ) ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2802]) ).
thf(2804,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( sk5 @ set_union2 ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2803:[bind(A,$thf( D @ ( sk5 @ set_union2 ) )),bind(B,$thf( D ))]]) ).
thf(2807,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( sk5 @ set_union2 ) @ subset ) ),
inference(simp,[status(thm)],[2804]) ).
thf(130,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(876,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(174,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
inference(cnf,[status(esa)],[173]) ).
thf(175,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
inference(lifteq,[status(thm)],[174]) ).
thf(4866,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk3 @ ( sk9 @ subset ) ) )
| ~ ( sk9 @ ( sk3 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[372,4524]) ).
thf(4867,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk3 @ ( sk9 @ subset ) ) )
| ~ ( sk9 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[4866]) ).
thf(4933,plain,
! [A: $i] :
( ( A != sk3 )
| ( sk9 != empty_set )
| ~ ( sk9 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[4867]) ).
thf(4944,plain,
( ( sk9 != empty_set )
| ~ ( sk9 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[4933]) ).
thf(118,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) )
<=> ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t106_zfmisc_1) ).
thf(759,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) )
=> ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) ) )
& ( ( D @ ( C @ cartesian_product2 ) @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( D @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(65,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) )
= ( B @ ( A @ set_difference ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_subset_1) ).
thf(368,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) )
= ( B @ ( A @ set_difference ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(957,plain,
! [A: $i] :
( ( ( A @ singleton @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[169,219]) ).
thf(958,plain,
! [A: $i] :
( ( A @ singleton @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[957]) ).
thf(961,plain,
! [A: $i] :
( ( A @ singleton )
!= sk4 ),
inference(simp,[status(thm)],[958]) ).
thf(95,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(517,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ subset ) ) )
& ( ( B @ ( A @ singleton @ subset ) )
=> ( B @ ( A @ in ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(52,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( C @ ( A @ set_union2 ) @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_xboole_1) ).
thf(319,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( C @ ( A @ set_union2 ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(59,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(351,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(73,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ meet_of_subsets ) )
= ( B @ set_meet ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k6_setfam_1) ).
thf(390,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ meet_of_subsets ) )
= ( B @ set_meet ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(139,plain,
sk3 @ relation,
inference(cnfConj,[status(thm)],[138]) ).
thf(994,plain,
! [B: $i,A: $i] :
( ( ( A @ sk6 @ empty )
!= ( B @ powerset @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[357,232]) ).
thf(995,plain,
! [B: $i,A: $i] :
( ( A @ sk6 @ empty )
!= ( B @ powerset @ empty ) ),
inference(simp,[status(thm)],[994]) ).
thf(999,plain,
! [B: $i,A: $i] :
( ( A @ sk6 )
!= ( B @ powerset ) ),
inference(simp,[status(thm)],[995]) ).
thf(3323,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[227,2390]) ).
thf(3324,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( A @ set_difference ) @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[3323]) ).
thf(3346,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( B @ ( A @ set_difference ) )
!= ( C @ singleton ) ) ),
inference(simp,[status(thm)],[3324]) ).
thf(3360,plain,
! [B: $i,A: $i] :
( ( A @ ( empty_set @ set_difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[3346]) ).
thf(7634,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(rewrite,[status(thm)],[3360,375]) ).
thf(86,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(432,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(146,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ proper_subset ) )
| ~ ( B @ ( A @ proper_subset ) ) ),
inference(cnf,[status(esa)],[145]) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( A @ powerset @ ( B @ ( A @ union_of_subsets ) @ element ) )
| ~ ( A @ powerset @ powerset @ ( B @ element ) ) ),
inference(cnf,[status(esa)],[189]) ).
thf(6578,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk5 @ ( sk4 @ subset ) ) )
| ~ $true
| ~ ( sk4 @ ( sk5 @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[372,4549]) ).
thf(6579,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk5 @ ( sk4 @ subset ) ) )
| ~ ( sk4 @ ( sk5 @ subset ) ) ),
inference(simp,[status(thm)],[6578]) ).
thf(6648,plain,
! [A: $i] :
( ( A != sk5 )
| ( sk4 != empty_set )
| ~ ( sk4 @ ( sk5 @ subset ) ) ),
inference(simp,[status(thm)],[6579]) ).
thf(6671,plain,
( ( sk4 != empty_set )
| ~ ( sk4 @ ( sk5 @ subset ) ) ),
inference(simp,[status(thm)],[6648]) ).
thf(126,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_field )
= ( A @ relation_rng @ ( A @ relation_dom @ set_union2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_relat_1) ).
thf(833,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_field )
= ( A @ relation_rng @ ( A @ relation_dom @ set_union2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(964,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk2 @ in ) ) )
| ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[140,943]) ).
thf(965,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_field @ ( sk2 @ in ) ) )
| ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 ) ),
inference(simp,[status(thm)],[964]) ).
thf(966,plain,
( ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk2 )
| ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 ) ),
inference(simp,[status(thm)],[965]) ).
thf(967,plain,
( ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk2 )
| ( ( sk3 @ relation_field )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 ) ),
inference(simp,[status(thm)],[966]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( B @ set_intersection2 ) @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t19_xboole_1) ).
thf(262,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( B @ set_intersection2 ) @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(6695,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk3 @ ( sk4 @ subset ) ) )
| ~ ( sk4 @ ( sk3 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[372,4722]) ).
thf(6696,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk3 @ ( sk4 @ subset ) ) )
| ~ ( sk4 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[6695]) ).
thf(6762,plain,
! [A: $i] :
( ( A != sk3 )
| ( sk4 != empty_set )
| ~ ( sk4 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[6696]) ).
thf(6782,plain,
( ( sk4 != empty_set )
| ~ ( sk4 @ ( sk3 @ subset ) ) ),
inference(simp,[status(thm)],[6762]) ).
thf(61,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) )
= ( B @ ( A @ set_union2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t39_xboole_1) ).
thf(358,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) )
= ( B @ ( A @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(437,plain,
sk9 @ relation,
inference(cnfConj,[status(thm)],[435]) ).
thf(91,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( A @ singleton @ ( B @ ( A @ unordered_pair ) @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_tarski) ).
thf(474,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( A @ singleton @ ( B @ ( A @ unordered_pair ) @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(5865,plain,
! [A: $i] :
( ( A
!= ( empty_set @ powerset ) )
| ( empty_set != empty_set )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(paramod_ordered,[status(thm)],[594,978]) ).
thf(5866,plain,
! [A: $i] :
( ( A
!= ( empty_set @ powerset ) )
| ( A @ ( A @ sk27 @ in ) ) ),
inference(simp,[status(thm)],[5865]) ).
thf(5867,plain,
empty_set @ powerset @ ( empty_set @ powerset @ sk27 @ in ),
inference(pattern_uni,[status(thm)],[5866:[bind(A,$thf( empty_set @ powerset ))]]) ).
thf(106,axiom,
! [A: $i,B: $i] :
( ~ ( ( B @ ( A @ disjoint ) )
& ? [C: $i] :
( ( B @ ( C @ in ) )
& ( A @ ( C @ in ) ) ) )
& ~ ( ! [C: $i] :
~ ( ( B @ ( C @ in ) )
& ( A @ ( C @ in ) ) )
& ~ ( B @ ( A @ disjoint ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_0) ).
thf(635,plain,
! [A: $i,B: $i] :
( ~ ( ( B @ ( A @ disjoint ) )
& ? [C: $i] :
( ( B @ ( C @ in ) )
& ( A @ ( C @ in ) ) ) )
& ~ ( ! [C: $i] :
~ ( ( B @ ( C @ in ) )
& ( A @ ( C @ in ) ) )
& ~ ( B @ ( A @ disjoint ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(41,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(283,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(981,plain,
! [A: $i] :
( ( ( A @ powerset @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[436,232]) ).
thf(982,plain,
! [A: $i] :
( ( A @ powerset @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[981]) ).
thf(987,plain,
! [A: $i] :
( ( A @ powerset )
!= sk9 ),
inference(simp,[status(thm)],[982]) ).
thf(1525,plain,
! [A: $i] :
( ( ( A @ sk6 @ empty )
!= ( sk3 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[357,1514]) ).
thf(1526,plain,
! [A: $i] :
( ( A @ sk6 @ empty )
!= ( sk3 @ empty ) ),
inference(simp,[status(thm)],[1525]) ).
thf(1532,plain,
! [A: $i] :
( ( A @ sk6 )
!= sk3 ),
inference(simp,[status(thm)],[1526]) ).
thf(39,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(277,plain,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(113,axiom,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(691,plain,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(1737,plain,
! [A: $i] :
( ( ( A @ sk11 @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[447,945]) ).
thf(1738,plain,
! [A: $i] :
( ( ( A @ sk11 @ ( A @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_field @ ( sk2 @ in ) )
!= ( sk3 @ relation_field @ ( sk1 @ in ) ) ) ),
inference(simp,[status(thm)],[1737]) ).
thf(1764,plain,
! [A: $i] :
( ( ( A @ sk11 )
!= ( sk3 @ relation_field ) )
| ( A != sk1 )
| ( ( sk3 @ relation_field )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1738]) ).
thf(1769,plain,
( ( ( sk1 @ sk11 )
!= ( sk3 @ relation_field ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[1764]) ).
thf(222,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ powerset @ ( C @ ( B @ ( A @ subset_difference ) ) @ element ) )
| ~ ( A @ powerset @ ( C @ element ) )
| ~ ( A @ powerset @ ( B @ element ) ) ),
inference(cnf,[status(esa)],[221]) ).
thf(63,axiom,
! [A: $i,B: $i] : ( A @ ( A @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(364,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(31,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_intersection2 ) )
= ( A @ ( B @ set_intersection2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k3_xboole_0) ).
thf(254,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_intersection2 ) )
= ( A @ ( B @ set_intersection2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(104478,plain,
$false,
inference(e,[status(thm)],[235,2630,2031,347,3581,1950,523,379,878,202,977,170,217,276,385,142,715,945,153,670,4803,301,436,257,4524,372,1860,196,934,357,559,421,3216,416,228,289,316,261,1514,189,216,924,789,179,321,376,253,238,937,821,2540,2034,766,147,280,221,453,396,660,998,988,1697,527,1696,2762,233,270,201,220,366,2589,952,349,424,630,429,137,165,229,361,583,324,650,3381,388,356,169,141,420,989,857,3218,725,488,225,339,642,1842,212,622,393,515,1530,2387,1947,371,3215,1776,5850,173,911,792,633,205,311,586,662,926,530,5295,425,1698,2033,264,161,1595,573,180,444,149,375,819,176,423,694,732,2791,843,286,204,259,413,191,402,382,313,2308,7534,144,335,236,181,947,350,159,979,2721,406,172,219,2415,419,230,245,434,303,5506,274,477,915,943,668,198,2842,594,140,959,768,223,306,1000,299,267,438,3490,226,7558,2390,922,167,426,548,1990,553,836,145,162,944,4722,823,282,150,2588,1531,409,199,4549,840,7662,2128,338,2675,2411,154,1008,595,2807,876,175,978,250,931,4944,3188,759,368,2591,961,517,319,351,390,139,999,447,7634,432,146,190,6671,833,967,168,262,373,6782,239,358,437,474,5867,635,283,928,395,987,1532,178,277,691,469,354,309,1769,222,592,364,232,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU180+2 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.07/0.40 % Computer : n007.cluster.edu
% 0.07/0.40 % Model : x86_64 x86_64
% 0.07/0.40 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.40 % Memory : 8046.5625MB
% 0.07/0.40 % OS : Linux 6.8.0-71-generic
% 0.07/0.41 % CPULimit : 300
% 0.07/0.41 % WCLimit : 300
% 0.07/0.41 % DateTime : Sat Sep 26 08:43:38 UTC 2026
% 0.07/0.41 % CPUTime :
% 0.07/0.41 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.28/1.16 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.51/1.60 % [INFO] Parsing done (439ms).
% 2.51/1.63 % [INFO] Running in sequential loop mode.
% 3.67/2.25 % [INFO] eprover registered as external prover.
% 3.67/2.26 % [INFO] Scanning for conjecture ...
% 4.12/2.46 % [INFO] Found a conjecture (or negated_conjecture) and 149 axioms. Running axiom selection ...
% 4.51/2.64 % [INFO] Axiom selection finished. Selected 134 axioms (removed 15 axioms).
% 5.09/2.86 % [INFO] Problem is first-order (TPTP FOF).
% 5.09/2.89 % [INFO] Type checking passed.
% 5.32/2.90 % [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 ...
% 216.02/56.57 % External prover 'e' found a proof!
% 216.02/56.57 % [INFO] Killing All external provers ...
% 216.02/56.57 % Time passed: 55999ms (effective reasoning time: 54935ms)
% 216.02/56.57 % 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)>
% 216.02/56.57 % Axioms used in derivation (134): fc4_subset_1, t2_boole, rc1_subset_1, involutiveness_k7_setfam_1, t19_xboole_1, t17_xboole_1, t43_subset_1, fc1_zfmisc_1, d1_tarski, d5_relat_1, idempotence_k3_xboole_0, t1_boole, t3_xboole_1, fc1_xboole_0, dt_k2_subset_1, d5_tarski, t1_subset, l3_subset_1, t99_zfmisc_1, t47_setfam_1, t37_xboole_1, redefinition_k6_setfam_1, commutativity_k3_xboole_0, d2_tarski, d4_subset_1, involutiveness_k3_subset_1, fc2_xboole_0, d1_xboole_0, l50_zfmisc_1, t118_zfmisc_1, t46_zfmisc_1, t6_boole, rc2_subset_1, dt_k6_subset_1, d1_relat_1, t1_xboole_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, t4_subset, fc3_xboole_0, d2_xboole_0, d4_tarski, t9_zfmisc_1, redefinition_k5_setfam_1, t7_boole, d2_subset_1, t48_xboole_1, fc1_subset_1, t33_zfmisc_1, d7_xboole_0, t83_xboole_1, l28_zfmisc_1, t46_setfam_1, l4_zfmisc_1, t3_subset, d2_zfmisc_1, existence_m1_subset_1, antisymmetry_r2_xboole_0, dt_k6_setfam_1, l3_zfmisc_1, commutativity_k2_tarski, t106_zfmisc_1, l2_zfmisc_1, t48_setfam_1, l71_subset_1, d5_subset_1, t92_zfmisc_1, d1_zfmisc_1, d8_setfam_1, t69_enumset1, t38_zfmisc_1, l55_zfmisc_1, t26_xboole_1, t63_xboole_1, t8_xboole_1, t28_xboole_1, d3_tarski, t39_zfmisc_1, t119_zfmisc_1, rc1_relat_1, reflexivity_r1_tarski, t6_zfmisc_1, t2_xboole_1, d4_xboole_0, dt_k5_setfam_1, t36_xboole_1, t20_relat_1, t54_subset_1, dt_k7_setfam_1, t136_zfmisc_1, d6_relat_1, l23_zfmisc_1, t8_boole, rc1_xboole_0, t12_xboole_1, t21_relat_1, t10_zfmisc_1, t2_subset, t37_zfmisc_1, fc2_relat_1, d1_setfam_1, symmetry_r1_xboole_0, idempotence_k2_xboole_0, t9_tarski, 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, t39_xboole_1, antisymmetry_r2_hidden, dt_k3_subset_1, l1_zfmisc_1
% 216.02/56.57 % No. of inferences in proof: 675
% 216.02/56.57 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 55999 ms resp. 54935 ms w/o parsing
% 213.08/56.81 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 213.08/56.82 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------