%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SEU177+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 : n009.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 61.62s 17.37s
% Output : Refutation 53.68s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 140
% Syntax : Number of formulae : 349 ( 119 unt; 0 typ; 12 def)
% Number of atoms : 1079 ( 322 equ; 0 cnn)
% Maximal formula atoms : 40 ( 3 avg)
% Number of connectives : 3149 ( 251 ~; 99 |; 224 &;2224 @)
% ( 46 <=>; 305 =>; 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 : 43 ( 40 usr; 9 con; 0-3 aty)
% Number of variables : 749 ( 11 ^; 673 !; 53 ?; 749 :)
% ( 0 !>; 0 ?*; 0 @-; 12 @+)
% 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_dom_decl,type,
relation_dom: $i > $i ).
thf(relation_rng_decl,type,
relation_rng: $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(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(sk9_decl,type,
sk9: $i ).
thf(sk27_decl,type,
sk27: $i > $i ).
thf(sk48_decl,type,
sk48: $i > $i > $i > $i ).
thf(sk49_decl,type,
sk49: $i > $i > $o ).
thf(sk50_decl,type,
sk50: $i > $i > $i ).
thf(sk51_decl,type,
sk51: $i > $i > $i ).
thf(sk52_decl,type,
sk52: $i > $i > $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ! [B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( C @ relation_rng @ ( B @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: $i] :
~ ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ ( sk1 @ ordered_pair ) @ in ) )
=> ( ( B @ relation_rng @ ( A @ in ) )
& ( B @ relation_dom @ ( sk1 @ in ) ) ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( @+[A: $i] :
~ ( ( A @ relation )
=> ( ( A @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
=> ( ( A @ relation_rng @ ( sk2 @ in ) )
& ( A @ relation_dom @ ( 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(sk9_def,definition,
( sk9
= ( @+[A: $i] :
( ( A @ relation )
& ( A @ empty ) ) ) ) ).
thf(sk27_def,definition,
( sk27
= ( ^ [A: $i] :
@+[B: $i] : ( A @ ( B @ in ) ) ) ) ).
thf(sk48_def,definition,
( sk48
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
~ ( ( C @ ( D @ in ) )
=> ( D @ ( A @ in ) ) ) ) ) ).
thf(sk50_def,definition,
( sk50
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ in ) )
=> ! [D: $i] :
( ( B @ ( D @ in ) )
=> ( D @ ( C @ in ) ) ) ) ) ) ).
thf(sk51_def,definition,
( sk51
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( B @ ( C @ in ) )
=> ( C @ ( B @ ( A @ sk50 ) @ in ) ) ) ) ) ).
thf(sk52_def,definition,
( sk52
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ! [D: $i] :
( ( B @ ( D @ in ) )
=> ( D @ ( C @ in ) ) )
=> ( A @ ( C @ in ) ) ) ) ) ).
thf(85,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(423,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(129,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(892,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
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(138,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
=> ~ ( A @ ( B @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(92,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(479,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)],[92]) ).
thf(1,conjecture,
! [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(2,negated_conjecture,
~ ! [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(neg_conjecture,[status(cth)],[1]) ).
thf(130,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)],[2]) ).
thf(131,plain,
( ( ~ ( sk3 @ relation_rng @ ( sk2 @ in ) )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
& ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
& ( sk3 @ relation ) ),
inference(cnf,[status(esa)],[130]) ).
thf(133,plain,
sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ),
inference(cnfConj,[status(thm)],[131]) ).
thf(134,plain,
( ~ ( sk3 @ relation_rng @ ( sk2 @ in ) )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(cnfConj,[status(thm)],[131]) ).
thf(904,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_rng @ ( sk2 @ in ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[133,134]) ).
thf(905,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_rng @ ( sk2 @ in ) ) ),
inference(simp,[status(thm)],[904]) ).
thf(909,plain,
( ( ( sk3 @ relation_dom )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 )
| ~ ( sk3 @ relation_rng @ ( sk2 @ in ) ) ),
inference(simp,[status(thm)],[905]) ).
thf(91,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(468,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)],[91]) ).
thf(119,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(777,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(61,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(352,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)],[61]) ).
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(276,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
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(247,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_intersection2 ) )
= ( A @ ( B @ set_intersection2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(71,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(379,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)],[71]) ).
thf(95,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(511,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)],[95]) ).
thf(102,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(614,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)],[102]) ).
thf(73,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_boole) ).
thf(384,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_subset_1) ).
thf(140,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(141,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(cnf,[status(esa)],[140]) ).
thf(142,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(lifteq,[status(thm)],[141]) ).
thf(57,axiom,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(340,plain,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(21,axiom,
empty_set @ empty,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(213,plain,
empty_set @ empty,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(26,axiom,
! [A: $i] :
~ ( A @ powerset @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(223,plain,
! [A: $i] :
~ ( A @ powerset @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(224,plain,
~ ? [A: $i] : ( A @ powerset @ empty ),
inference(miniscope,[status(thm)],[223]) ).
thf(225,plain,
! [A: $i] :
~ ( A @ powerset @ empty ),
inference(cnf,[status(esa)],[224]) ).
thf(924,plain,
! [A: $i] :
( ( ( A @ powerset @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[213,225]) ).
thf(925,plain,
! [A: $i] :
( ( A @ powerset @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[924]) ).
thf(930,plain,
! [A: $i] :
( ( A @ powerset )
!= empty_set ),
inference(simp,[status(thm)],[925]) ).
thf(111,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(709,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)],[111]) ).
thf(83,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(417,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( C @ empty )
& ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
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(257,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
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(174,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(78,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(404,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)],[78]) ).
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(184,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(63,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(357,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ disjoint ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
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(216,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(70,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(376,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)],[70]) ).
thf(104,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(623,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)],[104]) ).
thf(908,plain,
( ~ $true
| ( ( sk3 @ relation_rng @ ( sk2 @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(eqfactor_ordered,[status(thm)],[134]) ).
thf(912,plain,
( ( ( sk3 @ relation_rng )
!= ( sk3 @ relation_dom ) )
| ( sk2 != sk1 )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[908]) ).
thf(935,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_rng )
!= ( sk3 @ relation_dom ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[133,912]) ).
thf(936,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ( ( sk3 @ relation_rng )
!= ( sk3 @ relation_dom ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[935]) ).
thf(937,plain,
( ( ( sk3 @ relation_dom )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk1 )
| ( ( sk3 @ relation_rng )
!= ( sk3 @ relation_dom ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[936]) ).
thf(59,axiom,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(345,plain,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(346,plain,
! [A: $i] :
( ( A @ sk6 @ empty )
& ( A @ powerset @ ( A @ sk6 @ element ) ) ),
inference(cnf,[status(esa)],[345]) ).
thf(348,plain,
! [A: $i] : ( A @ sk6 @ empty ),
inference(cnfConj,[status(thm)],[346]) ).
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(147,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(76,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(397,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)],[76]) ).
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(221,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(132,plain,
sk3 @ relation,
inference(cnfConj,[status(thm)],[131]) ).
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(243,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(86,axiom,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(425,plain,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(426,plain,
( ( sk9 @ relation )
& ( sk9 @ empty ) ),
inference(cnf,[status(esa)],[425]) ).
thf(428,plain,
sk9 @ relation,
inference(cnfConj,[status(thm)],[426]) ).
thf(52,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(312,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)],[52]) ).
thf(50,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(307,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)],[50]) ).
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(292,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(72,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(381,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)],[72]) ).
thf(120,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(780,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)],[120]) ).
thf(781,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)],[780]) ).
thf(782,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 @ sk52 ) @ in ) )
| ( F @ ( A @ sk49 ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ( G @ ( A @ ( F @ sk52 ) @ in ) )
| ~ ( A @ ( G @ in ) )
| ( F @ ( A @ sk49 ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk49 ) )
| ~ ( A @ ( F @ sk51 ) @ ( A @ ( F @ sk50 ) @ in ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk49 ) )
| ( A @ ( A @ ( F @ sk51 ) @ in ) )
| ( A = empty_set ) )
& ( ( F
= ( A @ set_meet ) )
| ~ ( F @ ( A @ sk49 ) )
| ( F @ ( A @ ( F @ sk50 ) @ in ) )
| ( A = empty_set ) )
& ( ( B @ ( E @ in ) )
| ~ ( A @ ( B @ ( E @ sk48 ) ) @ ( E @ in ) )
| ( B
!= ( A @ set_meet ) )
| ( A = empty_set ) )
& ( ( B @ ( E @ in ) )
| ( A @ ( A @ ( B @ ( E @ sk48 ) ) @ 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)],[781]) ).
thf(788,plain,
! [B: $i,A: $i] :
( ( B = empty_set )
| ( B
!= ( A @ set_meet ) )
| ( A != empty_set ) ),
inference(cnfConj,[status(thm)],[782]) ).
thf(797,plain,
! [B: $i,A: $i] :
( ( B = empty_set )
| ( B
!= ( A @ set_meet ) )
| ( A != empty_set ) ),
inference(lifteq,[status(thm)],[788]) ).
thf(798,plain,
( ( empty_set @ set_meet )
= empty_set ),
inference(simp,[status(thm)],[797]) ).
thf(39,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(270,plain,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(96,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(529,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)],[96]) ).
thf(906,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_rng @ ( sk2 @ in ) ) )
| ~ $true
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(paramod_ordered,[status(thm)],[133,134]) ).
thf(907,plain,
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( sk3 @ relation_rng @ ( sk2 @ in ) ) )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[906]) ).
thf(910,plain,
( ( ( sk3 @ relation_rng )
!= sk3 )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= sk2 )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[907]) ).
thf(48,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(302,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(46,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(297,plain,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(87,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(429,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)],[87]) ).
thf(3,axiom,
! [A: $i,B: $i] :
~ ( A @ ( A @ proper_subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(135,plain,
! [A: $i] :
~ ( A @ ( A @ proper_subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(136,plain,
~ ? [A: $i] : ( A @ ( A @ proper_subset ) ),
inference(miniscope,[status(thm)],[135]) ).
thf(137,plain,
! [A: $i] :
~ ( A @ ( A @ proper_subset ) ),
inference(cnf,[status(esa)],[136]) ).
thf(122,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(815,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)],[122]) ).
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(260,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(60,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(349,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) )
= ( B @ ( A @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
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(192,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( A @ ( B @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(79,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(407,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B
= ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(28,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(229,plain,
! [A: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(33,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(252,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(18,axiom,
? [A: $i] :
~ ( A @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(195,plain,
? [A: $i] :
~ ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(196,plain,
~ ! [A: $i] : ( A @ empty ),
inference(miniscope,[status(thm)],[195]) ).
thf(197,plain,
~ ( sk5 @ empty ),
inference(cnf,[status(esa)],[196]) ).
thf(88,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(435,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)],[88]) ).
thf(54,axiom,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(326,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(327,plain,
~ ? [A: $i] :
( ( A @ singleton )
= empty_set ),
inference(miniscope,[status(thm)],[326]) ).
thf(328,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(cnf,[status(esa)],[327]) ).
thf(329,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(lifteq,[status(thm)],[328]) ).
thf(108,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(665,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)],[108]) ).
thf(110,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(682,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)],[110]) ).
thf(115,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(742,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)],[115]) ).
thf(12,axiom,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(169,plain,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(84,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(420,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(20,axiom,
! [A: $i] :
~ ( A @ singleton @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(210,plain,
! [A: $i] :
~ ( A @ singleton @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(211,plain,
~ ? [A: $i] : ( A @ singleton @ empty ),
inference(miniscope,[status(thm)],[210]) ).
thf(212,plain,
! [A: $i] :
~ ( A @ singleton @ empty ),
inference(cnf,[status(esa)],[211]) ).
thf(75,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(393,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)],[75]) ).
thf(911,plain,
( ( ( sk3 @ relation_rng @ ( sk2 @ in ) )
!= ( sk3 @ relation_dom @ ( sk1 @ in ) ) )
| ~ ( sk3 @ relation_dom @ ( sk1 @ in ) ) ),
inference(simp,[status(thm)],[908]) ).
thf(94,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(508,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)],[94]) ).
thf(118,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(773,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)],[118]) ).
thf(113,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(718,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)],[113]) ).
thf(127,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(888,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)],[127]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_enumset1) ).
thf(412,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(11,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_boole) ).
thf(166,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(9,axiom,
? [A: $i] : ( A @ empty ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(161,plain,
? [A: $i] : ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
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(279,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(98,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(567,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)],[98]) ).
thf(568,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)],[567]) ).
thf(569,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C = empty_set )
| ( C @ ( C @ sk27 @ in ) ) )
& ( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ) ),
inference(cnf,[status(esa)],[568]) ).
thf(570,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ),
inference(cnfConj,[status(thm)],[569]) ).
thf(572,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ in ) )
| ( A != empty_set ) ),
inference(lifteq,[status(thm)],[570]) ).
thf(573,plain,
! [A: $i] :
~ ( empty_set @ ( A @ in ) ),
inference(simp,[status(thm)],[572]) ).
thf(89,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(444,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)],[89]) ).
thf(106,axiom,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(641,plain,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(101,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(611,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(97,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(540,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)],[97]) ).
thf(898,plain,
( ( ( sk5 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[213,197]) ).
thf(899,plain,
( ( sk5 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[898]) ).
thf(900,plain,
sk5 != empty_set,
inference(simp,[status(thm)],[899]) ).
thf(13,axiom,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(172,plain,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(24,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(219,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(103,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(616,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)],[103]) ).
thf(65,axiom,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(362,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(128,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(890,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(74,axiom,
! [A: $i,B: $i] :
( ( B @ powerset @ ( A @ element ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(387,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)],[74]) ).
thf(105,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(631,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)],[105]) ).
thf(123,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(848,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)],[123]) ).
thf(116,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(769,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
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(155,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(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(198,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(55,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ( ( B @ ( A @ set_difference ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(330,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)],[55]) ).
thf(67,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(367,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(956,plain,
! [A: $i] :
( ( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( empty_set @ ( A @ in ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[133,573]) ).
thf(957,plain,
! [A: $i] :
( ( sk3 @ ( sk2 @ ( sk1 @ ordered_pair ) @ in ) )
!= ( empty_set @ ( A @ in ) ) ),
inference(simp,[status(thm)],[956]) ).
thf(958,plain,
! [A: $i] :
( ( sk3 != empty_set )
| ( ( sk2 @ ( sk1 @ ordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[957]) ).
thf(959,plain,
sk3 != empty_set,
inference(simp,[status(thm)],[958]) ).
thf(27,axiom,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(226,plain,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(267,plain,
! [A: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(162,plain,
sk4 @ empty,
inference(cnf,[status(esa)],[161]) ).
thf(917,plain,
! [A: $i] :
( ( ( A @ singleton @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[162,212]) ).
thf(918,plain,
! [A: $i] :
( ( A @ singleton @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[917]) ).
thf(922,plain,
! [A: $i] :
( ( A @ singleton )
!= sk4 ),
inference(simp,[status(thm)],[918]) ).
thf(109,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(675,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)],[109]) ).
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(255,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(58,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(342,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(62,axiom,
! [A: $i,B: $i] : ( A @ ( A @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(355,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
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(282,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(114,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(739,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)],[114]) ).
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(263,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(51,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(310,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)],[51]) ).
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(182,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(68,axiom,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(370,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(56,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(338,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)],[56]) ).
thf(99,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(576,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)],[99]) ).
thf(64,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(359,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)],[64]) ).
thf(341,plain,
( ( empty_set @ singleton )
= ( empty_set @ powerset ) ),
inference(lifteq,[status(thm)],[340]) ).
thf(427,plain,
sk9 @ empty,
inference(cnfConj,[status(thm)],[426]) ).
thf(919,plain,
! [A: $i] :
( ( ( A @ singleton @ empty )
!= ( sk9 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[427,212]) ).
thf(920,plain,
! [A: $i] :
( ( A @ singleton @ empty )
!= ( sk9 @ empty ) ),
inference(simp,[status(thm)],[919]) ).
thf(923,plain,
! [A: $i] :
( ( A @ singleton )
!= sk9 ),
inference(simp,[status(thm)],[920]) ).
thf(977,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= ( A @ singleton ) )
| ( ( empty_set @ powerset )
!= sk9 ) ),
inference(paramod_ordered,[status(thm)],[341,923]) ).
thf(978,plain,
( ( empty_set @ powerset )
!= sk9 ),
inference(pattern_uni,[status(thm)],[977:[bind(A,$thf( empty_set ))]]) ).
thf(6,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(143,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(363,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(cnf,[status(esa)],[362]) ).
thf(117,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(771,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)],[117]) ).
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(250,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( A @ ( B @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(90,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(465,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)],[90]) ).
thf(975,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= ( A @ singleton ) )
| ( ( empty_set @ powerset )
!= sk4 ) ),
inference(paramod_ordered,[status(thm)],[341,922]) ).
thf(976,plain,
( ( empty_set @ powerset )
!= sk4 ),
inference(pattern_uni,[status(thm)],[975:[bind(A,$thf( empty_set ))]]) ).
thf(901,plain,
( ( ( sk9 @ empty )
!= ( sk5 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[427,197]) ).
thf(902,plain,
( ( sk9 @ empty )
!= ( sk5 @ empty ) ),
inference(simp,[status(thm)],[901]) ).
thf(903,plain,
sk9 != sk5,
inference(simp,[status(thm)],[902]) ).
thf(121,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(813,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(49,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(304,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)],[49]) ).
thf(945,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= ( A @ singleton ) )
| ( ( empty_set @ powerset )
!= empty_set ) ),
inference(paramod_ordered,[status(thm)],[341,329]) ).
thf(946,plain,
( ( empty_set @ powerset )
!= empty_set ),
inference(pattern_uni,[status(thm)],[945:[bind(A,$thf( empty_set ))]]) ).
thf(139,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ proper_subset ) )
| ~ ( B @ ( A @ proper_subset ) ) ),
inference(cnf,[status(esa)],[138]) ).
thf(112,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(716,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
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(214,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(47,axiom,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(300,plain,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(53,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(315,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)],[53]) ).
thf(80,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(410,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(144,plain,
~ ? [A: $i] :
( ? [B: $i] :
( ( B @ empty )
& ( A != B ) )
& ( A @ empty ) ),
inference(miniscope,[status(thm)],[143]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ~ ( B @ empty )
| ( A = B )
| ~ ( A @ empty ) ),
inference(cnf,[status(esa)],[144]) ).
thf(146,plain,
! [B: $i,A: $i] :
( ~ ( B @ empty )
| ~ ( A @ empty )
| ( A = B ) ),
inference(lifteq,[status(thm)],[145]) ).
thf(124,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(854,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)],[124]) ).
thf(126,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(867,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)],[126]) ).
thf(40,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_boole) ).
thf(273,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(69,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(373,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_union2 ) )
= ( A @ ( B @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(100,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(603,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)],[100]) ).
thf(45,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(294,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)],[45]) ).
thf(93,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(506,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(895,plain,
( ( ( sk5 @ empty )
!= ( sk4 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[162,197]) ).
thf(896,plain,
( ( sk5 @ empty )
!= ( sk4 @ empty ) ),
inference(simp,[status(thm)],[895]) ).
thf(897,plain,
sk5 != sk4,
inference(simp,[status(thm)],[896]) ).
thf(77,axiom,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(400,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(82,axiom,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(415,plain,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(125,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(860,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)],[125]) ).
thf(107,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(644,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)],[107]) ).
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(163,plain,
! [A: $i,B: $i] :
~ ( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(66,axiom,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_boole) ).
thf(364,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
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(232,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(21383,plain,
$false,
inference(e,[status(thm)],[423,892,138,479,909,468,777,352,276,247,379,511,614,384,142,340,930,709,417,257,174,404,184,357,216,376,623,937,348,147,397,221,132,133,243,428,312,307,292,381,798,270,529,910,302,297,429,137,815,260,349,192,407,229,252,197,435,329,665,682,742,169,420,225,212,393,911,134,508,773,718,888,412,425,166,161,279,573,444,641,611,540,900,172,219,616,362,567,890,387,631,140,213,848,769,155,198,330,345,367,912,130,959,780,223,135,226,267,922,675,162,255,342,355,282,739,263,310,182,370,338,576,359,978,143,363,771,250,465,976,903,813,304,946,139,716,214,300,315,410,146,854,867,273,373,603,210,326,294,341,506,195,897,400,415,427,860,644,163,923,364,232]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU177+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
% 0.15/5.42 % Computer : n009.cluster.edu
% 0.15/5.42 % Model : x86_64 x86_64
% 0.15/5.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/5.42 % Memory : 8046.5625MB
% 0.15/5.42 % OS : Linux 6.8.0-71-generic
% 0.15/5.42 % CPULimit : 300
% 0.15/5.42 % WCLimit : 300
% 0.15/5.42 % DateTime : Sat Sep 26 08:45:04 UTC 2026
% 0.15/5.42 % CPUTime :
% 0.15/5.42 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
% 0.19/5.99 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.44/6.26 % [INFO] Parsing done (269ms).
% 1.44/6.28 % [INFO] Running in sequential loop mode.
% 2.40/6.68 % [INFO] eprover registered as external prover.
% 2.40/6.69 % [INFO] Scanning for conjecture ...
% 2.80/6.82 % [INFO] Found a conjecture (or negated_conjecture) and 141 axioms. Running axiom selection ...
% 3.03/6.93 % [INFO] Axiom selection finished. Selected 127 axioms (removed 14 axioms).
% 3.44/7.08 % [INFO] Problem is first-order (TPTP FOF).
% 3.44/7.10 % [INFO] Type checking passed.
% 3.44/7.11 % [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 ...
% 61.62/17.36 % External prover 'e' found a proof!
% 61.62/17.36 % [INFO] Killing All external provers ...
% 61.62/17.36 % Time passed: 11800ms (effective reasoning time: 11078ms)
% 61.62/17.36 % 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)>
% 61.62/17.37 % Axioms used in derivation (127): 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, 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, t54_subset_1, dt_k7_setfam_1, t136_zfmisc_1, l23_zfmisc_1, t8_boole, rc1_xboole_0, t12_xboole_1, t37_zfmisc_1, t10_zfmisc_1, t2_subset, 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, 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
% 61.62/17.37 % No. of inferences in proof: 337
% 61.62/17.37 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 11800 ms resp. 11078 ms w/o parsing
% 53.68/17.58 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 53.68/17.58 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------