%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SEU303+2 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : 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 08:38:23 AM UTC 2026
% Result : Theorem 160.98s 42.82s
% Output : Refutation 163.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 422
% Syntax : Number of formulae : 1026 ( 239 unt; 0 typ; 27 def)
% Number of atoms : 4679 (1005 equ; 0 cnn)
% Maximal formula atoms : 49 ( 4 avg)
% Number of connectives : 12962 ( 666 ~; 199 |;1723 &;8782 @)
% ( 154 <=>;1438 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 8 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 24 con; 0-3 aty)
% Number of variables : 2581 ( 15 ^;2189 !; 350 ?;2581 :)
% ( 0 !>; 0 ?*; 0 @-; 27 @+)
% Comments :
%------------------------------------------------------------------------------
thf(relation_decl,type,
relation: $i > $o ).
thf(function_decl,type,
function: $i > $o ).
thf(finite_decl,type,
finite: $i > $o ).
thf(relation_dom_decl,type,
relation_dom: $i > $i ).
thf(relation_rng_decl,type,
relation_rng: $i > $i ).
thf(cast_to_subset_decl,type,
cast_to_subset: $i > $i ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(proper_subset_decl,type,
proper_subset: $i > $i > $o ).
thf(union_decl,type,
union: $i > $i ).
thf(in_decl,type,
in: $i > $i > $o ).
thf(ordinal_decl,type,
ordinal: $i > $o ).
thf(ordinal_subset_decl,type,
ordinal_subset: $i > $i > $o ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(being_limit_ordinal_decl,type,
being_limit_ordinal: $i > $o ).
thf(succ_decl,type,
succ: $i > $i ).
thf(empty_decl,type,
empty: $i > $o ).
thf(pair_first_decl,type,
pair_first: $i > $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(pair_second_decl,type,
pair_second: $i > $i ).
thf(element_decl,type,
element: $i > $i > $o ).
thf(powerset_decl,type,
powerset: $i > $i ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(subset_difference_decl,type,
subset_difference: $i > $i > $i > $i ).
thf(union_of_subsets_decl,type,
union_of_subsets: $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_difference_decl,type,
set_difference: $i > $i > $i ).
thf(cartesian_product2_decl,type,
cartesian_product2: $i > $i > $i ).
thf(set_union2_decl,type,
set_union2: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(well_ordering_decl,type,
well_ordering: $i > $o ).
thf(inclusion_relation_decl,type,
inclusion_relation: $i > $i ).
thf(relation_of2_decl,type,
relation_of2: $i > $i > $i > $o ).
thf(epsilon_transitive_decl,type,
epsilon_transitive: $i > $o ).
thf(epsilon_connected_decl,type,
epsilon_connected: $i > $o ).
thf(disjoint_decl,type,
disjoint: $i > $i > $o ).
thf(subset_complement_decl,type,
subset_complement: $i > $i > $i ).
thf(natural_decl,type,
natural: $i > $o ).
thf(set_intersection2_decl,type,
set_intersection2: $i > $i > $i ).
thf(set_meet_decl,type,
set_meet: $i > $i ).
thf(relation_of2_as_subset_decl,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(relation_rng_as_subset_decl,type,
relation_rng_as_subset: $i > $i > $i > $i ).
thf(connected_decl,type,
connected: $i > $o ).
thf(are_equipotent_decl,type,
are_equipotent: $i > $i > $o ).
thf(relation_dom_as_subset_decl,type,
relation_dom_as_subset: $i > $i > $i > $i ).
thf(transitive_decl,type,
transitive: $i > $o ).
thf(well_founded_relation_decl,type,
well_founded_relation: $i > $o ).
thf(quasi_total_decl,type,
quasi_total: $i > $i > $i > $o ).
thf(antisymmetric_decl,type,
antisymmetric: $i > $o ).
thf(equipotent_decl,type,
equipotent: $i > $i > $o ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i > $i ).
thf(reflexive_decl,type,
reflexive: $i > $o ).
thf(unordered_triple_decl,type,
unordered_triple: $i > $i > $i > $i ).
thf(relation_dom_restriction_decl,type,
relation_dom_restriction: $i > $i > $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(relation_field_decl,type,
relation_field: $i > $i ).
thf(well_orders_decl,type,
well_orders: $i > $i > $o ).
thf(is_reflexive_in_decl,type,
is_reflexive_in: $i > $i > $o ).
thf(is_transitive_in_decl,type,
is_transitive_in: $i > $i > $o ).
thf(is_antisymmetric_in_decl,type,
is_antisymmetric_in: $i > $i > $o ).
thf(is_connected_in_decl,type,
is_connected_in: $i > $i > $o ).
thf(is_well_founded_in_decl,type,
is_well_founded_in: $i > $i > $o ).
thf(relation_image_decl,type,
relation_image: $i > $i > $i ).
thf(relation_empty_yielding_decl,type,
relation_empty_yielding: $i > $o ).
thf(function_inverse_decl,type,
function_inverse: $i > $i ).
thf(relation_inverse_decl,type,
relation_inverse: $i > $i ).
thf(relation_isomorphism_decl,type,
relation_isomorphism: $i > $i > $i > $o ).
thf(relation_inverse_image_decl,type,
relation_inverse_image: $i > $i > $i ).
thf(relation_restriction_decl,type,
relation_restriction: $i > $i > $i ).
thf(relation_composition_decl,type,
relation_composition: $i > $i > $i ).
thf(relation_rng_restriction_decl,type,
relation_rng_restriction: $i > $i > $i ).
thf(fiber_decl,type,
fiber: $i > $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i > $i > $i > $i ).
thf(sk3_decl,type,
sk3: $i > $i > $o ).
thf(sk4_decl,type,
sk4: $i > $i > $i ).
thf(sk5_decl,type,
sk5: $i > $i > $i ).
thf(sk6_decl,type,
sk6: $i > $i > $i ).
thf(sk12_decl,type,
sk12: $i ).
thf(sk15_decl,type,
sk15: $o ).
thf(sk16_decl,type,
sk16: $i ).
thf(sk17_decl,type,
sk17: $i > $i > $i ).
thf(sk18_decl,type,
sk18: $i ).
thf(sk19_decl,type,
sk19: $i > $i ).
thf(sk20_decl,type,
sk20: $i > $i > $i ).
thf(sk29_decl,type,
sk29: $i ).
thf(sk37_decl,type,
sk37: $i ).
thf(sk41_decl,type,
sk41: $i ).
thf(sk55_decl,type,
sk55: $i ).
thf(sk76_decl,type,
sk76: $i ).
thf(sk125_decl,type,
sk125: $i > $i ).
thf(sk176_decl,type,
sk176: $i ).
thf(sk205_decl,type,
sk205: $i ).
thf(sk215_decl,type,
sk215: $i ).
thf(sk226_decl,type,
sk226: $i ).
thf(sk273_decl,type,
sk273: $i ).
thf(sk285_decl,type,
sk285: $i ).
thf(sk322_decl,type,
sk322: $i ).
thf(sk332_decl,type,
sk332: $i ).
thf(sk358_decl,type,
sk358: $i ).
thf(sk379_decl,type,
sk379: $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_dom @ finite )
=> ( A @ relation_rng @ finite ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
( ( C @ ( D @ in ) )
& ( D @ ( A @ in ) ) ) ) ) ).
thf(sk4_def,definition,
( sk4
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ in ) )
=> ? [D: $i] :
( ( B @ ( D @ in ) )
& ( D @ ( C @ in ) ) ) ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ? [D: $i] :
( ( B @ ( D @ in ) )
& ( D @ ( C @ in ) ) )
=> ( A @ ( C @ in ) ) ) ) ) ).
thf(sk6_def,definition,
( sk6
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ( B @ ( C @ in ) )
& ( C @ ( B @ ( A @ sk5 ) @ in ) ) ) ) ) ).
thf(sk12_def,definition,
( sk12
= ( @+[A: $i] : ( A @ empty ) ) ) ).
thf(sk16_def,definition,
( sk16
= ( @+[A: $i] :
~ ( ( A @ ordinal )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ~ ? [D: $i] :
( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ~ ? [C: $i] :
( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ) ) ) ).
thf(sk17_def,definition,
( sk17
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( A @ ( D @ in ) ) )
=> ( D = C ) )
& ( A @ ( C @ in ) ) ) ) ) ).
thf(sk18_def,definition,
( sk18
= ( @+[A: $i] :
~ ( ( sk16 @ powerset @ powerset @ ( A @ element ) )
=> ~ ( ~ ? [B: $i] :
( ! [C: $i] :
( ( ( C @ ( B @ subset ) )
& ( A @ ( C @ in ) ) )
=> ( C = B ) )
& ( A @ ( B @ in ) ) )
& ( A != empty_set ) ) ) ) ) ).
thf(sk19_def,definition,
( sk19
= ( ^ [A: $i] :
@+[B: $i] :
~ ( ( ( B @ ( A @ subset ) )
& ( sk18 @ ( B @ in ) ) )
=> ( B = A ) ) ) ) ).
thf(sk20_def,definition,
( sk20
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( A @ ( D @ in ) ) )
=> ( D = C ) )
& ( A @ ( C @ in ) ) ) ) ) ).
thf(sk29_def,definition,
( sk29
= ( @+[A: $i] :
~ ( A @ empty ) ) ) ).
thf(sk37_def,definition,
( sk37
= ( @+[A: $i] :
( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ) ) ) ).
thf(sk41_def,definition,
( sk41
= ( @+[A: $i] :
( ( A @ being_limit_ordinal )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ) ) ).
thf(sk55_def,definition,
( sk55
= ( @+[A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ) ) ) ).
thf(sk76_def,definition,
( sk76
= ( @+[A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ) ) ).
thf(sk125_def,definition,
( sk125
= ( ^ [A: $i] :
@+[B: $i] :
~ ( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) ) ) ) ).
thf(sk176_def,definition,
( sk176
= ( @+[A: $i] :
( ( A @ relation_empty_yielding )
& ( A @ relation ) ) ) ) ).
thf(sk205_def,definition,
( sk205
= ( @+[A: $i] :
( ( A @ relation )
& ( A @ empty ) ) ) ) ).
thf(sk215_def,definition,
( sk215
= ( @+[A: $i] :
( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ) ) ) ).
thf(sk226_def,definition,
( sk226
= ( @+[A: $i] :
( ( A @ finite )
& ~ ( A @ empty ) ) ) ) ).
thf(sk273_def,definition,
( sk273
= ( @+[A: $i] :
( ( A @ function )
& ( A @ relation_empty_yielding )
& ( A @ relation ) ) ) ) ).
thf(sk285_def,definition,
( sk285
= ( @+[A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ) ) ) ).
thf(sk322_def,definition,
( sk322
= ( @+[A: $i] :
( ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ) ) ) ).
thf(sk332_def,definition,
( sk332
= ( @+[A: $i] :
( ( A @ function )
& ( A @ empty )
& ( A @ relation ) ) ) ) ).
thf(sk358_def,definition,
( sk358
= ( @+[A: $i] :
( ( A @ relation )
& ~ ( A @ empty ) ) ) ) ).
thf(sk379_def,definition,
( sk379
= ( @+[A: $i] :
( ( A @ function )
& ( A @ relation ) ) ) ) ).
thf(98,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ subset ) )
<=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t39_zfmisc_1) ).
thf(1036,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)],[98]) ).
thf(240,axiom,
! [A: $i] :
( ~ ( A @ empty )
=> ~ ( ! [B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ~ ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( C @ ( C @ ( B @ apply ) @ in ) ) )
& ( ( B @ relation_dom )
= A ) ) )
& ! [B: $i] :
~ ( ( B = empty_set )
& ( A @ ( B @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t28_wellord2) ).
thf(2094,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ~ ( ! [B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ~ ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( C @ ( C @ ( B @ apply ) @ in ) ) )
& ( ( B @ relation_dom )
= A ) ) )
& ! [B: $i] :
~ ( ( B = empty_set )
& ( A @ ( B @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc3_ordinal1) ).
thf(629,plain,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(313,axiom,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_ordinal1) ).
thf(2607,plain,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(2608,plain,
( ( sk285 @ ordinal )
& ( sk285 @ epsilon_connected )
& ( sk285 @ epsilon_transitive )
& ( sk285 @ empty )
& ( sk285 @ one_to_one )
& ( sk285 @ function )
& ( sk285 @ relation ) ),
inference(cnf,[status(esa)],[2607]) ).
thf(2612,plain,
sk285 @ epsilon_transitive,
inference(cnfConj,[status(thm)],[2608]) ).
thf(269,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( A @ relation_rng_restriction ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k8_relat_1) ).
thf(2263,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( A @ relation_rng_restriction ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_ordinal1) ).
thf(909,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(117,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_difference ) @ ( A @ set_difference ) )
= ( B @ ( A @ set_intersection2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t48_xboole_1) ).
thf(1295,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_difference ) @ ( A @ set_difference ) )
= ( B @ ( A @ set_intersection2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(265,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ well_orders ) )
=> ( ( A @ ( B @ relation_restriction ) @ well_ordering )
& ( ( A @ ( B @ relation_restriction ) @ relation_field )
= A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t25_wellord2) ).
thf(2250,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ well_orders ) )
=> ( ( A @ ( B @ relation_restriction ) @ well_ordering )
& ( ( A @ ( B @ relation_restriction ) @ relation_field )
= A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(110,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_tarski) ).
thf(1237,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)],[110]) ).
thf(76,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(846,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ~ ( A @ ( B @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(65,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/sandbox/benchmark/theBenchmark.p',redefinition_k6_subset_1) ).
thf(777,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)],[65]) ).
thf(29,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_xboole_1) ).
thf(555,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)],[29]) ).
thf(282,axiom,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_dom @ relation )
& ( A @ relation_dom @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc7_relat_1) ).
thf(2363,plain,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_dom @ relation )
& ( A @ relation_dom @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(4,axiom,
! [A: $i,B: $i] :
~ ( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(406,plain,
! [A: $i,B: $i] :
~ ( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(407,plain,
~ ? [A: $i,B: $i] :
( ( A @ ( B @ proper_subset ) )
& ( B @ ( A @ subset ) ) ),
inference(miniscope,[status(thm)],[406]) ).
thf(408,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ proper_subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnf,[status(esa)],[407]) ).
thf(21,axiom,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(523,plain,
! [A: $i] :
( ( A @ powerset @ union )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(255,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_dom )
= empty_set )
<=> ( ( A @ relation_rng )
= empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t65_relat_1) ).
thf(2210,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( ( A @ relation_rng )
= empty_set )
=> ( ( A @ relation_dom )
= empty_set ) )
& ( ( ( A @ relation_dom )
= empty_set )
=> ( ( A @ relation_rng )
= empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(58,axiom,
empty_set @ empty,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(748,plain,
empty_set @ empty,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(365,axiom,
? [A: $i] :
( ( A @ relation )
& ~ ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_relat_1) ).
thf(3076,plain,
? [A: $i] :
( ( A @ relation )
& ~ ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).
thf(3077,plain,
( ( sk358 @ relation )
& ~ ( sk358 @ empty ) ),
inference(cnf,[status(esa)],[3076]) ).
thf(3078,plain,
~ ( sk358 @ empty ),
inference(cnfConj,[status(thm)],[3077]) ).
thf(3314,plain,
( ( ( sk358 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,3078]) ).
thf(3315,plain,
( ( sk358 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3314]) ).
thf(3317,plain,
sk358 != empty_set,
inference(simp,[status(thm)],[3315]) ).
thf(55,axiom,
? [A: $i] :
~ ( A @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(730,plain,
? [A: $i] :
~ ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(259,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ reflexive )
=> ( A @ ( B @ relation_restriction ) @ reflexive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t22_wellord1) ).
thf(2231,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ reflexive )
=> ( A @ ( B @ relation_restriction ) @ reflexive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(369,axiom,
( ( empty_set @ ordinal )
& ( empty_set @ epsilon_connected )
& ( empty_set @ epsilon_transitive )
& ( empty_set @ empty )
& ( empty_set @ one_to_one )
& ( empty_set @ function )
& ( empty_set @ relation_empty_yielding )
& ( empty_set @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_ordinal1) ).
thf(3091,plain,
( ( empty_set @ ordinal )
& ( empty_set @ epsilon_connected )
& ( empty_set @ epsilon_transitive )
& ( empty_set @ empty )
& ( empty_set @ one_to_one )
& ( empty_set @ function )
& ( empty_set @ relation_empty_yielding )
& ( empty_set @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(3092,plain,
( ( empty_set @ ordinal )
& ( empty_set @ epsilon_connected )
& ( empty_set @ epsilon_transitive )
& ( empty_set @ empty )
& ( empty_set @ one_to_one )
& ( empty_set @ function )
& ( empty_set @ relation_empty_yielding )
& ( empty_set @ relation ) ),
inference(cnf,[status(esa)],[3091]) ).
thf(3098,plain,
empty_set @ relation_empty_yielding,
inference(cnfConj,[status(thm)],[3092]) ).
thf(318,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation )
& ( A @ function )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_composition ) @ function )
& ( B @ ( A @ relation_composition ) @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_funct_1) ).
thf(2671,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation )
& ( A @ function )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_composition ) @ function )
& ( B @ ( A @ relation_composition ) @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( B @ ( A @ in ) )
<=> ( B @ ( A @ succ @ ordinal_subset ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t33_ordinal1) ).
thf(511,plain,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ ( A @ succ @ ordinal_subset ) )
=> ( B @ ( A @ in ) ) )
& ( ( B @ ( A @ in ) )
=> ( B @ ( A @ succ @ ordinal_subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(200,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equipotent ) )
=> ( A @ ( B @ equipotent ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetry_r2_wellord2) ).
thf(1793,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ equipotent ) )
=> ( A @ ( B @ equipotent ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(183,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(1692,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(309,axiom,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_difference ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc12_finset_1) ).
thf(2590,plain,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_difference ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(136,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(1401,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)],[136]) ).
thf(232,axiom,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(2025,plain,
? [A: $i] :
( ( A @ relation )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(2026,plain,
( ( sk205 @ relation )
& ( sk205 @ empty ) ),
inference(cnf,[status(esa)],[2025]) ).
thf(2027,plain,
sk205 @ empty,
inference(cnfConj,[status(thm)],[2026]) ).
thf(243,axiom,
? [A: $i] :
( ( A @ finite )
& ~ ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_finset_1) ).
thf(2143,plain,
? [A: $i] :
( ( A @ finite )
& ~ ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(2144,plain,
( ( sk226 @ finite )
& ~ ( sk226 @ empty ) ),
inference(cnf,[status(esa)],[2143]) ).
thf(2145,plain,
~ ( sk226 @ empty ),
inference(cnfConj,[status(thm)],[2144]) ).
thf(3448,plain,
( ( ( sk226 @ empty )
!= ( sk205 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,2145]) ).
thf(3449,plain,
( ( sk226 @ empty )
!= ( sk205 @ empty ) ),
inference(simp,[status(thm)],[3448]) ).
thf(3460,plain,
sk226 != sk205,
inference(simp,[status(thm)],[3449]) ).
thf(141,axiom,
! [A: $i,B: $i] : ( A @ ( A @ equipotent ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity_r2_wellord2) ).
thf(1422,plain,
! [A: $i] : ( A @ ( A @ equipotent ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(165,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ ( C @ ( D @ relation_of2_as_subset ) ) )
=> ( ( B @ ( A @ subset ) )
=> ( B @ ( C @ ( D @ relation_of2_as_subset ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t16_relset_1) ).
thf(1569,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ ( C @ ( D @ relation_of2_as_subset ) ) )
=> ( ( B @ ( A @ subset ) )
=> ( B @ ( C @ ( D @ relation_of2_as_subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(188,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/sandbox/benchmark/theBenchmark.p',d4_xboole_0) ).
thf(1705,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)],[188]) ).
thf(180,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) )
= ( B @ ( A @ set_difference ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_subset_1) ).
thf(1671,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)],[180]) ).
thf(67,axiom,
? [A: $i] :
( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_arytm_3) ).
thf(791,plain,
? [A: $i] :
( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(792,plain,
( ( sk37 @ natural )
& ( sk37 @ ordinal )
& ( sk37 @ epsilon_connected )
& ( sk37 @ epsilon_transitive )
& ~ ( sk37 @ empty ) ),
inference(cnf,[status(esa)],[791]) ).
thf(797,plain,
sk37 @ ordinal,
inference(cnfConj,[status(thm)],[792]) ).
thf(354,axiom,
? [A: $i] :
( ( A @ function )
& ( A @ empty )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_funct_1) ).
thf(2933,plain,
? [A: $i] :
( ( A @ function )
& ( A @ empty )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).
thf(82,axiom,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc3_ordinal1) ).
thf(913,plain,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive )
& ~ ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(101,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(1063,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(2146,plain,
sk226 @ finite,
inference(cnfConj,[status(thm)],[2144]) ).
thf(88,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ singleton @ subset ) )
<=> ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(930,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)],[88]) ).
thf(47,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/sandbox/benchmark/theBenchmark.p',d1_setfam_1) ).
thf(670,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)],[47]) ).
thf(256,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( ( C @ ( B @ ( A @ relation_rng_as_subset ) ) )
= ( C @ relation_rng ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k5_relset_1) ).
thf(2216,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( ( C @ ( B @ ( A @ relation_rng_as_subset ) ) )
= ( C @ relation_rng ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(357,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ relation_rng @ ( A @ relation_dom @ subset ) )
=> ( ( A @ ( B @ relation_composition ) @ relation_rng )
= ( A @ relation_rng ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t47_relat_1) ).
thf(2985,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ relation_rng @ ( A @ relation_dom @ subset ) )
=> ( ( A @ ( B @ relation_composition ) @ relation_rng )
= ( A @ relation_rng ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).
thf(50,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ union_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k5_setfam_1) ).
thf(709,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)],[50]) ).
thf(31,axiom,
? [A: $i] : ( A @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(569,plain,
? [A: $i] : ( A @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(570,plain,
sk12 @ empty,
inference(cnf,[status(esa)],[569]) ).
thf(731,plain,
~ ! [A: $i] : ( A @ empty ),
inference(miniscope,[status(thm)],[730]) ).
thf(732,plain,
~ ( sk29 @ empty ),
inference(cnf,[status(esa)],[731]) ).
thf(3279,plain,
( ( ( sk29 @ empty )
!= ( sk12 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,732]) ).
thf(3280,plain,
( ( sk29 @ empty )
!= ( sk12 @ empty ) ),
inference(simp,[status(thm)],[3279]) ).
thf(3281,plain,
sk29 != sk12,
inference(simp,[status(thm)],[3280]) ).
thf(6,axiom,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( ( A @ ( B @ ordinal_subset ) )
| ( B @ ( A @ ordinal_subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',connectedness_r1_ordinal1) ).
thf(436,plain,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( ( A @ ( B @ ordinal_subset ) )
| ( B @ ( A @ ordinal_subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(345,axiom,
? [A: $i] :
( ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_partfun1) ).
thf(2869,plain,
? [A: $i] :
( ( A @ empty )
& ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).
thf(215,axiom,
? [A: $i] :
( ( A @ relation_empty_yielding )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc3_relat_1) ).
thf(1887,plain,
? [A: $i] :
( ( A @ relation_empty_yielding )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(230,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ relation_restriction ) )
= ( B @ ( B @ cartesian_product2 ) @ ( A @ set_intersection2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d6_wellord1) ).
thf(2020,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ relation_restriction ) )
= ( B @ ( B @ cartesian_product2 ) @ ( A @ set_intersection2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(74,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( B @ powerset @ ( C @ ( B @ ( A @ relation_rng_as_subset ) ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k5_relset_1) ).
thf(841,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( B @ powerset @ ( C @ ( B @ ( A @ relation_rng_as_subset ) ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(285,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/sandbox/benchmark/theBenchmark.p',t25_relat_1) ).
thf(2400,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)],[285]) ).
thf(156,axiom,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ( ( B @ ( A @ in ) )
=> ( A @ ordinal ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t23_ordinal1) ).
thf(1529,plain,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ( ( B @ ( A @ in ) )
=> ( A @ ordinal ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(54,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/sandbox/benchmark/theBenchmark.p',t4_xboole_0) ).
thf(724,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)],[54]) ).
thf(333,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_field @ ( A @ well_orders ) )
<=> ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_wellord1) ).
thf(2814,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ well_ordering )
=> ( A @ relation_field @ ( A @ well_orders ) ) )
& ( ( A @ relation_field @ ( A @ well_orders ) )
=> ( A @ well_ordering ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(52,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_subset) ).
thf(719,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ ( A @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(364,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( ~ ( A @ relation_dom @ ( B @ in ) )
=> ( ( C
= ( B @ ( A @ apply ) ) )
<=> ( C = empty_set ) ) )
& ( ( A @ relation_dom @ ( B @ in ) )
=> ( ( C
= ( B @ ( A @ apply ) ) )
<=> ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_funct_1) ).
thf(3061,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( ~ ( A @ relation_dom @ ( B @ in ) )
=> ( ( ( C = empty_set )
=> ( C
= ( B @ ( A @ apply ) ) ) )
& ( ( C
= ( B @ ( A @ apply ) ) )
=> ( C = empty_set ) ) ) )
& ( ( A @ relation_dom @ ( B @ in ) )
=> ( ( ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
=> ( C
= ( B @ ( A @ apply ) ) ) )
& ( ( C
= ( B @ ( A @ apply ) ) )
=> ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(914,plain,
( ( sk55 @ ordinal )
& ( sk55 @ epsilon_connected )
& ( sk55 @ epsilon_transitive )
& ~ ( sk55 @ empty ) ),
inference(cnf,[status(esa)],[913]) ).
thf(917,plain,
sk55 @ epsilon_connected,
inference(cnfConj,[status(thm)],[914]) ).
thf(145,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ( ( B @ ( A @ set_difference ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(1455,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)],[145]) ).
thf(368,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( B @ ( A @ relation_composition ) @ relation_rng @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t45_relat_1) ).
thf(3089,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( B @ ( A @ relation_composition ) @ relation_rng @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).
thf(2870,plain,
( ( sk322 @ empty )
& ( sk322 @ one_to_one )
& ( sk322 @ function )
& ( sk322 @ relation ) ),
inference(cnf,[status(esa)],[2869]) ).
thf(2874,plain,
sk322 @ empty,
inference(cnfConj,[status(thm)],[2870]) ).
thf(915,plain,
~ ( sk55 @ empty ),
inference(cnfConj,[status(thm)],[914]) ).
thf(3797,plain,
( ( ( sk322 @ empty )
!= ( sk55 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2874,915]) ).
thf(3798,plain,
( ( sk322 @ empty )
!= ( sk55 @ empty ) ),
inference(simp,[status(thm)],[3797]) ).
thf(3803,plain,
sk322 != sk55,
inference(simp,[status(thm)],[3798]) ).
thf(394,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( A @ ( A @ ( B @ relation_inverse_image ) @ ( B @ relation_image ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t145_funct_1) ).
thf(3263,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( A @ ( A @ ( B @ relation_inverse_image ) @ ( B @ relation_image ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).
thf(115,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( A @ ( B @ element ) )
=> ( ( B @ ordinal )
& ( B @ epsilon_connected )
& ( B @ epsilon_transitive ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_arytm_3) ).
thf(1287,plain,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( A @ ( B @ element ) )
=> ( ( B @ ordinal )
& ( B @ epsilon_connected )
& ( B @ epsilon_transitive ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(360,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_dom @ ( B @ ( A @ relation_rng_restriction ) @ relation_dom @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l29_wellord1) ).
thf(3029,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_dom @ ( B @ ( A @ relation_rng_restriction ) @ relation_dom @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(373,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_image ) @ ( A @ in ) )
<=> ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( A @ ( D @ ordered_pair ) @ in ) )
& ( C @ relation_dom @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t143_relat_1) ).
thf(3131,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( A @ ( D @ ordered_pair ) @ in ) )
& ( C @ relation_dom @ ( D @ in ) ) )
=> ( B @ ( C @ relation_image ) @ ( A @ in ) ) )
& ( ( B @ ( C @ relation_image ) @ ( A @ in ) )
=> ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( A @ ( D @ ordered_pair ) @ in ) )
& ( C @ relation_dom @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).
thf(118,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(1298,plain,
! [A: $i] :
( ( A @ ( A @ set_intersection2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(377,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ relation_composition ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k5_relat_1) ).
thf(3148,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ relation_composition ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).
thf(72,axiom,
? [A: $i] :
( ( A @ being_limit_ordinal )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_ordinal2) ).
thf(833,plain,
? [A: $i] :
( ( A @ being_limit_ordinal )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(834,plain,
( ( sk41 @ being_limit_ordinal )
& ( sk41 @ ordinal )
& ( sk41 @ epsilon_connected )
& ( sk41 @ epsilon_transitive ) ),
inference(cnf,[status(esa)],[833]) ).
thf(838,plain,
sk41 @ being_limit_ordinal,
inference(cnfConj,[status(thm)],[834]) ).
thf(25,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/sandbox/benchmark/theBenchmark.p',t48_setfam_1) ).
thf(542,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)],[25]) ).
thf(322,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_restriction ) @ relation_field @ ( A @ in ) )
=> ( ( B @ ( A @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t19_wellord1) ).
thf(2703,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_restriction ) @ relation_field @ ( A @ in ) )
=> ( ( B @ ( A @ in ) )
& ( C @ relation_field @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).
thf(80,axiom,
! [A: $i] :
( ( A @ ordinal )
<=> ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_ordinal1) ).
thf(902,plain,
! [A: $i] :
( ( ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) )
=> ( A @ ordinal ) )
& ( ( A @ ordinal )
=> ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(3099,plain,
empty_set @ epsilon_transitive,
inference(cnfConj,[status(thm)],[3092]) ).
thf(341,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( C @ ( B @ relation_rng_restriction ) @ relation_rng @ ( A @ in ) )
<=> ( ( C @ relation_rng @ ( A @ in ) )
& ( B @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t115_relat_1) ).
thf(2857,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( ( ( C @ relation_rng @ ( A @ in ) )
& ( B @ ( A @ in ) ) )
=> ( C @ ( B @ relation_rng_restriction ) @ relation_rng @ ( A @ in ) ) )
& ( ( C @ ( B @ relation_rng_restriction ) @ relation_rng @ ( A @ in ) )
=> ( ( C @ relation_rng @ ( A @ in ) )
& ( B @ ( A @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(154,axiom,
! [A: $i,B: $i] :
? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ? [E: $i,F: $i] :
( ( F
= ( E @ singleton ) )
& ( A @ ( E @ in ) )
& ( ( F @ ( E @ ordered_pair ) )
= D ) )
& ( B @ ( A @ cartesian_product2 ) @ ( D @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e16_22__wellord2__1) ).
thf(1514,plain,
! [A: $i,B: $i] :
? [C: $i] :
! [D: $i] :
( ( ( ? [E: $i,F: $i] :
( ( F
= ( E @ singleton ) )
& ( A @ ( E @ in ) )
& ( ( F @ ( E @ ordered_pair ) )
= D ) )
& ( B @ ( A @ cartesian_product2 ) @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ? [E: $i,F: $i] :
( ( F
= ( E @ singleton ) )
& ( A @ ( E @ in ) )
& ( ( F @ ( E @ ordered_pair ) )
= D ) )
& ( B @ ( A @ cartesian_product2 ) @ ( D @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(135,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ( ( B @ ( A @ complements_of_subsets ) )
= empty_set )
& ( B != empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_setfam_1) ).
thf(1398,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)],[135]) ).
thf(193,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( B @ ( C @ in ) ) )
=> ( B @ powerset @ ( A @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l71_subset_1) ).
thf(1773,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)],[193]) ).
thf(275,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( B @ ( C @ fiber ) @ ( B @ ( A @ ( C @ relation_restriction ) @ fiber ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_wellord1) ).
thf(2301,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( B @ ( C @ fiber ) @ ( B @ ( A @ ( C @ relation_restriction ) @ fiber ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(336,axiom,
! [A: $i] :
( ( ( A @ relation )
& ~ ( A @ empty ) )
=> ~ ( A @ relation_rng @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc6_relat_1) ).
thf(2829,plain,
! [A: $i] :
( ( ( A @ relation )
& ~ ( A @ empty ) )
=> ~ ( A @ relation_rng @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(3454,plain,
( ( ( sk205 @ empty )
!= ( sk55 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,915]) ).
thf(3455,plain,
( ( sk205 @ empty )
!= ( sk55 @ empty ) ),
inference(simp,[status(thm)],[3454]) ).
thf(3463,plain,
sk205 != sk55,
inference(simp,[status(thm)],[3455]) ).
thf(129,axiom,
! [A: $i] : ( A @ inclusion_relation @ antisymmetric ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_wellord2) ).
thf(1379,plain,
! [A: $i] : ( A @ inclusion_relation @ antisymmetric ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(5,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ union ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ( A @ ( D @ in ) )
& ( D @ ( C @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_tarski) ).
thf(409,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)],[5]) ).
thf(410,plain,
( ! [A: $i,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 @ union ) ) )
& ! [A: $i,B: $i] :
( ( B
= ( A @ union ) )
=> ( ! [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)],[409]) ).
thf(411,plain,
! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( G
= ( F @ union ) )
| ~ ( G @ ( F @ ( G @ sk5 ) @ in ) )
| ( G @ ( F @ sk3 ) ) )
& ( ( G
= ( F @ union ) )
| ( F @ ( F @ ( G @ sk6 ) @ in ) )
| ( G @ ( F @ sk3 ) ) )
& ( ( G
= ( F @ union ) )
| ( F @ ( G @ sk6 ) @ ( F @ ( G @ sk5 ) @ in ) )
| ( G @ ( F @ sk3 ) ) )
& ( ( G
= ( F @ union ) )
| ~ ( G @ ( F @ sk3 ) )
| ~ ( F @ ( H @ in ) )
| ~ ( H @ ( F @ ( G @ sk4 ) @ in ) ) )
& ( ( G
= ( F @ union ) )
| ~ ( G @ ( F @ sk3 ) )
| ( G @ ( F @ ( G @ sk4 ) @ in ) ) )
& ( ( B @ ( D @ in ) )
| ~ ( A @ ( E @ in ) )
| ~ ( E @ ( D @ in ) )
| ( B
!= ( A @ union ) ) )
& ( ( A @ ( A @ ( B @ ( C @ sk2 ) ) @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) )
& ( ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ) ),
inference(cnf,[status(esa)],[410]) ).
thf(417,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ union ) )
| ( A @ ( A @ ( B @ sk6 ) @ in ) )
| ( B @ ( A @ sk3 ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(420,plain,
! [B: $i,A: $i] :
( ( A @ ( A @ ( B @ sk6 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[417]) ).
thf(421,plain,
! [B: $i,A: $i] :
( ( A @ ( A @ ( B @ sk6 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(simp,[status(thm)],[420]) ).
thf(1888,plain,
( ( sk176 @ relation_empty_yielding )
& ( sk176 @ relation ) ),
inference(cnf,[status(esa)],[1887]) ).
thf(1890,plain,
sk176 @ relation_empty_yielding,
inference(cnfConj,[status(thm)],[1888]) ).
thf(340,axiom,
! [A: $i,B: $i] :
? [C: $i] :
( ( C @ function )
& ( C @ relation )
& ( B @ ( A @ ( C @ relation_of2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_partfun1) ).
thf(2852,plain,
! [A: $i,B: $i] :
? [C: $i] :
( ( C @ function )
& ( C @ relation )
& ( B @ ( A @ ( C @ relation_of2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).
thf(387,axiom,
! [A: $i] :
( ( A @ relation )
<=> ! [B: $i] :
~ ( ! [C: $i,D: $i] :
( B
!= ( D @ ( C @ ordered_pair ) ) )
& ( A @ ( B @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_relat_1) ).
thf(3231,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)],[387]) ).
thf(140,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ unordered_pair ) @ subset ) )
<=> ( ( C @ ( B @ in ) )
& ( C @ ( A @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t38_zfmisc_1) ).
thf(1415,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)],[140]) ).
thf(241,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ connected )
<=> ! [B: $i,C: $i] :
~ ( ~ ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ~ ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
& ( B != C )
& ( A @ relation_field @ ( C @ in ) )
& ( A @ relation_field @ ( B @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l4_wellord1) ).
thf(2105,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ! [B: $i,C: $i] :
~ ( ~ ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ~ ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
& ( B != C )
& ( A @ relation_field @ ( C @ in ) )
& ( A @ relation_field @ ( B @ in ) ) )
=> ( A @ connected ) )
& ( ( A @ connected )
=> ! [B: $i,C: $i] :
~ ( ~ ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ~ ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
& ( B != C )
& ( A @ relation_field @ ( C @ in ) )
& ( A @ relation_field @ ( B @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(239,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ empty ) )
=> ( ( A @ ( B @ relation_composition ) @ relation )
& ( A @ ( B @ relation_composition ) @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc10_relat_1) ).
thf(2090,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ empty ) )
=> ( ( A @ ( B @ relation_composition ) @ relation )
& ( A @ ( B @ relation_composition ) @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(359,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ( ! [C: $i] :
~ ( ! [D: $i] :
~ ? [E: $i] :
( ! [F: $i] :
( ( E @ ( F @ in ) )
=> ( B @ ( F @ ( D @ ordered_pair ) @ in ) ) )
& ( E @ ( D @ in ) )
& ( C = E ) )
& ( A @ ( C @ in ) ) )
& ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) ) )
=> ? [C: $i] :
( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ? [E: $i] :
( ! [F: $i] :
( ( E @ ( F @ in ) )
=> ( B @ ( F @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) ) )
& ( E @ ( D @ ( C @ apply ) @ in ) )
& ( D = E ) ) )
& ( ( C @ relation_dom )
= A )
& ( C @ function )
& ( C @ relation ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s2_funct_1__e10_24__wellord2) ).
thf(2996,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ( ! [C: $i] :
~ ( ! [D: $i] :
~ ? [E: $i] :
( ! [F: $i] :
( ( E @ ( F @ in ) )
=> ( B @ ( F @ ( D @ ordered_pair ) @ in ) ) )
& ( E @ ( D @ in ) )
& ( C = E ) )
& ( A @ ( C @ in ) ) )
& ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) ) )
=> ? [C: $i] :
( ! [D: $i] :
( ( A @ ( D @ in ) )
=> ? [E: $i] :
( ! [F: $i] :
( ( E @ ( F @ in ) )
=> ( B @ ( F @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) ) )
& ( E @ ( D @ ( C @ apply ) @ in ) )
& ( D = E ) ) )
& ( ( C @ relation_dom )
= A )
& ( C @ function )
& ( C @ relation ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).
thf(174,axiom,
! [A: $i] :
( ( ( A @ ordinal )
& ( A @ empty ) )
=> ( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc2_arytm_3) ).
thf(1632,plain,
! [A: $i] :
( ( ( A @ ordinal )
& ( A @ empty ) )
=> ( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(268,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ reflexive )
<=> ( A @ relation_field @ ( A @ is_reflexive_in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d9_relat_2) ).
thf(2259,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_field @ ( A @ is_reflexive_in ) )
=> ( A @ reflexive ) )
& ( ( A @ reflexive )
=> ( A @ relation_field @ ( A @ is_reflexive_in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(39,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
<=> ( B @ ( A @ cartesian_product2 ) @ ( C @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_relset_1) ).
thf(623,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ cartesian_product2 ) @ ( C @ subset ) )
=> ( B @ ( A @ ( C @ relation_of2 ) ) ) )
& ( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( B @ ( A @ cartesian_product2 ) @ ( C @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(13,axiom,
! [A: $i,B: $i] :
~ ( A @ ( A @ proper_subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(485,plain,
! [A: $i] :
~ ( A @ ( A @ proper_subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( B @ powerset @ ( A @ element ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_subset) ).
thf(574,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)],[33]) ).
thf(3093,plain,
empty_set @ ordinal,
inference(cnfConj,[status(thm)],[3092]) ).
thf(314,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( A @ relation_dom @ ( B @ ( A @ relation_composition ) @ relation_dom @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t44_relat_1) ).
thf(2616,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( A @ relation_dom @ ( B @ ( A @ relation_composition ) @ relation_dom @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(224,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ transitive )
<=> ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( A @ ( D @ ( B @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l2_wellord1) ).
thf(1946,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( A @ ( D @ ( B @ ordered_pair ) @ in ) ) )
=> ( A @ transitive ) )
& ( ( A @ transitive )
=> ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( A @ ( D @ ( B @ ordered_pair ) @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(147,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/sandbox/benchmark/theBenchmark.p',t119_zfmisc_1) ).
thf(1465,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)],[147]) ).
thf(3290,plain,
( ( ( sk55 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,915]) ).
thf(3291,plain,
( ( sk55 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3290]) ).
thf(3295,plain,
sk55 != empty_set,
inference(simp,[status(thm)],[3291]) ).
thf(329,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
& ( A @ well_ordering ) )
=> ( B @ well_ordering ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t54_wellord1) ).
thf(2782,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
& ( A @ well_ordering ) )
=> ( B @ well_ordering ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(2609,plain,
sk285 @ empty,
inference(cnfConj,[status(thm)],[2608]) ).
thf(3615,plain,
( ( ( sk285 @ empty )
!= ( sk55 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,915]) ).
thf(3616,plain,
( ( sk285 @ empty )
!= ( sk55 @ empty ) ),
inference(simp,[status(thm)],[3615]) ).
thf(3621,plain,
sk285 != sk55,
inference(simp,[status(thm)],[3616]) ).
thf(351,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ! [E: $i] :
( ( ( E @ function )
& ( E @ relation ) )
=> ( ( A @ ( C @ in ) )
=> ( ( ( C @ ( E @ ( D @ relation_composition ) @ apply ) )
= ( C @ ( D @ apply ) @ ( E @ apply ) ) )
| ( B = empty_set ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_funct_2) ).
thf(2925,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ! [E: $i] :
( ( ( E @ function )
& ( E @ relation ) )
=> ( ( A @ ( C @ in ) )
=> ( ( ( C @ ( E @ ( D @ relation_composition ) @ apply ) )
= ( C @ ( D @ apply ) @ ( E @ apply ) ) )
| ( B = empty_set ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).
thf(1,conjecture,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_dom @ finite )
=> ( A @ relation_rng @ finite ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t26_finset_1) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_dom @ finite )
=> ( A @ relation_rng @ finite ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(397,plain,
~ ! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_dom @ finite )
=> ( A @ relation_rng @ finite ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(283,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_dom_restriction ) )
= ( B @ ( A @ identity_relation @ relation_composition ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t94_relat_1) ).
thf(2367,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_dom_restriction ) )
= ( B @ ( A @ identity_relation @ relation_composition ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(128,axiom,
! [A: $i,B: $i] :
( ( ~ ( B @ empty )
& ~ ( A @ empty ) )
=> ~ ( B @ ( A @ cartesian_product2 ) @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_subset_1) ).
thf(1377,plain,
! [A: $i,B: $i] :
( ( ~ ( B @ empty )
& ~ ( A @ empty ) )
=> ~ ( B @ ( A @ cartesian_product2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(315,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B @ ( C @ relation_composition ) @ relation_dom @ ( A @ in ) )
=> ( ( A @ ( B @ ( C @ relation_composition ) @ apply ) )
= ( A @ ( C @ apply ) @ ( B @ apply ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t22_funct_1) ).
thf(2618,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B @ ( C @ relation_composition ) @ relation_dom @ ( A @ in ) )
=> ( ( A @ ( B @ ( C @ relation_composition ) @ apply ) )
= ( A @ ( C @ apply ) @ ( B @ apply ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(41,axiom,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(634,plain,
! [A: $i,B: $i] : ( B @ ( A @ set_union2 ) @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(155,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(1526,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(45,axiom,
! [A: $i] :
( ( ( A @ natural )
& ( A @ ordinal ) )
=> ( ( A @ succ @ natural )
& ( A @ succ @ ordinal )
& ( A @ succ @ epsilon_connected )
& ( A @ succ @ epsilon_transitive )
& ~ ( A @ succ @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_arytm_3) ).
thf(660,plain,
! [A: $i] :
( ( ( A @ natural )
& ( A @ ordinal ) )
=> ( ( A @ succ @ natural )
& ( A @ succ @ ordinal )
& ( A @ succ @ epsilon_connected )
& ( A @ succ @ epsilon_transitive )
& ~ ( A @ succ @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(385,axiom,
? [A: $i] :
( ( A @ function )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_funct_1) ).
thf(3211,plain,
? [A: $i] :
( ( A @ function )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).
thf(307,axiom,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_rng @ ( A @ relation_dom @ cartesian_product2 ) @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_relat_1) ).
thf(2586,plain,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_rng @ ( A @ relation_dom @ cartesian_product2 ) @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
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/sandbox/benchmark/theBenchmark.p',d2_xboole_0) ).
thf(1072,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(362,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( A @ finite )
=> ( A @ ( B @ relation_image ) @ finite ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_finset_1) ).
thf(3036,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( A @ finite )
=> ( A @ ( B @ relation_image ) @ finite ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(105,axiom,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( C @ ( B @ ordered_pair ) ) )
=> ! [B: $i] :
( ( B
= ( A @ pair_second ) )
<=> ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_mcart_1) ).
thf(1104,plain,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( C @ ( B @ ordered_pair ) ) )
=> ! [B: $i] :
( ( ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = D ) )
=> ( B
= ( A @ pair_second ) ) )
& ( ( B
= ( A @ pair_second ) )
=> ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(185,axiom,
! [A: $i] :
( ( A @ epsilon_transitive )
=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( B @ ( A @ proper_subset ) )
=> ( B @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_ordinal1) ).
thf(1697,plain,
! [A: $i] :
( ( A @ epsilon_transitive )
=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( B @ ( A @ proper_subset ) )
=> ( B @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(3097,plain,
empty_set @ epsilon_connected,
inference(cnfConj,[status(thm)],[3092]) ).
thf(254,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( ( B @ relation_rng @ ( A @ in ) )
& ( B @ one_to_one ) )
=> ( ( A
= ( A @ ( B @ ( B @ function_inverse @ relation_composition ) @ apply ) ) )
& ( A
= ( A @ ( B @ function_inverse @ apply ) @ ( B @ apply ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t57_funct_1) ).
thf(2204,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( ( B @ relation_rng @ ( A @ in ) )
& ( B @ one_to_one ) )
=> ( ( A
= ( A @ ( B @ ( B @ function_inverse @ relation_composition ) @ apply ) ) )
& ( A
= ( A @ ( B @ function_inverse @ apply ) @ ( B @ apply ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(23,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
<=> ( ( A != B )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_xboole_0) ).
thf(529,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)],[23]) ).
thf(793,plain,
sk37 @ epsilon_transitive,
inference(cnfConj,[status(thm)],[792]) ).
thf(61,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( B @ ( A @ ( C @ relation_of2_as_subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m2_relset_1) ).
thf(761,plain,
! [A: $i,B: $i] :
? [C: $i] : ( B @ ( A @ ( C @ relation_of2_as_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(196,axiom,
! [A: $i,B: $i,C: $i] :
( ( A @ powerset @ ( C @ element ) )
=> ~ ( ( C @ ( B @ in ) )
& ( C @ ( A @ subset_complement ) @ ( B @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t54_subset_1) ).
thf(1783,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)],[196]) ).
thf(278,axiom,
! [A: $i] :
( ( A @ function )
<=> ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( B @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( C = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_funct_1) ).
thf(2307,plain,
! [A: $i] :
( ( ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( B @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( C = D ) )
=> ( A @ function ) )
& ( ( A @ function )
=> ! [B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( B @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( C = D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(395,axiom,
! [A: $i] :
? [B: $i] :
( ( A @ ( B @ well_orders ) )
& ( B @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t26_wellord2) ).
thf(3265,plain,
! [A: $i] :
? [B: $i] :
( ( A @ ( B @ well_orders ) )
& ( B @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).
thf(86,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B
= ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t45_xboole_1) ).
thf(925,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B
= ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(237,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( A @ function_inverse @ one_to_one ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t62_funct_1) ).
thf(2083,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( A @ function_inverse @ one_to_one ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(191,axiom,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( ( B @ ( A @ ordinal_subset ) )
<=> ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_r1_ordinal1) ).
thf(1736,plain,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( ( ( B @ ( A @ subset ) )
=> ( B @ ( A @ ordinal_subset ) ) )
& ( ( B @ ( A @ ordinal_subset ) )
=> ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(316,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= F ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= E ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ? [F: $i] :
( ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= E ) )
& ( F = E )
& ( C @ ( A @ cartesian_product2 ) @ ( F @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e10_24__wellord2__2) ).
thf(2621,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= F ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= E ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ? [F: $i] :
( ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= E ) )
& ( F = E )
& ( C @ ( A @ cartesian_product2 ) @ ( F @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ? [F: $i] :
( ? [G: $i,H: $i] :
( ? [I: $i] :
( ! [J: $i] :
( ( I @ ( J @ in ) )
=> ( B @ ( J @ ( H @ ordered_pair ) @ in ) ) )
& ( I @ ( H @ in ) )
& ( G = I ) )
& ( A @ ( G @ in ) )
& ( ( H @ ( G @ ordered_pair ) )
= E ) )
& ( F = E )
& ( C @ ( A @ cartesian_product2 ) @ ( F @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(331,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( C @ relation )
=> ( ( C
= ( B @ ( A @ relation_dom_restriction ) ) )
<=> ! [D: $i,E: $i] :
( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
<=> ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d11_relat_1) ).
thf(2790,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( C @ relation )
=> ( ( ! [D: $i,E: $i] :
( ( ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_dom_restriction ) ) ) )
& ( ( C
= ( B @ ( A @ relation_dom_restriction ) ) )
=> ! [D: $i,E: $i] :
( ( ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(99,axiom,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ? [E: $i] :
( ( A @ ( E @ in ) )
& ( D = E )
& ( E @ ordinal ) )
& ( B @ succ @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e8_6__wellord2__1) ).
thf(1047,plain,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ? [C: $i] :
! [D: $i] :
( ( ( ? [E: $i] :
( ( A @ ( E @ in ) )
& ( D = E )
& ( E @ ordinal ) )
& ( B @ succ @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ? [E: $i] :
( ( A @ ( E @ in ) )
& ( D = E )
& ( E @ ordinal ) )
& ( B @ succ @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(68,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/sandbox/benchmark/theBenchmark.p',t136_zfmisc_1) ).
thf(798,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)],[68]) ).
thf(288,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( A @ ( B @ ( A @ relation_rng_restriction ) @ relation_rng @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t116_relat_1) ).
thf(2409,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( A @ ( B @ ( A @ relation_rng_restriction ) @ relation_rng @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(132,axiom,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(1387,plain,
! [A: $i] : ( A @ powerset @ ( A @ cast_to_subset @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(182,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( A @ ( B @ in ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ( C
= ( D @ singleton ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e16_22__wellord2__1) ).
thf(1676,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( A @ ( B @ in ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ? [D: $i] :
( ( C
= ( D @ singleton ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( C
= ( D @ singleton ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(244,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( A @ ( B @ in ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) )
=> ? [B: $i] :
( ! [C: $i,D: $i] :
( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
<=> ( ( D
= ( C @ singleton ) )
& ( A @ ( C @ in ) )
& ( A @ ( C @ in ) ) ) )
& ( B @ function )
& ( B @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_funct_1__e16_22__wellord2__1) ).
thf(2147,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( A @ ( B @ in ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) )
=> ? [B: $i] :
( ! [C: $i,D: $i] :
( ( ( ( D
= ( C @ singleton ) )
& ( A @ ( C @ in ) )
& ( A @ ( C @ in ) ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( ( D
= ( C @ singleton ) )
& ( A @ ( C @ in ) )
& ( A @ ( C @ in ) ) ) ) )
& ( B @ function )
& ( B @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(2934,plain,
( ( sk332 @ function )
& ( sk332 @ empty )
& ( sk332 @ relation ) ),
inference(cnf,[status(esa)],[2933]) ).
thf(2937,plain,
sk332 @ function,
inference(cnfConj,[status(thm)],[2934]) ).
thf(419,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( C @ in ) )
| ~ ( A @ ( D @ in ) )
| ~ ( D @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(428,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( C @ in ) )
| ~ ( A @ ( D @ in ) )
| ~ ( D @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[419]) ).
thf(429,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ union @ ( B @ in ) )
| ~ ( A @ ( C @ in ) )
| ~ ( C @ ( B @ in ) ) ),
inference(simp,[status(thm)],[428]) ).
thf(390,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( A @ ( B @ relation_image ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t144_relat_1) ).
thf(3250,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( A @ ( B @ relation_image ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(301,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ~ ( ! [C: $i] :
( ( C @ relation )
=> ~ ( A @ ( C @ well_orders ) ) )
& ( B @ relation_field @ ( A @ equipotent ) )
& ( B @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l30_wellord2) ).
thf(2528,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ~ ( ! [C: $i] :
( ( C @ relation )
=> ~ ( A @ ( C @ well_orders ) ) )
& ( B @ relation_field @ ( A @ equipotent ) )
& ( B @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(229,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_dom_restriction ) @ relation_dom @ ( A @ in ) )
<=> ( ( C @ relation_dom @ ( A @ in ) )
& ( B @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t86_relat_1) ).
thf(2015,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( ( ( C @ relation_dom @ ( A @ in ) )
& ( B @ ( A @ in ) ) )
=> ( B @ ( C @ relation_dom_restriction ) @ relation_dom @ ( A @ in ) ) )
& ( ( B @ ( C @ relation_dom_restriction ) @ relation_dom @ ( A @ in ) )
=> ( ( C @ relation_dom @ ( A @ in ) )
& ( B @ ( A @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(163,axiom,
( ~ ( omega @ empty )
& ( omega @ ordinal )
& ( omega @ epsilon_connected )
& ( omega @ epsilon_transitive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_ordinal2) ).
thf(1560,plain,
( ~ ( omega @ empty )
& ( omega @ ordinal )
& ( omega @ epsilon_connected )
& ( omega @ epsilon_transitive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(2873,plain,
sk322 @ one_to_one,
inference(cnfConj,[status(thm)],[2870]) ).
thf(396,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( C @ ( B @ subset ) )
=> ( ( ( C @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( C @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
| ( ( A != empty_set )
& ( B = empty_set ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t9_funct_2) ).
thf(3269,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( C @ ( B @ subset ) )
=> ( ( ( C @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( C @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
| ( ( A != empty_set )
& ( B = empty_set ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).
thf(415,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ union ) )
| ( A @ ( B @ sk6 ) @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(434,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk6 ) @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[415]) ).
thf(435,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk6 ) @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(simp,[status(thm)],[434]) ).
thf(374,axiom,
( ( empty_set @ relation_empty_yielding )
& ( empty_set @ relation )
& ( empty_set @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc12_relat_1) ).
thf(3137,plain,
( ( empty_set @ relation_empty_yielding )
& ( empty_set @ relation )
& ( empty_set @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(2872,plain,
sk322 @ function,
inference(cnfConj,[status(thm)],[2870]) ).
thf(172,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( B @ ( A @ ( C @ relation_of2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_relset_1) ).
thf(1624,plain,
! [A: $i,B: $i] :
? [C: $i] : ( B @ ( A @ ( C @ relation_of2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(376,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ connected )
<=> ( A @ relation_field @ ( A @ is_connected_in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d14_relat_2) ).
thf(3144,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_field @ ( A @ is_connected_in ) )
=> ( A @ connected ) )
& ( ( A @ connected )
=> ( A @ relation_field @ ( A @ is_connected_in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).
thf(137,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ complements_of_subsets ) @ ( A @ complements_of_subsets ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k7_setfam_1) ).
thf(1407,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)],[137]) ).
thf(28,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_boole) ).
thf(551,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( A != B )
& ( A @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(92,axiom,
! [A: $i] :
( ! [B: $i] :
( ( A @ ( B @ in ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ordinal ) ) )
=> ( A @ ordinal ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t31_ordinal1) ).
thf(974,plain,
! [A: $i] :
( ! [B: $i] :
( ( A @ ( B @ in ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ordinal ) ) )
=> ( A @ ordinal ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(916,plain,
sk55 @ epsilon_transitive,
inference(cnfConj,[status(thm)],[914]) ).
thf(379,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( A @ ( B @ relation_dom_restriction ) @ relation_rng @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t99_relat_1) ).
thf(3166,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( A @ ( B @ relation_dom_restriction ) @ relation_rng @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(253,axiom,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_inverse @ relation )
& ( A @ relation_inverse @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc11_relat_1) ).
thf(2200,plain,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_inverse @ relation )
& ( A @ relation_inverse @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(71,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( A @ singleton @ ( B @ ( A @ unordered_pair ) @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_tarski) ).
thf(830,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)],[71]) ).
thf(206,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ well_ordering )
<=> ( ( A @ well_founded_relation )
& ( A @ connected )
& ( A @ antisymmetric )
& ( A @ transitive )
& ( A @ reflexive ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_wellord1) ).
thf(1836,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( ( A @ well_founded_relation )
& ( A @ connected )
& ( A @ antisymmetric )
& ( A @ transitive )
& ( A @ reflexive ) )
=> ( A @ well_ordering ) )
& ( ( A @ well_ordering )
=> ( ( A @ well_founded_relation )
& ( A @ connected )
& ( A @ antisymmetric )
& ( A @ transitive )
& ( A @ reflexive ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(198,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(1787,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ ordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(248,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ relation )
=> ( ( D @ ( C @ identity_relation @ relation_composition ) @ ( B @ ( A @ ordered_pair ) @ in ) )
<=> ( ( D @ ( B @ ( A @ ordered_pair ) @ in ) )
& ( C @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t74_relat_1) ).
thf(2181,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ relation )
=> ( ( ( ( D @ ( B @ ( A @ ordered_pair ) @ in ) )
& ( C @ ( A @ in ) ) )
=> ( D @ ( C @ identity_relation @ relation_composition ) @ ( B @ ( A @ ordered_pair ) @ in ) ) )
& ( ( D @ ( C @ identity_relation @ relation_composition ) @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( D @ ( B @ ( A @ ordered_pair ) @ in ) )
& ( C @ ( A @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(192,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D
= ( C @ ( B @ ( A @ unordered_triple ) ) ) )
<=> ! [E: $i] :
( ( D @ ( E @ in ) )
<=> ~ ( ( E != C )
& ( E != B )
& ( E != A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_enumset1) ).
thf(1740,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ! [E: $i] :
( ( ~ ( ( E != C )
& ( E != B )
& ( E != A ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ~ ( ( E != C )
& ( E != B )
& ( E != A ) ) ) )
=> ( D
= ( C @ ( B @ ( A @ unordered_triple ) ) ) ) )
& ( ( D
= ( C @ ( B @ ( A @ unordered_triple ) ) ) )
=> ! [E: $i] :
( ( ~ ( ( E != C )
& ( E != B )
& ( E != A ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ~ ( ( E != C )
& ( E != B )
& ( E != A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(220,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( ( A @ function_inverse )
= ( A @ relation_inverse ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d9_funct_1) ).
thf(1910,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( ( A @ function_inverse )
= ( A @ relation_inverse ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(371,axiom,
! [A: $i] : ( A @ identity_relation @ relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k6_relat_1) ).
thf(3105,plain,
! [A: $i] : ( A @ identity_relation @ relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).
thf(392,axiom,
! [A: $i,B: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_dom_restriction ) @ function )
& ( B @ ( A @ relation_dom_restriction ) @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_funct_1) ).
thf(3254,plain,
! [A: $i,B: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_dom_restriction ) @ function )
& ( B @ ( A @ relation_dom_restriction ) @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(158,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(1534,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_intersection2 ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(127,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ meet_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k6_setfam_1) ).
thf(1375,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)],[127]) ).
thf(3,axiom,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_subset_1) ).
thf(403,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(213,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( A @ powerset @ powerset @ ( B @ element ) ) )
=> ? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( C @ relation_dom @ powerset @ ( E @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e6_27__finset_1) ).
thf(1878,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( A @ powerset @ powerset @ ( B @ element ) ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( C @ relation_dom @ powerset @ ( E @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( C @ relation_dom @ powerset @ ( E @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(303,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i] :
( ( B
= ( A @ relation_rng ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_funct_1) ).
thf(2537,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i] :
( ( ! [C: $i] :
( ( ? [D: $i] :
( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) )
=> ( B
= ( A @ relation_rng ) ) )
& ( ( B
= ( A @ relation_rng ) )
=> ! [C: $i] :
( ( ? [D: $i] :
( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(79,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/sandbox/benchmark/theBenchmark.p',d2_zfmisc_1) ).
thf(869,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)],[79]) ).
thf(107,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ! [B: $i,C: $i,D: $i] :
( ( ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( D = E )
& ( E @ ordinal ) )
& ( B = D )
& ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( C = E )
& ( E @ ordinal ) )
& ( B = C ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( C = E )
& ( E @ ordinal ) )
& ( D = C )
& ( A @ succ @ ( D @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e18_27__finset_1__1) ).
thf(1118,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ! [B: $i,C: $i,D: $i] :
( ( ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( D = E )
& ( E @ ordinal ) )
& ( B = D )
& ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( C = E )
& ( E @ ordinal ) )
& ( B = C ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ? [D: $i] :
( ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( C = E )
& ( E @ ordinal ) )
& ( D = C )
& ( A @ succ @ ( D @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ? [E: $i] :
( ( ( omega @ ( E @ in ) )
=> ! [F: $i] :
( ( E @ powerset @ powerset @ ( F @ element ) )
=> ~ ( ! [G: $i] :
~ ( ! [H: $i] :
( ( ( H @ ( G @ subset ) )
& ( F @ ( H @ in ) ) )
=> ( H = G ) )
& ( F @ ( G @ in ) ) )
& ( F != empty_set ) ) ) )
& ( C = E )
& ( E @ ordinal ) )
& ( D = C )
& ( A @ succ @ ( D @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(367,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_relat_1) ).
thf(3087,plain,
! [A: $i] :
( ( A @ empty )
=> ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).
thf(293,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_difference ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_relat_1) ).
thf(2498,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_difference ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(160,axiom,
! [A: $i,B: $i] :
( ( ( A @ empty )
=> ( ( A @ ( B @ element ) )
<=> ( B @ empty ) ) )
& ( ~ ( A @ empty )
=> ( ( A @ ( B @ element ) )
<=> ( A @ ( B @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_subset_1) ).
thf(1541,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)],[160]) ).
thf(204,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
<=> ! [B: $i,C: $i] :
( ( ( ( B @ ( A @ apply ) )
= ( C @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( C @ in ) )
& ( A @ relation_dom @ ( B @ in ) ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_funct_1) ).
thf(1804,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( ! [B: $i,C: $i] :
( ( ( ( B @ ( A @ apply ) )
= ( C @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( C @ in ) )
& ( A @ relation_dom @ ( B @ in ) ) )
=> ( B = C ) )
=> ( A @ one_to_one ) )
& ( ( A @ one_to_one )
=> ! [B: $i,C: $i] :
( ( ( ( B @ ( A @ apply ) )
= ( C @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( C @ in ) )
& ( A @ relation_dom @ ( B @ in ) ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(36,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_boole) ).
thf(615,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(231,axiom,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( A @ function )
& ( A @ relation ) )
=> ( B @ ( A @ relation_image ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc13_finset_1) ).
thf(2023,plain,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( A @ function )
& ( A @ relation ) )
=> ( B @ ( A @ relation_image ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
=> ~ ( A @ ( B @ proper_subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',antisymmetry_r2_xboole_0) ).
thf(488,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ proper_subset ) )
=> ~ ( A @ ( B @ proper_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(223,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ! [B: $i,C: $i] :
~ ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t56_relat_1) ).
thf(1942,plain,
! [A: $i] :
( ( A @ relation )
=> ( ! [B: $i,C: $i] :
~ ( A @ ( C @ ( B @ ordered_pair ) @ in ) )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(3094,plain,
empty_set @ one_to_one,
inference(cnfConj,[status(thm)],[3092]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ being_limit_ordinal )
<=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( A @ ( B @ succ @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t41_ordinal1) ).
thf(476,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( A @ ( B @ succ @ in ) ) ) )
=> ( A @ being_limit_ordinal ) )
& ( ( A @ being_limit_ordinal )
=> ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( A @ ( B @ succ @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(142,axiom,
! [A: $i] :
( ( A @ epsilon_connected )
<=> ! [B: $i,C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ( B != C )
& ~ ( C @ ( B @ in ) )
& ( A @ ( C @ in ) )
& ( A @ ( B @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_ordinal1) ).
thf(1424,plain,
! [A: $i] :
( ( ! [B: $i,C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ( B != C )
& ~ ( C @ ( B @ in ) )
& ( A @ ( C @ in ) )
& ( A @ ( B @ in ) ) )
=> ( A @ epsilon_connected ) )
& ( ( A @ epsilon_connected )
=> ! [B: $i,C: $i] :
~ ( ~ ( B @ ( C @ in ) )
& ( B != C )
& ~ ( C @ ( B @ in ) )
& ( A @ ( C @ in ) )
& ( A @ ( B @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(22,axiom,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(526,plain,
! [A: $i,B: $i] :
~ ( B @ ( A @ unordered_pair ) @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(2613,plain,
sk285 @ one_to_one,
inference(cnfConj,[status(thm)],[2608]) ).
thf(3288,plain,
( ( ( sk29 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,732]) ).
thf(3289,plain,
( ( sk29 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3288]) ).
thf(3294,plain,
sk29 != empty_set,
inference(simp,[status(thm)],[3289]) ).
thf(49,axiom,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_boole) ).
thf(706,plain,
! [A: $i,B: $i] :
~ ( ( B @ empty )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(3308,plain,
( ( ( sk226 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,2145]) ).
thf(3309,plain,
( ( sk226 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3308]) ).
thf(3311,plain,
sk226 != empty_set,
inference(simp,[status(thm)],[3309]) ).
thf(347,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/sandbox/benchmark/theBenchmark.p',d5_relat_1) ).
thf(2877,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)],[347]) ).
thf(116,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(1292,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( ( B @ ( A @ set_intersection2 ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(796,plain,
~ ( sk37 @ empty ),
inference(cnfConj,[status(thm)],[792]) ).
thf(3617,plain,
( ( ( sk285 @ empty )
!= ( sk37 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,796]) ).
thf(3618,plain,
( ( sk285 @ empty )
!= ( sk37 @ empty ) ),
inference(simp,[status(thm)],[3617]) ).
thf(3622,plain,
sk285 != sk37,
inference(simp,[status(thm)],[3618]) ).
thf(35,axiom,
( ! [A: $i] :
( ( A @ ordinal )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) )
=> ! [A: $i] :
( ( A @ ordinal )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s2_ordinal1__e18_27__finset_1__1) ).
thf(590,plain,
( ! [A: $i] :
( ( A @ ordinal )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) )
=> ! [A: $i] :
( ( A @ ordinal )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(238,axiom,
? [A: $i] :
( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc3_funct_1) ).
thf(2085,plain,
? [A: $i] :
( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(1889,plain,
sk176 @ relation,
inference(cnfConj,[status(thm)],[1888]) ).
thf(176,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(1644,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ in ) )
=> ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(837,plain,
sk41 @ ordinal,
inference(cnfConj,[status(thm)],[834]) ).
thf(103,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_intersection2 ) )
= ( A @ ( B @ set_intersection2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k3_xboole_0) ).
thf(1099,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_intersection2 ) )
= ( A @ ( B @ set_intersection2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(144,axiom,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(1451,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(100,axiom,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_ordinal1) ).
thf(1058,plain,
? [A: $i] :
( ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(1059,plain,
( ( sk76 @ ordinal )
& ( sk76 @ epsilon_connected )
& ( sk76 @ epsilon_transitive ) ),
inference(cnf,[status(esa)],[1058]) ).
thf(1060,plain,
sk76 @ epsilon_transitive,
inference(cnfConj,[status(thm)],[1059]) ).
thf(295,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ well_ordering )
=> ( A @ ( B @ relation_restriction ) @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t32_wellord1) ).
thf(2505,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ well_ordering )
=> ( A @ ( B @ relation_restriction ) @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(53,axiom,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( A @ ( B @ set_union2 ) @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(721,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( A @ ( B @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(382,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B @ ( C @ relation_composition ) @ relation_dom @ ( A @ in ) )
<=> ( ( B @ relation_dom @ ( A @ ( C @ apply ) @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t21_funct_1) ).
thf(3201,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( ( B @ relation_dom @ ( A @ ( C @ apply ) @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) )
=> ( B @ ( C @ relation_composition ) @ relation_dom @ ( A @ in ) ) )
& ( ( B @ ( C @ relation_composition ) @ relation_dom @ ( A @ in ) )
=> ( ( B @ relation_dom @ ( A @ ( C @ apply ) @ in ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).
thf(20,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ~ ( ( A @ being_limit_ordinal )
& ? [B: $i] :
( ( A
= ( B @ succ ) )
& ( B @ ordinal ) ) )
& ~ ( ! [B: $i] :
( ( B @ ordinal )
=> ( A
!= ( B @ succ ) ) )
& ~ ( A @ being_limit_ordinal ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t42_ordinal1) ).
thf(515,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ~ ( ( A @ being_limit_ordinal )
& ? [B: $i] :
( ( A
= ( B @ succ ) )
& ( B @ ordinal ) ) )
& ~ ( ! [B: $i] :
( ( B @ ordinal )
=> ( A
!= ( B @ succ ) ) )
& ~ ( A @ being_limit_ordinal ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(298,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( ( B @ relation_field @ ( A @ subset ) )
& ( B @ well_ordering ) )
=> ( ( A @ ( B @ relation_restriction ) @ relation_field )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t39_wellord1) ).
thf(2519,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( ( B @ relation_field @ ( A @ subset ) )
& ( B @ well_ordering ) )
=> ( ( A @ ( B @ relation_restriction ) @ relation_field )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(302,axiom,
? [A: $i] :
( ( A @ function )
& ( A @ relation_empty_yielding )
& ( A @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc4_funct_1) ).
thf(2532,plain,
? [A: $i] :
( ( A @ function )
& ( A @ relation_empty_yielding )
& ( A @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(2533,plain,
( ( sk273 @ function )
& ( sk273 @ relation_empty_yielding )
& ( sk273 @ relation ) ),
inference(cnf,[status(esa)],[2532]) ).
thf(2536,plain,
sk273 @ function,
inference(cnfConj,[status(thm)],[2533]) ).
thf(321,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( C @ relation )
=> ( ( C
= ( B @ ( A @ relation_rng_restriction ) ) )
<=> ! [D: $i,E: $i] :
( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
<=> ( ( B @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d12_relat_1) ).
thf(2681,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( C @ relation )
=> ( ( ! [D: $i,E: $i] :
( ( ( ( B @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( B @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( E @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_rng_restriction ) ) ) )
& ( ( C
= ( B @ ( A @ relation_rng_restriction ) ) )
=> ! [D: $i,E: $i] :
( ( ( ( B @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( B @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).
thf(342,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
=> ( C @ function_inverse @ ( A @ ( B @ relation_isomorphism ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t49_wellord1) ).
thf(2862,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
=> ( C @ function_inverse @ ( A @ ( B @ relation_isomorphism ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).
thf(284,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( E = F ) )
& ( A @ ( E @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e10_24__wellord2__1) ).
thf(2370,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ? [E: $i] :
( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( E = F ) )
& ( A @ ( E @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( E = F ) )
& ( A @ ( E @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(386,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ is_transitive_in ) )
<=> ! [C: $i,D: $i,E: $i] :
( ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( E @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( A @ ( E @ ( C @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_relat_2) ).
thf(3215,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i,D: $i,E: $i] :
( ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( E @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( A @ ( E @ ( C @ ordered_pair ) @ in ) ) )
=> ( B @ ( A @ is_transitive_in ) ) )
& ( ( B @ ( A @ is_transitive_in ) )
=> ! [C: $i,D: $i,E: $i] :
( ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( E @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( A @ ( E @ ( C @ ordered_pair ) @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(209,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ empty )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc2_funct_1) ).
thf(1857,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ empty )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(205,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B
= ( A @ identity_relation ) )
<=> ! [C: $i,D: $i] :
( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
<=> ( ( C = D )
& ( A @ ( C @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d10_relat_1) ).
thf(1814,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( ! [C: $i,D: $i] :
( ( ( ( C = D )
& ( A @ ( C @ in ) ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( ( C = D )
& ( A @ ( C @ in ) ) ) ) )
=> ( B
= ( A @ identity_relation ) ) )
& ( ( B
= ( A @ identity_relation ) )
=> ! [C: $i,D: $i] :
( ( ( ( C = D )
& ( A @ ( C @ in ) ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( ( C = D )
& ( A @ ( C @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(3608,plain,
( ( ( sk285 @ empty )
!= ( sk226 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,2145]) ).
thf(3609,plain,
( ( sk285 @ empty )
!= ( sk226 @ empty ) ),
inference(simp,[status(thm)],[3608]) ).
thf(3624,plain,
sk285 != sk226,
inference(simp,[status(thm)],[3609]) ).
thf(84,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ subset_complement ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k3_subset_1) ).
thf(921,plain,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( A @ powerset @ ( B @ ( A @ subset_complement ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(1561,plain,
( ~ ( omega @ empty )
& ( omega @ ordinal )
& ( omega @ epsilon_connected )
& ( omega @ epsilon_transitive ) ),
inference(cnf,[status(esa)],[1560]) ).
thf(1563,plain,
omega @ epsilon_connected,
inference(cnfConj,[status(thm)],[1561]) ).
thf(63,axiom,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(765,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(2086,plain,
( ( sk215 @ one_to_one )
& ( sk215 @ function )
& ( sk215 @ relation ) ),
inference(cnf,[status(esa)],[2085]) ).
thf(2089,plain,
sk215 @ one_to_one,
inference(cnfConj,[status(thm)],[2086]) ).
thf(794,plain,
sk37 @ natural,
inference(cnfConj,[status(thm)],[792]) ).
thf(189,axiom,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(1732,plain,
! [A: $i] :
? [B: $i] : ( A @ ( B @ element ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(109,axiom,
( ( ! [A: $i] :
( ( A @ ordinal )
=> ( ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
& ( A @ being_limit_ordinal ) )
=> ( ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) )
| ( A = empty_set ) ) ) )
& ! [A: $i] :
( ( A @ ordinal )
=> ( ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) )
=> ( ( omega @ ( A @ succ @ in ) )
=> ! [B: $i] :
( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) )
& ( ( omega @ ( empty_set @ in ) )
=> ! [A: $i] :
( ( empty_set @ powerset @ powerset @ ( A @ element ) )
=> ~ ( ! [B: $i] :
~ ( ! [C: $i] :
( ( ( C @ ( B @ subset ) )
& ( A @ ( C @ in ) ) )
=> ( C = B ) )
& ( A @ ( B @ in ) ) )
& ( A != empty_set ) ) ) ) )
=> ! [A: $i] :
( ( A @ ordinal )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_ordinal2__e18_27__finset_1) ).
thf(1187,plain,
( ( ! [A: $i] :
( ( A @ ordinal )
=> ( ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ! [D: $i] :
~ ( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
& ( A @ being_limit_ordinal ) )
=> ( ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) )
| ( A = empty_set ) ) ) )
& ! [A: $i] :
( ( A @ ordinal )
=> ( ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) )
=> ( ( omega @ ( A @ succ @ in ) )
=> ! [B: $i] :
( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) )
& ( ( omega @ ( empty_set @ in ) )
=> ! [A: $i] :
( ( empty_set @ powerset @ powerset @ ( A @ element ) )
=> ~ ( ! [B: $i] :
~ ( ! [C: $i] :
( ( ( C @ ( B @ subset ) )
& ( A @ ( C @ in ) ) )
=> ( C = B ) )
& ( A @ ( B @ in ) ) )
& ( A != empty_set ) ) ) ) )
=> ! [A: $i] :
( ( A @ ordinal )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(290,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( C
= ( B @ ( A @ relation_image ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( D @ ( E @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d13_relat_1) ).
thf(2440,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( ! [D: $i] :
( ( ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( D @ ( E @ ordered_pair ) @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( D @ ( E @ ordered_pair ) @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_image ) ) ) )
& ( ( C
= ( B @ ( A @ relation_image ) ) )
=> ! [D: $i] :
( ( ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( D @ ( E @ ordered_pair ) @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( D @ ( E @ ordered_pair ) @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(250,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_dom @ ( A @ relation_image ) )
= ( A @ relation_rng ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t146_relat_1) ).
thf(2189,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_dom @ ( A @ relation_image ) )
= ( A @ relation_rng ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(171,axiom,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ~ ( ! [C: $i] :
( ( C @ ordinal )
=> ~ ( ! [D: $i] :
( ( D @ ordinal )
=> ( ( A @ ( D @ in ) )
=> ( D @ ( C @ ordinal_subset ) ) ) )
& ( A @ ( C @ in ) ) ) )
& ( A != empty_set )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t32_ordinal1) ).
thf(1616,plain,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ~ ( ! [C: $i] :
( ( C @ ordinal )
=> ~ ( ! [D: $i] :
( ( D @ ordinal )
=> ( ( A @ ( D @ in ) )
=> ( D @ ( C @ ordinal_subset ) ) ) )
& ( A @ ( C @ in ) ) ) )
& ( A != empty_set )
& ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(261,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( ( ( A @ relation_rng )
= empty_set )
| ( ( A @ relation_dom )
= empty_set ) )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t64_relat_1) ).
thf(2235,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( ( A @ relation_rng )
= empty_set )
| ( ( A @ relation_dom )
= empty_set ) )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(227,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ~ ( ( ( A @ ( B @ relation_inverse_image ) )
= empty_set )
& ( B @ relation_rng @ ( A @ subset ) )
& ( A != empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t174_relat_1) ).
thf(2001,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ~ ( ( ( A @ ( B @ relation_inverse_image ) )
= empty_set )
& ( B @ relation_rng @ ( A @ subset ) )
& ( A != empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(37,axiom,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(618,plain,
! [A: $i] :
( ( empty_set @ ( A @ subset ) )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(272,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ well_founded_relation )
<=> ! [B: $i] :
~ ( ! [C: $i] :
~ ( ( B @ ( C @ ( A @ fiber ) @ disjoint ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set )
& ( A @ relation_field @ ( B @ subset ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_wellord1) ).
thf(2275,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ! [B: $i] :
~ ( ! [C: $i] :
~ ( ( B @ ( C @ ( A @ fiber ) @ disjoint ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set )
& ( A @ relation_field @ ( B @ subset ) ) )
=> ( A @ well_founded_relation ) )
& ( ( A @ well_founded_relation )
=> ! [B: $i] :
~ ( ! [C: $i] :
~ ( ( B @ ( C @ ( A @ fiber ) @ disjoint ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set )
& ( A @ relation_field @ ( B @ subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(324,axiom,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t15_finset_1) ).
thf(2717,plain,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).
thf(3079,plain,
sk358 @ relation,
inference(cnfConj,[status(thm)],[3077]) ).
thf(57,axiom,
! [A: $i] :
~ ( A @ singleton @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(745,plain,
! [A: $i] :
~ ( A @ singleton @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(56,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(733,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)],[56]) ).
thf(2936,plain,
sk332 @ empty,
inference(cnfConj,[status(thm)],[2934]) ).
thf(178,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ powerset ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( A @ ( C @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_zfmisc_1) ).
thf(1648,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)],[178]) ).
thf(412,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ union ) )
| ~ ( B @ ( A @ sk3 ) )
| ( B @ ( A @ ( B @ sk4 ) @ in ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(424,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ sk3 ) )
| ( B @ ( A @ ( B @ sk4 ) @ in ) )
| ( B
= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[412]) ).
thf(425,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ sk3 ) )
| ( B @ ( A @ ( B @ sk4 ) @ in ) )
| ( B
= ( A @ union ) ) ),
inference(simp,[status(thm)],[424]) ).
thf(320,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ function_inverse @ function )
& ( A @ function_inverse @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k2_funct_1) ).
thf(2677,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ function_inverse @ function )
& ( A @ function_inverse @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(2028,plain,
sk205 @ relation,
inference(cnfConj,[status(thm)],[2026]) ).
thf(119,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_boole) ).
thf(1301,plain,
! [A: $i] :
( ( A @ empty )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(416,plain,
! [C: $i,B: $i,A: $i] :
( ( B
= ( A @ union ) )
| ~ ( B @ ( A @ sk3 ) )
| ~ ( A @ ( C @ in ) )
| ~ ( C @ ( A @ ( B @ sk4 ) @ in ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(422,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ sk3 ) )
| ~ ( A @ ( C @ in ) )
| ~ ( C @ ( A @ ( B @ sk4 ) @ in ) )
| ( B
= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[416]) ).
thf(423,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ sk3 ) )
| ~ ( A @ ( C @ in ) )
| ~ ( C @ ( A @ ( B @ sk4 ) @ in ) )
| ( B
= ( A @ union ) ) ),
inference(simp,[status(thm)],[422]) ).
thf(97,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ? [B: $i] :
! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( ? [D: $i] :
( ( ( omega @ ( D @ in ) )
=> ! [E: $i] :
( ( D @ powerset @ powerset @ ( E @ element ) )
=> ~ ( ! [F: $i] :
~ ( ! [G: $i] :
( ( ( G @ ( F @ subset ) )
& ( E @ ( G @ in ) ) )
=> ( G = F ) )
& ( E @ ( F @ in ) ) )
& ( E != empty_set ) ) ) )
& ( C = D )
& ( D @ ordinal ) )
& ( A @ succ @ ( C @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e18_27__finset_1__1) ).
thf(1007,plain,
! [A: $i] :
( ( A @ ordinal )
=> ? [B: $i] :
! [C: $i] :
( ( ( ? [D: $i] :
( ( ( omega @ ( D @ in ) )
=> ! [E: $i] :
( ( D @ powerset @ powerset @ ( E @ element ) )
=> ~ ( ! [F: $i] :
~ ( ! [G: $i] :
( ( ( G @ ( F @ subset ) )
& ( E @ ( G @ in ) ) )
=> ( G = F ) )
& ( E @ ( F @ in ) ) )
& ( E != empty_set ) ) ) )
& ( C = D )
& ( D @ ordinal ) )
& ( A @ succ @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( ? [D: $i] :
( ( ( omega @ ( D @ in ) )
=> ! [E: $i] :
( ( D @ powerset @ powerset @ ( E @ element ) )
=> ~ ( ! [F: $i] :
~ ( ! [G: $i] :
( ( ( G @ ( F @ subset ) )
& ( E @ ( G @ in ) ) )
=> ( G = F ) )
& ( E @ ( F @ in ) ) )
& ( E != empty_set ) ) ) )
& ( C = D )
& ( D @ ordinal ) )
& ( A @ succ @ ( C @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(398,plain,
( ~ ( sk1 @ relation_rng @ finite )
& ( sk1 @ relation_dom @ finite )
& ( sk1 @ function )
& ( sk1 @ relation ) ),
inference(cnf,[status(esa)],[397]) ).
thf(402,plain,
~ ( sk1 @ relation_rng @ finite ),
inference(cnfConj,[status(thm)],[398]) ).
thf(355,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( C
= ( B @ ( A @ relation_image ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ( D
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ in ) )
& ( A @ relation_dom @ ( E @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d12_funct_1) ).
thf(2938,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( ! [D: $i] :
( ( ? [E: $i] :
( ( D
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ in ) )
& ( A @ relation_dom @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( D
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ in ) )
& ( A @ relation_dom @ ( E @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_image ) ) ) )
& ( ( C
= ( B @ ( A @ relation_image ) ) )
=> ! [D: $i] :
( ( ? [E: $i] :
( ( D
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ in ) )
& ( A @ relation_dom @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( D
= ( E @ ( A @ apply ) ) )
& ( B @ ( E @ in ) )
& ( A @ relation_dom @ ( E @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).
thf(133,axiom,
! [A: $i] :
? [B: $i] :
! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( ( C @ ordinal )
& ( A @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e6_22__wellord2) ).
thf(1389,plain,
! [A: $i] :
? [B: $i] :
! [C: $i] :
( ( ( ( C @ ordinal )
& ( A @ ( C @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ( ( C @ ordinal )
& ( A @ ( C @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(75,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ union_of_subsets ) )
= ( B @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k5_setfam_1) ).
thf(843,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)],[75]) ).
thf(51,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ( ( B @ ( A @ set_intersection2 ) )
= empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_xboole_0) ).
thf(711,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)],[51]) ).
thf(201,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_boole) ).
thf(1795,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_intersection2 ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(26,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t69_enumset1) ).
thf(545,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(3212,plain,
( ( sk379 @ function )
& ( sk379 @ relation ) ),
inference(cnf,[status(esa)],[3211]) ).
thf(3213,plain,
sk379 @ relation,
inference(cnfConj,[status(thm)],[3212]) ).
thf(151,axiom,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(1503,plain,
( ( empty_set @ powerset )
= ( empty_set @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(179,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ disjoint ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ disjoint ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t63_xboole_1) ).
thf(1669,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ disjoint ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(89,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/sandbox/benchmark/theBenchmark.p',t9_tarski) ).
thf(936,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)],[89]) ).
thf(233,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ is_antisymmetric_in ) )
<=> ! [C: $i,D: $i] :
( ( ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( C = D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_relat_2) ).
thf(2029,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i,D: $i] :
( ( ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( C = D ) )
=> ( B @ ( A @ is_antisymmetric_in ) ) )
& ( ( B @ ( A @ is_antisymmetric_in ) )
=> ! [C: $i,D: $i] :
( ( ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( C = D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(274,axiom,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= ( C @ singleton ) ) )
& ( ( B @ relation_dom )
= A )
& ( B @ function )
& ( B @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s3_funct_1__e16_22__wellord2) ).
thf(2293,plain,
! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= ( C @ singleton ) ) )
& ( ( B @ relation_dom )
= A )
& ( B @ function )
& ( B @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(2935,plain,
sk332 @ relation,
inference(cnfConj,[status(thm)],[2934]) ).
thf(1565,plain,
~ ( omega @ empty ),
inference(cnfConj,[status(thm)],[1561]) ).
thf(3300,plain,
( ( ( sk12 @ empty )
!= ( omega @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,1565]) ).
thf(3301,plain,
( ( sk12 @ empty )
!= ( omega @ empty ) ),
inference(simp,[status(thm)],[3300]) ).
thf(3304,plain,
sk12 != omega,
inference(simp,[status(thm)],[3301]) ).
thf(175,axiom,
! [A: $i] :
( ( omega @ ( A @ element ) )
=> ( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc3_arytm_3) ).
thf(1638,plain,
! [A: $i] :
( ( omega @ ( A @ element ) )
=> ( ( A @ natural )
& ( A @ ordinal )
& ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(279,axiom,
! [A: $i,B: $i] :
( ( A @ relation )
=> ( B @ ( A @ relation_dom_restriction ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k7_relat_1) ).
thf(2319,plain,
! [A: $i,B: $i] :
( ( A @ relation )
=> ( B @ ( A @ relation_dom_restriction ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(210,axiom,
( ( empty_set @ relation )
& ( empty_set @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_relat_1) ).
thf(1862,plain,
( ( empty_set @ relation )
& ( empty_set @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(1863,plain,
( ( empty_set @ relation )
& ( empty_set @ empty ) ),
inference(cnf,[status(esa)],[1862]) ).
thf(1865,plain,
empty_set @ relation,
inference(cnfConj,[status(thm)],[1863]) ).
thf(16,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/sandbox/benchmark/theBenchmark.p',t47_setfam_1) ).
thf(498,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)],[16]) ).
thf(218,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ subset ) )
<=> ! [C: $i,D: $i] :
( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_relat_1) ).
thf(1897,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( ! [C: $i,D: $i] :
( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
=> ( B @ ( A @ subset ) ) )
& ( ( B @ ( A @ subset ) )
=> ! [C: $i,D: $i] :
( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(83,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(919,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ set_difference ) @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
& ( A @ ordinal ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( E
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( C = E )
& ? [F: $i] :
( ( D
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ? [F: $i] :
( ( D
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( E = D )
& ( A @ powerset @ ( E @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e4_27_3_1__finset_1__1) ).
thf(947,plain,
! [A: $i,B: $i] :
( ( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
& ( A @ ordinal ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( E
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( C = E )
& ? [F: $i] :
( ( D
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ? [E: $i] :
( ? [F: $i] :
( ( D
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( E = D )
& ( A @ powerset @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ? [F: $i] :
( ( D
= ( A @ singleton @ ( F @ set_difference ) ) )
& ( B @ ( F @ in ) ) )
& ( E = D )
& ( A @ powerset @ ( E @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(370,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B @ ( A @ relation_rng_restriction ) @ function )
& ( B @ ( A @ relation_rng_restriction ) @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc5_funct_1) ).
thf(3101,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B @ ( A @ relation_rng_restriction ) @ function )
& ( B @ ( A @ relation_rng_restriction ) @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).
thf(217,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( A @ ( C @ relation_dom_restriction ) @ relation_dom @ ( B @ in ) )
=> ( ( B @ ( A @ ( C @ relation_dom_restriction ) @ apply ) )
= ( B @ ( C @ apply ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t70_funct_1) ).
thf(1894,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( A @ ( C @ relation_dom_restriction ) @ relation_dom @ ( B @ in ) )
=> ( ( B @ ( A @ ( C @ relation_dom_restriction ) @ apply ) )
= ( B @ ( C @ apply ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(363,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( C @ relation )
=> ( ( C
= ( B @ ( A @ relation_composition ) ) )
<=> ! [D: $i,E: $i] :
( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
<=> ? [F: $i] :
( ( B @ ( E @ ( F @ ordered_pair ) @ in ) )
& ( A @ ( F @ ( D @ ordered_pair ) @ in ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_relat_1) ).
thf(3038,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( C @ relation )
=> ( ( ! [D: $i,E: $i] :
( ( ? [F: $i] :
( ( B @ ( E @ ( F @ ordered_pair ) @ in ) )
& ( A @ ( F @ ( D @ ordered_pair ) @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ? [F: $i] :
( ( B @ ( E @ ( F @ ordered_pair ) @ in ) )
& ( A @ ( F @ ( D @ ordered_pair ) @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_composition ) ) ) )
& ( ( C
= ( B @ ( A @ relation_composition ) ) )
=> ! [D: $i,E: $i] :
( ( ? [F: $i] :
( ( B @ ( E @ ( F @ ordered_pair ) @ in ) )
& ( A @ ( F @ ( D @ ordered_pair ) @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ? [F: $i] :
( ( B @ ( E @ ( F @ ordered_pair ) @ in ) )
& ( A @ ( F @ ( D @ ordered_pair ) @ in ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(3285,plain,
( ( ( sk55 @ empty )
!= ( sk12 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,915]) ).
thf(3286,plain,
( ( sk55 @ empty )
!= ( sk12 @ empty ) ),
inference(simp,[status(thm)],[3285]) ).
thf(3287,plain,
sk55 != sk12,
inference(simp,[status(thm)],[3286]) ).
thf(125,axiom,
! [A: $i,B: $i] :
( ( ( B @ singleton @ ( A @ set_difference ) )
= A )
<=> ~ ( A @ ( B @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t65_zfmisc_1) ).
thf(1349,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)],[125]) ).
thf(242,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( C
= ( B @ ( A @ relation_inverse_image ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d14_relat_1) ).
thf(2116,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( ! [D: $i] :
( ( ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_inverse_image ) ) ) )
& ( ( C
= ( B @ ( A @ relation_inverse_image ) ) )
=> ! [D: $i] :
( ( ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ( B @ ( E @ in ) )
& ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(267,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ connected )
=> ( A @ ( B @ relation_restriction ) @ connected ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t23_wellord1) ).
thf(2257,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ connected )
=> ( A @ ( B @ relation_restriction ) @ connected ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(94,axiom,
! [A: $i] :
~ ( A @ powerset @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(996,plain,
! [A: $i] :
~ ( A @ powerset @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(18,axiom,
! [A: $i] : ( A @ succ @ ( A @ in ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t10_ordinal1) ).
thf(509,plain,
! [A: $i] : ( A @ succ @ ( A @ in ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(203,axiom,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc11_finset_1) ).
thf(1801,plain,
! [A: $i,B: $i] :
( ( A @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(166,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/sandbox/benchmark/theBenchmark.p',l55_zfmisc_1) ).
thf(1571,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)],[166]) ).
thf(2871,plain,
sk322 @ relation,
inference(cnfConj,[status(thm)],[2870]) ).
thf(252,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_dom_restriction ) @ relation_dom )
= ( A @ ( B @ relation_dom @ set_intersection2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t90_relat_1) ).
thf(2197,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_dom_restriction ) @ relation_dom )
= ( A @ ( B @ relation_dom @ set_intersection2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(226,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ! [D: $i,E: $i] :
( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
<=> ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( D = F ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) )
& ( C @ function )
& ( C @ relation ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_funct_1__e10_24__wellord2__1) ).
thf(1969,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) )
& ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( D @ ordered_pair ) @ in ) ) )
& ( F @ ( D @ in ) )
& ( C = F ) )
& ( A @ ( C @ in ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ! [D: $i,E: $i] :
( ( ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( D = F ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) )
=> ( C @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ? [F: $i] :
( ! [G: $i] :
( ( F @ ( G @ in ) )
=> ( B @ ( G @ ( E @ ordered_pair ) @ in ) ) )
& ( F @ ( E @ in ) )
& ( D = F ) )
& ( A @ ( D @ in ) )
& ( A @ ( D @ in ) ) ) ) )
& ( C @ function )
& ( C @ relation ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(121,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ well_founded_relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_wellord2) ).
thf(1307,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ well_founded_relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(2610,plain,
sk285 @ function,
inference(cnfConj,[status(thm)],[2608]) ).
thf(162,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) )
= ( B @ ( A @ set_union2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t39_xboole_1) ).
thf(1557,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ set_difference ) @ ( A @ set_union2 ) )
= ( B @ ( A @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(38,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ well_ordering ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_wellord2) ).
thf(621,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ well_ordering ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(77,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( A @ ( C @ in ) )
& ( C @ ( B @ in ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_ordinal1) ).
thf(848,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( A @ ( C @ in ) )
& ( C @ ( B @ in ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(310,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( A @ ( B @ relation_restriction ) @ relation_field @ subset ) )
& ( B @ relation_field @ ( A @ ( B @ relation_restriction ) @ relation_field @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t20_wellord1) ).
thf(2593,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( A @ ( B @ relation_restriction ) @ relation_field @ subset ) )
& ( B @ relation_field @ ( A @ ( B @ relation_restriction ) @ relation_field @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(64,axiom,
! [A: $i] :
( ( A @ epsilon_transitive )
<=> ! [B: $i] :
( ( A @ ( B @ in ) )
=> ( A @ ( B @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_ordinal1) ).
thf(769,plain,
! [A: $i] :
( ( ! [B: $i] :
( ( A @ ( B @ in ) )
=> ( A @ ( B @ subset ) ) )
=> ( A @ epsilon_transitive ) )
& ( ( A @ epsilon_transitive )
=> ! [B: $i] :
( ( A @ ( B @ in ) )
=> ( A @ ( B @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(332,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ transitive )
=> ( A @ ( B @ relation_restriction ) @ transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t24_wellord1) ).
thf(2812,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ transitive )
=> ( A @ ( B @ relation_restriction ) @ transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).
thf(352,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ relation_rng_restriction ) @ relation_rng )
= ( A @ ( B @ relation_rng @ set_intersection2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t119_relat_1) ).
thf(2928,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ relation_rng_restriction ) @ relation_rng )
= ( A @ ( B @ relation_rng @ set_intersection2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).
thf(108,axiom,
! [A: $i,B: $i,C: $i] :
( ( A @ ( B @ ( C @ relation_of2_as_subset ) ) )
=> ( ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
<=> ( ( C @ ( A @ ( B @ relation_dom_as_subset ) ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t22_relset_1) ).
thf(1177,plain,
! [A: $i,B: $i,C: $i] :
( ( A @ ( B @ ( C @ relation_of2_as_subset ) ) )
=> ( ( ( ( C @ ( A @ ( B @ relation_dom_as_subset ) ) )
= B )
=> ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) ) )
& ( ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( E @ ( D @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
=> ( ( C @ ( A @ ( B @ relation_dom_as_subset ) ) )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(130,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_difference ) @ ( C @ ( A @ set_difference ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t33_xboole_1) ).
thf(1381,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)],[130]) ).
thf(378,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ is_connected_in ) )
<=> ! [C: $i,D: $i] :
~ ( ~ ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ~ ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( C != D )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d6_relat_2) ).
thf(3150,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i,D: $i] :
~ ( ~ ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ~ ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( C != D )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) )
=> ( B @ ( A @ is_connected_in ) ) )
& ( ( B @ ( A @ is_connected_in ) )
=> ! [C: $i,D: $i] :
~ ( ~ ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
& ~ ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
& ( C != D )
& ( B @ ( D @ in ) )
& ( B @ ( C @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).
thf(1061,plain,
sk76 @ epsilon_connected,
inference(cnfConj,[status(thm)],[1059]) ).
thf(225,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( A = B )
<=> ! [C: $i,D: $i] :
( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
<=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_relat_1) ).
thf(1952,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( ! [C: $i,D: $i] :
( ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( A @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) ) )
=> ( A = B ) )
& ( ( A = B )
=> ! [C: $i,D: $i] :
( ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( A @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( A @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(328,axiom,
! [A: $i] :
? [B: $i] :
( ( B @ finite )
& ( B @ natural )
& ( B @ ordinal )
& ( B @ epsilon_connected )
& ( B @ epsilon_transitive )
& ( B @ one_to_one )
& ( B @ function )
& ( B @ relation )
& ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_finset_1) ).
thf(2770,plain,
! [A: $i] :
? [B: $i] :
( ( B @ finite )
& ( B @ natural )
& ( B @ ordinal )
& ( B @ epsilon_connected )
& ( B @ epsilon_transitive )
& ( B @ one_to_one )
& ( B @ function )
& ( B @ relation )
& ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(134,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t1_xboole_1) ).
thf(1396,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(399,plain,
sk1 @ relation,
inference(cnfConj,[status(thm)],[398]) ).
thf(186,axiom,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(1699,plain,
! [A: $i,B: $i] :
( ~ ( A @ empty )
=> ~ ( B @ ( A @ set_union2 ) @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(338,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equipotent ) )
<=> ? [C: $i] :
( ( ( C @ relation_rng )
= B )
& ( ( C @ relation_dom )
= A )
& ( C @ one_to_one )
& ( C @ function )
& ( C @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d4_wellord2) ).
thf(2835,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( ( ( C @ relation_rng )
= B )
& ( ( C @ relation_dom )
= A )
& ( C @ one_to_one )
& ( C @ function )
& ( C @ relation ) )
=> ( B @ ( A @ equipotent ) ) )
& ( ( B @ ( A @ equipotent ) )
=> ? [C: $i] :
( ( ( C @ relation_rng )
= B )
& ( ( C @ relation_dom )
= A )
& ( C @ one_to_one )
& ( C @ function )
& ( C @ relation ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).
thf(66,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
<=> ( B @ ( C @ in ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_tarski) ).
thf(780,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)],[66]) ).
thf(353,axiom,
! [A: $i] :
( ( ( A @ relation )
& ~ ( A @ empty ) )
=> ~ ( A @ relation_dom @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc5_relat_1) ).
thf(2931,plain,
! [A: $i] :
( ( ( A @ relation )
& ~ ( A @ empty ) )
=> ~ ( A @ relation_dom @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).
thf(263,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_inverse @ relation_inverse )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k4_relat_1) ).
thf(2244,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_inverse @ relation_inverse )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(3306,plain,
( ( ( sk226 @ empty )
!= ( sk12 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,2145]) ).
thf(3307,plain,
( ( sk226 @ empty )
!= ( sk12 @ empty ) ),
inference(simp,[status(thm)],[3306]) ).
thf(3310,plain,
sk226 != sk12,
inference(simp,[status(thm)],[3307]) ).
thf(375,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( B @ ( A @ relation_rng_restriction ) @ relation_rng @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t118_relat_1) ).
thf(3142,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_rng @ ( B @ ( A @ relation_rng_restriction ) @ relation_rng @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(2535,plain,
sk273 @ relation_empty_yielding,
inference(cnfConj,[status(thm)],[2533]) ).
thf(113,axiom,
! [A: $i,B: $i] :
( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= E ) )
& ( C = E )
& ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= D ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= D ) )
& ( E = D )
& ( B @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e16_22__wellord2__2) ).
thf(1250,plain,
! [A: $i,B: $i] :
( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= E ) )
& ( C = E )
& ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= D ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ? [E: $i] :
( ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= D ) )
& ( E = D )
& ( B @ ( A @ cartesian_product2 ) @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ? [F: $i,G: $i] :
( ( G
= ( F @ singleton ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= D ) )
& ( E = D )
& ( B @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(276,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_intersection2 ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_relat_1) ).
thf(2303,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_intersection2 ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(7,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( B @ ( A @ unordered_pair ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( D = B )
| ( D = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d2_tarski) ).
thf(438,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)],[7]) ).
thf(414,plain,
! [B: $i,A: $i] :
( ( B
= ( A @ union ) )
| ~ ( B @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(430,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[414]) ).
thf(431,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ ( B @ sk5 ) @ in ) )
| ( B @ ( A @ sk3 ) )
| ( B
= ( A @ union ) ) ),
inference(simp,[status(thm)],[430]) ).
thf(184,axiom,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_boole) ).
thf(1694,plain,
! [A: $i] :
( ( A @ ( empty_set @ set_difference ) )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(1562,plain,
omega @ epsilon_transitive,
inference(cnfConj,[status(thm)],[1561]) ).
thf(235,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( A @ powerset @ powerset @ ( B @ element ) ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( B @ ( F @ ( C @ relation_image ) @ in ) )
& ( D = F )
& ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ? [F: $i] :
( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( F = E )
& ( C @ relation_dom @ powerset @ ( F @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e6_27__finset_1__1) ).
thf(2051,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( A @ powerset @ powerset @ ( B @ element ) ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( B @ ( F @ ( C @ relation_image ) @ in ) )
& ( D = F )
& ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ? [F: $i] :
( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( F = E )
& ( C @ relation_dom @ powerset @ ( F @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ? [F: $i] :
( ( B @ ( E @ ( C @ relation_image ) @ in ) )
& ( F = E )
& ( C @ relation_dom @ powerset @ ( F @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(42,axiom,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( C @ ( B @ ordered_pair ) ) )
=> ! [B: $i] :
( ( B
= ( A @ pair_first ) )
<=> ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_mcart_1) ).
thf(636,plain,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( C @ ( B @ ordered_pair ) ) )
=> ! [B: $i] :
( ( ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = C ) )
=> ( B
= ( A @ pair_first ) ) )
& ( ( B
= ( A @ pair_first ) )
=> ! [C: $i,D: $i] :
( ( A
= ( D @ ( C @ ordered_pair ) ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(111,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_zfmisc_1) ).
thf(1245,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(388,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_field )
= ( A @ relation_rng @ ( A @ relation_dom @ set_union2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d6_relat_1) ).
thf(3241,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ relation_field )
= ( A @ relation_rng @ ( A @ relation_dom @ set_union2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(30,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/sandbox/benchmark/theBenchmark.p',t118_zfmisc_1) ).
thf(563,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)],[30]) ).
thf(78,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D @ ordinal )
& ( B = D )
& ( C @ ordinal )
& ( B = C ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( B @ ( C @ in ) )
<=> ? [D: $i] :
( ( C @ ordinal )
& ( D = C )
& ( A @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e6_22__wellord2__1) ).
thf(851,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( D @ ordinal )
& ( B = D )
& ( C @ ordinal )
& ( B = C ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ? [D: $i] :
( ( C @ ordinal )
& ( D = C )
& ( A @ ( D @ in ) ) )
=> ( B @ ( C @ in ) ) )
& ( ( B @ ( C @ in ) )
=> ? [D: $i] :
( ( C @ ordinal )
& ( D = C )
& ( A @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(104,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( A @ ( B @ disjoint ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(1102,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
=> ( A @ ( B @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(236,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B
= ( A @ identity_relation ) )
<=> ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= C ) )
& ( ( B @ relation_dom )
= A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t34_funct_1) ).
thf(2069,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= C ) )
& ( ( B @ relation_dom )
= A ) )
=> ( B
= ( A @ identity_relation ) ) )
& ( ( B
= ( A @ identity_relation ) )
=> ( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= C ) )
& ( ( B @ relation_dom )
= A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(344,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_dom @ ( A @ ( B @ relation_inverse_image ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t167_relat_1) ).
thf(2867,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ relation_dom @ ( A @ ( B @ relation_inverse_image ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(473,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(173,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ succ @ ordinal )
& ( A @ succ @ epsilon_connected )
& ( A @ succ @ epsilon_transitive )
& ~ ( A @ succ @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_ordinal1) ).
thf(1626,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ succ @ ordinal )
& ( A @ succ @ epsilon_connected )
& ( A @ succ @ epsilon_transitive )
& ~ ( A @ succ @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(2614,plain,
sk285 @ epsilon_connected,
inference(cnfConj,[status(thm)],[2608]) ).
thf(120,axiom,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_boole) ).
thf(1304,plain,
! [A: $i] :
( ( empty_set @ ( A @ set_difference ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(148,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equipotent ) )
<=> ( B @ ( A @ are_equipotent ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_r2_wellord2) ).
thf(1467,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ are_equipotent ) )
=> ( B @ ( A @ equipotent ) ) )
& ( ( B @ ( A @ equipotent ) )
=> ( B @ ( A @ are_equipotent ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(27,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t9_zfmisc_1) ).
thf(548,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(43,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ set_difference ) )
= empty_set )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l32_xboole_1) ).
thf(648,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)],[43]) ).
thf(153,axiom,
! [A: $i] :
( ? [B: $i] :
( ( A @ ( B @ in ) )
& ( B @ ordinal ) )
=> ? [B: $i] :
( ! [C: $i] :
( ( C @ ordinal )
=> ( ( A @ ( C @ in ) )
=> ( C @ ( B @ ordinal_subset ) ) ) )
& ( A @ ( B @ in ) )
& ( B @ ordinal ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_ordinal1__e8_6__wellord2) ).
thf(1509,plain,
! [A: $i] :
( ? [B: $i] :
( ( A @ ( B @ in ) )
& ( B @ ordinal ) )
=> ? [B: $i] :
( ! [C: $i] :
( ( C @ ordinal )
=> ( ( A @ ( C @ in ) )
=> ( C @ ( B @ ordinal_subset ) ) ) )
& ( A @ ( B @ in ) )
& ( B @ ordinal ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(59,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( D @ ( E @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
<=> ( ( C @ ( B @ ( A @ relation_rng_as_subset ) ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t23_relset_1) ).
thf(749,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ( ( ( C @ ( B @ ( A @ relation_rng_as_subset ) ) )
= B )
=> ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( D @ ( E @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) ) )
& ( ! [D: $i] :
~ ( ! [E: $i] :
~ ( C @ ( D @ ( E @ ordered_pair ) @ in ) )
& ( B @ ( D @ in ) ) )
=> ( ( C @ ( B @ ( A @ relation_rng_as_subset ) ) )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(304,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_image ) )
= ( A @ ( B @ relation_dom @ set_intersection2 ) @ ( B @ relation_image ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t145_relat_1) ).
thf(2564,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_image ) )
= ( A @ ( B @ relation_dom @ set_intersection2 ) @ ( B @ relation_image ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(311,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B @ relation_rng @ ( A @ subset ) )
=> ( ( A @ ( B @ relation_inverse_image ) @ ( B @ relation_image ) )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t147_funct_1) ).
thf(2597,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B @ relation_rng @ ( A @ subset ) )
=> ( ( A @ ( B @ relation_inverse_image ) @ ( B @ relation_image ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(15,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ordered_pair ) @ pair_second )
= B )
& ( ( B @ ( A @ ordered_pair ) @ pair_first )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_mcart_1) ).
thf(490,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ordered_pair ) @ pair_second )
= B )
& ( ( B @ ( A @ ordered_pair ) @ pair_first )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(361,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_restriction ) @ ( A @ in ) )
<=> ( ( B @ ( B @ cartesian_product2 ) @ ( A @ in ) )
& ( C @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t16_wellord1) ).
thf(3031,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( ( ( B @ ( B @ cartesian_product2 ) @ ( A @ in ) )
& ( C @ ( A @ in ) ) )
=> ( B @ ( C @ relation_restriction ) @ ( A @ in ) ) )
& ( ( B @ ( C @ relation_restriction ) @ ( A @ in ) )
=> ( ( B @ ( B @ cartesian_product2 ) @ ( A @ in ) )
& ( C @ ( A @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).
thf(795,plain,
sk37 @ epsilon_connected,
inference(cnfConj,[status(thm)],[792]) ).
thf(330,axiom,
! [A: $i,B: $i] :
( ( ( A @ relation_empty_yielding )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_dom_restriction ) @ relation_empty_yielding )
& ( B @ ( A @ relation_dom_restriction ) @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc13_relat_1) ).
thf(2784,plain,
! [A: $i,B: $i] :
( ( ( A @ relation_empty_yielding )
& ( A @ relation ) )
=> ( ( B @ ( A @ relation_dom_restriction ) @ relation_empty_yielding )
& ( B @ ( A @ relation_dom_restriction ) @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(271,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/sandbox/benchmark/theBenchmark.p',t20_relat_1) ).
thf(2271,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)],[271]) ).
thf(383,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B @ relation_dom @ ( A @ in ) )
=> ( ( A @ ( C @ ( B @ relation_composition ) @ apply ) )
= ( A @ ( B @ apply ) @ ( C @ apply ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t23_funct_1) ).
thf(3206,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B @ relation_dom @ ( A @ in ) )
=> ( ( A @ ( C @ ( B @ relation_composition ) @ apply ) )
= ( A @ ( B @ apply ) @ ( C @ apply ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
thf(836,plain,
sk41 @ epsilon_connected,
inference(cnfConj,[status(thm)],[834]) ).
thf(161,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
<=> ( B @ ( A @ ( C @ relation_of2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_m2_relset_1) ).
thf(1551,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( B @ ( A @ ( C @ relation_of2_as_subset ) ) ) )
& ( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( B @ ( A @ ( C @ relation_of2 ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(305,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ( ? [F: $i,G: $i] :
( ? [H: $i] :
( ! [I: $i] :
( ( H @ ( I @ in ) )
=> ( B @ ( I @ ( G @ ordered_pair ) @ in ) ) )
& ( H @ ( G @ in ) )
& ( F = H ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= E ) )
& ( C @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e10_24__wellord2__1) ).
thf(2567,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ~ ( A @ empty ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( ( ? [F: $i,G: $i] :
( ? [H: $i] :
( ! [I: $i] :
( ( H @ ( I @ in ) )
=> ( B @ ( I @ ( G @ ordered_pair ) @ in ) ) )
& ( H @ ( G @ in ) )
& ( F = H ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= E ) )
& ( C @ ( A @ cartesian_product2 ) @ ( E @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ( ? [F: $i,G: $i] :
( ? [H: $i] :
( ! [I: $i] :
( ( H @ ( I @ in ) )
=> ( B @ ( I @ ( G @ ordered_pair ) @ in ) ) )
& ( H @ ( G @ in ) )
& ( F = H ) )
& ( A @ ( F @ in ) )
& ( ( G @ ( F @ ordered_pair ) )
= E ) )
& ( C @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(202,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( A @ ( B @ in ) )
=> ( ( B @ ( A @ ( C @ relation_dom_restriction ) @ apply ) )
= ( B @ ( C @ apply ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t72_funct_1) ).
thf(1798,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( A @ ( B @ in ) )
=> ( ( B @ ( A @ ( C @ relation_dom_restriction ) @ apply ) )
= ( B @ ( C @ apply ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(34,axiom,
! [A: $i,B: $i] :
( ( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
& ( A @ ordinal ) )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ? [E: $i] :
( ( D
= ( A @ singleton @ ( E @ set_difference ) ) )
& ( B @ ( E @ in ) ) )
& ( A @ powerset @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e4_27_3_1__finset_1) ).
thf(580,plain,
! [A: $i,B: $i] :
( ( ( A @ succ @ powerset @ powerset @ ( B @ element ) )
& ( A @ ordinal ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( ? [E: $i] :
( ( D
= ( A @ singleton @ ( E @ set_difference ) ) )
& ( B @ ( E @ in ) ) )
& ( A @ powerset @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ? [E: $i] :
( ( D
= ( A @ singleton @ ( E @ set_difference ) ) )
& ( B @ ( E @ in ) ) )
& ( A @ powerset @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(251,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ reflexive )
<=> ! [B: $i] :
( ( A @ relation_field @ ( B @ in ) )
=> ( A @ ( B @ ( B @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l1_wellord1) ).
thf(2192,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ! [B: $i] :
( ( A @ relation_field @ ( B @ in ) )
=> ( A @ ( B @ ( B @ ordered_pair ) @ in ) ) )
=> ( A @ reflexive ) )
& ( ( A @ reflexive )
=> ! [B: $i] :
( ( A @ relation_field @ ( B @ in ) )
=> ( A @ ( B @ ( B @ ordered_pair ) @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(3458,plain,
( ( ( sk205 @ empty )
!= ( omega @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,1565]) ).
thf(3459,plain,
( ( sk205 @ empty )
!= ( omega @ empty ) ),
inference(simp,[status(thm)],[3458]) ).
thf(3462,plain,
sk205 != omega,
inference(simp,[status(thm)],[3459]) ).
thf(247,axiom,
! [A: $i,B: $i] :
? [C: $i] :
( ( B @ ( A @ ( C @ quasi_total ) ) )
& ( C @ function )
& ( C @ relation )
& ( B @ ( A @ ( C @ relation_of2 ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc1_funct_2) ).
thf(2175,plain,
! [A: $i,B: $i] :
? [C: $i] :
( ( B @ ( A @ ( C @ quasi_total ) ) )
& ( C @ function )
& ( C @ relation )
& ( B @ ( A @ ( C @ relation_of2 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(287,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( A @ subset ) )
=> ( B @ ( C @ relation_inverse_image ) @ ( A @ ( C @ relation_inverse_image ) @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t178_relat_1) ).
thf(2407,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( A @ subset ) )
=> ( B @ ( C @ relation_inverse_image ) @ ( A @ ( C @ relation_inverse_image ) @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(325,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_union2 ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_relat_1) ).
thf(2720,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ relation ) )
=> ( B @ ( A @ set_union2 ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(157,axiom,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(1531,plain,
! [A: $i,B: $i] :
~ ( ( B @ ( A @ in ) )
& ( B @ ( A @ singleton @ disjoint ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(337,axiom,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ empty ) )
=> ( ( B @ ( A @ relation_composition ) @ relation )
& ( B @ ( A @ relation_composition ) @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc9_relat_1) ).
thf(2831,plain,
! [A: $i,B: $i] :
( ( ( B @ relation )
& ( A @ empty ) )
=> ( ( B @ ( A @ relation_composition ) @ relation )
& ( B @ ( A @ relation_composition ) @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(3214,plain,
sk379 @ function,
inference(cnfConj,[status(thm)],[3212]) ).
thf(91,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( ( C @ singleton @ ( B @ set_difference ) @ ( A @ subset ) )
| ( A @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3_zfmisc_1) ).
thf(971,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)],[91]) ).
thf(164,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ in ) )
=> ( ( B @ ( A @ identity_relation @ apply ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t35_funct_1) ).
thf(1566,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ in ) )
=> ( ( B @ ( A @ identity_relation @ apply ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(168,axiom,
! [A: $i] : ( A @ inclusion_relation @ reflexive ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_wellord2) ).
thf(1605,plain,
! [A: $i] : ( A @ inclusion_relation @ reflexive ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(393,axiom,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( B @ ( A @ subset ) ) )
=> ( A @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t13_finset_1) ).
thf(3260,plain,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( B @ ( A @ subset ) ) )
=> ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).
thf(308,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ antisymmetric )
=> ( A @ ( B @ relation_restriction ) @ antisymmetric ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t25_wellord1) ).
thf(2588,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ antisymmetric )
=> ( A @ ( B @ relation_restriction ) @ antisymmetric ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(195,axiom,
! [A: $i] :
( ( A @ succ )
= ( A @ singleton @ ( A @ set_union2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_ordinal1) ).
thf(1780,plain,
! [A: $i] :
( ( A @ succ )
= ( A @ singleton @ ( A @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(384,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ well_founded_relation )
=> ( A @ ( B @ relation_restriction ) @ well_founded_relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t31_wellord1) ).
thf(3209,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ well_founded_relation )
=> ( A @ ( B @ relation_restriction ) @ well_founded_relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).
thf(366,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
=> ( ( ( A @ well_founded_relation )
=> ( B @ well_founded_relation ) )
& ( ( A @ antisymmetric )
=> ( B @ antisymmetric ) )
& ( ( A @ connected )
=> ( B @ connected ) )
& ( ( A @ transitive )
=> ( B @ transitive ) )
& ( ( A @ reflexive )
=> ( B @ reflexive ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t53_wellord1) ).
thf(3080,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
=> ( ( ( A @ well_founded_relation )
=> ( B @ well_founded_relation ) )
& ( ( A @ antisymmetric )
=> ( B @ antisymmetric ) )
& ( ( A @ connected )
=> ( B @ connected ) )
& ( ( A @ transitive )
=> ( B @ transitive ) )
& ( ( A @ reflexive )
=> ( B @ reflexive ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(199,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ meet_of_subsets ) )
= ( B @ set_meet ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k6_setfam_1) ).
thf(1790,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)],[199]) ).
thf(197,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ( A @ powerset @ powerset @ ( B @ ( A @ complements_of_subsets ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k7_setfam_1) ).
thf(1785,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)],[197]) ).
thf(257,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ well_founded_relation )
<=> ( A @ relation_field @ ( A @ is_well_founded_in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_wellord1) ).
thf(2219,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_field @ ( A @ is_well_founded_in ) )
=> ( A @ well_founded_relation ) )
& ( ( A @ well_founded_relation )
=> ( A @ relation_field @ ( A @ is_well_founded_in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(280,axiom,
! [A: $i] :
( ( ! [B: $i] :
~ ( ! [C: $i] :
( C
!= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
& ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) ) )
=> ? [B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= ( C @ singleton ) ) )
& ( ( B @ relation_dom )
= A )
& ( B @ function )
& ( B @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s2_funct_1__e16_22__wellord2__1) ).
thf(2322,plain,
! [A: $i] :
( ( ! [B: $i] :
~ ( ! [C: $i] :
( C
!= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
& ! [B: $i,C: $i,D: $i] :
( ( ( D
= ( B @ singleton ) )
& ( C
= ( B @ singleton ) )
& ( A @ ( B @ in ) ) )
=> ( C = D ) ) )
=> ? [B: $i] :
( ! [C: $i] :
( ( A @ ( C @ in ) )
=> ( ( C @ ( B @ apply ) )
= ( C @ singleton ) ) )
& ( ( B @ relation_dom )
= A )
& ( B @ function )
& ( B @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(3606,plain,
( ( ( sk358 @ empty )
!= ( sk285 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,3078]) ).
thf(3607,plain,
( ( sk358 @ empty )
!= ( sk285 @ empty ) ),
inference(simp,[status(thm)],[3606]) ).
thf(3623,plain,
sk358 != sk285,
inference(simp,[status(thm)],[3607]) ).
thf(3095,plain,
empty_set @ function,
inference(cnfConj,[status(thm)],[3092]) ).
thf(123,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ordered_pair ) )
= ( D @ ( C @ ordered_pair ) ) )
=> ( ( B = D )
& ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t33_zfmisc_1) ).
thf(1336,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)],[123]) ).
thf(3302,plain,
( ( ( omega @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,1565]) ).
thf(3303,plain,
( ( omega @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3302]) ).
thf(3305,plain,
omega != empty_set,
inference(simp,[status(thm)],[3303]) ).
thf(245,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( A @ ( B @ relation_dom_restriction ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t88_relat_1) ).
thf(2165,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( A @ ( B @ relation_dom_restriction ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(372,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( F
= ( H @ ( G @ ordered_pair ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( H @ ( G @ ordered_pair ) ) ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ? [F: $i] :
( ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( H @ ( G @ ordered_pair ) ) ) )
& ( F = E )
& ( A @ ( A @ cartesian_product2 ) @ ( F @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e6_21__wellord2__1) ).
thf(3107,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( F
= ( H @ ( G @ ordered_pair ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( H @ ( G @ ordered_pair ) ) ) )
& ( D = E ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ? [F: $i] :
( ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( H @ ( G @ ordered_pair ) ) ) )
& ( F = E )
& ( A @ ( A @ cartesian_product2 ) @ ( F @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ? [F: $i] :
( ? [G: $i,H: $i] :
( ( B @ ( H @ ( C @ apply ) @ ( G @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( H @ ( G @ ordered_pair ) ) ) )
& ( F = E )
& ( A @ ( A @ cartesian_product2 ) @ ( F @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(48,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( C @ empty )
& ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_subset) ).
thf(703,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( C @ empty )
& ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(159,axiom,
! [A: $i,B: $i] :
~ ( ! [C: $i] :
~ ( ! [D: $i] :
~ ( ( C @ ( D @ in ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ( B @ ( A @ in ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t7_tarski) ).
thf(1536,plain,
! [A: $i,B: $i] :
~ ( ! [C: $i] :
~ ( ! [D: $i] :
~ ( ( C @ ( D @ in ) )
& ( B @ ( D @ in ) ) )
& ( B @ ( C @ in ) ) )
& ( B @ ( A @ in ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(44,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/sandbox/benchmark/theBenchmark.p',t43_subset_1) ).
thf(656,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)],[44]) ).
thf(187,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/sandbox/benchmark/theBenchmark.p',t50_subset_1) ).
thf(1702,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)],[187]) ).
thf(93,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/sandbox/benchmark/theBenchmark.p',d8_setfam_1) ).
thf(978,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)],[93]) ).
thf(326,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/sandbox/benchmark/theBenchmark.p',d4_relat_1) ).
thf(2722,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)],[326]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l23_zfmisc_1) ).
thf(571,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( ( B @ ( A @ singleton @ set_union2 ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(339,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ( B @ ( C @ relation_rng @ subset ) )
& ( A @ ( C @ relation_dom @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t12_relset_1) ).
thf(2848,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ( B @ ( C @ relation_rng @ subset ) )
& ( A @ ( C @ relation_dom @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(306,axiom,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_inverse @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k4_relat_1) ).
thf(2584,plain,
! [A: $i] :
( ( A @ relation )
=> ( A @ relation_inverse @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(319,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_funct_1) ).
thf(2675,plain,
! [A: $i] :
( ( A @ empty )
=> ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(264,axiom,
! [A: $i,B: $i] :
( ( A @ relation )
=> ( B @ ( A @ relation_restriction ) @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k2_wellord1) ).
thf(2247,plain,
! [A: $i,B: $i] :
( ( A @ relation )
=> ( B @ ( A @ relation_restriction ) @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(289,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B
= ( A @ inclusion_relation ) )
<=> ( ! [C: $i,D: $i] :
( ( ( A @ ( D @ in ) )
& ( A @ ( C @ in ) ) )
=> ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
<=> ( D @ ( C @ subset ) ) ) )
& ( ( B @ relation_field )
= A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_wellord2) ).
thf(2411,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( ( ! [C: $i,D: $i] :
( ( ( A @ ( D @ in ) )
& ( A @ ( C @ in ) ) )
=> ( ( ( D @ ( C @ subset ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( D @ ( C @ subset ) ) ) ) )
& ( ( B @ relation_field )
= A ) )
=> ( B
= ( A @ inclusion_relation ) ) )
& ( ( B
= ( A @ inclusion_relation ) )
=> ( ! [C: $i,D: $i] :
( ( ( A @ ( D @ in ) )
& ( A @ ( C @ in ) ) )
=> ( ( ( D @ ( C @ subset ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( D @ ( C @ subset ) ) ) ) )
& ( ( B @ relation_field )
= A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(294,axiom,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ( B @ finite )
& ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc3_finset_1) ).
thf(2500,plain,
! [A: $i] :
( ~ ( A @ empty )
=> ? [B: $i] :
( ( B @ finite )
& ~ ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(3282,plain,
( ( ( sk37 @ empty )
!= ( sk12 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,796]) ).
thf(3283,plain,
( ( sk37 @ empty )
!= ( sk12 @ empty ) ),
inference(simp,[status(thm)],[3282]) ).
thf(3284,plain,
sk37 != sk12,
inference(simp,[status(thm)],[3283]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_zfmisc_1) ).
thf(465,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A @ singleton )
= ( C @ ( B @ unordered_pair ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(3582,plain,
( ( ( sk1 @ relation_rng @ finite )
!= ( sk226 @ finite ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2146,402]) ).
thf(3583,plain,
( ( sk1 @ relation_rng @ finite )
!= ( sk226 @ finite ) ),
inference(simp,[status(thm)],[3582]) ).
thf(3584,plain,
( ( sk1 @ relation_rng )
!= sk226 ),
inference(simp,[status(thm)],[3583]) ).
thf(2088,plain,
sk215 @ function,
inference(cnfConj,[status(thm)],[2086]) ).
thf(234,axiom,
( ( ( empty_set @ relation_rng )
= empty_set )
& ( ( empty_set @ relation_dom )
= empty_set ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t60_relat_1) ).
thf(2045,plain,
( ( ( empty_set @ relation_rng )
= empty_set )
& ( ( empty_set @ relation_dom )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(334,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( A @ ( C @ relation_dom_restriction ) @ relation_dom @ ( B @ in ) )
<=> ( ( A @ ( B @ in ) )
& ( C @ relation_dom @ ( B @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l82_funct_1) ).
thf(2818,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( ( A @ ( B @ in ) )
& ( C @ relation_dom @ ( B @ in ) ) )
=> ( A @ ( C @ relation_dom_restriction ) @ relation_dom @ ( B @ in ) ) )
& ( ( A @ ( C @ relation_dom_restriction ) @ relation_dom @ ( B @ in ) )
=> ( ( A @ ( B @ in ) )
& ( C @ relation_dom @ ( B @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(24,axiom,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( A @ ( A @ ordinal_subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity_r1_ordinal1) ).
thf(539,plain,
! [A: $i,B: $i] :
( ( ( B @ ordinal )
& ( A @ ordinal ) )
=> ( A @ ( A @ ordinal_subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(12,axiom,
! [A: $i] :
~ ( A @ succ @ empty ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_ordinal1) ).
thf(482,plain,
! [A: $i] :
~ ( A @ succ @ empty ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(381,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ! [B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( B
= ( A @ function_inverse ) )
<=> ( ! [C: $i,D: $i] :
( ( ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) ) )
& ( ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) )
=> ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) )
& ( ( B @ relation_dom )
= ( A @ relation_rng ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t54_funct_1) ).
thf(3171,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ! [B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ( ( ( ! [C: $i,D: $i] :
( ( ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) ) )
& ( ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) )
=> ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) )
& ( ( B @ relation_dom )
= ( A @ relation_rng ) ) )
=> ( B
= ( A @ function_inverse ) ) )
& ( ( B
= ( A @ function_inverse ) )
=> ( ! [C: $i,D: $i] :
( ( ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) ) )
& ( ( ( D
= ( C @ ( B @ apply ) ) )
& ( A @ relation_rng @ ( C @ in ) ) )
=> ( ( C
= ( D @ ( A @ apply ) ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) )
& ( ( B @ relation_dom )
= ( A @ relation_rng ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).
thf(207,axiom,
! [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/sandbox/benchmark/theBenchmark.p',t30_relat_1) ).
thf(1844,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)],[207]) ).
thf(139,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( ( B @ ( A @ subset_complement ) @ ( A @ subset_complement ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',involutiveness_k3_subset_1) ).
thf(1412,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)],[139]) ).
thf(3613,plain,
( ( ( sk285 @ empty )
!= ( sk29 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,732]) ).
thf(3614,plain,
( ( sk285 @ empty )
!= ( sk29 @ empty ) ),
inference(simp,[status(thm)],[3613]) ).
thf(3625,plain,
sk285 != sk29,
inference(simp,[status(thm)],[3614]) ).
thf(327,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( C
= ( B @ ( A @ relation_inverse_image ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( B @ ( D @ ( A @ apply ) @ in ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d13_funct_1) ).
thf(2743,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ! [B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( B @ ( D @ ( A @ apply ) @ in ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ ( A @ apply ) @ in ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) )
=> ( C
= ( B @ ( A @ relation_inverse_image ) ) ) )
& ( ( C
= ( B @ ( A @ relation_inverse_image ) ) )
=> ! [D: $i] :
( ( ( ( B @ ( D @ ( A @ apply ) @ in ) )
& ( A @ relation_dom @ ( D @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( B @ ( D @ ( A @ apply ) @ in ) )
& ( A @ relation_dom @ ( D @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(211,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ well_orders ) )
<=> ( ( B @ ( A @ is_well_founded_in ) )
& ( B @ ( A @ is_connected_in ) )
& ( B @ ( A @ is_antisymmetric_in ) )
& ( B @ ( A @ is_transitive_in ) )
& ( B @ ( A @ is_reflexive_in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_wellord1) ).
thf(1866,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ( ( B @ ( A @ is_well_founded_in ) )
& ( B @ ( A @ is_connected_in ) )
& ( B @ ( A @ is_antisymmetric_in ) )
& ( B @ ( A @ is_transitive_in ) )
& ( B @ ( A @ is_reflexive_in ) ) )
=> ( B @ ( A @ well_orders ) ) )
& ( ( B @ ( A @ well_orders ) )
=> ( ( B @ ( A @ is_well_founded_in ) )
& ( B @ ( A @ is_connected_in ) )
& ( B @ ( A @ is_antisymmetric_in ) )
& ( B @ ( A @ is_transitive_in ) )
& ( B @ ( A @ is_reflexive_in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(190,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(1734,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ in ) )
=> ( B @ union @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(346,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ relation_dom @ ( A @ subset ) )
=> ( A @ ( B @ relation_image ) @ ( B @ relation_inverse_image ) @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t146_funct_1) ).
thf(2875,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( B @ relation_dom @ ( A @ subset ) )
=> ( A @ ( B @ relation_image ) @ ( B @ relation_inverse_image ) @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(380,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ ( A @ relation_rng_restriction ) @ relation_dom_restriction ) )
= ( B @ ( C @ relation_dom_restriction ) @ ( A @ relation_rng_restriction ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t140_relat_1) ).
thf(3168,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ ( A @ relation_rng_restriction ) @ relation_dom_restriction ) )
= ( B @ ( C @ relation_dom_restriction ) @ ( A @ relation_rng_restriction ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).
thf(122,axiom,
! [A: $i] :
( ( A = omega )
<=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) )
& ( A @ ordinal )
& ( A @ being_limit_ordinal )
& ( A @ ( empty_set @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d5_ordinal2) ).
thf(1309,plain,
! [A: $i] :
( ( ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) )
& ( A @ ordinal )
& ( A @ being_limit_ordinal )
& ( A @ ( empty_set @ in ) ) )
=> ( A = omega ) )
& ( ( A = omega )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) )
& ( A @ ordinal )
& ( A @ being_limit_ordinal )
& ( A @ ( empty_set @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(1310,plain,
( ! [A: $i] :
( ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) )
& ( A @ ordinal )
& ( A @ being_limit_ordinal )
& ( A @ ( empty_set @ in ) ) )
=> ( A = omega ) )
& ! [A: $i] :
( ( A = omega )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( ( B @ being_limit_ordinal )
& ( B @ ( empty_set @ in ) ) )
=> ( B @ ( A @ subset ) ) ) )
& ( A @ ordinal )
& ( A @ being_limit_ordinal )
& ( A @ ( empty_set @ in ) ) ) ) ),
inference(miniscope,[status(thm)],[1309]) ).
thf(1311,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C = omega )
| ~ ( C @ sk125 @ ( C @ subset ) )
| ~ ( C @ ordinal )
| ~ ( C @ being_limit_ordinal )
| ~ ( C @ ( empty_set @ in ) ) )
& ( ( C = omega )
| ( C @ sk125 @ being_limit_ordinal )
| ~ ( C @ ordinal )
| ~ ( C @ being_limit_ordinal )
| ~ ( C @ ( empty_set @ in ) ) )
& ( ( C = omega )
| ( C @ sk125 @ ( empty_set @ in ) )
| ~ ( C @ ordinal )
| ~ ( C @ being_limit_ordinal )
| ~ ( C @ ( empty_set @ in ) ) )
& ( ( C = omega )
| ( C @ sk125 @ ordinal )
| ~ ( C @ ordinal )
| ~ ( C @ being_limit_ordinal )
| ~ ( C @ ( empty_set @ in ) ) )
& ( ( B @ ( A @ subset ) )
| ~ ( B @ being_limit_ordinal )
| ~ ( B @ ( empty_set @ in ) )
| ~ ( B @ ordinal )
| ( A != omega ) )
& ( ( A @ ordinal )
| ( A != omega ) )
& ( ( A @ being_limit_ordinal )
| ( A != omega ) )
& ( ( A @ ( empty_set @ in ) )
| ( A != omega ) ) ),
inference(cnf,[status(esa)],[1310]) ).
thf(1315,plain,
! [A: $i] :
( ( A @ ordinal )
| ( A != omega ) ),
inference(cnfConj,[status(thm)],[1311]) ).
thf(1320,plain,
! [A: $i] :
( ( A @ ordinal )
| ( A != omega ) ),
inference(lifteq,[status(thm)],[1315]) ).
thf(1321,plain,
omega @ ordinal,
inference(simp,[status(thm)],[1320]) ).
thf(221,axiom,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_rng @ relation )
& ( A @ relation_rng @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc8_relat_1) ).
thf(1913,plain,
! [A: $i] :
( ( A @ empty )
=> ( ( A @ relation_rng @ relation )
& ( A @ relation_rng @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(2611,plain,
sk285 @ ordinal,
inference(cnfConj,[status(thm)],[2608]) ).
thf(60,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ connected ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_wellord2) ).
thf(759,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( A @ inclusion_relation @ connected ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(277,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( B @ ( A @ relation_rng_restriction ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t117_relat_1) ).
thf(2305,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( B @ ( B @ ( A @ relation_rng_restriction ) @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(350,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( B @ ( C @ relation_inverse_image ) @ ( A @ in ) )
<=> ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( D @ ( A @ ordered_pair ) @ in ) )
& ( C @ relation_rng @ ( D @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t166_relat_1) ).
thf(2919,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ relation )
=> ( ( ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( D @ ( A @ ordered_pair ) @ in ) )
& ( C @ relation_rng @ ( D @ in ) ) )
=> ( B @ ( C @ relation_inverse_image ) @ ( A @ in ) ) )
& ( ( B @ ( C @ relation_inverse_image ) @ ( A @ in ) )
=> ? [D: $i] :
( ( B @ ( D @ in ) )
& ( C @ ( D @ ( A @ ordered_pair ) @ in ) )
& ( C @ relation_rng @ ( D @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).
thf(3446,plain,
( ( ( sk358 @ empty )
!= ( sk205 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,3078]) ).
thf(3447,plain,
( ( sk358 @ empty )
!= ( sk205 @ empty ) ),
inference(simp,[status(thm)],[3446]) ).
thf(3465,plain,
sk358 != sk205,
inference(simp,[status(thm)],[3447]) ).
thf(177,axiom,
! [A: $i,B: $i] : ( A @ ( A @ subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(1646,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(323,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( B != empty_set )
=> ! [E: $i] :
( ( C @ ( D @ relation_inverse_image ) @ ( E @ in ) )
<=> ( ( C @ ( E @ ( D @ apply ) @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_funct_2) ).
thf(2707,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( B != empty_set )
=> ! [E: $i] :
( ( ( ( C @ ( E @ ( D @ apply ) @ in ) )
& ( A @ ( E @ in ) ) )
=> ( C @ ( D @ relation_inverse_image ) @ ( E @ in ) ) )
& ( ( C @ ( D @ relation_inverse_image ) @ ( E @ in ) )
=> ( ( C @ ( E @ ( D @ apply ) @ in ) )
& ( A @ ( E @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(358,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
<=> ( ( B
= ( A @ ( C @ apply ) ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_funct_1) ).
thf(2988,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( ( B
= ( A @ ( C @ apply ) ) )
& ( C @ relation_dom @ ( A @ in ) ) )
=> ( C @ ( B @ ( A @ ordered_pair ) @ in ) ) )
& ( ( C @ ( B @ ( A @ ordered_pair ) @ in ) )
=> ( ( B
= ( A @ ( C @ apply ) ) )
& ( C @ relation_dom @ ( A @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).
thf(246,axiom,
! [A: $i] :
( ( ( A @ identity_relation @ relation_rng )
= A )
& ( ( A @ identity_relation @ relation_dom )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t71_relat_1) ).
thf(2167,plain,
! [A: $i] :
( ( ( A @ identity_relation @ relation_rng )
= A )
& ( ( A @ identity_relation @ relation_dom )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(181,axiom,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(1674,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(335,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_dom )
= ( A @ relation_inverse @ relation_rng ) )
& ( ( A @ relation_rng )
= ( A @ relation_inverse @ relation_dom ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t37_relat_1) ).
thf(2823,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_dom )
= ( A @ relation_inverse @ relation_rng ) )
& ( ( A @ relation_rng )
= ( A @ relation_inverse @ relation_dom ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).
thf(228,axiom,
! [A: $i] :
( ( A @ finite )
<=> ? [B: $i] :
( ( omega @ ( B @ relation_dom @ in ) )
& ( ( B @ relation_rng )
= A )
& ( B @ function )
& ( B @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_finset_1) ).
thf(2004,plain,
! [A: $i] :
( ( ? [B: $i] :
( ( omega @ ( B @ relation_dom @ in ) )
& ( ( B @ relation_rng )
= A )
& ( B @ function )
& ( B @ relation ) )
=> ( A @ finite ) )
& ( ( A @ finite )
=> ? [B: $i] :
( ( omega @ ( B @ relation_dom @ in ) )
& ( ( B @ relation_rng )
= A )
& ( B @ function )
& ( B @ relation ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(356,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ is_well_founded_in ) )
<=> ! [C: $i] :
~ ( ! [D: $i] :
~ ( ( C @ ( D @ ( A @ fiber ) @ disjoint ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set )
& ( B @ ( C @ subset ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d3_wellord1) ).
thf(2971,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i] :
~ ( ! [D: $i] :
~ ( ( C @ ( D @ ( A @ fiber ) @ disjoint ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set )
& ( B @ ( C @ subset ) ) )
=> ( B @ ( A @ is_well_founded_in ) ) )
& ( ( B @ ( A @ is_well_founded_in ) )
=> ! [C: $i] :
~ ( ! [D: $i] :
~ ( ( C @ ( D @ ( A @ fiber ) @ disjoint ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set )
& ( B @ ( C @ subset ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(401,plain,
sk1 @ relation_dom @ finite,
inference(cnfConj,[status(thm)],[398]) ).
thf(391,axiom,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( A @ finite ) )
=> ( B @ ( A @ set_union2 ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc9_finset_1) ).
thf(3252,plain,
! [A: $i,B: $i] :
( ( ( B @ finite )
& ( A @ finite ) )
=> ( B @ ( A @ set_union2 ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).
thf(300,axiom,
! [A: $i,B: $i] :
( ( B @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc10_finset_1) ).
thf(2526,plain,
! [A: $i,B: $i] :
( ( B @ finite )
=> ( B @ ( A @ set_intersection2 ) @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(1062,plain,
sk76 @ ordinal,
inference(cnfConj,[status(thm)],[1059]) ).
thf(418,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( A @ ( B @ ( C @ sk2 ) ) @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(432,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( A @ ( B @ ( C @ sk2 ) ) @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[418]) ).
thf(433,plain,
! [B: $i,A: $i] :
( ( A @ ( A @ ( A @ union @ ( B @ sk2 ) ) @ in ) )
| ~ ( A @ union @ ( B @ in ) ) ),
inference(simp,[status(thm)],[432]) ).
thf(194,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ set_union2 ) )
= ( A @ ( B @ set_union2 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).
thf(1777,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ set_union2 ) )
= ( A @ ( B @ set_union2 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(46,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ subset ) )
=> ( C @ ( B @ set_intersection2 ) @ ( C @ ( A @ set_intersection2 ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t26_xboole_1) ).
thf(667,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)],[46]) ).
thf(143,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ subset ) )
<=> ( ( A
= ( B @ singleton ) )
| ( A = empty_set ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l4_zfmisc_1) ).
thf(1440,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)],[143]) ).
thf(349,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B
= ( A @ relation_inverse ) )
<=> ! [C: $i,D: $i] :
( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
<=> ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_relat_1) ).
thf(2902,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( ! [C: $i,D: $i] :
( ( ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) )
=> ( B
= ( A @ relation_inverse ) ) )
& ( ( B
= ( A @ relation_inverse ) )
=> ! [C: $i,D: $i] :
( ( ( A @ ( C @ ( D @ ordered_pair ) @ in ) )
=> ( B @ ( D @ ( C @ ordered_pair ) @ in ) ) )
& ( ( B @ ( D @ ( C @ ordered_pair ) @ in ) )
=> ( A @ ( C @ ( D @ ordered_pair ) @ in ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).
thf(292,axiom,
! [A: $i] :
( ( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_inverse @ function )
& ( A @ relation_inverse @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc3_funct_1) ).
thf(2494,plain,
! [A: $i] :
( ( ( A @ one_to_one )
& ( A @ function )
& ( A @ relation ) )
=> ( ( A @ relation_inverse @ function )
& ( A @ relation_inverse @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(170,axiom,
! [A: $i,B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3_subset_1) ).
thf(1614,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)],[170]) ).
thf(169,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/sandbox/benchmark/theBenchmark.p',t106_zfmisc_1) ).
thf(1607,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)],[169]) ).
thf(62,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_subset) ).
thf(763,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ element ) )
=> ( ( B @ ( A @ in ) )
| ( B @ empty ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(106,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( B @ set_intersection2 ) @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t19_xboole_1) ).
thf(1116,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)],[106]) ).
thf(297,axiom,
! [A: $i] :
( ( A @ empty )
=> ( A @ finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_finset_1) ).
thf(2517,plain,
! [A: $i] :
( ( A @ empty )
=> ( A @ finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(114,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/sandbox/benchmark/theBenchmark.p',t3_xboole_0) ).
thf(1280,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)],[114]) ).
thf(167,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/sandbox/benchmark/theBenchmark.p',d3_xboole_0) ).
thf(1578,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)],[167]) ).
thf(835,plain,
sk41 @ epsilon_transitive,
inference(cnfConj,[status(thm)],[834]) ).
thf(260,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ ( C @ ( D @ relation_of2_as_subset ) ) )
=> ( ( B @ ( D @ relation_rng @ subset ) )
=> ( B @ ( C @ ( D @ relation_of2_as_subset ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t14_relset_1) ).
thf(2233,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ ( C @ ( D @ relation_of2_as_subset ) ) )
=> ( ( B @ ( D @ relation_rng @ subset ) )
=> ( B @ ( C @ ( D @ relation_of2_as_subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(299,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ antisymmetric )
<=> ( A @ relation_field @ ( A @ is_antisymmetric_in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d12_relat_2) ).
thf(2522,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_field @ ( A @ is_antisymmetric_in ) )
=> ( A @ antisymmetric ) )
& ( ( A @ antisymmetric )
=> ( A @ relation_field @ ( A @ is_antisymmetric_in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(3456,plain,
( ( ( sk205 @ empty )
!= ( sk37 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,796]) ).
thf(3457,plain,
( ( sk205 @ empty )
!= ( sk37 @ empty ) ),
inference(simp,[status(thm)],[3456]) ).
thf(3464,plain,
sk205 != sk37,
inference(simp,[status(thm)],[3457]) ).
thf(404,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(cnf,[status(esa)],[403]) ).
thf(405,plain,
! [A: $i] :
( ( A @ cast_to_subset )
= A ),
inference(lifteq,[status(thm)],[404]) ).
thf(212,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ cartesian_product2 ) @ powerset @ ( C @ element ) )
=> ( C @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc1_relset_1) ).
thf(1876,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ cartesian_product2 ) @ powerset @ ( C @ element ) )
=> ( C @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(112,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ powerset @ ( B @ element ) )
& ( B @ ( A @ in ) ) )
=> ( C @ ( A @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t4_subset) ).
thf(1248,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)],[112]) ).
thf(591,plain,
( ! [A: $i] :
( ( A @ ordinal )
=> ( ! [B: $i] :
( ( B @ ordinal )
=> ( ( A @ ( B @ in ) )
=> ( ( omega @ ( B @ in ) )
=> ! [C: $i] :
( ( B @ powerset @ powerset @ ( C @ element ) )
=> ~ ( ~ ? [D: $i] :
( ! [E: $i] :
( ( ( E @ ( D @ subset ) )
& ( C @ ( E @ in ) ) )
=> ( E = D ) )
& ( C @ ( D @ in ) ) )
& ( C != empty_set ) ) ) ) ) )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ~ ? [C: $i] :
( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) )
=> ! [A: $i] :
( ( A @ ordinal )
=> ( ( omega @ ( A @ in ) )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ~ ? [C: $i] :
( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ) ) ),
inference(miniscope,[status(thm)],[590]) ).
thf(592,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( G
= ( E @ ( F @ sk20 ) ) )
| ~ ( G @ ( E @ ( F @ sk20 ) @ subset ) )
| ~ ( F @ ( G @ in ) )
| ( F = empty_set )
| ~ ( E @ powerset @ powerset @ ( F @ element ) )
| ~ ( omega @ ( E @ in ) )
| ~ ( E @ ordinal )
| ~ sk15 )
& ( ( F @ ( E @ ( F @ sk20 ) @ in ) )
| ( F = empty_set )
| ~ ( E @ powerset @ powerset @ ( F @ element ) )
| ~ ( omega @ ( E @ in ) )
| ~ ( E @ ordinal )
| ~ sk15 )
& ( sk15
| ( ( D @ sk19 )
!= D )
| ~ ( sk18 @ ( D @ in ) ) )
& ( sk15
| ( D @ sk19 @ ( D @ subset ) )
| ~ ( sk18 @ ( D @ in ) ) )
& ( sk15
| ( sk18 @ ( D @ sk19 @ in ) )
| ~ ( sk18 @ ( D @ in ) ) )
& ( sk15
| ( sk18 != empty_set ) )
& ( sk15
| ( sk16 @ powerset @ powerset @ ( sk18 @ element ) ) )
& ( sk15
| ( omega @ ( sk16 @ in ) ) )
& ( sk15
| ( C
= ( A @ ( B @ sk17 ) ) )
| ~ ( C @ ( A @ ( B @ sk17 ) @ subset ) )
| ~ ( B @ ( C @ in ) )
| ( B = empty_set )
| ~ ( A @ powerset @ powerset @ ( B @ element ) )
| ~ ( omega @ ( A @ in ) )
| ~ ( sk16 @ ( A @ in ) )
| ~ ( A @ ordinal ) )
& ( sk15
| ( B @ ( A @ ( B @ sk17 ) @ in ) )
| ( B = empty_set )
| ~ ( A @ powerset @ powerset @ ( B @ element ) )
| ~ ( omega @ ( A @ in ) )
| ~ ( sk16 @ ( A @ in ) )
| ~ ( A @ ordinal ) )
& ( sk15
| ( sk16 @ ordinal ) ) ),
inference(cnf,[status(esa)],[591]) ).
thf(603,plain,
( sk15
| ( sk16 @ ordinal ) ),
inference(cnfConj,[status(thm)],[592]) ).
thf(95,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ union @ ordinal )
& ( A @ union @ epsilon_connected )
& ( A @ union @ epsilon_transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc4_ordinal1) ).
thf(999,plain,
! [A: $i] :
( ( A @ ordinal )
=> ( ( A @ union @ ordinal )
& ( A @ union @ epsilon_connected )
& ( A @ union @ epsilon_transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(273,axiom,
! [A: $i] :
( ( A @ identity_relation @ function )
& ( A @ identity_relation @ relation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_funct_1) ).
thf(2287,plain,
! [A: $i] :
( ( A @ identity_relation @ function )
& ( A @ identity_relation @ relation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(2534,plain,
sk273 @ relation,
inference(cnfConj,[status(thm)],[2533]) ).
thf(266,axiom,
! [A: $i] : ( A @ inclusion_relation @ relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k1_wellord2) ).
thf(2255,plain,
! [A: $i] : ( A @ inclusion_relation @ relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(918,plain,
sk55 @ ordinal,
inference(cnfConj,[status(thm)],[914]) ).
thf(249,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_restriction ) )
= ( A @ ( B @ ( A @ relation_rng_restriction ) @ relation_dom_restriction ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_wellord1) ).
thf(2186,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_restriction ) )
= ( A @ ( B @ ( A @ relation_rng_restriction ) @ relation_dom_restriction ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(437,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ ordinal_subset ) )
| ( B @ ( A @ ordinal_subset ) )
| ~ ( B @ ordinal )
| ~ ( A @ ordinal ) ),
inference(cnf,[status(esa)],[436]) ).
thf(138,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( C @ ( A @ set_union2 ) @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t8_xboole_1) ).
thf(1410,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)],[138]) ).
thf(400,plain,
sk1 @ function,
inference(cnfConj,[status(thm)],[398]) ).
thf(286,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ relation_composition ) @ relation_rng )
= ( A @ relation_rng @ ( B @ relation_image ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t160_relat_1) ).
thf(2404,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ ( A @ relation_composition ) @ relation_rng )
= ( A @ relation_rng @ ( B @ relation_image ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(216,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( ( C @ ( B @ ( A @ relation_dom_as_subset ) ) )
= ( C @ relation_dom ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',redefinition_k4_relset_1) ).
thf(1891,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( ( C @ ( B @ ( A @ relation_dom_as_subset ) ) )
= ( C @ relation_dom ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(70,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/sandbox/benchmark/theBenchmark.p',dt_k6_subset_1) ).
thf(828,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)],[70]) ).
thf(317,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( A @ ( C @ in ) )
=> ( ( D @ relation_rng @ ( C @ ( D @ apply ) @ in ) )
| ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_funct_2) ).
thf(2668,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( A @ ( D @ relation_of2_as_subset ) ) )
& ( B @ ( A @ ( D @ quasi_total ) ) )
& ( D @ function ) )
=> ( ( A @ ( C @ in ) )
=> ( ( D @ relation_rng @ ( C @ ( D @ apply ) @ in ) )
| ( B = empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(124,axiom,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ set_union2 ) @ set_difference ) )
= ( B @ ( A @ set_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t40_xboole_1) ).
thf(1346,plain,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ set_union2 ) @ set_difference ) )
= ( B @ ( A @ set_difference ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(87,axiom,
! [A: $i] :
( ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) )
=> ( A @ ordinal ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc2_ordinal1) ).
thf(928,plain,
! [A: $i] :
( ( ( A @ epsilon_connected )
& ( A @ epsilon_transitive ) )
=> ( A @ ordinal ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(69,axiom,
! [A: $i,B: $i] :
( ( B
= ( A @ singleton ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_tarski) ).
thf(807,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)],[69]) ).
thf(3801,plain,
( ( ( sk322 @ empty )
!= ( omega @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2874,1565]) ).
thf(3802,plain,
( ( sk322 @ empty )
!= ( omega @ empty ) ),
inference(simp,[status(thm)],[3801]) ).
thf(3804,plain,
sk322 != omega,
inference(simp,[status(thm)],[3802]) ).
thf(258,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ ( A @ is_reflexive_in ) )
<=> ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ ( C @ ordered_pair ) @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_relat_2) ).
thf(2223,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ ( C @ ordered_pair ) @ in ) ) )
=> ( B @ ( A @ is_reflexive_in ) ) )
& ( ( B @ ( A @ is_reflexive_in ) )
=> ! [C: $i] :
( ( B @ ( C @ in ) )
=> ( A @ ( C @ ( C @ ordered_pair ) @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(149,axiom,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( A @ ( F @ in ) )
& ( E = F )
& ( F @ ordinal ) )
& ( C = E )
& ? [F: $i] :
( ( A @ ( F @ in ) )
& ( D = F )
& ( F @ ordinal ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( C @ ( D @ in ) )
<=> ? [E: $i] :
( ? [F: $i] :
( ( A @ ( F @ in ) )
& ( D = F )
& ( F @ ordinal ) )
& ( E = D )
& ( B @ succ @ ( E @ in ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_tarski__e8_6__wellord2__1) ).
thf(1473,plain,
! [A: $i,B: $i] :
( ( B @ ordinal )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( A @ ( F @ in ) )
& ( E = F )
& ( F @ ordinal ) )
& ( C = E )
& ? [F: $i] :
( ( A @ ( F @ in ) )
& ( D = F )
& ( F @ ordinal ) )
& ( C = D ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ? [E: $i] :
( ? [F: $i] :
( ( A @ ( F @ in ) )
& ( D = F )
& ( F @ ordinal ) )
& ( E = D )
& ( B @ succ @ ( E @ in ) ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ? [E: $i] :
( ? [F: $i] :
( ( A @ ( F @ in ) )
& ( D = F )
& ( F @ ordinal ) )
& ( E = D )
& ( B @ succ @ ( E @ in ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(73,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( A @ powerset @ ( C @ ( B @ ( A @ relation_dom_as_subset ) ) @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_k4_relset_1) ).
thf(839,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2 ) ) )
=> ( A @ powerset @ ( C @ ( B @ ( A @ relation_dom_as_subset ) ) @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(389,axiom,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( ( ( A @ relation_dom )
= ( A @ function_inverse @ relation_rng ) )
& ( ( A @ relation_rng )
= ( A @ function_inverse @ relation_dom ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t55_funct_1) ).
thf(3244,plain,
! [A: $i] :
( ( ( A @ function )
& ( A @ relation ) )
=> ( ( A @ one_to_one )
=> ( ( ( A @ relation_dom )
= ( A @ function_inverse @ relation_rng ) )
& ( ( A @ relation_rng )
= ( A @ function_inverse @ relation_dom ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).
thf(262,axiom,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_restriction ) )
= ( A @ ( B @ relation_dom_restriction ) @ ( A @ relation_rng_restriction ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t18_wellord1) ).
thf(2241,plain,
! [A: $i,B: $i] :
( ( B @ relation )
=> ( ( A @ ( B @ relation_restriction ) )
= ( A @ ( B @ relation_dom_restriction ) @ ( A @ relation_rng_restriction ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(413,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(cnfConj,[status(thm)],[411]) ).
thf(426,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ in ) )
| ~ ( B @ ( C @ in ) )
| ( B
!= ( A @ union ) ) ),
inference(lifteq,[status(thm)],[413]) ).
thf(427,plain,
! [B: $i,A: $i] :
( ( A @ ( A @ union @ ( B @ sk2 ) ) @ ( B @ in ) )
| ~ ( A @ union @ ( B @ in ) ) ),
inference(simp,[status(thm)],[426]) ).
thf(270,axiom,
! [A: $i] :
( ( A @ singleton @ finite )
& ~ ( A @ singleton @ empty ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc1_finset_1) ).
thf(2265,plain,
! [A: $i] :
( ( A @ singleton @ finite )
& ~ ( A @ singleton @ empty ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(291,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( C
= ( B @ ( A @ fiber ) ) )
<=> ! [D: $i] :
( ( C @ ( D @ in ) )
<=> ( ( A @ ( B @ ( D @ ordered_pair ) @ in ) )
& ( D != B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_wellord1) ).
thf(2467,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i,C: $i] :
( ( ! [D: $i] :
( ( ( ( A @ ( B @ ( D @ ordered_pair ) @ in ) )
& ( D != B ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( A @ ( B @ ( D @ ordered_pair ) @ in ) )
& ( D != B ) ) ) )
=> ( C
= ( B @ ( A @ fiber ) ) ) )
& ( ( C
= ( B @ ( A @ fiber ) ) )
=> ! [D: $i] :
( ( ( ( A @ ( B @ ( D @ ordered_pair ) @ in ) )
& ( D != B ) )
=> ( C @ ( D @ in ) ) )
& ( ( C @ ( D @ in ) )
=> ( ( A @ ( B @ ( D @ ordered_pair ) @ in ) )
& ( D != B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(126,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ( ( B = empty_set )
=> ( ( ( B @ ( A @ ( C @ quasi_total ) ) )
<=> ( C = empty_set ) )
| ( A = empty_set ) ) )
& ( ( ( B = empty_set )
=> ( A = empty_set ) )
=> ( ( B @ ( A @ ( C @ quasi_total ) ) )
<=> ( A
= ( C @ ( B @ ( A @ relation_dom_as_subset ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_funct_2) ).
thf(1357,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( ( ( B = empty_set )
=> ( ( ( ( C = empty_set )
=> ( B @ ( A @ ( C @ quasi_total ) ) ) )
& ( ( B @ ( A @ ( C @ quasi_total ) ) )
=> ( C = empty_set ) ) )
| ( A = empty_set ) ) )
& ( ( ( B = empty_set )
=> ( A = empty_set ) )
=> ( ( ( A
= ( C @ ( B @ ( A @ relation_dom_as_subset ) ) ) )
=> ( B @ ( A @ ( C @ quasi_total ) ) ) )
& ( ( B @ ( A @ ( C @ quasi_total ) ) )
=> ( A
= ( C @ ( B @ ( A @ relation_dom_as_subset ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(150,axiom,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( B @ ordinal )
=> ~ ( ~ ( A @ ( B @ in ) )
& ( A != B )
& ~ ( B @ ( A @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t24_ordinal1) ).
thf(1500,plain,
! [A: $i] :
( ( A @ ordinal )
=> ! [B: $i] :
( ( B @ ordinal )
=> ~ ( ~ ( A @ ( B @ in ) )
& ( A != B )
& ~ ( B @ ( A @ in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(296,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ? [D: $i] :
! [E: $i] :
( ( D @ ( E @ in ) )
<=> ( ? [F: $i,G: $i] :
( ( B @ ( G @ ( C @ apply ) @ ( F @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( G @ ( F @ ordered_pair ) ) ) )
& ( A @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_xboole_0__e6_21__wellord2__1) ).
thf(2507,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( ? [F: $i,G: $i] :
( ( B @ ( G @ ( C @ apply ) @ ( F @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( G @ ( F @ ordered_pair ) ) ) )
& ( A @ ( A @ cartesian_product2 ) @ ( E @ in ) ) )
=> ( D @ ( E @ in ) ) )
& ( ( D @ ( E @ in ) )
=> ( ? [F: $i,G: $i] :
( ( B @ ( G @ ( C @ apply ) @ ( F @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( E
= ( G @ ( F @ ordered_pair ) ) ) )
& ( A @ ( A @ cartesian_product2 ) @ ( E @ in ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(214,axiom,
! [A: $i] :
( ( A @ finite )
=> ! [B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( B @ finite ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cc2_finset_1) ).
thf(1885,plain,
! [A: $i] :
( ( A @ finite )
=> ! [B: $i] :
( ( A @ powerset @ ( B @ element ) )
=> ( B @ finite ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(152,axiom,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(1505,plain,
! [A: $i] :
? [B: $i] :
( ( B @ empty )
& ( A @ powerset @ ( B @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ being_limit_ordinal )
<=> ( A
= ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d6_ordinal1) ).
thf(501,plain,
! [A: $i] :
( ( ( A
= ( A @ union ) )
=> ( A @ being_limit_ordinal ) )
& ( ( A @ being_limit_ordinal )
=> ( A
= ( A @ union ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(312,axiom,
! [A: $i] :
( ( A @ finite )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t18_finset_1) ).
thf(2600,plain,
! [A: $i] :
( ( A @ finite )
=> ! [B: $i] :
( ( A @ powerset @ powerset @ ( B @ element ) )
=> ~ ( ! [C: $i] :
~ ( ! [D: $i] :
( ( ( D @ ( C @ subset ) )
& ( B @ ( D @ in ) ) )
=> ( D = C ) )
& ( B @ ( C @ in ) ) )
& ( B != empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(348,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ transitive )
<=> ( A @ relation_field @ ( A @ is_transitive_in ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d16_relat_2) ).
thf(2898,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ( A @ relation_field @ ( A @ is_transitive_in ) )
=> ( A @ transitive ) )
& ( ( A @ transitive )
=> ( A @ relation_field @ ( A @ is_transitive_in ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).
thf(131,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(1384,plain,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ singleton @ subset ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(343,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ relation_dom @ ( A @ relation_rng @ subset ) )
=> ( ( B @ ( A @ relation_composition ) @ relation_dom )
= ( A @ relation_dom ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t46_relat_1) ).
thf(2864,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ( ( B @ relation_dom @ ( A @ relation_rng @ subset ) )
=> ( ( B @ ( A @ relation_composition ) @ relation_dom )
= ( A @ relation_dom ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).
thf(96,axiom,
! [A: $i,B: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(1004,plain,
! [A: $i] :
( ( A @ ( A @ set_union2 ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(219,axiom,
! [A: $i] :
( ( A @ relation )
=> ( ( A @ antisymmetric )
<=> ! [B: $i,C: $i] :
( ( ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3_wellord1) ).
thf(1902,plain,
! [A: $i] :
( ( A @ relation )
=> ( ( ! [B: $i,C: $i] :
( ( ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( B = C ) )
=> ( A @ antisymmetric ) )
& ( ( A @ antisymmetric )
=> ! [B: $i,C: $i] :
( ( ( A @ ( B @ ( C @ ordered_pair ) @ in ) )
& ( A @ ( C @ ( B @ ordered_pair ) @ in ) ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(1317,plain,
! [A: $i] :
( ( A @ being_limit_ordinal )
| ( A != omega ) ),
inference(cnfConj,[status(thm)],[1311]) ).
thf(1330,plain,
! [A: $i] :
( ( A @ being_limit_ordinal )
| ( A != omega ) ),
inference(lifteq,[status(thm)],[1317]) ).
thf(1331,plain,
omega @ being_limit_ordinal,
inference(simp,[status(thm)],[1330]) ).
thf(3452,plain,
( ( ( sk205 @ empty )
!= ( sk29 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2027,732]) ).
thf(3453,plain,
( ( sk205 @ empty )
!= ( sk29 @ empty ) ),
inference(simp,[status(thm)],[3452]) ).
thf(3461,plain,
sk205 != sk29,
inference(simp,[status(thm)],[3453]) ).
thf(9,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( A != D )
& ( A != C )
& ( ( B @ ( A @ unordered_pair ) )
= ( D @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t10_zfmisc_1) ).
thf(469,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)],[9]) ).
thf(3619,plain,
( ( ( sk285 @ empty )
!= ( omega @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2609,1565]) ).
thf(3620,plain,
( ( sk285 @ empty )
!= ( omega @ empty ) ),
inference(simp,[status(thm)],[3619]) ).
thf(3626,plain,
sk285 != omega,
inference(simp,[status(thm)],[3620]) ).
thf(3297,plain,
( ( ( sk1 @ relation_rng @ finite )
!= ( sk1 @ relation_dom @ finite ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[401,402]) ).
thf(3298,plain,
( ( sk1 @ relation_rng @ finite )
!= ( sk1 @ relation_dom @ finite ) ),
inference(simp,[status(thm)],[3297]) ).
thf(3299,plain,
( ( sk1 @ relation_rng )
!= ( sk1 @ relation_dom ) ),
inference(simp,[status(thm)],[3298]) ).
thf(85,axiom,
! [A: $i] : ( A @ inclusion_relation @ transitive ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t3_wellord2) ).
thf(923,plain,
! [A: $i] : ( A @ inclusion_relation @ transitive ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(208,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ? [D: $i] :
( ! [E: $i,F: $i] :
( ( D @ ( F @ ( E @ ordered_pair ) @ in ) )
<=> ( ( B @ ( F @ ( C @ apply ) @ ( E @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ ( F @ in ) )
& ( A @ ( E @ in ) ) ) )
& ( D @ relation ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s1_relat_1__e6_21__wellord2) ).
thf(1848,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ function )
& ( C @ relation )
& ( B @ relation ) )
=> ? [D: $i] :
( ! [E: $i,F: $i] :
( ( ( ( B @ ( F @ ( C @ apply ) @ ( E @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ ( F @ in ) )
& ( A @ ( E @ in ) ) )
=> ( D @ ( F @ ( E @ ordered_pair ) @ in ) ) )
& ( ( D @ ( F @ ( E @ ordered_pair ) @ in ) )
=> ( ( B @ ( F @ ( C @ apply ) @ ( E @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ ( F @ in ) )
& ( A @ ( E @ in ) ) ) ) )
& ( D @ relation ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(2087,plain,
sk215 @ relation,
inference(cnfConj,[status(thm)],[2086]) ).
thf(3292,plain,
( ( ( sk37 @ empty )
!= ( empty_set @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[748,796]) ).
thf(3293,plain,
( ( sk37 @ empty )
!= ( empty_set @ empty ) ),
inference(simp,[status(thm)],[3292]) ).
thf(3296,plain,
sk37 != empty_set,
inference(simp,[status(thm)],[3293]) ).
thf(146,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( B @ ( A @ cartesian_product2 ) @ powerset @ ( C @ element ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',dt_m2_relset_1) ).
thf(1463,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ relation_of2_as_subset ) ) )
=> ( B @ ( A @ cartesian_product2 ) @ powerset @ ( C @ element ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(2615,plain,
sk285 @ relation,
inference(cnfConj,[status(thm)],[2608]) ).
thf(222,axiom,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
<=> ( ! [D: $i,E: $i] :
( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
<=> ( ( B @ ( E @ ( C @ apply ) @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ relation_field @ ( E @ in ) )
& ( A @ relation_field @ ( D @ in ) ) ) )
& ( C @ one_to_one )
& ( ( C @ relation_rng )
= ( B @ relation_field ) )
& ( ( C @ relation_dom )
= ( A @ relation_field ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_wellord1) ).
thf(1917,plain,
! [A: $i] :
( ( A @ relation )
=> ! [B: $i] :
( ( B @ relation )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( ! [D: $i,E: $i] :
( ( ( ( B @ ( E @ ( C @ apply ) @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ relation_field @ ( E @ in ) )
& ( A @ relation_field @ ( D @ in ) ) )
=> ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( B @ ( E @ ( C @ apply ) @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ relation_field @ ( E @ in ) )
& ( A @ relation_field @ ( D @ in ) ) ) ) )
& ( C @ one_to_one )
& ( ( C @ relation_rng )
= ( B @ relation_field ) )
& ( ( C @ relation_dom )
= ( A @ relation_field ) ) )
=> ( C @ ( B @ ( A @ relation_isomorphism ) ) ) )
& ( ( C @ ( B @ ( A @ relation_isomorphism ) ) )
=> ( ! [D: $i,E: $i] :
( ( ( ( B @ ( E @ ( C @ apply ) @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ relation_field @ ( E @ in ) )
& ( A @ relation_field @ ( D @ in ) ) )
=> ( A @ ( E @ ( D @ ordered_pair ) @ in ) ) )
& ( ( A @ ( E @ ( D @ ordered_pair ) @ in ) )
=> ( ( B @ ( E @ ( C @ apply ) @ ( D @ ( C @ apply ) @ ordered_pair ) @ in ) )
& ( A @ relation_field @ ( E @ in ) )
& ( A @ relation_field @ ( D @ in ) ) ) ) )
& ( C @ one_to_one )
& ( ( C @ relation_rng )
= ( B @ relation_field ) )
& ( ( C @ relation_dom )
= ( A @ relation_field ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(3312,plain,
( ( ( sk358 @ empty )
!= ( sk12 @ empty ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[570,3078]) ).
thf(3313,plain,
( ( sk358 @ empty )
!= ( sk12 @ empty ) ),
inference(simp,[status(thm)],[3312]) ).
thf(3316,plain,
sk358 != sk12,
inference(simp,[status(thm)],[3313]) ).
thf(281,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( B
= ( A @ ( C @ relation_dom_restriction ) ) )
<=> ( ! [D: $i] :
( ( B @ relation_dom @ ( D @ in ) )
=> ( ( D @ ( B @ apply ) )
= ( D @ ( C @ apply ) ) ) )
& ( ( B @ relation_dom )
= ( A @ ( C @ relation_dom @ set_intersection2 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t68_funct_1) ).
thf(2351,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( B @ relation ) )
=> ! [C: $i] :
( ( ( C @ function )
& ( C @ relation ) )
=> ( ( ( ! [D: $i] :
( ( B @ relation_dom @ ( D @ in ) )
=> ( ( D @ ( B @ apply ) )
= ( D @ ( C @ apply ) ) ) )
& ( ( B @ relation_dom )
= ( A @ ( C @ relation_dom @ set_intersection2 ) ) ) )
=> ( B
= ( A @ ( C @ relation_dom_restriction ) ) ) )
& ( ( B
= ( A @ ( C @ relation_dom_restriction ) ) )
=> ( ! [D: $i] :
( ( B @ relation_dom @ ( D @ in ) )
=> ( ( D @ ( B @ apply ) )
= ( D @ ( C @ apply ) ) ) )
& ( ( B @ relation_dom )
= ( A @ ( C @ relation_dom @ set_intersection2 ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(99411,plain,
$false,
inference(e,[status(thm)],[1036,2094,629,2612,2263,909,1295,2250,1237,846,777,555,2363,408,523,2210,3317,730,2231,3098,2671,511,1793,1692,2590,1401,3460,1422,1569,1705,1671,2607,797,2933,913,1063,2146,930,670,2216,2985,709,3281,436,2869,1887,2020,841,2400,1529,724,2814,719,3061,917,1455,3089,3803,3263,1287,3029,570,3131,1298,3148,838,542,2703,902,3099,2857,1514,1398,1773,2301,2829,3463,1379,421,1890,2852,3231,1415,2027,2105,2090,2996,1632,2259,623,485,574,3093,2616,1946,1465,3295,2782,3621,2925,397,2367,1377,2618,634,1526,660,3211,2586,1072,3036,1104,1697,3097,2204,529,793,761,1783,2307,3265,925,2083,1736,2621,2790,1047,798,2409,1387,1676,2147,2937,429,3250,2528,2015,1560,2873,3269,435,3137,2872,1624,3144,1407,551,974,916,3166,2200,2609,830,1836,1787,2181,1740,1910,3105,3254,1534,1375,403,1878,2537,3078,869,1118,3087,2498,1541,1804,615,2023,488,1942,3094,476,1424,526,2613,3294,706,3311,2877,1292,3622,590,2085,1889,1644,837,1099,1451,1060,2505,721,3201,515,2519,2536,2681,569,2862,2370,3215,1857,1814,3624,921,1563,765,2089,794,1732,1187,2440,2189,1616,2235,2001,618,2275,2717,3079,745,733,2936,1648,425,2677,2028,1301,423,1007,402,2938,732,1389,843,711,1795,545,3213,1503,1669,936,2029,2293,2935,1058,3304,1638,2319,1865,498,1897,919,3091,947,3101,406,1894,3038,3287,1349,2116,2257,996,509,1801,1571,2871,2197,1969,1307,2610,1557,621,1862,848,2593,2874,915,769,2025,2812,2928,1177,1381,3150,1061,1952,2770,1396,399,1699,2835,780,2931,2244,3310,3142,2535,1250,2303,438,431,1694,1562,2051,636,1245,3241,563,851,1102,2069,2867,473,1626,2614,1304,1467,548,648,1509,749,2564,2597,490,3031,795,2784,2271,3206,836,1551,2567,1798,580,2192,3462,2175,2407,2720,1531,409,2831,3214,971,1566,2143,1605,3260,2588,1780,3209,3080,1790,791,1785,2219,2322,3623,3095,1336,3305,2165,3107,703,1536,656,1702,2145,978,2722,571,2848,2584,2675,2247,2411,2500,3284,465,3584,2088,2045,2818,539,482,3171,1844,1412,3625,2743,1866,1734,2875,3168,1321,1913,2611,759,2305,2919,3465,1646,2707,2988,2167,748,1674,2823,2004,2971,401,3252,2526,1062,2532,433,1777,667,1440,2902,1309,2494,1614,1607,763,1116,2517,1280,1578,833,835,2233,2522,3464,405,1876,1248,603,999,2287,2534,2255,918,2186,437,1410,400,2404,1891,828,2668,1346,796,928,807,3804,2223,1473,839,3244,2241,427,2265,2467,1357,1500,2507,1885,1505,501,2600,2898,1384,2864,1004,3076,1902,1331,3461,469,3626,3299,923,1848,2087,3296,1463,1565,2615,1917,3316,2351]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU303+2 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.15/0.42 % Computer : n009.cluster.edu
% 0.15/0.42 % Model : x86_64 x86_64
% 0.15/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.42 % Memory : 8046.5625MB
% 0.15/0.42 % OS : Linux 6.8.0-71-generic
% 0.15/0.42 % CPULimit : 300
% 0.15/0.42 % WCLimit : 300
% 0.15/0.42 % DateTime : Sat Sep 26 09:09:14 UTC 2026
% 0.15/0.42 % CPUTime :
% 0.15/0.42 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.84/0.99 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.83/1.34 % [INFO] Parsing done (351ms).
% 1.83/1.35 % [INFO] Running in sequential loop mode.
% 3.01/1.73 % [INFO] eprover registered as external prover.
% 3.01/1.74 % [INFO] Scanning for conjecture ...
% 3.22/1.89 % [INFO] Found a conjecture (or negated_conjecture) and 419 axioms. Running axiom selection ...
% 3.90/2.06 % [INFO] Axiom selection finished. Selected 394 axioms (removed 25 axioms).
% 4.99/2.40 % [INFO] Problem is first-order (TPTP FOF).
% 4.99/2.43 % [INFO] Type checking passed.
% 4.99/2.44 % [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 ...
% 160.98/42.81 % External prover 'e' found a proof!
% 160.98/42.81 % [INFO] Killing All external provers ...
% 160.98/42.81 % Time passed: 42257ms (effective reasoning time: 41453ms)
% 160.98/42.81 % 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)>
% 160.98/42.82 % Axioms used in derivation (394): t15_finset_1, d1_enumset1, redefinition_m2_relset_1, redefinition_r1_ordinal1, d1_wellord2, l3_wellord1, s3_funct_1__e16_22__wellord2, cc2_finset_1, fc1_zfmisc_1, d1_finset_1, d1_tarski, s1_xboole_0__e4_27_3_1__finset_1, t30_relat_1, t147_funct_1, d4_funct_1, t32_wellord1, t19_xboole_1, t99_relat_1, t17_xboole_1, t90_relat_1, t43_subset_1, t54_wellord1, fc1_ordinal2, d6_relat_2, l82_funct_1, idempotence_k3_xboole_0, d4_ordinal1, t1_boole, t39_wellord1, t3_xboole_1, dt_k4_relset_1, dt_k5_relset_1, d5_relat_1, l1_wellord1, l2_wellord1, fc1_xboole_0, t7_mcart_1, s1_tarski__e4_27_3_1__finset_1__1, dt_k2_subset_1, cc2_arytm_3, d5_tarski, fc4_relat_1, d12_relat_1, s1_xboole_0__e18_27__finset_1__1, t44_relat_1, fc4_funct_1, t1_subset, t144_relat_1, fc4_subset_1, t22_funct_1, fc3_relat_1, d8_funct_1, t117_relat_1, t2_boole, d5_ordinal2, t32_ordinal1, t47_relat_1, fc9_relat_1, l3_subset_1, t99_zfmisc_1, t47_setfam_1, fc1_funct_1, s1_tarski__e10_24__wellord2__2, d1_wellord1, rc2_partfun1, d9_funct_1, rc3_funct_1, t56_relat_1, t37_xboole_1, cc1_relat_1, redefinition_k6_setfam_1, t55_funct_1, dt_m2_relset_1, t18_finset_1, t21_ordinal1, t26_wellord2, commutativity_k3_xboole_0, s1_tarski__e6_22__wellord2__1, t62_funct_1, fc11_finset_1, d2_tarski, d4_subset_1, involutiveness_k3_subset_1, rc1_funct_1, fc2_xboole_0, d1_xboole_0, t64_relat_1, d6_ordinal1, l50_zfmisc_1, t23_funct_1, t46_funct_2, s1_tarski__e6_21__wellord2__1, t34_funct_1, t118_zfmisc_1, fc10_finset_1, d3_wellord1, t14_relset_1, t16_relset_1, rc3_relat_1, t6_boole, t4_wellord2, dt_k6_subset_1, d1_relat_1, d13_relat_1, t46_zfmisc_1, rc2_subset_1, s1_xboole_0__e6_21__wellord2__1, fc9_finset_1, t1_xboole_1, d9_relat_2, t28_wellord2, t31_ordinal1, t116_relat_1, cc2_ordinal1, commutativity_k2_xboole_0, fc3_subset_1, t5_subset, d4_relat_1, t4_xboole_0, d12_relat_2, cc1_arytm_3, d4_wellord2, existence_m1_relset_1, l25_zfmisc_1, t1_zfmisc_1, t17_wellord1, fc2_subset_1, d8_xboole_0, t5_wellord2, d5_wellord1, d1_ordinal1, s1_relat_1__e6_21__wellord2, t3_wellord2, rc4_funct_1, fc2_funct_1, t5_wellord1, t8_funct_1, t166_relat_1, l32_xboole_1, t8_zfmisc_1, t23_wellord1, t2_tarski, s1_ordinal1__e8_6__wellord2, t33_xboole_1, t4_boole, t146_funct_1, d13_funct_1, dt_k4_relat_1, t4_subset, fc3_xboole_0, t178_relat_1, d2_xboole_0, dt_k2_funct_1, t20_wellord1, rc1_arytm_3, t19_wellord1, fc13_relat_1, d4_tarski, t18_wellord1, d16_relat_2, t9_zfmisc_1, reflexivity_r2_wellord2, t174_relat_1, redefinition_k5_setfam_1, s1_tarski__e6_27__finset_1__1, involutiveness_k4_relat_1, cc1_ordinal1, t7_boole, t21_wellord1, cc1_funct_1, t23_ordinal1, s1_tarski__e10_24__wellord2__1, t118_relat_1, s2_ordinal1__e18_27__finset_1__1, d10_relat_1, s1_tarski__e16_22__wellord2__2, t143_relat_1, d2_subset_1, t48_xboole_1, t3_ordinal1, l4_wellord1, dt_k1_wellord2, fc1_subset_1, t33_zfmisc_1, fc5_funct_1, t13_finset_1, t24_wellord1, fc3_ordinal1, rc2_funct_1, d11_relat_1, connectedness_r1_ordinal1, t21_funct_2, t8_wellord1, t9_funct_2, t160_relat_1, fc10_relat_1, rc1_subset_1, cc3_ordinal1, t74_relat_1, rc1_funct_2, t2_wellord2, rc3_ordinal1, t83_xboole_1, s1_xboole_0__e6_27__finset_1, t45_relat_1, l28_zfmisc_1, t140_relat_1, d14_relat_1, s1_funct_1__e10_24__wellord2__1, s1_xboole_0__e16_22__wellord2__1, d7_xboole_0, fc5_relat_1, cc2_funct_1, t46_setfam_1, d2_wellord1, t17_finset_1, l4_zfmisc_1, t3_subset, cc1_relset_1, t22_relset_1, t33_ordinal1, t115_relat_1, existence_m1_subset_1, d2_mcart_1, t24_ordinal1, s1_ordinal2__e18_27__finset_1, antisymmetry_r2_xboole_0, dt_k6_setfam_1, d2_zfmisc_1, fc11_relat_1, l3_zfmisc_1, commutativity_k2_tarski, d7_relat_1, rc1_ordinal1, t106_zfmisc_1, t25_wellord1, t145_funct_1, fc1_finset_1, l71_subset_1, dt_k7_relat_1, t7_wellord2, l2_zfmisc_1, d8_relat_1, t42_ordinal1, t94_relat_1, t48_setfam_1, d4_wellord1, d5_subset_1, t92_zfmisc_1, t146_relat_1, t70_funct_1, t31_wellord1, d1_zfmisc_1, t69_enumset1, t41_ordinal1, d7_wellord1, existence_m2_relset_1, t38_zfmisc_1, l55_zfmisc_1, s1_tarski__e8_6__wellord2__1, t10_ordinal1, dt_k8_relat_1, d8_setfam_1, t26_xboole_1, t63_xboole_1, t22_wellord1, rc2_relat_1, t72_funct_1, t6_wellord2, t25_wellord2, involutiveness_k7_setfam_1, s1_tarski__e16_22__wellord2__1, t6_funct_2, t8_xboole_1, t37_relat_1, d3_tarski, t39_zfmisc_1, t119_zfmisc_1, s1_xboole_0__e10_24__wellord2__1, rc1_relat_1, t60_relat_1, t54_funct_1, rc2_finset_1, dt_k5_relat_1, reflexivity_r1_tarski, t28_xboole_1, d1_relset_1, t6_zfmisc_1, dt_k6_relat_1, reflexivity_r1_ordinal1, s1_tarski__e18_27__finset_1__1, t2_xboole_1, s1_xboole_0__e6_22__wellord2, fc6_relat_1, d1_funct_2, d12_funct_1, d3_ordinal1, t46_relat_1, rc2_ordinal1, d4_xboole_0, d14_relat_2, t35_funct_1, t36_xboole_1, t20_relat_1, dt_k5_setfam_1, rc3_finset_1, t68_funct_1, rc1_ordinal2, dt_k7_setfam_1, t21_funct_1, fc2_arytm_3, d1_relat_2, fc1_relat_1, d6_wellord1, d6_relat_1, l23_zfmisc_1, t54_subset_1, rc1_finset_1, cc1_finset_1, s2_funct_1__e10_24__wellord2, fc12_finset_1, dt_k2_wellord1, s1_funct_1__e16_22__wellord2__1, t136_zfmisc_1, t8_boole, rc1_xboole_0, fc3_funct_1, t16_wellord1, t12_xboole_1, d3_relat_1, fc7_relat_1, fc1_ordinal1, t49_wellord1, t86_relat_1, fc8_relat_1, s1_xboole_0__e8_6__wellord2__1, t10_zfmisc_1, t2_subset, t37_zfmisc_1, fc2_relat_1, d2_ordinal1, fc12_relat_1, t65_relat_1, t21_relat_1, d1_setfam_1, symmetry_r1_xboole_0, redefinition_r2_wellord2, idempotence_k2_xboole_0, d1_funct_1, l30_wellord2, t9_tarski, t145_relat_1, cc3_arytm_3, d10_xboole_0, symmetry_r2_wellord2, t7_xboole_1, t50_subset_1, fc2_ordinal1, t7_tarski, t60_xboole_1, rc1_partfun1, redefinition_k5_relset_1, t57_funct_1, t3_boole, d5_funct_1, redefinition_k6_subset_1, t12_relset_1, redefinition_k4_relset_1, d8_relat_2, t25_relat_1, d4_relat_2, t3_xboole_0, l29_wellord1, fc13_finset_1, s2_funct_1__e16_22__wellord2__1, rc2_xboole_0, t45_xboole_1, fc4_ordinal1, t40_xboole_1, irreflexivity_r2_xboole_0, t53_wellord1, t65_zfmisc_1, d3_xboole_0, d2_relat_1, t39_xboole_1, t71_relat_1, t23_relset_1, d1_mcart_1, t88_relat_1, t167_relat_1, dt_k3_subset_1, l1_zfmisc_1, t119_relat_1, antisymmetry_r2_hidden
% 160.98/42.82 % No. of inferences in proof: 999
% 160.98/42.82 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 42257 ms resp. 41453 ms w/o parsing
% 163.63/43.21 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 163.63/43.22 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------