TSTP Solution File: SEU280+2 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SEU280+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri May 19 11:57:59 EDT 2023
% Result : Theorem 16.29s 4.83s
% Output : Refutation 16.80s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 410
% Syntax : Number of formulae : 752 ( 89 unt; 70 typ; 0 def)
% Number of atoms : 2859 ( 554 equ; 0 cnn)
% Maximal formula atoms : 28 ( 4 avg)
% Number of connectives : 8542 ( 277 ~; 24 |; 973 &;6088 @)
% ( 135 <=>;1045 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 115 ( 115 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 70 usr; 2 con; 0-3 aty)
% Number of variables : 1793 ( 0 ^;1624 !; 169 ?;1793 :)
% Comments :
%------------------------------------------------------------------------------
thf(in_type,type,
in: $i > $i > $o ).
thf(ordinal_type,type,
ordinal: $i > $o ).
thf(cast_to_subset_type,type,
cast_to_subset: $i > $i ).
thf(equipotent_type,type,
equipotent: $i > $i > $o ).
thf(relation_of2_type,type,
relation_of2: $i > $i > $i > $o ).
thf(empty_type,type,
empty: $i > $o ).
thf(set_difference_type,type,
set_difference: $i > $i > $i ).
thf(empty_set_type,type,
empty_set: $i ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(cartesian_product2_type,type,
cartesian_product2: $i > $i > $i ).
thf(being_limit_ordinal_type,type,
being_limit_ordinal: $i > $o ).
thf(union_type,type,
union: $i > $i ).
thf(relation_type,type,
relation: $i > $o ).
thf(well_ordering_type,type,
well_ordering: $i > $o ).
thf(reflexive_type,type,
reflexive: $i > $o ).
thf(transitive_type,type,
transitive: $i > $o ).
thf(antisymmetric_type,type,
antisymmetric: $i > $o ).
thf(connected_type,type,
connected: $i > $o ).
thf(well_founded_relation_type,type,
well_founded_relation: $i > $o ).
thf(element_type,type,
element: $i > $i > $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(proper_subset_type,type,
proper_subset: $i > $i > $o ).
thf(function_type,type,
function: $i > $o ).
thf(one_to_one_type,type,
one_to_one: $i > $o ).
thf(well_orders_type,type,
well_orders: $i > $i > $o ).
thf(is_reflexive_in_type,type,
is_reflexive_in: $i > $i > $o ).
thf(is_transitive_in_type,type,
is_transitive_in: $i > $i > $o ).
thf(is_antisymmetric_in_type,type,
is_antisymmetric_in: $i > $i > $o ).
thf(is_connected_in_type,type,
is_connected_in: $i > $i > $o ).
thf(is_well_founded_in_type,type,
is_well_founded_in: $i > $i > $o ).
thf(set_union2_type,type,
set_union2: $i > $i > $i ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(pair_first_type,type,
pair_first: $i > $i ).
thf(disjoint_type,type,
disjoint: $i > $i > $o ).
thf(subset_complement_type,type,
subset_complement: $i > $i > $i ).
thf(relation_empty_yielding_type,type,
relation_empty_yielding: $i > $o ).
thf(relation_dom_as_subset_type,type,
relation_dom_as_subset: $i > $i > $i > $i ).
thf(relation_dom_type,type,
relation_dom: $i > $i ).
thf(function_inverse_type,type,
function_inverse: $i > $i ).
thf(relation_inverse_type,type,
relation_inverse: $i > $i ).
thf(relation_rng_type,type,
relation_rng: $i > $i ).
thf(relation_inverse_image_type,type,
relation_inverse_image: $i > $i > $i ).
thf(relation_restriction_type,type,
relation_restriction: $i > $i > $i ).
thf(set_intersection2_type,type,
set_intersection2: $i > $i > $i ).
thf(union_of_subsets_type,type,
union_of_subsets: $i > $i > $i ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(relation_composition_type,type,
relation_composition: $i > $i > $i ).
thf(relation_of2_as_subset_type,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(relation_dom_restriction_type,type,
relation_dom_restriction: $i > $i > $i ).
thf(relation_rng_restriction_type,type,
relation_rng_restriction: $i > $i > $i ).
thf(identity_relation_type,type,
identity_relation: $i > $i ).
thf(subset_difference_type,type,
subset_difference: $i > $i > $i > $i ).
thf(relation_image_type,type,
relation_image: $i > $i > $i ).
thf(relation_rng_as_subset_type,type,
relation_rng_as_subset: $i > $i > $i > $i ).
thf(relation_field_type,type,
relation_field: $i > $i ).
thf(inclusion_relation_type,type,
inclusion_relation: $i > $i ).
thf(fiber_type,type,
fiber: $i > $i > $i ).
thf(pair_second_type,type,
pair_second: $i > $i ).
thf(meet_of_subsets_type,type,
meet_of_subsets: $i > $i > $i ).
thf(complements_of_subsets_type,type,
complements_of_subsets: $i > $i > $i ).
thf(succ_type,type,
succ: $i > $i ).
thf(relation_isomorphism_type,type,
relation_isomorphism: $i > $i > $i > $o ).
thf(are_equipotent_type,type,
are_equipotent: $i > $i > $o ).
thf(set_meet_type,type,
set_meet: $i > $i ).
thf(apply_type,type,
apply: $i > $i > $i ).
thf(epsilon_transitive_type,type,
epsilon_transitive: $i > $o ).
thf(epsilon_connected_type,type,
epsilon_connected: $i > $o ).
thf(ordinal_subset_type,type,
ordinal_subset: $i > $i > $o ).
thf(unordered_triple_type,type,
unordered_triple: $i > $i > $i > $i ).
thf(285,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ A @ ( relation_dom @ ( relation_composition @ C @ B ) ) )
=> ( ( apply @ ( relation_composition @ C @ B ) @ A )
= ( apply @ B @ ( apply @ C @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_funct_1) ).
thf(1718,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ A @ ( relation_dom @ ( relation_composition @ C @ B ) ) )
=> ( ( apply @ ( relation_composition @ C @ B ) @ A )
= ( apply @ B @ ( apply @ C @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(162,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t39_xboole_1) ).
thf(893,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_setfam_1) ).
thf(468,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( union_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(4,axiom,
! [A: $i,B: $i] : ( equipotent @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r2_wellord2) ).
thf(352,plain,
! [A: $i] : ( equipotent @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(236,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( connected @ A )
<=> ! [B: $i,C: $i] :
~ ( ( in @ B @ ( relation_field @ A ) )
& ( in @ C @ ( relation_field @ A ) )
& ( B != C )
& ~ ( in @ ( ordered_pair @ B @ C ) @ A )
& ~ ( in @ ( ordered_pair @ C @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l4_wellord1) ).
thf(1237,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( connected @ A )
=> ! [B: $i,C: $i] :
~ ( ( in @ B @ ( relation_field @ A ) )
& ( in @ C @ ( relation_field @ A ) )
& ( B != C )
& ~ ( in @ ( ordered_pair @ B @ C ) @ A )
& ~ ( in @ ( ordered_pair @ C @ B ) @ A ) ) )
& ( ! [B: $i,C: $i] :
~ ( ( in @ B @ ( relation_field @ A ) )
& ( in @ C @ ( relation_field @ A ) )
& ( B != C )
& ~ ( in @ ( ordered_pair @ B @ C ) @ A )
& ~ ( in @ ( ordered_pair @ C @ B ) @ A ) )
=> ( connected @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(149,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
<=> ( are_equipotent @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r2_wellord2) ).
thf(846,plain,
! [A: $i,B: $i] :
( ( ( equipotent @ A @ B )
=> ( are_equipotent @ A @ B ) )
& ( ( are_equipotent @ A @ B )
=> ( equipotent @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(80,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( subset @ ( fiber @ ( relation_restriction @ C @ A ) @ B ) @ ( fiber @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_wellord1) ).
thf(628,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( subset @ ( fiber @ ( relation_restriction @ C @ A ) @ B ) @ ( fiber @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(210,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ B @ A )
=> ( ( apply @ ( relation_dom_restriction @ C @ A ) @ B )
= ( apply @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t72_funct_1) ).
thf(1031,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ B @ A )
=> ( ( apply @ ( relation_dom_restriction @ C @ A ) @ B )
= ( apply @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(318,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ~ ( empty @ ( succ @ A ) )
& ( epsilon_transitive @ ( succ @ A ) )
& ( epsilon_connected @ ( succ @ A ) )
& ( ordinal @ ( succ @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_ordinal1) ).
thf(2077,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ~ ( empty @ ( succ @ A ) )
& ( epsilon_transitive @ ( succ @ A ) )
& ( epsilon_connected @ ( succ @ A ) )
& ( ordinal @ ( succ @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).
thf(289,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( B
= ( relation_dom @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_relat_1) ).
thf(1750,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( B
= ( relation_dom @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( relation_dom @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(306,axiom,
! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( ( apply @ ( identity_relation @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t35_funct_1) ).
thf(1939,plain,
! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( ( apply @ ( identity_relation @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(270,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(1544,plain,
! [A: $i] :
( ( ( A = empty_set )
=> ! [B: $i] :
~ ( in @ B @ A ) )
& ( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(95,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( subset @ A @ B )
=> ( subset @ ( relation_inverse_image @ C @ A ) @ ( relation_inverse_image @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t178_relat_1) ).
thf(677,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( subset @ A @ B )
=> ( subset @ ( relation_inverse_image @ C @ A ) @ ( relation_inverse_image @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(47,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_restriction @ B @ A )
= ( relation_dom_restriction @ ( relation_rng_restriction @ A @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_wellord1) ).
thf(523,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_restriction @ B @ A )
= ( relation_dom_restriction @ ( relation_rng_restriction @ A @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(66,axiom,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_wellord1) ).
thf(582,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(274,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_zfmisc_1) ).
thf(1596,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(321,axiom,
( ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set )
& ( function @ empty_set )
& ( one_to_one @ empty_set )
& ( empty @ empty_set )
& ( epsilon_transitive @ empty_set )
& ( epsilon_connected @ empty_set )
& ( ordinal @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_ordinal1) ).
thf(2104,plain,
( ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set )
& ( function @ empty_set )
& ( one_to_one @ empty_set )
& ( empty @ empty_set )
& ( epsilon_transitive @ empty_set )
& ( epsilon_connected @ empty_set )
& ( ordinal @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).
thf(241,axiom,
! [A: $i,B: $i] :
( ( B
= ( singleton @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ( C = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_tarski) ).
thf(1290,plain,
! [A: $i,B: $i] :
( ( ( B
= ( singleton @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ( C = A ) )
& ( ( C = A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ( C = A ) )
& ( ( C = A )
=> ( in @ C @ B ) ) )
=> ( B
= ( singleton @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(301,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( B
= ( relation_rng @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_relat_1) ).
thf(1882,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( B
= ( relation_rng @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ? [D: $i] : ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(157,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_rng @ A ) @ ( relation_dom @ B ) )
=> ( ( relation_dom @ ( relation_composition @ A @ B ) )
= ( relation_dom @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_relat_1) ).
thf(878,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_rng @ A ) @ ( relation_dom @ B ) )
=> ( ( relation_dom @ ( relation_composition @ A @ B ) )
= ( relation_dom @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(24,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ B )
= empty_set )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l32_xboole_1) ).
thf(440,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ B )
= empty_set )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( ( set_difference @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(283,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ( being_limit_ordinal @ A )
<=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( in @ ( succ @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t41_ordinal1) ).
thf(1705,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( ( being_limit_ordinal @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( in @ ( succ @ B ) @ A ) ) ) )
& ( ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( in @ ( succ @ B ) @ A ) ) )
=> ( being_limit_ordinal @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(216,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_wellord2) ).
thf(1078,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(242,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation @ D )
=> ( ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) )
<=> ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t74_relat_1) ).
thf(1310,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation @ D )
=> ( ( ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) )
=> ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ ( ordered_pair @ A @ B ) @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( relation_composition @ ( identity_relation @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(136,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation_empty_yielding @ A ) )
=> ( ( relation @ ( relation_dom_restriction @ A @ B ) )
& ( relation_empty_yielding @ ( relation_dom_restriction @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc13_relat_1) ).
thf(797,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation_empty_yielding @ A ) )
=> ( ( relation @ ( relation_dom_restriction @ A @ B ) )
& ( relation_empty_yielding @ ( relation_dom_restriction @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(239,axiom,
! [A: $i] :
( ( epsilon_transitive @ A )
<=> ! [B: $i] :
( ( in @ B @ A )
=> ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_ordinal1) ).
thf(1275,plain,
! [A: $i] :
( ( ( epsilon_transitive @ A )
=> ! [B: $i] :
( ( in @ B @ A )
=> ( subset @ B @ A ) ) )
& ( ! [B: $i] :
( ( in @ B @ A )
=> ( subset @ B @ A ) )
=> ( epsilon_transitive @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(139,axiom,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(814,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(169,axiom,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_wellord2) ).
thf(913,plain,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(181,axiom,
! [A: $i] : ( subset @ empty_set @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(945,plain,
! [A: $i] : ( subset @ empty_set @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(246,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( ( one_to_one @ B )
& ( in @ A @ ( relation_rng @ B ) ) )
=> ( ( A
= ( apply @ B @ ( apply @ ( function_inverse @ B ) @ A ) ) )
& ( A
= ( apply @ ( relation_composition @ ( function_inverse @ B ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t57_funct_1) ).
thf(1354,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( ( one_to_one @ B )
& ( in @ A @ ( relation_rng @ B ) ) )
=> ( ( A
= ( apply @ B @ ( apply @ ( function_inverse @ B ) @ A ) ) )
& ( A
= ( apply @ ( relation_composition @ ( function_inverse @ B ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(164,axiom,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_funct_1) ).
thf(898,plain,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(200,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( union_of_subsets @ A @ B ) )
= ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t47_setfam_1) ).
thf(1001,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( union_of_subsets @ A @ B ) )
= ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(258,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(1441,plain,
! [A: $i,B: $i] :
( ( ( subset @ ( singleton @ A ) @ B )
=> ( in @ A @ B ) )
& ( ( in @ A @ B )
=> ( subset @ ( singleton @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(147,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc6_relat_1) ).
thf(841,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(160,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( transitive @ A )
<=> ( is_transitive_in @ A @ ( relation_field @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d16_relat_2) ).
thf(885,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( transitive @ A )
=> ( is_transitive_in @ A @ ( relation_field @ A ) ) )
& ( ( is_transitive_in @ A @ ( relation_field @ A ) )
=> ( transitive @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(247,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ( ordinal_subset @ A @ B )
| ( ordinal_subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connectedness_r1_ordinal1) ).
thf(1359,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ( ordinal_subset @ A @ B )
| ( ordinal_subset @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(319,axiom,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(2082,plain,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(250,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( is_reflexive_in @ A @ B )
<=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( ordered_pair @ C @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_relat_2) ).
thf(1366,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( is_reflexive_in @ A @ B )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( ordered_pair @ C @ C ) @ A ) ) )
& ( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( ordered_pair @ C @ C ) @ A ) )
=> ( is_reflexive_in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(143,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ D ) )
=> ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t119_zfmisc_1) ).
thf(829,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ D ) )
=> ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(220,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ B @ ( relation_dom @ ( relation_dom_restriction @ C @ A ) ) )
=> ( ( apply @ ( relation_dom_restriction @ C @ A ) @ B )
= ( apply @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t70_funct_1) ).
thf(1095,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ B @ ( relation_dom @ ( relation_dom_restriction @ C @ A ) ) )
=> ( ( apply @ ( relation_dom_restriction @ C @ A ) @ B )
= ( apply @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(132,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k7_setfam_1) ).
thf(785,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( complements_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(40,axiom,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(504,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(146,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( well_orders @ A @ ( relation_field @ A ) )
<=> ( well_ordering @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_wellord1) ).
thf(838,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( well_orders @ A @ ( relation_field @ A ) )
=> ( well_ordering @ A ) )
& ( ( well_ordering @ A )
=> ( well_orders @ A @ ( relation_field @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(118,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t33_xboole_1) ).
thf(751,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(137,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_m2_relset_1) ).
thf(802,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(293,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ B @ ( relation_dom @ ( relation_dom_restriction @ C @ A ) ) )
<=> ( ( in @ B @ ( relation_dom @ C ) )
& ( in @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l82_funct_1) ).
thf(1832,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( in @ B @ ( relation_dom @ ( relation_dom_restriction @ C @ A ) ) )
=> ( ( in @ B @ ( relation_dom @ C ) )
& ( in @ B @ A ) ) )
& ( ( ( in @ B @ ( relation_dom @ C ) )
& ( in @ B @ A ) )
=> ( in @ B @ ( relation_dom @ ( relation_dom_restriction @ C @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(230,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(1191,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(165,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_rng_restriction @ A @ B ) )
= ( set_intersection2 @ ( relation_rng @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t119_relat_1) ).
thf(902,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_rng_restriction @ A @ B ) )
= ( set_intersection2 @ ( relation_rng @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(151,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
<=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( one_to_one @ C )
& ( ( relation_dom @ C )
= A )
& ( ( relation_rng @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_wellord2) ).
thf(853,plain,
! [A: $i,B: $i] :
( ( ( equipotent @ A @ B )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( one_to_one @ C )
& ( ( relation_dom @ C )
= A )
& ( ( relation_rng @ C )
= B ) ) )
& ( ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( one_to_one @ C )
& ( ( relation_dom @ C )
= A )
& ( ( relation_rng @ C )
= B ) )
=> ( equipotent @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(327,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_image @ C @ B ) )
<=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t143_relat_1) ).
thf(2169,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_image @ C @ B ) )
=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_dom @ C ) )
& ( in @ ( ordered_pair @ D @ A ) @ C )
& ( in @ D @ B ) )
=> ( in @ A @ ( relation_image @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(237,axiom,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(1247,plain,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(20,axiom,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_boole) ).
thf(421,plain,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(190,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( connected @ A )
<=> ( is_connected_in @ A @ ( relation_field @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d14_relat_2) ).
thf(969,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( connected @ A )
=> ( is_connected_in @ A @ ( relation_field @ A ) ) )
& ( ( is_connected_in @ A @ ( relation_field @ A ) )
=> ( connected @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(126,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_composition @ A @ B ) ) @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t44_relat_1) ).
thf(770,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_composition @ A @ B ) ) @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(158,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_inverse_image @ B @ A ) @ ( relation_dom @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t167_relat_1) ).
thf(881,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_inverse_image @ B @ A ) @ ( relation_dom @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(299,axiom,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(1875,plain,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(174,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_relat_1) ).
thf(924,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(120,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(756,plain,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(292,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_dom_restriction @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d11_relat_1) ).
thf(1811,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_dom_restriction @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) )
& ( ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) ) )
& ( ( ( in @ D @ B )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_dom_restriction @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(69,axiom,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(591,plain,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(251,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( in @ B @ C )
& ( in @ C @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_ordinal1) ).
thf(1373,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( in @ B @ C )
& ( in @ C @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(11,axiom,
! [A: $i] :
( ( being_limit_ordinal @ A )
<=> ( A
= ( union @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_ordinal1) ).
thf(376,plain,
! [A: $i] :
( ( ( being_limit_ordinal @ A )
=> ( A
= ( union @ A ) ) )
& ( ( A
= ( union @ A ) )
=> ( being_limit_ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(335,axiom,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ~ ( ( in @ B @ ( subset_complement @ A @ C ) )
& ( in @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t54_subset_1) ).
thf(2236,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ~ ( ( in @ B @ ( subset_complement @ A @ C ) )
& ( in @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).
thf(87,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc7_relat_1) ).
thf(646,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(339,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ B @ A )
=> ( ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ D @ E ) @ C ) )
<=> ( ( relation_dom_as_subset @ B @ A @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_relset_1) ).
thf(2291,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ B @ A )
=> ( ( ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ D @ E ) @ C ) )
=> ( ( relation_dom_as_subset @ B @ A @ C )
= B ) )
& ( ( ( relation_dom_as_subset @ B @ A @ C )
= B )
=> ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(206,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( meet_of_subsets @ A @ B )
= ( set_meet @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k6_setfam_1) ).
thf(1020,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( meet_of_subsets @ A @ B )
= ( set_meet @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(166,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation_of2_as_subset @ D @ C @ A )
=> ( ( subset @ A @ B )
=> ( relation_of2_as_subset @ D @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t16_relset_1) ).
thf(905,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation_of2_as_subset @ D @ C @ A )
=> ( ( subset @ A @ B )
=> ( relation_of2_as_subset @ D @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(35,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(480,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(74,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k8_relat_1) ).
thf(602,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(99,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( ( unordered_pair @ A @ B )
= ( unordered_pair @ C @ D ) )
& ( A != C )
& ( A != D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_zfmisc_1) ).
thf(687,plain,
! [A: $i,B: $i,C: $i,D: $i] :
~ ( ( ( unordered_pair @ A @ B )
= ( unordered_pair @ C @ D ) )
& ( A != C )
& ( A != D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(83,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(634,plain,
! [A: $i] :
( ( set_union2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(199,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_funct_1) ).
thf(998,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(41,axiom,
empty @ empty_set,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(507,plain,
empty @ empty_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(291,axiom,
! [A: $i] :
( ( epsilon_connected @ A )
<=> ! [B: $i,C: $i] :
~ ( ( in @ B @ A )
& ( in @ C @ A )
& ~ ( in @ B @ C )
& ( B != C )
& ~ ( in @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_ordinal1) ).
thf(1796,plain,
! [A: $i] :
( ( ( epsilon_connected @ A )
=> ! [B: $i,C: $i] :
~ ( ( in @ B @ A )
& ( in @ C @ A )
& ~ ( in @ B @ C )
& ( B != C )
& ~ ( in @ C @ B ) ) )
& ( ! [B: $i,C: $i] :
~ ( ( in @ B @ A )
& ( in @ C @ A )
& ~ ( in @ B @ C )
& ( B != C )
& ~ ( in @ C @ B ) )
=> ( epsilon_connected @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(27,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( relation_dom_as_subset @ A @ B @ C )
= ( relation_dom @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k4_relset_1) ).
thf(453,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( relation_dom_as_subset @ A @ B @ C )
= ( relation_dom @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(172,axiom,
! [A: $i,B: $i] : ( subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(920,plain,
! [A: $i] : ( subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(122,axiom,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_relat_1) ).
thf(761,plain,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(263,axiom,
! [A: $i] :
( ! [B: $i] :
( ( in @ B @ A )
=> ( ( ordinal @ B )
& ( subset @ B @ A ) ) )
=> ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t31_ordinal1) ).
thf(1470,plain,
! [A: $i] :
( ! [B: $i] :
( ( in @ B @ A )
=> ( ( ordinal @ B )
& ( subset @ B @ A ) ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(214,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ B @ ( relation_field @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t30_relat_1) ).
thf(1067,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ B @ ( relation_field @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(108,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( antisymmetric @ A )
<=> ( is_antisymmetric_in @ A @ ( relation_field @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d12_relat_2) ).
thf(714,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( antisymmetric @ A )
=> ( is_antisymmetric_in @ A @ ( relation_field @ A ) ) )
& ( ( is_antisymmetric_in @ A @ ( relation_field @ A ) )
=> ( antisymmetric @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(43,axiom,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ B @ A ) )
& ( relation @ ( relation_composition @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc10_relat_1) ).
thf(512,plain,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ B @ A ) )
& ( relation @ ( relation_composition @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(180,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ B )
= ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_subset_1) ).
thf(942,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ B )
= ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(168,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_rng @ B ) )
=> ( ( relation_rng @ ( relation_composition @ B @ A ) )
= ( relation_rng @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t47_relat_1) ).
thf(910,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_rng @ B ) )
=> ( ( relation_rng @ ( relation_composition @ B @ A ) )
= ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(116,axiom,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_wellord2) ).
thf(746,plain,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(141,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_relat_1) ).
thf(825,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(129,axiom,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_ordinal1) ).
thf(778,plain,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(3,axiom,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_subset_1) ).
thf(349,plain,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(72,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_subset_1) ).
thf(598,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(59,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( relation_rng_as_subset @ A @ B @ C )
= ( relation_rng @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k5_relset_1) ).
thf(561,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( relation_rng_as_subset @ A @ B @ C )
= ( relation_rng @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(21,axiom,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(424,plain,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(23,axiom,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( ordered_pair @ B @ C ) )
=> ! [B: $i] :
( ( B
= ( pair_first @ A ) )
<=> ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_mcart_1) ).
thf(429,plain,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( ordered_pair @ B @ C ) )
=> ! [B: $i] :
( ( ( B
= ( pair_first @ A ) )
=> ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = C ) ) )
& ( ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = C ) )
=> ( B
= ( pair_first @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(133,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( relation @ B )
& ( function @ B ) )
=> ( ( relation @ ( relation_composition @ A @ B ) )
& ( function @ ( relation_composition @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_funct_1) ).
thf(788,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( relation @ B )
& ( function @ B ) )
=> ( ( relation @ ( relation_composition @ A @ B ) )
& ( function @ ( relation_composition @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(311,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
<=> ( ( in @ A @ C )
& ( in @ B @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t106_zfmisc_1) ).
thf(2015,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
=> ( ( in @ A @ C )
& ( in @ B @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(17,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_relat_1) ).
thf(407,plain,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(81,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_relat_1) ).
thf(630,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(308,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( relation_dom @ A ) )
& ( in @ E @ B )
& ( D
= ( apply @ A @ E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d12_funct_1) ).
thf(1951,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( relation_dom @ A ) )
& ( in @ E @ B )
& ( D
= ( apply @ A @ E ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( relation_dom @ A ) )
& ( in @ E @ B )
& ( D
= ( apply @ A @ E ) ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( relation_dom @ A ) )
& ( in @ E @ B )
& ( D
= ( apply @ A @ E ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( relation_dom @ A ) )
& ( in @ E @ B )
& ( D
= ( apply @ A @ E ) ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(280,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_wellord2) ).
thf(1687,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(330,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( is_connected_in @ A @ B )
<=> ! [C: $i,D: $i] :
~ ( ( in @ C @ B )
& ( in @ D @ B )
& ( C != D )
& ~ ( in @ ( ordered_pair @ C @ D ) @ A )
& ~ ( in @ ( ordered_pair @ D @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_relat_2) ).
thf(2179,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( is_connected_in @ A @ B )
=> ! [C: $i,D: $i] :
~ ( ( in @ C @ B )
& ( in @ D @ B )
& ( C != D )
& ~ ( in @ ( ordered_pair @ C @ D ) @ A )
& ~ ( in @ ( ordered_pair @ D @ C ) @ A ) ) )
& ( ! [C: $i,D: $i] :
~ ( ( in @ C @ B )
& ( in @ D @ B )
& ( C != D )
& ~ ( in @ ( ordered_pair @ C @ D ) @ A )
& ~ ( in @ ( ordered_pair @ D @ C ) @ A ) )
=> ( is_connected_in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(294,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( ordinal @ F )
& ( D = F )
& ( in @ F @ A ) )
& ( C = E )
& ? [F: $i] :
( ( ordinal @ F )
& ( E = F )
& ( in @ F @ A ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( succ @ B ) )
& ( E = D )
& ? [F: $i] :
( ( ordinal @ F )
& ( D = F )
& ( in @ F @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e8_6__wellord2__1) ).
thf(1836,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( ordinal @ F )
& ( D = F )
& ( in @ F @ A ) )
& ( C = E )
& ? [F: $i] :
( ( ordinal @ F )
& ( E = F )
& ( in @ F @ A ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( succ @ B ) )
& ( E = D )
& ? [F: $i] :
( ( ordinal @ F )
& ( D = F )
& ( in @ F @ A ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( succ @ B ) )
& ( E = D )
& ? [F: $i] :
( ( ordinal @ F )
& ( D = F )
& ( in @ F @ A ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(56,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( element @ ( relation_rng_as_subset @ A @ B @ C ) @ ( powerset @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_relset_1) ).
thf(551,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( element @ ( relation_rng_as_subset @ A @ B @ C ) @ ( powerset @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(28,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( ( function_inverse @ A )
= ( relation_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d9_funct_1) ).
thf(456,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( ( function_inverse @ A )
= ( relation_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(192,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_dom_restriction @ B @ A ) ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_relat_1) ).
thf(974,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_dom_restriction @ B @ A ) ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(204,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(1012,plain,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(102,axiom,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_boole) ).
thf(697,plain,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(309,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
<=> ( ( in @ A @ C )
& ( in @ B @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l55_zfmisc_1) ).
thf(1983,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
=> ( ( in @ A @ C )
& ( in @ B @ D ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ D ) )
=> ( in @ ( ordered_pair @ A @ B ) @ ( cartesian_product2 @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(78,axiom,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(615,plain,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(245,axiom,
! [A: $i,B: $i] :
( ( B
= ( union @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_tarski) ).
thf(1328,plain,
! [A: $i,B: $i] :
( ( ( B
= ( union @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) )
& ( ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) ) )
& ( ? [D: $i] :
( ( in @ C @ D )
& ( in @ D @ A ) )
=> ( in @ C @ B ) ) )
=> ( B
= ( union @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(213,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l23_zfmisc_1) ).
thf(1064,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(16,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( empty @ A )
& ( function @ A ) )
=> ( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_funct_1) ).
thf(403,plain,
! [A: $i] :
( ( ( relation @ A )
& ( empty @ A )
& ( function @ A ) )
=> ( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(254,axiom,
! [A: $i] :
( ( ordinal @ A )
<=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_ordinal1) ).
thf(1425,plain,
! [A: $i] :
( ( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) )
& ( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(269,axiom,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_ordinal1) ).
thf(1540,plain,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(300,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ( in @ D @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_tarski) ).
thf(1878,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(6,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r2_wellord2) ).
thf(356,plain,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(70,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_wellord1) ).
thf(593,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(131,axiom,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_funct_1) ).
thf(783,plain,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(114,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_setfam_1) ).
thf(742,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( meet_of_subsets @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(340,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i] :
( ( B
= ( relation_rng @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_funct_1) ).
thf(2300,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i] :
( ( ( B
= ( relation_rng @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) )
=> ( in @ C @ B ) ) )
=> ( B
= ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).
thf(155,axiom,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(874,plain,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(323,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( E
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( F
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ A ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( E
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e6_21__wellord2__1) ).
thf(2115,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( E
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( F
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ A ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( E
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) ) ) )
& ( ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ A ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( E
= ( ordered_pair @ G @ H ) )
& ( in @ ( ordered_pair @ ( apply @ C @ G ) @ ( apply @ C @ H ) ) @ B ) ) )
=> ( in @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(37,axiom,
? [A: $i] :
~ ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(488,plain,
? [A: $i] :
~ ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(248,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(1361,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(307,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
=> ( ( element @ B @ A )
<=> ( in @ B @ A ) ) )
& ( ( empty @ A )
=> ( ( element @ B @ A )
<=> ( empty @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_subset_1) ).
thf(1942,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
=> ( ( ( element @ B @ A )
=> ( in @ B @ A ) )
& ( ( in @ B @ A )
=> ( element @ B @ A ) ) ) )
& ( ( empty @ A )
=> ( ( ( element @ B @ A )
=> ( empty @ B ) )
& ( ( empty @ B )
=> ( element @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(176,axiom,
! [A: $i,B: $i] :
~ ( proper_subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(933,plain,
! [A: $i] :
~ ( proper_subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(51,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t146_relat_1) ).
thf(537,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(48,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t88_relat_1) ).
thf(526,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(84,axiom,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k7_relat_1) ).
thf(637,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(259,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ~ ( ~ ( being_limit_ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( A
!= ( succ @ B ) ) ) )
& ~ ( ? [B: $i] :
( ( ordinal @ B )
& ( A
= ( succ @ B ) ) )
& ( being_limit_ordinal @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t42_ordinal1) ).
thf(1446,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ~ ( ~ ( being_limit_ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( A
!= ( succ @ B ) ) ) )
& ~ ( ? [B: $i] :
( ( ordinal @ B )
& ( A
= ( succ @ B ) ) )
& ( being_limit_ordinal @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(96,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t116_relat_1) ).
thf(679,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(105,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_relat_1) ).
thf(706,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(209,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_intersection2 @ A @ C ) @ ( set_intersection2 @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_xboole_1) ).
thf(1028,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( subset @ ( set_intersection2 @ A @ C ) @ ( set_intersection2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(313,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ~ ( ( subset @ A @ B )
& ( A != empty_set )
& ! [C: $i] :
( ( ordinal @ C )
=> ~ ( ( in @ C @ A )
& ! [D: $i] :
( ( ordinal @ D )
=> ( ( in @ D @ A )
=> ( ordinal_subset @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t32_ordinal1) ).
thf(2023,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ~ ( ( subset @ A @ B )
& ( A != empty_set )
& ! [C: $i] :
( ( ordinal @ C )
=> ~ ( ( in @ C @ A )
& ! [D: $i] :
( ( ordinal @ D )
=> ( ( in @ D @ A )
=> ( ordinal_subset @ C @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(244,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( reflexive @ A )
<=> ! [B: $i] :
( ( in @ B @ ( relation_field @ A ) )
=> ( in @ ( ordered_pair @ B @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_wellord1) ).
thf(1324,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( reflexive @ A )
=> ! [B: $i] :
( ( in @ B @ ( relation_field @ A ) )
=> ( in @ ( ordered_pair @ B @ B ) @ A ) ) )
& ( ! [B: $i] :
( ( in @ B @ ( relation_field @ A ) )
=> ( in @ ( ordered_pair @ B @ B ) @ A ) )
=> ( reflexive @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(113,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc4_funct_1) ).
thf(738,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(271,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( B
= ( relation_dom_restriction @ C @ A ) )
<=> ( ( ( relation_dom @ B )
= ( set_intersection2 @ ( relation_dom @ C ) @ A ) )
& ! [D: $i] :
( ( in @ D @ ( relation_dom @ B ) )
=> ( ( apply @ B @ D )
= ( apply @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t68_funct_1) ).
thf(1552,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( B
= ( relation_dom_restriction @ C @ A ) )
=> ( ( ( relation_dom @ B )
= ( set_intersection2 @ ( relation_dom @ C ) @ A ) )
& ! [D: $i] :
( ( in @ D @ ( relation_dom @ B ) )
=> ( ( apply @ B @ D )
= ( apply @ C @ D ) ) ) ) )
& ( ( ( ( relation_dom @ B )
= ( set_intersection2 @ ( relation_dom @ C ) @ A ) )
& ! [D: $i] :
( ( in @ D @ ( relation_dom @ B ) )
=> ( ( apply @ B @ D )
= ( apply @ C @ D ) ) ) )
=> ( B
= ( relation_dom_restriction @ C @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(62,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation_of2_as_subset @ D @ C @ A )
=> ( ( subset @ ( relation_rng @ D ) @ B )
=> ( relation_of2_as_subset @ D @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t14_relset_1) ).
thf(569,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( relation_of2_as_subset @ D @ C @ A )
=> ( ( subset @ ( relation_rng @ D ) @ B )
=> ( relation_of2_as_subset @ D @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(325,axiom,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( subset_complement @ A @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t50_subset_1) ).
thf(2140,plain,
! [A: $i] :
( ( A != empty_set )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( subset_complement @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(253,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( cartesian_product2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_zfmisc_1) ).
thf(1393,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( cartesian_product2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) )
& ( ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) ) )
& ( ? [E: $i,F: $i] :
( ( in @ E @ A )
& ( in @ F @ B )
& ( D
= ( ordered_pair @ E @ F ) ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( cartesian_product2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(276,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d13_relat_1) ).
thf(1627,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ E @ D ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(44,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m2_relset_1) ).
thf(515,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(238,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_inverse_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d14_relat_1) ).
thf(1249,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_inverse_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) ) )
& ( ? [E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
& ( in @ E @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_inverse_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(268,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( succ @ B ) )
& ? [E: $i] :
( ( ordinal @ E )
& ( D = E )
& ( in @ E @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e8_6__wellord2__1) ).
thf(1530,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( succ @ B ) )
& ? [E: $i] :
( ( ordinal @ E )
& ( D = E )
& ( in @ E @ A ) ) ) )
& ( ( ( in @ D @ ( succ @ B ) )
& ? [E: $i] :
( ( ordinal @ E )
& ( D = E )
& ( in @ E @ A ) ) )
=> ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(261,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
| ( subset @ A @ ( set_difference @ B @ ( singleton @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_zfmisc_1) ).
thf(1456,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
| ( subset @ A @ ( set_difference @ B @ ( singleton @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(224,axiom,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_ordinal1) ).
thf(1131,plain,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k3_xboole_0) ).
thf(654,plain,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(315,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
<=> ( ( in @ A @ ( relation_dom @ C ) )
& ( B
= ( apply @ C @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_funct_1) ).
thf(2052,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( B
= ( apply @ C @ A ) ) ) )
& ( ( ( in @ A @ ( relation_dom @ C ) )
& ( B
= ( apply @ C @ A ) ) )
=> ( in @ ( ordered_pair @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(275,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( B
= ( inclusion_relation @ A ) )
<=> ( ( ( relation_field @ B )
= A )
& ! [C: $i,D: $i] :
( ( ( in @ C @ A )
& ( in @ D @ A ) )
=> ( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( subset @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_wellord2) ).
thf(1599,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( ( B
= ( inclusion_relation @ A ) )
=> ( ( ( relation_field @ B )
= A )
& ! [C: $i,D: $i] :
( ( ( in @ C @ A )
& ( in @ D @ A ) )
=> ( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( subset @ C @ D ) )
& ( ( subset @ C @ D )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ) )
& ( ( ( ( relation_field @ B )
= A )
& ! [C: $i,D: $i] :
( ( ( in @ C @ A )
& ( in @ D @ A ) )
=> ( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( subset @ C @ D ) )
& ( ( subset @ C @ D )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) )
=> ( B
= ( inclusion_relation @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(178,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t63_xboole_1) ).
thf(938,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(42,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_funct_1) ).
thf(508,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(124,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( subset @ A @ ( relation_rng @ B ) )
=> ( ( relation_image @ B @ ( relation_inverse_image @ B @ A ) )
= A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t147_funct_1) ).
thf(765,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( subset @ A @ ( relation_rng @ B ) )
=> ( ( relation_image @ B @ ( relation_inverse_image @ B @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(184,axiom,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_relat_1) ).
thf(953,plain,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(135,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( function_inverse @ A ) )
& ( function @ ( function_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_funct_1) ).
thf(794,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( function_inverse @ A ) )
& ( function @ ( function_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(161,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
<=> ( relation_of2 @ C @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_m2_relset_1) ).
thf(888,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation_of2_as_subset @ C @ A @ B )
=> ( relation_of2 @ C @ A @ B ) )
& ( ( relation_of2 @ C @ A @ B )
=> ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(272,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_union2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
| ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_xboole_0) ).
thf(1563,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_union2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
| ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
| ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
| ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
| ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_union2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(91,axiom,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( ordered_pair @ B @ C ) )
=> ! [B: $i] :
( ( B
= ( pair_second @ A ) )
<=> ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_mcart_1) ).
thf(657,plain,
! [A: $i] :
( ? [B: $i,C: $i] :
( A
= ( ordered_pair @ B @ C ) )
=> ! [B: $i] :
( ( ( B
= ( pair_second @ A ) )
=> ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = D ) ) )
& ( ! [C: $i,D: $i] :
( ( A
= ( ordered_pair @ C @ D ) )
=> ( B = D ) )
=> ( B
= ( pair_second @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(229,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) )
<=> ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t86_relat_1) ).
thf(1187,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) )
=> ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) ) )
& ( ( ( in @ A @ B )
& ( in @ A @ ( relation_dom @ C ) ) )
=> ( in @ A @ ( relation_dom @ ( relation_dom_restriction @ C @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(94,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_composition @ A @ B ) )
= ( relation_image @ B @ ( relation_rng @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t160_relat_1) ).
thf(674,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( relation_rng @ ( relation_composition @ A @ B ) )
= ( relation_image @ B @ ( relation_rng @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(79,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ ( singleton @ B ) )
<=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t39_zfmisc_1) ).
thf(618,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ ( singleton @ B ) )
=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) )
& ( ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) )
=> ( subset @ A @ ( singleton @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(329,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ( ordinal_subset @ A @ B )
<=> ( subset @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r1_ordinal1) ).
thf(2176,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ( ( ordinal_subset @ A @ B )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( ordinal_subset @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(282,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ ( singleton @ B ) )
= A )
<=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t65_zfmisc_1) ).
thf(1698,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ ( singleton @ B ) )
= A )
=> ~ ( in @ B @ A ) )
& ( ~ ( in @ B @ A )
=> ( ( set_difference @ A @ ( singleton @ B ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(337,axiom,
! [A: $i] :
( ( relation @ A )
<=> ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_relat_1) ).
thf(2253,plain,
! [A: $i] :
( ( ( relation @ A )
=> ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) ) )
& ( ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i,D: $i] :
( B
!= ( ordered_pair @ C @ D ) ) )
=> ( relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(288,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_field @ ( relation_restriction @ C @ B ) ) )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t19_wellord1) ).
thf(1747,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_field @ ( relation_restriction @ C @ B ) ) )
=> ( ( in @ A @ ( relation_field @ C ) )
& ( in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(148,axiom,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ A @ B ) )
& ( relation @ ( relation_composition @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc9_relat_1) ).
thf(843,plain,
! [A: $i,B: $i] :
( ( ( empty @ A )
& ( relation @ B ) )
=> ( ( empty @ ( relation_composition @ A @ B ) )
& ( relation @ ( relation_composition @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(127,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_field @ ( relation_restriction @ B @ A ) ) @ ( relation_field @ B ) )
& ( subset @ ( relation_field @ ( relation_restriction @ B @ A ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t20_wellord1) ).
thf(772,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( subset @ ( relation_field @ ( relation_restriction @ B @ A ) ) @ ( relation_field @ B ) )
& ( subset @ ( relation_field @ ( relation_restriction @ B @ A ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(107,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( ( well_ordering @ B )
& ( subset @ A @ ( relation_field @ B ) ) )
=> ( ( relation_field @ ( relation_restriction @ B @ A ) )
= A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t39_wellord1) ).
thf(711,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( ( well_ordering @ B )
& ( subset @ A @ ( relation_field @ B ) ) )
=> ( ( relation_field @ ( relation_restriction @ B @ A ) )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(65,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k4_relat_1) ).
thf(579,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(186,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(958,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(202,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t144_relat_1) ).
thf(1007,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(101,axiom,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(694,plain,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(234,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ E @ D ) @ C ) )
<=> ( ( relation_rng_as_subset @ A @ B @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_relset_1) ).
thf(1226,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ( ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ E @ D ) @ C ) )
=> ( ( relation_rng_as_subset @ A @ B @ C )
= B ) )
& ( ( ( relation_rng_as_subset @ A @ B @ C )
= B )
=> ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
~ ( in @ ( ordered_pair @ E @ D ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(54,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom @ ( relation_dom_restriction @ B @ A ) )
= ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t90_relat_1) ).
thf(545,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom @ ( relation_dom_restriction @ B @ A ) )
= ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(177,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_composition @ A @ B ) ) @ ( relation_rng @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t45_relat_1) ).
thf(936,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_composition @ A @ B ) ) @ ( relation_rng @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(341,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ ( unordered_pair @ A @ B ) @ C )
<=> ( ( in @ A @ C )
& ( in @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t38_zfmisc_1) ).
thf(2326,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ ( unordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ C )
& ( in @ B @ C ) ) )
& ( ( ( in @ A @ C )
& ( in @ B @ C ) )
=> ( subset @ ( unordered_pair @ A @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(296,axiom,
! [A: $i] :
( ? [B: $i] :
( ( ordinal @ B )
& ( in @ B @ A ) )
=> ? [B: $i] :
( ( ordinal @ B )
& ( in @ B @ A )
& ! [C: $i] :
( ( ordinal @ C )
=> ( ( in @ C @ A )
=> ( ordinal_subset @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_ordinal1__e8_6__wellord2) ).
thf(1865,plain,
! [A: $i] :
( ? [B: $i] :
( ( ordinal @ B )
& ( in @ B @ A ) )
=> ? [B: $i] :
( ( ordinal @ B )
& ( in @ B @ A )
& ! [C: $i] :
( ( ordinal @ C )
=> ( ( in @ C @ A )
=> ( ordinal_subset @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(138,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ ( singleton @ B ) )
<=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l4_zfmisc_1) ).
thf(804,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ ( singleton @ B ) )
=> ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) ) )
& ( ( ( A = empty_set )
| ( A
= ( singleton @ B ) ) )
=> ( subset @ A @ ( singleton @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(182,axiom,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_boole) ).
thf(947,plain,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(52,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_difference @ A @ B @ C )
= ( set_difference @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k6_subset_1) ).
thf(540,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_difference @ A @ B @ C )
= ( set_difference @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(144,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t24_wellord1) ).
thf(831,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(97,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(681,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(240,axiom,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( powerset @ C ) @ B ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t136_zfmisc_1) ).
thf(1282,plain,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ( in @ ( powerset @ C ) @ B ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(333,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ A @ ( relation_dom @ ( relation_composition @ C @ B ) ) )
<=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ ( apply @ C @ A ) @ ( relation_dom @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_funct_1) ).
thf(2229,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( in @ A @ ( relation_dom @ ( relation_composition @ C @ B ) ) )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ ( apply @ C @ A ) @ ( relation_dom @ B ) ) ) )
& ( ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ ( apply @ C @ A ) @ ( relation_dom @ B ) ) )
=> ( in @ A @ ( relation_dom @ ( relation_composition @ C @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(320,axiom,
! [A: $i,B: $i] :
( ( B
= ( powerset @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ( subset @ C @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_zfmisc_1) ).
thf(2084,plain,
! [A: $i,B: $i] :
( ( ( B
= ( powerset @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ( subset @ C @ A ) )
& ( ( subset @ C @ A )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ( subset @ C @ A ) )
& ( ( subset @ C @ A )
=> ( in @ C @ B ) ) )
=> ( B
= ( powerset @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_relset_1) ).
thf(419,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(222,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ! [B: $i,C: $i] :
~ ( in @ ( ordered_pair @ B @ C ) @ A )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t56_relat_1) ).
thf(1122,plain,
! [A: $i] :
( ( relation @ A )
=> ( ! [B: $i,C: $i] :
~ ( in @ ( ordered_pair @ B @ C ) @ A )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(194,axiom,
! [A: $i,B: $i] :
( ( ( pair_first @ ( ordered_pair @ A @ B ) )
= A )
& ( ( pair_second @ ( ordered_pair @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_mcart_1) ).
thf(979,plain,
! [A: $i,B: $i] :
( ( ( pair_first @ ( ordered_pair @ A @ B ) )
= A )
& ( ( pair_second @ ( ordered_pair @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(75,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t45_xboole_1) ).
thf(604,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(273,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
<=> ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_tarski) ).
thf(1589,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
=> ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) ) )
& ( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(198,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_founded_relation @ B )
=> ( well_founded_relation @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t31_wellord1) ).
thf(996,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_founded_relation @ B )
=> ( well_founded_relation @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( ( set_difference @ A @ B )
= empty_set )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_xboole_1) ).
thf(362,plain,
! [A: $i,B: $i] :
( ( ( ( set_difference @ A @ B )
= empty_set )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( ( set_difference @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(61,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_wellord1) ).
thf(567,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(88,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom_restriction @ B @ A )
= ( relation_composition @ ( identity_relation @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t94_relat_1) ).
thf(649,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_dom_restriction @ B @ A )
= ( relation_composition @ ( identity_relation @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(212,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( B
= ( identity_relation @ A ) )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( ( in @ C @ A )
& ( C = D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d10_relat_1) ).
thf(1043,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( ( B
= ( identity_relation @ A ) )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( ( in @ C @ A )
& ( C = D ) ) )
& ( ( ( in @ C @ A )
& ( C = D ) )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( ( in @ C @ A )
& ( C = D ) ) )
& ( ( ( in @ C @ A )
& ( C = D ) )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
=> ( B
= ( identity_relation @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(252,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( B = C )
& ( ordinal @ C )
& ( B = D )
& ( ordinal @ D ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ D @ A )
& ( D = C )
& ( ordinal @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e6_22__wellord2__1) ).
thf(1376,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( B = C )
& ( ordinal @ C )
& ( B = D )
& ( ordinal @ D ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ D @ A )
& ( D = C )
& ( ordinal @ C ) ) )
& ( ? [D: $i] :
( ( in @ D @ A )
& ( D = C )
& ( ordinal @ C ) )
=> ( in @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(257,axiom,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_ordinal1) ).
thf(1439,plain,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(34,axiom,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(477,plain,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(217,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_ordinal1) ).
thf(1080,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(50,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_subset_1) ).
thf(535,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_difference @ A @ B @ C ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(110,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_ordering @ B )
=> ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t32_wellord1) ).
thf(726,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_ordering @ B )
=> ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(226,axiom,
! [A: $i,B: $i] :
( ( ( A != empty_set )
=> ( ( B
= ( set_meet @ A ) )
<=> ! [C: $i] :
( ( in @ C @ B )
<=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) ) ) )
& ( ( A = empty_set )
=> ( ( B
= ( set_meet @ A ) )
<=> ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_setfam_1) ).
thf(1149,plain,
! [A: $i,B: $i] :
( ( ( A != empty_set )
=> ( ( ( B
= ( set_meet @ A ) )
=> ! [C: $i] :
( ( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) )
& ( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) )
=> ( in @ C @ B ) ) ) )
& ( ! [C: $i] :
( ( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) ) )
& ( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ C @ D ) )
=> ( in @ C @ B ) ) )
=> ( B
= ( set_meet @ A ) ) ) ) )
& ( ( A = empty_set )
=> ( ( ( B
= ( set_meet @ A ) )
=> ( B = empty_set ) )
& ( ( B = empty_set )
=> ( B
= ( set_meet @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(170,axiom,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_enumset1) ).
thf(915,plain,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(92,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(668,plain,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(30,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ~ ( ( A != empty_set )
& ( subset @ A @ ( relation_rng @ B ) )
& ( ( relation_inverse_image @ B @ A )
= empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t174_relat_1) ).
thf(462,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ~ ( ( A != empty_set )
& ( subset @ A @ ( relation_rng @ B ) )
& ( ( relation_inverse_image @ B @ A )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(295,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ~ ( ~ ( in @ A @ B )
& ( A != B )
& ~ ( in @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t24_ordinal1) ).
thf(1862,plain,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ~ ( ~ ( in @ A @ B )
& ( A != B )
& ~ ( in @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(36,axiom,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_relat_1) ).
thf(483,plain,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(103,axiom,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(700,plain,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(46,axiom,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_tarski) ).
thf(520,plain,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(305,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( is_well_founded_in @ A @ B )
<=> ! [C: $i] :
~ ( ( subset @ C @ B )
& ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ( disjoint @ ( fiber @ A @ D ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_wellord1) ).
thf(1926,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( is_well_founded_in @ A @ B )
=> ! [C: $i] :
~ ( ( subset @ C @ B )
& ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ( disjoint @ ( fiber @ A @ D ) @ C ) ) ) )
& ( ! [C: $i] :
~ ( ( subset @ C @ B )
& ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ( disjoint @ ( fiber @ A @ D ) @ C ) ) )
=> ( is_well_founded_in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(153,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ( subset @ ( relation_dom @ C ) @ A )
& ( subset @ ( relation_rng @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_relset_1) ).
thf(868,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ( subset @ ( relation_dom @ C ) @ A )
& ( subset @ ( relation_rng @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(159,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ ( relation_dom @ B ) )
=> ( subset @ A @ ( relation_inverse_image @ B @ ( relation_image @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t146_funct_1) ).
thf(883,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ ( relation_dom @ B ) )
=> ( subset @ A @ ( relation_inverse_image @ B @ ( relation_image @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(205,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( relation_dom_restriction @ A @ B ) )
& ( function @ ( relation_dom_restriction @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_funct_1) ).
thf(1015,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( relation_dom_restriction @ A @ B ) )
& ( function @ ( relation_dom_restriction @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(195,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( function_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( function_inverse @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t55_funct_1) ).
thf(986,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( function_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( function_inverse @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(287,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_rng_restriction @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d12_relat_1) ).
thf(1726,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_rng_restriction @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) )
& ( ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) ) )
& ( ( ( in @ E @ A )
& ( in @ ( ordered_pair @ D @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_rng_restriction @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(150,axiom,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(851,plain,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(173,axiom,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_relat_1) ).
thf(922,plain,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(233,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( B
= ( identity_relation @ A ) )
<=> ( ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t34_funct_1) ).
thf(1213,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( ( B
= ( identity_relation @ A ) )
=> ( ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= C ) ) ) )
& ( ( ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= C ) ) )
=> ( B
= ( identity_relation @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(125,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_xboole_1) ).
thf(768,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(142,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( well_ordering @ A )
& ( relation_isomorphism @ A @ B @ C ) )
=> ( well_ordering @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t54_wellord1) ).
thf(827,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( well_ordering @ A )
& ( relation_isomorphism @ A @ B @ C ) )
=> ( well_ordering @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(211,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
<=> ! [B: $i,C: $i] :
( ( ( in @ B @ ( relation_dom @ A ) )
& ( in @ C @ ( relation_dom @ A ) )
& ( ( apply @ A @ B )
= ( apply @ A @ C ) ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_funct_1) ).
thf(1034,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( ( one_to_one @ A )
=> ! [B: $i,C: $i] :
( ( ( in @ B @ ( relation_dom @ A ) )
& ( in @ C @ ( relation_dom @ A ) )
& ( ( apply @ A @ B )
= ( apply @ A @ C ) ) )
=> ( B = C ) ) )
& ( ! [B: $i,C: $i] :
( ( ( in @ B @ ( relation_dom @ A ) )
& ( in @ C @ ( relation_dom @ A ) )
& ( ( apply @ A @ B )
= ( apply @ A @ C ) ) )
=> ( B = C ) )
=> ( one_to_one @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(298,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_ordinal1) ).
thf(1873,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(55,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc11_relat_1) ).
thf(548,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(1,conjecture,
! [A: $i] :
? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ( ( in @ C @ A )
& ( ordinal @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e6_22__wellord2) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ( ( in @ C @ A )
& ( ordinal @ C ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(342,plain,
~ ! [A: $i] :
? [B: $i] :
! [C: $i] :
( ( ( in @ C @ B )
=> ( ( in @ C @ A )
& ( ordinal @ C ) ) )
& ( ( ( in @ C @ A )
& ( ordinal @ C ) )
=> ( in @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(119,axiom,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_relat_1) ).
thf(754,plain,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(57,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( union_of_subsets @ A @ B )
= ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k5_setfam_1) ).
thf(553,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( union_of_subsets @ A @ B )
= ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(227,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_subset) ).
thf(1181,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(232,axiom,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( in @ C @ ( set_intersection2 @ A @ B ) ) )
& ~ ( ? [C: $i] : ( in @ C @ ( set_intersection2 @ A @ B ) )
& ( disjoint @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_xboole_0) ).
thf(1208,plain,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( in @ C @ ( set_intersection2 @ A @ B ) ) )
& ~ ( ? [C: $i] : ( in @ C @ ( set_intersection2 @ A @ B ) )
& ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(68,axiom,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_wellord2) ).
thf(589,plain,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(175,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( relation_isomorphism @ A @ B @ C )
=> ( ( ( reflexive @ A )
=> ( reflexive @ B ) )
& ( ( transitive @ A )
=> ( transitive @ B ) )
& ( ( connected @ A )
=> ( connected @ B ) )
& ( ( antisymmetric @ A )
=> ( antisymmetric @ B ) )
& ( ( well_founded_relation @ A )
=> ( well_founded_relation @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t53_wellord1) ).
thf(927,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( relation_isomorphism @ A @ B @ C )
=> ( ( ( reflexive @ A )
=> ( reflexive @ B ) )
& ( ( transitive @ A )
=> ( transitive @ B ) )
& ( ( connected @ A )
=> ( connected @ B ) )
& ( ( antisymmetric @ A )
=> ( antisymmetric @ B ) )
& ( ( well_founded_relation @ A )
=> ( well_founded_relation @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(10,axiom,
? [A: $i] : ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(374,plain,
? [A: $i] : ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(290,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( C
= ( relation_inverse_image @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( relation_dom @ A ) )
& ( in @ ( apply @ A @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d13_funct_1) ).
thf(1770,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( ( C
= ( relation_inverse_image @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( relation_dom @ A ) )
& ( in @ ( apply @ A @ D ) @ B ) ) )
& ( ( ( in @ D @ ( relation_dom @ A ) )
& ( in @ ( apply @ A @ D ) @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( relation_dom @ A ) )
& ( in @ ( apply @ A @ D ) @ B ) ) )
& ( ( ( in @ D @ ( relation_dom @ A ) )
& ( in @ ( apply @ A @ D ) @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( relation_inverse_image @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(85,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_zfmisc_1) ).
thf(640,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(208,axiom,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_boole) ).
thf(1025,plain,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(215,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
( ( relation @ D )
& ! [E: $i,F: $i] :
( ( in @ ( ordered_pair @ E @ F ) @ D )
<=> ( ( in @ E @ A )
& ( in @ F @ A )
& ( in @ ( ordered_pair @ ( apply @ C @ E ) @ ( apply @ C @ F ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_relat_1__e6_21__wellord2) ).
thf(1070,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
( ( relation @ D )
& ! [E: $i,F: $i] :
( ( ( in @ ( ordered_pair @ E @ F ) @ D )
=> ( ( in @ E @ A )
& ( in @ F @ A )
& ( in @ ( ordered_pair @ ( apply @ C @ E ) @ ( apply @ C @ F ) ) @ B ) ) )
& ( ( ( in @ E @ A )
& ( in @ F @ A )
& ( in @ ( ordered_pair @ ( apply @ C @ E ) @ ( apply @ C @ F ) ) @ B ) )
=> ( in @ ( ordered_pair @ E @ F ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(326,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_difference @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_xboole_0) ).
thf(2143,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_difference @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ~ ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ~ ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ~ ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).
thf(221,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( relation_isomorphism @ A @ B @ C )
<=> ( ( ( relation_dom @ C )
= ( relation_field @ A ) )
& ( ( relation_rng @ C )
= ( relation_field @ B ) )
& ( one_to_one @ C )
& ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ A )
<=> ( ( in @ D @ ( relation_field @ A ) )
& ( in @ E @ ( relation_field @ A ) )
& ( in @ ( ordered_pair @ ( apply @ C @ D ) @ ( apply @ C @ E ) ) @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d7_wellord1) ).
thf(1098,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( ( relation_isomorphism @ A @ B @ C )
=> ( ( ( relation_dom @ C )
= ( relation_field @ A ) )
& ( ( relation_rng @ C )
= ( relation_field @ B ) )
& ( one_to_one @ C )
& ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ A )
=> ( ( in @ D @ ( relation_field @ A ) )
& ( in @ E @ ( relation_field @ A ) )
& ( in @ ( ordered_pair @ ( apply @ C @ D ) @ ( apply @ C @ E ) ) @ B ) ) )
& ( ( ( in @ D @ ( relation_field @ A ) )
& ( in @ E @ ( relation_field @ A ) )
& ( in @ ( ordered_pair @ ( apply @ C @ D ) @ ( apply @ C @ E ) ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ A ) ) ) ) )
& ( ( ( ( relation_dom @ C )
= ( relation_field @ A ) )
& ( ( relation_rng @ C )
= ( relation_field @ B ) )
& ( one_to_one @ C )
& ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ A )
=> ( ( in @ D @ ( relation_field @ A ) )
& ( in @ E @ ( relation_field @ A ) )
& ( in @ ( ordered_pair @ ( apply @ C @ D ) @ ( apply @ C @ E ) ) @ B ) ) )
& ( ( ( in @ D @ ( relation_field @ A ) )
& ( in @ E @ ( relation_field @ A ) )
& ( in @ ( ordered_pair @ ( apply @ C @ D ) @ ( apply @ C @ E ) ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ A ) ) ) )
=> ( relation_isomorphism @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(197,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).
thf(993,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(134,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k3_subset_1) ).
thf(791,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( subset_complement @ A @ ( subset_complement @ A @ B ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(67,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_orders @ B @ A )
=> ( ( ( relation_field @ ( relation_restriction @ B @ A ) )
= A )
& ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_wellord2) ).
thf(585,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_orders @ B @ A )
=> ( ( ( relation_field @ ( relation_restriction @ B @ A ) )
= A )
& ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(109,axiom,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
<=> ( ( subset @ A @ B )
& ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_xboole_0) ).
thf(717,plain,
! [A: $i,B: $i] :
( ( ( proper_subset @ A @ B )
=> ( ( subset @ A @ B )
& ( A != B ) ) )
& ( ( ( subset @ A @ B )
& ( A != B ) )
=> ( proper_subset @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(235,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_wellord2) ).
thf(1235,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(278,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_subset) ).
thf(1679,plain,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(167,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( meet_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t48_setfam_1) ).
thf(907,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_difference @ A @ ( cast_to_subset @ A ) @ ( meet_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(39,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( one_to_one @ ( function_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t62_funct_1) ).
thf(502,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ( one_to_one @ ( function_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(297,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) )
<=> ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t115_relat_1) ).
thf(1869,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) )
=> ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) ) )
& ( ( ( in @ A @ B )
& ( in @ A @ ( relation_rng @ C ) ) )
=> ( in @ A @ ( relation_rng @ ( relation_rng_restriction @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_intersection2 @ A @ B )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d7_xboole_0) ).
thf(470,plain,
! [A: $i,B: $i] :
( ( ( disjoint @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= empty_set ) )
& ( ( ( set_intersection2 @ A @ B )
= empty_set )
=> ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(64,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_restriction @ B @ A )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t18_wellord1) ).
thf(576,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_restriction @ B @ A )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(115,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_subset_1) ).
thf(744,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(207,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( subset @ ( relation_image @ B @ ( relation_inverse_image @ B @ A ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t145_funct_1) ).
thf(1023,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( subset @ ( relation_image @ B @ ( relation_inverse_image @ B @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(73,axiom,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_wellord2) ).
thf(600,plain,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(267,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( unordered_pair @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( D = A )
| ( D = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_tarski) ).
thf(1504,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( unordered_pair @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D = A )
| ( D = B ) ) )
& ( ( ( D = A )
| ( D = B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D = A )
| ( D = B ) ) )
& ( ( ( D = A )
| ( D = B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( unordered_pair @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(314,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( C
= ( relation_composition @ A @ B ) )
<=> ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_relat_1) ).
thf(2030,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( relation @ C )
=> ( ( ( C
= ( relation_composition @ A @ B ) )
=> ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) )
& ( ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) )
& ( ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) ) )
& ( ? [F: $i] :
( ( in @ ( ordered_pair @ D @ F ) @ A )
& ( in @ ( ordered_pair @ F @ E ) @ B ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) )
=> ( C
= ( relation_composition @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(63,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
| ( ( relation_rng @ A )
= empty_set ) )
=> ( A = empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t64_relat_1) ).
thf(571,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
| ( ( relation_rng @ A )
= empty_set ) )
=> ( A = empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(71,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( reflexive @ A )
<=> ( is_reflexive_in @ A @ ( relation_field @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d9_relat_2) ).
thf(595,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( reflexive @ A )
=> ( is_reflexive_in @ A @ ( relation_field @ A ) ) )
& ( ( is_reflexive_in @ A @ ( relation_field @ A ) )
=> ( reflexive @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(77,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(612,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(156,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( relation_isomorphism @ A @ B @ C )
=> ( relation_isomorphism @ B @ A @ ( function_inverse @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t49_wellord1) ).
thf(876,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( relation_isomorphism @ A @ B @ C )
=> ( relation_isomorphism @ B @ A @ ( function_inverse @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(196,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k7_setfam_1) ).
thf(991,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( element @ ( complements_of_subsets @ A @ B ) @ ( powerset @ ( powerset @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(322,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(2113,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).
thf(104,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) )
=> ( ( relation @ ( relation_inverse @ A ) )
& ( function @ ( relation_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_funct_1) ).
thf(703,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) )
=> ( ( relation @ ( relation_inverse @ A ) )
& ( function @ ( relation_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(31,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation_restriction @ A @ B )
= ( set_intersection2 @ A @ ( cartesian_product2 @ B @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_wellord1) ).
thf(465,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation_restriction @ A @ B )
= ( set_intersection2 @ A @ ( cartesian_product2 @ B @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(262,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( well_founded_relation @ A )
<=> ! [B: $i] :
~ ( ( subset @ B @ ( relation_field @ A ) )
& ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ( disjoint @ ( fiber @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_wellord1) ).
thf(1459,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( well_founded_relation @ A )
=> ! [B: $i] :
~ ( ( subset @ B @ ( relation_field @ A ) )
& ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ( disjoint @ ( fiber @ A @ C ) @ B ) ) ) )
& ( ! [B: $i] :
~ ( ( subset @ B @ ( relation_field @ A ) )
& ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ( disjoint @ ( fiber @ A @ C ) @ B ) ) )
=> ( well_founded_relation @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(58,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_dom @ A )
= empty_set )
<=> ( ( relation_rng @ A )
= empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t65_relat_1) ).
thf(556,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( ( relation_dom @ A )
= empty_set )
=> ( ( relation_rng @ A )
= empty_set ) )
& ( ( ( relation_rng @ A )
= empty_set )
=> ( ( relation_dom @ A )
= empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(93,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_dom @ B ) )
& ( subset @ ( relation_rng @ A ) @ ( relation_rng @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_relat_1) ).
thf(671,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
=> ( ( subset @ ( relation_dom @ A ) @ ( relation_dom @ B ) )
& ( subset @ ( relation_rng @ A ) @ ( relation_rng @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(193,axiom,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_ordinal1) ).
thf(976,plain,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(331,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D
= ( unordered_triple @ A @ B @ C ) )
<=> ! [E: $i] :
( ( in @ E @ D )
<=> ~ ( ( E != A )
& ( E != B )
& ( E != C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_enumset1) ).
thf(2194,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D
= ( unordered_triple @ A @ B @ C ) )
=> ! [E: $i] :
( ( ( in @ E @ D )
=> ~ ( ( E != A )
& ( E != B )
& ( E != C ) ) )
& ( ~ ( ( E != A )
& ( E != B )
& ( E != C ) )
=> ( in @ E @ D ) ) ) )
& ( ! [E: $i] :
( ( ( in @ E @ D )
=> ~ ( ( E != A )
& ( E != B )
& ( E != C ) ) )
& ( ~ ( ( E != A )
& ( E != B )
& ( E != C ) )
=> ( in @ E @ D ) ) )
=> ( D
= ( unordered_triple @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(121,axiom,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(759,plain,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(223,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( transitive @ A )
<=> ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ D ) @ A ) )
=> ( in @ ( ordered_pair @ B @ D ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_wellord1) ).
thf(1126,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( transitive @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ D ) @ A ) )
=> ( in @ ( ordered_pair @ B @ D ) @ A ) ) )
& ( ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ D ) @ A ) )
=> ( in @ ( ordered_pair @ B @ D ) @ A ) )
=> ( transitive @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(303,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_inverse_image @ C @ B ) )
<=> ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t166_relat_1) ).
thf(1918,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_inverse_image @ C @ B ) )
=> ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( relation_rng @ C ) )
& ( in @ ( ordered_pair @ A @ D ) @ C )
& ( in @ D @ B ) )
=> ( in @ A @ ( relation_inverse_image @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(26,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_relat_1) ).
thf(450,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(218,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( subset @ A @ B )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_relat_1) ).
thf(1084,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( subset @ A @ B )
=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
& ( ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
=> ( subset @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(332,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l71_subset_1) ).
thf(2226,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( element @ A @ ( powerset @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).
thf(260,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ B @ ( relation_rng @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t20_relat_1) ).
thf(1453,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ ( ordered_pair @ A @ B ) @ C )
=> ( ( in @ A @ ( relation_dom @ C ) )
& ( in @ B @ ( relation_rng @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(328,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(2174,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(89,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ A @ C ) )
=> ( subset @ A @ ( set_intersection2 @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t19_xboole_1) ).
thf(652,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ A @ C ) )
=> ( subset @ A @ ( set_intersection2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(13,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(390,plain,
! [A: $i,B: $i] :
( ( ( element @ A @ ( powerset @ B ) )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( element @ A @ ( powerset @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(82,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t117_relat_1) ).
thf(632,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(112,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ordered_pair @ A @ B )
= ( ordered_pair @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t33_zfmisc_1) ).
thf(731,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ordered_pair @ A @ B )
= ( ordered_pair @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(130,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ B ) )
=> ( subset @ ( set_union2 @ A @ C ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_xboole_1) ).
thf(781,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ C @ B ) )
=> ( subset @ ( set_union2 @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(338,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ! [B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( B
= ( function_inverse @ A ) )
<=> ( ( ( relation_dom @ B )
= ( relation_rng @ A ) )
& ! [C: $i,D: $i] :
( ( ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) )
=> ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) )
& ( ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) )
=> ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t54_funct_1) ).
thf(2262,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( one_to_one @ A )
=> ! [B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( ( B
= ( function_inverse @ A ) )
=> ( ( ( relation_dom @ B )
= ( relation_rng @ A ) )
& ! [C: $i,D: $i] :
( ( ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) )
=> ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) )
& ( ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) )
=> ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) ) ) ) ) )
& ( ( ( ( relation_dom @ B )
= ( relation_rng @ A ) )
& ! [C: $i,D: $i] :
( ( ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) )
=> ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) ) )
& ( ( ( in @ D @ ( relation_dom @ A ) )
& ( C
= ( apply @ A @ D ) ) )
=> ( ( in @ C @ ( relation_rng @ A ) )
& ( D
= ( apply @ B @ C ) ) ) ) ) )
=> ( B
= ( function_inverse @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).
thf(187,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(961,plain,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(286,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(1721,plain,
! [A: $i,B: $i] :
( ( ( subset @ ( singleton @ A ) @ B )
=> ( in @ A @ B ) )
& ( ( in @ A @ B )
=> ( subset @ ( singleton @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(188,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_relat_1) ).
thf(963,plain,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(45,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(517,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(163,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_relat_1) ).
thf(896,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(117,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_image @ B @ A )
= ( relation_image @ B @ ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t145_relat_1) ).
thf(748,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( relation_image @ B @ A )
= ( relation_image @ B @ ( set_intersection2 @ ( relation_dom @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(281,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ( ( in @ E @ ( cartesian_product2 @ A @ A ) )
& ? [F: $i,G: $i] :
( ( E
= ( ordered_pair @ F @ G ) )
& ( in @ ( ordered_pair @ ( apply @ C @ F ) @ ( apply @ C @ G ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e6_21__wellord2__1) ).
thf(1689,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation @ B )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ( ( in @ E @ ( cartesian_product2 @ A @ A ) )
& ? [F: $i,G: $i] :
( ( E
= ( ordered_pair @ F @ G ) )
& ( in @ ( ordered_pair @ ( apply @ C @ F ) @ ( apply @ C @ G ) ) @ B ) ) ) )
& ( ( ( in @ E @ ( cartesian_product2 @ A @ A ) )
& ? [F: $i,G: $i] :
( ( E
= ( ordered_pair @ F @ G ) )
& ( in @ ( ordered_pair @ ( apply @ C @ F ) @ ( apply @ C @ G ) ) @ B ) ) )
=> ( in @ E @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(231,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( is_antisymmetric_in @ A @ B )
<=> ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ C ) @ A ) )
=> ( C = D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_relat_2) ).
thf(1193,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( is_antisymmetric_in @ A @ B )
=> ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ C ) @ A ) )
=> ( C = D ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ C ) @ A ) )
=> ( C = D ) )
=> ( is_antisymmetric_in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(123,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_wellord1) ).
thf(763,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(12,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( well_ordering @ A )
<=> ( ( reflexive @ A )
& ( transitive @ A )
& ( antisymmetric @ A )
& ( connected @ A )
& ( well_founded_relation @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_wellord1) ).
thf(383,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( well_ordering @ A )
=> ( ( reflexive @ A )
& ( transitive @ A )
& ( antisymmetric @ A )
& ( connected @ A )
& ( well_founded_relation @ A ) ) )
& ( ( ( reflexive @ A )
& ( transitive @ A )
& ( antisymmetric @ A )
& ( connected @ A )
& ( well_founded_relation @ A ) )
=> ( well_ordering @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(98,axiom,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t48_xboole_1) ).
thf(684,plain,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(225,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( A = B )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ A )
<=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_relat_1) ).
thf(1133,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( A = B )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
& ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ C @ D ) @ A ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) )
& ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ C @ D ) @ A ) ) )
=> ( A = B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(25,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ( ( disjoint @ B @ C )
<=> ( subset @ B @ ( subset_complement @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t43_subset_1) ).
thf(447,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ( ( ( disjoint @ B @ C )
=> ( subset @ B @ ( subset_complement @ A @ C ) ) )
& ( ( subset @ B @ ( subset_complement @ A @ C ) )
=> ( disjoint @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(183,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( relation @ ( relation_rng_restriction @ A @ B ) )
& ( function @ ( relation_rng_restriction @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_funct_1) ).
thf(950,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( relation @ ( relation_rng_restriction @ A @ B ) )
& ( function @ ( relation_rng_restriction @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(334,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ A @ ( relation_dom @ B ) )
=> ( ( apply @ ( relation_composition @ B @ C ) @ A )
= ( apply @ C @ ( apply @ B @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_funct_1) ).
thf(2233,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ! [C: $i] :
( ( ( relation @ C )
& ( function @ C ) )
=> ( ( in @ A @ ( relation_dom @ B ) )
=> ( ( apply @ ( relation_composition @ B @ C ) @ A )
= ( apply @ C @ ( apply @ B @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(140,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_difference @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(818,plain,
! [A: $i,B: $i] :
( ( ( disjoint @ A @ B )
=> ( ( set_difference @ A @ B )
= A ) )
& ( ( ( set_difference @ A @ B )
= A )
=> ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(152,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(865,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(145,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( relation_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( relation_inverse @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_relat_1) ).
thf(833,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( relation_rng @ A )
= ( relation_dom @ ( relation_inverse @ A ) ) )
& ( ( relation_dom @ A )
= ( relation_rng @ ( relation_inverse @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(18,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( well_orders @ A @ B )
<=> ( ( is_reflexive_in @ A @ B )
& ( is_transitive_in @ A @ B )
& ( is_antisymmetric_in @ A @ B )
& ( is_connected_in @ A @ B )
& ( is_well_founded_in @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_wellord1) ).
thf(410,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( well_orders @ A @ B )
=> ( ( is_reflexive_in @ A @ B )
& ( is_transitive_in @ A @ B )
& ( is_antisymmetric_in @ A @ B )
& ( is_connected_in @ A @ B )
& ( is_well_founded_in @ A @ B ) ) )
& ( ( ( is_reflexive_in @ A @ B )
& ( is_transitive_in @ A @ B )
& ( is_antisymmetric_in @ A @ B )
& ( is_connected_in @ A @ B )
& ( is_well_founded_in @ A @ B ) )
=> ( well_orders @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(189,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_rng @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t118_relat_1) ).
thf(967,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(60,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( well_founded_relation @ A )
<=> ( is_well_founded_in @ A @ ( relation_field @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_wellord1) ).
thf(564,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( well_founded_relation @ A )
=> ( is_well_founded_in @ A @ ( relation_field @ A ) ) )
& ( ( is_well_founded_in @ A @ ( relation_field @ A ) )
=> ( well_founded_relation @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(256,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
<=> ( ordinal_subset @ ( succ @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t33_ordinal1) ).
thf(1436,plain,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( ( in @ A @ B )
=> ( ordinal_subset @ ( succ @ A ) @ B ) )
& ( ( ordinal_subset @ ( succ @ A ) @ B )
=> ( in @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(7,axiom,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(358,plain,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(264,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ A ) ) )
=> ( ( C
= ( complements_of_subsets @ A @ B ) )
<=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( in @ D @ C )
<=> ( in @ ( subset_complement @ A @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_setfam_1) ).
thf(1473,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ A ) ) )
=> ( ( ( C
= ( complements_of_subsets @ A @ B ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( in @ D @ C )
=> ( in @ ( subset_complement @ A @ D ) @ B ) )
& ( ( in @ ( subset_complement @ A @ D ) @ B )
=> ( in @ D @ C ) ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( in @ D @ C )
=> ( in @ ( subset_complement @ A @ D ) @ B ) )
& ( ( in @ ( subset_complement @ A @ D ) @ B )
=> ( in @ D @ C ) ) ) )
=> ( C
= ( complements_of_subsets @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(185,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t9_zfmisc_1) ).
thf(955,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(171,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_dom @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l29_wellord1) ).
thf(918,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom @ ( relation_rng_restriction @ A @ B ) ) @ ( relation_dom @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(310,axiom,
! [A: $i,B: $i,C: $i] :
( ( C
= ( set_intersection2 @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ A )
& ( in @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_xboole_0) ).
thf(1989,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C
= ( set_intersection2 @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ( in @ D @ B ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ A )
& ( in @ D @ B ) ) )
& ( ( ( in @ D @ A )
& ( in @ D @ B ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(249,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_ordinal1) ).
thf(1363,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(243,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
<=> ( in @ C @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_tarski) ).
thf(1314,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( ( in @ C @ A )
=> ( in @ C @ B ) )
& ( ( in @ C @ B )
=> ( in @ C @ A ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(284,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_ordinal1) ).
thf(1710,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(15,axiom,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(400,plain,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(312,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_subset_1) ).
thf(2021,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(317,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( ( in @ B @ ( relation_dom @ A ) )
=> ( ( C
= ( apply @ A @ B ) )
<=> ( in @ ( ordered_pair @ B @ C ) @ A ) ) )
& ( ~ ( in @ B @ ( relation_dom @ A ) )
=> ( ( C
= ( apply @ A @ B ) )
<=> ( C = empty_set ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_funct_1) ).
thf(2063,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( ( in @ B @ ( relation_dom @ A ) )
=> ( ( ( C
= ( apply @ A @ B ) )
=> ( in @ ( ordered_pair @ B @ C ) @ A ) )
& ( ( in @ ( ordered_pair @ B @ C ) @ A )
=> ( C
= ( apply @ A @ B ) ) ) ) )
& ( ~ ( in @ B @ ( relation_dom @ A ) )
=> ( ( ( C
= ( apply @ A @ B ) )
=> ( C = empty_set ) )
& ( ( C = empty_set )
=> ( C
= ( apply @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(14,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
<=> ( subset @ C @ ( cartesian_product2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_relset_1) ).
thf(395,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation_of2 @ C @ A @ B )
=> ( subset @ C @ ( cartesian_product2 @ A @ B ) ) )
& ( ( subset @ C @ ( cartesian_product2 @ A @ B ) )
=> ( relation_of2 @ C @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(111,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ~ ( ( well_ordering @ B )
& ( equipotent @ A @ ( relation_field @ B ) )
& ! [C: $i] :
( ( relation @ C )
=> ~ ( well_orders @ C @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l30_wellord2) ).
thf(728,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ~ ( ( well_ordering @ B )
& ( equipotent @ A @ ( relation_field @ B ) )
& ! [C: $i] :
( ( relation @ C )
=> ~ ( well_orders @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(22,axiom,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(427,plain,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(49,axiom,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t71_relat_1) ).
thf(528,plain,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(86,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(644,plain,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(29,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc8_relat_1) ).
thf(459,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(154,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(871,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(266,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ ( union @ A ) )
& ( epsilon_connected @ ( union @ A ) )
& ( ordinal @ ( union @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_ordinal1) ).
thf(1500,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ ( union @ A ) )
& ( epsilon_connected @ ( union @ A ) )
& ( ordinal @ ( union @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(304,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_ordinal1) ).
thf(1923,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(255,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_ordinal1) ).
thf(1431,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(201,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_relat_1) ).
thf(1004,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(302,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( B
= ( relation_inverse @ A ) )
<=> ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( in @ ( ordered_pair @ D @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d7_relat_1) ).
thf(1902,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( relation @ B )
=> ( ( ( B
= ( relation_inverse @ A ) )
=> ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) )
& ( ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( in @ ( ordered_pair @ D @ C ) @ A ) )
& ( ( in @ ( ordered_pair @ D @ C ) @ A )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) )
=> ( B
= ( relation_inverse @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(100,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(691,plain,
! [A: $i] :
( ( set_intersection2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(9,axiom,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ C ) )
& ( subset @ ( cartesian_product2 @ C @ A ) @ ( cartesian_product2 @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t118_zfmisc_1) ).
thf(369,plain,
! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ C ) )
& ( subset @ ( cartesian_product2 @ C @ A ) @ ( cartesian_product2 @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(76,axiom,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_funct_1) ).
thf(607,plain,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(324,axiom,
! [A: $i] :
( ( epsilon_transitive @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( proper_subset @ A @ B )
=> ( in @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_ordinal1) ).
thf(2138,plain,
! [A: $i] :
( ( epsilon_transitive @ A )
=> ! [B: $i] :
( ( ordinal @ B )
=> ( ( proper_subset @ A @ B )
=> ( in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).
thf(5,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_relset_1) ).
thf(354,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(203,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( relation_dom_restriction @ ( relation_rng_restriction @ A @ C ) @ B )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t140_relat_1) ).
thf(1009,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( relation_dom_restriction @ ( relation_rng_restriction @ A @ C ) @ B )
= ( relation_rng_restriction @ A @ ( relation_dom_restriction @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(53,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( element @ ( relation_dom_as_subset @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_relset_1) ).
thf(543,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( element @ ( relation_dom_as_subset @ A @ B @ C ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(316,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( in @ A @ ( relation_restriction @ C @ B ) )
<=> ( ( in @ A @ C )
& ( in @ A @ ( cartesian_product2 @ B @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t16_wellord1) ).
thf(2059,plain,
! [A: $i,B: $i,C: $i] :
( ( relation @ C )
=> ( ( ( in @ A @ ( relation_restriction @ C @ B ) )
=> ( ( in @ A @ C )
& ( in @ A @ ( cartesian_product2 @ B @ B ) ) ) )
& ( ( ( in @ A @ C )
& ( in @ A @ ( cartesian_product2 @ B @ B ) ) )
=> ( in @ A @ ( relation_restriction @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(106,axiom,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t40_xboole_1) ).
thf(708,plain,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(191,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_relat_1) ).
thf(972,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(277,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( C
= ( fiber @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ C )
<=> ( ( D != B )
& ( in @ ( ordered_pair @ D @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_wellord1) ).
thf(1653,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i,C: $i] :
( ( ( C
= ( fiber @ A @ B ) )
=> ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D != B )
& ( in @ ( ordered_pair @ D @ B ) @ A ) ) )
& ( ( ( D != B )
& ( in @ ( ordered_pair @ D @ B ) @ A ) )
=> ( in @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( ( in @ D @ C )
=> ( ( D != B )
& ( in @ ( ordered_pair @ D @ B ) @ A ) ) )
& ( ( ( D != B )
& ( in @ ( ordered_pair @ D @ B ) @ A ) )
=> ( in @ D @ C ) ) )
=> ( C
= ( fiber @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(179,axiom,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_xboole_0) ).
thf(940,plain,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(279,axiom,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ( in @ C @ B ) ) )
& ~ ( ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
& ( disjoint @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_0) ).
thf(1681,plain,
! [A: $i,B: $i] :
( ~ ( ~ ( disjoint @ A @ B )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ( in @ C @ B ) ) )
& ~ ( ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
& ( disjoint @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(228,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(1184,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(336,axiom,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( is_transitive_in @ A @ B )
<=> ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ E @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ C @ E ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_relat_2) ).
thf(2238,plain,
! [A: $i] :
( ( relation @ A )
=> ! [B: $i] :
( ( ( is_transitive_in @ A @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ E @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ C @ E ) @ A ) ) )
& ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B )
& ( in @ E @ B )
& ( in @ ( ordered_pair @ C @ D ) @ A )
& ( in @ ( ordered_pair @ D @ E ) @ A ) )
=> ( in @ ( ordered_pair @ C @ E ) @ A ) )
=> ( is_transitive_in @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(265,axiom,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
( ( subset @ E @ C )
=> ( in @ E @ D ) ) ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t9_tarski) ).
thf(1490,plain,
! [A: $i] :
? [B: $i] :
( ( in @ A @ B )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( subset @ D @ C ) )
=> ( in @ D @ B ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ( ( in @ D @ B )
& ! [E: $i] :
( ( subset @ E @ C )
=> ( in @ E @ D ) ) ) )
& ! [C: $i] :
~ ( ( subset @ C @ B )
& ~ ( are_equipotent @ C @ B )
& ~ ( in @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(128,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_setfam_1) ).
thf(775,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(491,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) )
& ( ( ( subset @ A @ B )
& ( subset @ B @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(219,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( antisymmetric @ A )
<=> ! [B: $i,C: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ B ) @ A ) )
=> ( B = C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3_wellord1) ).
thf(1088,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( ( antisymmetric @ A )
=> ! [B: $i,C: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ B ) @ A ) )
=> ( B = C ) ) )
& ( ! [B: $i,C: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ C @ B ) @ A ) )
=> ( B = C ) )
=> ( antisymmetric @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(2392,plain,
$false,
inference(cvc4,[status(thm)],[1718,893,468,352,1237,846,628,1031,2077,1750,1939,1544,677,523,582,1596,2104,1290,1882,878,440,1705,1078,1310,797,1275,814,913,945,1354,898,1001,1441,841,885,1359,2082,1366,829,1095,785,504,838,751,802,1832,1191,902,853,2169,1247,421,969,770,881,1875,924,756,1811,591,1373,376,2236,646,2291,1020,905,480,602,687,634,998,507,1796,453,920,761,1470,1067,714,512,942,910,746,825,778,349,598,561,424,429,788,2015,407,630,1951,1687,2179,1836,551,456,974,1012,697,1983,615,1328,1064,403,1425,1540,1878,356,593,783,742,2300,874,2115,488,1361,1942,933,537,526,637,1446,679,706,1028,2023,1324,738,1552,569,2140,1393,1627,515,1249,1530,1456,1131,654,2052,1599,938,508,765,953,794,888,1563,657,1187,674,618,2176,1698,2253,1747,843,772,711,579,958,1007,694,1226,545,936,2326,1865,804,947,540,831,681,1282,2229,2084,419,1122,979,604,1589,996,362,567,649,1043,1376,1439,477,1080,535,726,1149,915,668,462,1862,483,700,520,1926,868,883,1015,986,1726,851,922,1213,768,827,1034,1873,548,342,754,553,1181,1208,589,927,374,1770,640,1025,1070,2143,1098,993,791,585,717,1235,1679,907,502,1869,470,576,744,1023,600,1504,2030,571,595,612,876,991,2113,703,465,1459,556,671,976,2194,759,1126,1918,450,1084,2226,1453,2174,652,390,632,731,781,2262,961,1721,963,517,896,748,1689,1193,763,383,684,1133,447,950,2233,818,865,833,410,967,564,1436,358,1473,955,918,1989,1363,1314,1710,400,2021,2063,395,728,427,528,644,459,871,1500,1923,1431,1004,1902,691,369,607,2138,354,1009,543,2059,708,972,1653,940,1681,1184,2238,1490,775,491,1088]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.12 % Problem : SEU280+2 : TPTP v8.1.2. Released v3.3.0.
% 0.05/0.14 % Command : run_Leo-III %s %d
% 0.14/0.36 % Computer : n029.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu May 18 12:42:59 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.82/0.81 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.55/1.06 % [INFO] Parsing done (252ms).
% 1.55/1.07 % [INFO] Running in sequential loop mode.
% 2.11/1.32 % [INFO] eprover registered as external prover.
% 2.11/1.33 % [INFO] cvc4 registered as external prover.
% 2.11/1.33 % [INFO] Scanning for conjecture ...
% 2.48/1.43 % [INFO] Found a conjecture and 363 axioms. Running axiom selection ...
% 2.86/1.53 % [INFO] Axiom selection finished. Selected 339 axioms (removed 24 axioms).
% 3.47/1.72 % [INFO] Problem is first-order (TPTP FOF).
% 3.47/1.74 % [INFO] Type checking passed.
% 3.47/1.75 % [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 ...
% 16.29/4.82 % External prover 'cvc4' found a proof!
% 16.29/4.82 % [INFO] Killing All external provers ...
% 16.29/4.83 % Time passed: 4330ms (effective reasoning time: 3748ms)
% 16.29/4.83 % 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)>
% 16.29/4.83 % Axioms used in derivation (339): d1_enumset1, redefinition_m2_relset_1, redefinition_r1_ordinal1, d1_wellord2, l3_wellord1, fc1_zfmisc_1, d1_tarski, 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, 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, dt_k2_subset_1, d5_tarski, fc4_relat_1, d12_relat_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, t32_ordinal1, t47_relat_1, fc9_relat_1, l3_subset_1, t99_zfmisc_1, d1_wellord1, t47_setfam_1, fc1_funct_1, d9_funct_1, rc3_funct_1, t56_relat_1, t37_xboole_1, dt_m2_relset_1, cc1_relat_1, t21_ordinal1, redefinition_k6_setfam_1, t55_funct_1, commutativity_k3_xboole_0, s1_tarski__e6_22__wellord2__1, t62_funct_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, s1_tarski__e6_21__wellord2__1, t34_funct_1, t118_zfmisc_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, t1_xboole_1, d9_relat_2, 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, 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, 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, involutiveness_k4_relat_1, cc1_ordinal1, t7_boole, t21_wellord1, cc1_funct_1, t23_ordinal1, t118_relat_1, d10_relat_1, t143_relat_1, d2_subset_1, t24_wellord1, t48_xboole_1, t3_ordinal1, l4_wellord1, dt_k1_wellord2, fc1_subset_1, t33_zfmisc_1, fc5_funct_1, fc3_ordinal1, rc2_funct_1, t160_relat_1, d11_relat_1, connectedness_r1_ordinal1, t8_wellord1, t2_wellord2, fc10_relat_1, rc1_subset_1, cc3_ordinal1, t74_relat_1, rc3_ordinal1, t83_xboole_1, t45_relat_1, l28_zfmisc_1, t140_relat_1, d14_relat_1, d7_xboole_0, fc5_relat_1, cc2_funct_1, t46_setfam_1, d2_wellord1, l4_zfmisc_1, t3_subset, cc1_relset_1, t22_relset_1, t24_ordinal1, t33_ordinal1, t115_relat_1, existence_m1_subset_1, d2_mcart_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, 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, t8_xboole_1, t37_relat_1, d3_tarski, t39_zfmisc_1, t119_zfmisc_1, rc1_relat_1, t60_relat_1, t54_funct_1, dt_k5_relat_1, reflexivity_r1_tarski, t28_xboole_1, d1_relset_1, t6_zfmisc_1, dt_k6_relat_1, reflexivity_r1_ordinal1, t2_xboole_1, fc6_relat_1, 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, t68_funct_1, t54_subset_1, dt_k7_setfam_1, t21_funct_1, dt_k2_wellord1, d1_relat_2, t136_zfmisc_1, fc1_relat_1, d6_wellord1, d6_relat_1, l23_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, l30_wellord2, t9_tarski, t145_relat_1, d10_xboole_0, symmetry_r2_wellord2, t7_xboole_1, t50_subset_1, fc2_ordinal1, t7_tarski, t60_xboole_1, 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, 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
% 16.29/4.83 % No. of inferences in proof: 682
% 16.29/4.83 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 4330 ms resp. 3748 ms w/o parsing
% 16.80/4.99 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 16.80/5.00 % [INFO] Killing All external provers ...
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