TSTP Solution File: SEU326+2 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SEU326+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n001.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 11:58:17 EDT 2023
% Result : Theorem 26.22s 6.21s
% Output : Refutation 26.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 626
% Syntax : Number of formulae : 1139 ( 99 unt; 115 typ; 0 def)
% Number of atoms : 5102 ( 927 equ; 0 cnn)
% Maximal formula atoms : 49 ( 4 avg)
% Number of connectives : 15280 ( 491 ~; 34 |;2070 &;10711 @)
% ( 174 <=>;1800 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 185 ( 185 >; 0 *; 0 +; 0 <<)
% Number of symbols : 117 ( 115 usr; 3 con; 0-6 aty)
% Number of variables : 2947 ( 0 ^;2556 !; 391 ?;2947 :)
% Comments :
%------------------------------------------------------------------------------
thf(element_type,type,
element: $i > $i > $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(empty_set_type,type,
empty_set: $i ).
thf(complements_of_subsets_type,type,
complements_of_subsets: $i > $i > $i ).
thf(relation_type,type,
relation: $i > $o ).
thf(well_orders_type,type,
well_orders: $i > $i > $o ).
thf(relation_field_type,type,
relation_field: $i > $i ).
thf(well_ordering_type,type,
well_ordering: $i > $o ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(proper_subset_type,type,
proper_subset: $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(ordinal_type,type,
ordinal: $i > $o ).
thf(inclusion_relation_type,type,
inclusion_relation: $i > $i ).
thf(reflexive_type,type,
reflexive: $i > $o ).
thf(relation_restriction_type,type,
relation_restriction: $i > $i > $i ).
thf(relation_rng_restriction_type,type,
relation_rng_restriction: $i > $i > $i ).
thf(relation_dom_restriction_type,type,
relation_dom_restriction: $i > $i > $i ).
thf(relation_inverse_type,type,
relation_inverse: $i > $i ).
thf(connected_type,type,
connected: $i > $o ).
thf(transitive_type,type,
transitive: $i > $o ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(in_type,type,
in: $i > $i > $o ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(empty_type,type,
empty: $i > $o ).
thf(relation_composition_type,type,
relation_composition: $i > $i > $i ).
thf(identity_relation_type,type,
identity_relation: $i > $i ).
thf(antisymmetric_type,type,
antisymmetric: $i > $o ).
thf(equipotent_type,type,
equipotent: $i > $i > $o ).
thf(function_type,type,
function: $i > $o ).
thf(relation_isomorphism_type,type,
relation_isomorphism: $i > $i > $i > $o ).
thf(function_inverse_type,type,
function_inverse: $i > $i ).
thf(well_founded_relation_type,type,
well_founded_relation: $i > $o ).
thf(pair_first_type,type,
pair_first: $i > $i ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(pair_second_type,type,
pair_second: $i > $i ).
thf(finite_type,type,
finite: $i > $o ).
thf(relation_dom_type,type,
relation_dom: $i > $i ).
thf(relation_rng_type,type,
relation_rng: $i > $i ).
thf(apply_type,type,
apply: $i > $i > $i ).
thf(set_intersection2_type,type,
set_intersection2: $i > $i > $i ).
thf(v1_membered_type,type,
v1_membered: $i > $o ).
thf(v2_membered_type,type,
v2_membered: $i > $o ).
thf(v3_membered_type,type,
v3_membered: $i > $o ).
thf(v4_membered_type,type,
v4_membered: $i > $o ).
thf(v5_membered_type,type,
v5_membered: $i > $o ).
thf(cartesian_product2_type,type,
cartesian_product2: $i > $i > $i ).
thf(one_to_one_type,type,
one_to_one: $i > $o ).
thf(set_union2_type,type,
set_union2: $i > $i > $i ).
thf(top_str_type,type,
top_str: $i > $o ).
thf(being_limit_ordinal_type,type,
being_limit_ordinal: $i > $o ).
thf(union_type,type,
union: $i > $i ).
thf(set_difference_type,type,
set_difference: $i > $i > $i ).
thf(empty_carrier_type,type,
empty_carrier: $i > $o ).
thf(one_sorted_str_type,type,
one_sorted_str: $i > $o ).
thf(the_carrier_type,type,
the_carrier: $i > $i ).
thf(cast_to_subset_type,type,
cast_to_subset: $i > $i ).
thf(relation_of2_type,type,
relation_of2: $i > $i > $i > $o ).
thf(epsilon_transitive_type,type,
epsilon_transitive: $i > $o ).
thf(epsilon_connected_type,type,
epsilon_connected: $i > $o ).
thf(relation_dom_as_subset_type,type,
relation_dom_as_subset: $i > $i > $i > $i ).
thf(natural_type,type,
natural: $i > $o ).
thf(succ_type,type,
succ: $i > $i ).
thf(relation_empty_yielding_type,type,
relation_empty_yielding: $i > $o ).
thf(omega_type,type,
omega: $i ).
thf(relation_image_type,type,
relation_image: $i > $i > $i ).
thf(meet_semilatt_str_type,type,
meet_semilatt_str: $i > $o ).
thf(the_L_meet_type,type,
the_L_meet: $i > $i ).
thf(quasi_total_type,type,
quasi_total: $i > $i > $i > $o ).
thf(relation_of2_as_subset_type,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(disjoint_type,type,
disjoint: $i > $i > $o ).
thf(relation_rng_as_subset_type,type,
relation_rng_as_subset: $i > $i > $i > $i ).
thf(relation_inverse_image_type,type,
relation_inverse_image: $i > $i > $i ).
thf(cast_as_carrier_subset_type,type,
cast_as_carrier_subset: $i > $i ).
thf(cup_closed_type,type,
cup_closed: $i > $o ).
thf(diff_closed_type,type,
diff_closed: $i > $o ).
thf(preboolean_type,type,
preboolean: $i > $o ).
thf(ordinal_subset_type,type,
ordinal_subset: $i > $i > $o ).
thf(are_equipotent_type,type,
are_equipotent: $i > $i > $o ).
thf(fiber_type,type,
fiber: $i > $i > $i ).
thf(topological_space_type,type,
topological_space: $i > $o ).
thf(closed_subset_type,type,
closed_subset: $i > $i > $o ).
thf(join_semilatt_str_type,type,
join_semilatt_str: $i > $o ).
thf(the_L_join_type,type,
the_L_join: $i > $i ).
thf(empty_carrier_subset_type,type,
empty_carrier_subset: $i > $i ).
thf(latt_str_type,type,
latt_str: $i > $o ).
thf(unordered_triple_type,type,
unordered_triple: $i > $i > $i > $i ).
thf(join_commutative_type,type,
join_commutative: $i > $o ).
thf(join_commut_type,type,
join_commut: $i > $i > $i > $i ).
thf(join_type,type,
join: $i > $i > $i > $i ).
thf(subset_complement_type,type,
subset_complement: $i > $i > $i ).
thf(apply_binary_as_element_type,type,
apply_binary_as_element: $i > $i > $i > $i > $i > $i > $i ).
thf(meet_commutative_type,type,
meet_commutative: $i > $o ).
thf(meet_commut_type,type,
meet_commut: $i > $i > $i > $i ).
thf(open_subset_type,type,
open_subset: $i > $i > $o ).
thf(meet_type,type,
meet: $i > $i > $i > $i ).
thf(topstr_closure_type,type,
topstr_closure: $i > $i > $i ).
thf(set_meet_type,type,
set_meet: $i > $i ).
thf(union_of_subsets_type,type,
union_of_subsets: $i > $i > $i ).
thf(meet_absorbing_type,type,
meet_absorbing: $i > $o ).
thf(subset_difference_type,type,
subset_difference: $i > $i > $i > $i ).
thf(interior_type,type,
interior: $i > $i > $i ).
thf(the_topology_type,type,
the_topology: $i > $i ).
thf(subset_intersection2_type,type,
subset_intersection2: $i > $i > $i > $i ).
thf(apply_binary_type,type,
apply_binary: $i > $i > $i > $i ).
thf(meet_of_subsets_type,type,
meet_of_subsets: $i > $i > $i ).
thf(below_type,type,
below: $i > $i > $i > $o ).
thf(v1_xcmplx_0_type,type,
v1_xcmplx_0: $i > $o ).
thf(v1_xreal_0_type,type,
v1_xreal_0: $i > $o ).
thf(v1_int_1_type,type,
v1_int_1: $i > $o ).
thf(v1_rat_1_type,type,
v1_rat_1: $i > $o ).
thf(is_a_cover_of_carrier_type,type,
is_a_cover_of_carrier: $i > $i > $o ).
thf(107,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(1036,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)],[107]) ).
thf(72,axiom,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(809,plain,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(149,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(1337,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)],[149]) ).
thf(28,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(629,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)],[28]) ).
thf(209,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(1767,plain,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(452,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(3434,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)],[452]) ).
thf(472,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ! [E: $i] :
( ( ( relation @ E )
& ( function @ E ) )
=> ( ( in @ C @ A )
=> ( ( B = empty_set )
| ( ( apply @ ( relation_composition @ D @ E ) @ C )
= ( apply @ E @ ( apply @ D @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t21_funct_2) ).
thf(3566,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ! [E: $i] :
( ( ( relation @ E )
& ( function @ E ) )
=> ( ( in @ C @ A )
=> ( ( B = empty_set )
| ( ( apply @ ( relation_composition @ D @ E ) @ C )
= ( apply @ E @ ( apply @ D @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).
thf(235,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(1950,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)],[235]) ).
thf(354,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(2744,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)],[354]) ).
thf(126,axiom,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc30_membered) ).
thf(1190,plain,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(290,axiom,
( ( epsilon_transitive @ omega )
& ( epsilon_connected @ omega )
& ( ordinal @ omega )
& ~ ( empty @ omega ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_ordinal2) ).
thf(2295,plain,
( ( epsilon_transitive @ omega )
& ( epsilon_connected @ omega )
& ( ordinal @ omega )
& ~ ( empty @ omega ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(63,axiom,
! [A: $i] :
( ( being_limit_ordinal @ A )
<=> ( A
= ( union @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_ordinal1) ).
thf(760,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)],[63]) ).
thf(286,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(2280,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)],[286]) ).
thf(427,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( subset @ ( interior @ A @ B ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t44_tops_1) ).
thf(3172,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( subset @ ( interior @ A @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).
thf(3,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( well_orders @ A @ ( relation_field @ A ) )
<=> ( well_ordering @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_wellord1) ).
thf(523,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)],[3]) ).
thf(274,axiom,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(2210,plain,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(228,axiom,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_lattices) ).
thf(1882,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(341,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(2644,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)],[341]) ).
thf(217,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(1808,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(195,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(1618,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)],[195]) ).
thf(392,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(2953,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(281,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(2248,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)],[281]) ).
thf(420,axiom,
! [A: $i] :
( ( v4_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B )
& ( v4_membered @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc19_membered) ).
thf(3098,plain,
! [A: $i] :
( ( v4_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B )
& ( v4_membered @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).
thf(189,axiom,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) )
& ( v4_membered @ ( set_intersection2 @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc34_membered) ).
thf(1591,plain,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) )
& ( v4_membered @ ( set_intersection2 @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(406,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_pre_topc) ).
thf(3012,plain,
! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).
thf(130,axiom,
! [A: $i] :
( ( ( cup_closed @ A )
& ( diff_closed @ A ) )
=> ( preboolean @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_finsub_1) ).
thf(1205,plain,
! [A: $i] :
( ( ( cup_closed @ A )
& ( diff_closed @ A ) )
=> ( preboolean @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(83,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(878,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)],[83]) ).
thf(461,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( B != empty_set )
=> ! [E: $i] :
( ( in @ E @ ( relation_inverse_image @ D @ C ) )
<=> ( ( in @ E @ A )
& ( in @ ( apply @ D @ E ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_funct_2) ).
thf(3498,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( B != empty_set )
=> ! [E: $i] :
( ( ( in @ E @ ( relation_inverse_image @ D @ C ) )
=> ( ( in @ E @ A )
& ( in @ ( apply @ D @ E ) @ C ) ) )
& ( ( ( in @ E @ A )
& ( in @ ( apply @ D @ E ) @ C ) )
=> ( in @ E @ ( relation_inverse_image @ D @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).
thf(214,axiom,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u2_lattices) ).
thf(1793,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(375,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( meet @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_lattices) ).
thf(2854,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( meet @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(69,axiom,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_subset_1) ).
thf(797,plain,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(56,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(715,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)],[56]) ).
thf(504,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( subset @ B @ C )
=> ( ( ( B = empty_set )
& ( A != empty_set ) )
| ( ( function @ D )
& ( quasi_total @ D @ A @ C )
& ( relation_of2_as_subset @ D @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t9_funct_2) ).
thf(3693,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( subset @ B @ C )
=> ( ( ( B = empty_set )
& ( A != empty_set ) )
| ( ( function @ D )
& ( quasi_total @ D @ A @ C )
& ( relation_of2_as_subset @ D @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).
thf(246,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(1988,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)],[246]) ).
thf(404,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(3004,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)],[404]) ).
thf(250,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(2035,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)],[250]) ).
thf(42,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(670,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)],[42]) ).
thf(328,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc27_membered) ).
thf(2575,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(276,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(2216,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)],[276]) ).
thf(135,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(1243,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(55,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( v1_membered @ A )
& ( v2_membered @ A )
& ( v3_membered @ A )
& ( v4_membered @ A )
& ( v5_membered @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc15_membered) ).
thf(709,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( v1_membered @ A )
& ( v2_membered @ A )
& ( v3_membered @ A )
& ( v4_membered @ A )
& ( v5_membered @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(232,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(1935,plain,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(162,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(1386,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)],[162]) ).
thf(16,axiom,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_wellord2) ).
thf(565,plain,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(239,axiom,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_ordinal1) ).
thf(1967,plain,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(138,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_ordinal1) ).
thf(1254,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(502,axiom,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc16_membered) ).
thf(3688,plain,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).
thf(205,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(1746,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)],[205]) ).
thf(421,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_lattices) ).
thf(3103,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).
thf(334,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(2603,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(245,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(1985,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( function_inverse @ A ) )
& ( function @ ( function_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(208,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( v1_membered @ A )
& ( v2_membered @ A )
& ( v3_membered @ A )
& ( v4_membered @ A )
& ( v5_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_membered) ).
thf(1760,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( v1_membered @ A )
& ( v2_membered @ A )
& ( v3_membered @ A )
& ( v4_membered @ A )
& ( v5_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(283,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_ordinal1) ).
thf(2269,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(415,axiom,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(3061,plain,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).
thf(484,axiom,
! [A: $i] :
( ( element @ A @ omega )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_arytm_3) ).
thf(3612,plain,
! [A: $i] :
( ( element @ A @ omega )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).
thf(89,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ D @ A )
& ( in @ D @ A )
& ? [F: $i] :
( ( D = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_funct_1__e10_24__wellord2__1) ).
thf(917,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ A )
& ( in @ D @ A )
& ? [F: $i] :
( ( D = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) ) )
& ( ( ( in @ D @ A )
& ( in @ D @ A )
& ? [F: $i] :
( ( D = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(309,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(2466,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)],[309]) ).
thf(110,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(1059,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(485,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( below @ A @ ( meet_commut @ A @ B @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_lattices) ).
thf(3617,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( below @ A @ ( meet_commut @ A @ B @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).
thf(304,axiom,
! [A: $i] :
( ( v3_membered @ A )
=> ( v2_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_membered) ).
thf(2444,plain,
! [A: $i] :
( ( v3_membered @ A )
=> ( v2_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(140,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(1264,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( in @ B @ C )
& ( in @ C @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(186,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(1577,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)],[186]) ).
thf(170,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ( closed_subset @ ( cast_as_carrier_subset @ A ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_pre_topc) ).
thf(1455,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ( closed_subset @ ( cast_as_carrier_subset @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(326,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(2571,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)],[326]) ).
thf(508,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( open_subset @ B @ A )
<=> ( closed_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t30_tops_1) ).
thf(3713,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( open_subset @ B @ A )
=> ( closed_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) )
& ( ( closed_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A )
=> ( open_subset @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).
thf(302,axiom,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( finite @ A )
=> ( finite @ ( relation_image @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_finset_1) ).
thf(2420,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( finite @ A )
=> ( finite @ ( relation_image @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(32,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(638,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)],[32]) ).
thf(160,axiom,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_funct_1) ).
thf(1379,plain,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(376,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ B @ A )
=> ( ( topstr_closure @ A @ B )
= B ) )
& ( ( ( topological_space @ A )
& ( ( topstr_closure @ A @ B )
= B ) )
=> ( closed_subset @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t52_pre_topc) ).
thf(2857,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ B @ A )
=> ( ( topstr_closure @ A @ B )
= B ) )
& ( ( ( topological_space @ A )
& ( ( topstr_closure @ A @ B )
= B ) )
=> ( closed_subset @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).
thf(101,axiom,
? [A: $i] :
~ ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(1013,plain,
? [A: $i] :
~ ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(388,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_pre_topc) ).
thf(2929,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(407,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( in @ C @ ( the_carrier @ A ) )
=> ( ( in @ C @ ( topstr_closure @ A @ B ) )
<=> ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) )
=> ( in @ C @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t45_pre_topc) ).
thf(3014,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( in @ C @ ( the_carrier @ A ) )
=> ( ( ( in @ C @ ( topstr_closure @ A @ B ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) )
=> ( in @ C @ D ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) )
=> ( in @ C @ D ) ) )
=> ( in @ C @ ( topstr_closure @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).
thf(263,axiom,
! [A: $i,B: $i] :
? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( cartesian_product2 @ A @ B ) )
& ? [E: $i,F: $i] :
( ( ( ordered_pair @ E @ F )
= D )
& ( in @ E @ A )
& ( F
= ( singleton @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e16_22__wellord2__1) ).
thf(2158,plain,
! [A: $i,B: $i] :
? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( cartesian_product2 @ A @ B ) )
& ? [E: $i,F: $i] :
( ( ( ordered_pair @ E @ F )
= D )
& ( in @ E @ A )
& ( F
= ( singleton @ E ) ) ) ) )
& ( ( ( in @ D @ ( cartesian_product2 @ A @ B ) )
& ? [E: $i,F: $i] :
( ( ( ordered_pair @ E @ F )
= D )
& ( in @ E @ A )
& ( F
= ( singleton @ E ) ) ) )
=> ( in @ D @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(211,axiom,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) )
& ( v4_membered @ ( set_intersection2 @ A @ B ) )
& ( v5_membered @ ( set_intersection2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc35_membered) ).
thf(1773,plain,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) )
& ( v4_membered @ ( set_intersection2 @ A @ B ) )
& ( v5_membered @ ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(264,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(2169,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)],[264]) ).
thf(112,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_finset_1) ).
thf(1087,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(144,axiom,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc31_membered) ).
thf(1319,plain,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(200,axiom,
! [A: $i,B: $i] :
( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= D )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) )
& ( C = E )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( cartesian_product2 @ A @ B ) )
& ( E = D )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= D )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e16_22__wellord2__2) ).
thf(1704,plain,
! [A: $i,B: $i] :
( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= D )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) )
& ( C = E )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( cartesian_product2 @ A @ B ) )
& ( E = D )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= D )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( cartesian_product2 @ A @ B ) )
& ( E = D )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= D )
& ( in @ F @ A )
& ( G
= ( singleton @ F ) ) ) )
=> ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(203,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(1741,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)],[203]) ).
thf(84,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(881,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)],[84]) ).
thf(177,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(1489,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)],[177]) ).
thf(451,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( open_subset @ B @ A )
<=> ( in @ B @ ( the_topology @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_pre_topc) ).
thf(3431,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( open_subset @ B @ A )
=> ( in @ B @ ( the_topology @ A ) ) )
& ( ( in @ B @ ( the_topology @ A ) )
=> ( open_subset @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).
thf(417,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(3075,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)],[417]) ).
thf(158,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(1373,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)],[158]) ).
thf(434,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( C
= ( topstr_closure @ A @ B ) )
<=> ! [D: $i] :
( ( in @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
<=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ~ ( ( open_subset @ E @ A )
& ( in @ D @ E )
& ( disjoint @ B @ E ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d13_pre_topc) ).
thf(3295,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( C
= ( topstr_closure @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ ( the_carrier @ A ) )
=> ( ( ( in @ D @ C )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ~ ( ( open_subset @ E @ A )
& ( in @ D @ E )
& ( disjoint @ B @ E ) ) ) )
& ( ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ~ ( ( open_subset @ E @ A )
& ( in @ D @ E )
& ( disjoint @ B @ E ) ) )
=> ( in @ D @ C ) ) ) ) )
& ( ! [D: $i] :
( ( in @ D @ ( the_carrier @ A ) )
=> ( ( ( in @ D @ C )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ~ ( ( open_subset @ E @ A )
& ( in @ D @ E )
& ( disjoint @ B @ E ) ) ) )
& ( ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ~ ( ( open_subset @ E @ A )
& ( in @ D @ E )
& ( disjoint @ B @ E ) ) )
=> ( in @ D @ C ) ) ) )
=> ( C
= ( topstr_closure @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).
thf(495,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(3670,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)],[495]) ).
thf(505,axiom,
! [A: $i] :
( ( v4_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B )
& ( v1_int_1 @ B )
& ( v1_rat_1 @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc13_membered) ).
thf(3702,plain,
! [A: $i] :
( ( v4_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B )
& ( v1_int_1 @ B )
& ( v1_rat_1 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).
thf(118,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(1136,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(289,axiom,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_funct_1) ).
thf(2291,plain,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(293,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(2335,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)],[293]) ).
thf(473,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(3569,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)],[473]) ).
thf(324,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(2540,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)],[324]) ).
thf(220,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(1843,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)],[220]) ).
thf(467,axiom,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(3537,plain,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).
thf(370,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(2833,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)],[370]) ).
thf(30,axiom,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_relat_1) ).
thf(634,plain,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(314,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(2483,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)],[314]) ).
thf(40,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(660,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(364,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( in @ C @ ( subset_complement @ ( the_carrier @ A ) @ B ) )
<=> ~ ( in @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l40_tops_1) ).
thf(2809,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( in @ C @ ( subset_complement @ ( the_carrier @ A ) @ B ) )
=> ~ ( in @ C @ B ) )
& ( ~ ( in @ C @ B )
=> ( in @ C @ ( subset_complement @ ( the_carrier @ A ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(362,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_relat_1) ).
thf(2804,plain,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(190,axiom,
! [A: $i] :
( ( preboolean @ A )
=> ( ( cup_closed @ A )
& ( diff_closed @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_finsub_1) ).
thf(1600,plain,
! [A: $i] :
( ( preboolean @ A )
=> ( ( cup_closed @ A )
& ( diff_closed @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(478,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(3588,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)],[478]) ).
thf(272,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_ordinal1) ).
thf(2205,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(332,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(2586,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)],[332]) ).
thf(21,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(602,plain,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(394,axiom,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( topstr_closure @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_pre_topc) ).
thf(2964,plain,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( topstr_closure @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).
thf(122,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(1169,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)],[122]) ).
thf(47,axiom,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_wellord2) ).
thf(687,plain,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(49,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(692,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)],[49]) ).
thf(268,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(2190,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)],[268]) ).
thf(449,axiom,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t18_finset_1) ).
thf(3421,plain,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).
thf(413,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(3036,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).
thf(320,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(2530,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(97,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(988,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(287,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(2286,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)],[287]) ).
thf(5,axiom,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(529,plain,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(68,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(793,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)],[68]) ).
thf(342,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_funct_1) ).
thf(2646,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).
thf(340,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(2641,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)],[340]) ).
thf(337,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_pre_topc) ).
thf(2621,plain,
! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(224,axiom,
! [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(1868,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)],[224]) ).
thf(498,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(3679,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)],[498]) ).
thf(425,axiom,
! [A: $i] :
( ( v3_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc18_membered) ).
thf(3151,plain,
! [A: $i] :
( ( v3_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).
thf(248,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(2011,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)],[248]) ).
thf(465,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(3531,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)],[465]) ).
thf(128,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_pre_topc) ).
thf(1199,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(282,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(2264,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)],[282]) ).
thf(93,axiom,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_lattices) ).
thf(957,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(333,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(2601,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)],[333]) ).
thf(91,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(952,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)],[91]) ).
thf(176,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( cast_as_carrier_subset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_pre_topc) ).
thf(1487,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( cast_as_carrier_subset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(335,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(2606,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)],[335]) ).
thf(249,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( meet_semilatt_str @ A )
& ( join_semilatt_str @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l3_lattices) ).
thf(2032,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( meet_semilatt_str @ A )
& ( join_semilatt_str @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(432,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( subset_intersection2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k5_subset_1) ).
thf(3286,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( subset_intersection2 @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).
thf(75,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(825,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)],[75]) ).
thf(151,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(1343,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)],[151]) ).
thf(104,axiom,
! [A: $i] :
( ( v4_membered @ A )
=> ( v3_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_membered) ).
thf(1021,plain,
! [A: $i] :
( ( v4_membered @ A )
=> ( v3_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(481,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_intersection2 @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_subset_1) ).
thf(3603,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_intersection2 @ A @ B @ C ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).
thf(285,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(2277,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(154,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(1355,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)],[154]) ).
thf(187,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(1580,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)],[187]) ).
thf(20,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(598,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(260,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(2147,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)],[260]) ).
thf(463,axiom,
! [A: $i] :
( ( v5_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( natural @ B )
& ( v1_xreal_0 @ B )
& ( v1_int_1 @ B )
& ( v1_rat_1 @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc14_membered) ).
thf(3514,plain,
! [A: $i] :
( ( v5_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( natural @ B )
& ( v1_xreal_0 @ B )
& ( v1_int_1 @ B )
& ( v1_rat_1 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).
thf(499,axiom,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( v1_xcmplx_0 @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc10_membered) ).
thf(3682,plain,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( v1_xcmplx_0 @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).
thf(168,axiom,
! [A: $i] :
( ( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
& ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i] :
( C
!= ( singleton @ B ) ) ) )
=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= ( singleton @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s2_funct_1__e16_22__wellord2__1) ).
thf(1423,plain,
! [A: $i] :
( ( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
& ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i] :
( C
!= ( singleton @ B ) ) ) )
=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= ( singleton @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(308,axiom,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_relat_1) ).
thf(2464,plain,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(106,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_funct_1) ).
thf(1032,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(156,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(1365,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)],[156]) ).
thf(292,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(2332,plain,
! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( ( apply @ ( identity_relation @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(171,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(1457,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ ( union @ A ) )
& ( epsilon_connected @ ( union @ A ) )
& ( ordinal @ ( union @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(132,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(1210,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)],[132]) ).
thf(410,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( set_intersection2 @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k5_subset_1) ).
thf(3028,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ C )
= ( set_intersection2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).
thf(280,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(2228,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)],[280]) ).
thf(490,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(3647,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)],[490]) ).
thf(479,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( open_subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_tops_1) ).
thf(3595,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( open_subset @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).
thf(45,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(682,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)],[45]) ).
thf(127,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(1195,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)],[127]) ).
thf(11,axiom,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_wellord1) ).
thf(551,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(318,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(2492,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)],[318]) ).
thf(51,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( finite @ ( relation_dom @ A ) )
=> ( finite @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_finset_1) ).
thf(697,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( finite @ ( relation_dom @ A ) )
=> ( finite @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(401,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ D @ C )
<=> ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s3_subset_1__e1_40__pre_topc) ).
thf(2990,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ D @ C )
=> ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) ) )
& ( ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) )
=> ( in @ D @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).
thf(182,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(1534,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)],[182]) ).
thf(398,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e1_40__pre_topc__1) ).
thf(2973,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) ) ) )
& ( ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ? [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( closed_subset @ E @ A )
& ( subset @ B @ D ) ) )
=> ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).
thf(240,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(1970,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)],[240]) ).
thf(99,axiom,
! [A: $i] :
( ( v2_membered @ A )
=> ( v1_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc4_membered) ).
thf(1006,plain,
! [A: $i] :
( ( v2_membered @ A )
=> ( v1_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(206,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty_carrier @ A )
<=> ( empty @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_struct_0) ).
thf(1755,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( ( empty_carrier @ A )
=> ( empty @ ( the_carrier @ A ) ) )
& ( ( empty @ ( the_carrier @ A ) )
=> ( empty_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(454,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( in @ C @ A )
=> ( ( B = empty_set )
| ( in @ ( apply @ D @ C ) @ ( relation_rng @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_funct_2) ).
thf(3440,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( function @ D )
& ( quasi_total @ D @ A @ B )
& ( relation_of2_as_subset @ D @ A @ B ) )
=> ( ( in @ C @ A )
=> ( ( B = empty_set )
| ( in @ ( apply @ D @ C ) @ ( relation_rng @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).
thf(155,axiom,
! [A: $i] :
( ~ ( empty @ ( singleton @ A ) )
& ( finite @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_finset_1) ).
thf(1360,plain,
! [A: $i] :
( ~ ( empty @ ( singleton @ A ) )
& ( finite @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(152,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(1350,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(422,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ( ( in @ C @ ( succ @ A ) )
& ? [D: $i] :
( ( ordinal @ D )
& ( C = D )
& ( ( in @ D @ omega )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( powerset @ D ) ) )
=> ~ ( ( E != empty_set )
& ! [F: $i] :
~ ( ( in @ F @ E )
& ! [G: $i] :
( ( ( in @ G @ E )
& ( subset @ F @ G ) )
=> ( G = F ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e18_27__finset_1__1) ).
thf(3105,plain,
! [A: $i] :
( ( ordinal @ A )
=> ? [B: $i] :
! [C: $i] :
( ( ( in @ C @ B )
=> ( ( in @ C @ ( succ @ A ) )
& ? [D: $i] :
( ( ordinal @ D )
& ( C = D )
& ( ( in @ D @ omega )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( powerset @ D ) ) )
=> ~ ( ( E != empty_set )
& ! [F: $i] :
~ ( ( in @ F @ E )
& ! [G: $i] :
( ( ( in @ G @ E )
& ( subset @ F @ G ) )
=> ( G = F ) ) ) ) ) ) ) ) )
& ( ( ( in @ C @ ( succ @ A ) )
& ? [D: $i] :
( ( ordinal @ D )
& ( C = D )
& ( ( in @ D @ omega )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( powerset @ D ) ) )
=> ~ ( ( E != empty_set )
& ! [F: $i] :
~ ( ( in @ F @ E )
& ! [G: $i] :
( ( ( in @ G @ E )
& ( subset @ F @ G ) )
=> ( G = F ) ) ) ) ) ) ) )
=> ( in @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).
thf(291,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(2300,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)],[291]) ).
thf(319,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( in @ B @ A )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ D @ A )
& ( in @ D @ A )
& ( C
= ( singleton @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e16_22__wellord2__1) ).
thf(2515,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( in @ B @ A )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ D @ A )
& ( in @ D @ A )
& ( C
= ( singleton @ D ) ) ) )
& ( ? [D: $i] :
( ( in @ D @ A )
& ( in @ D @ A )
& ( C
= ( singleton @ D ) ) )
=> ( in @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(446,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below @ A @ B @ C )
& ( below @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_lattices) ).
thf(3413,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below @ A @ B @ C )
& ( below @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).
thf(90,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(948,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)],[90]) ).
thf(361,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( closed_subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc6_pre_topc) ).
thf(2801,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( closed_subset @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).
thf(216,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc4_funct_1) ).
thf(1804,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(25,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(615,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)],[25]) ).
thf(373,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(2841,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)],[373]) ).
thf(146,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_ordinal1) ).
thf(1328,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(119,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(1138,plain,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(323,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(2537,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(66,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(783,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)],[66]) ).
thf(339,axiom,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_ordinal1) ).
thf(2638,plain,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(424,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(3143,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)],[424]) ).
thf(234,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(1942,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)],[234]) ).
thf(402,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(3000,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)],[402]) ).
thf(12,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(554,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)],[12]) ).
thf(369,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k4_lattices) ).
thf(2830,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(34,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(642,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)],[34]) ).
thf(233,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(1937,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)],[233]) ).
thf(251,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_difference @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(2038,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)],[251]) ).
thf(204,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(1744,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(54,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc11_finset_1) ).
thf(706,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(166,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(1419,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)],[166]) ).
thf(306,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_relset_1) ).
thf(2460,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(360,axiom,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_boole) ).
thf(2798,plain,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(202,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_wellord2) ).
thf(1739,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(172,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(1461,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(165,axiom,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc38_membered) ).
thf(1414,plain,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(4,axiom,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( well_orders @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_wellord2) ).
thf(526,plain,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( well_orders @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(218,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty @ ( empty_carrier_subset @ A ) )
& ( v1_membered @ ( empty_carrier_subset @ A ) )
& ( v2_membered @ ( empty_carrier_subset @ A ) )
& ( v3_membered @ ( empty_carrier_subset @ A ) )
& ( v4_membered @ ( empty_carrier_subset @ A ) )
& ( v5_membered @ ( empty_carrier_subset @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_pre_topc) ).
thf(1810,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty @ ( empty_carrier_subset @ A ) )
& ( v1_membered @ ( empty_carrier_subset @ A ) )
& ( v2_membered @ ( empty_carrier_subset @ A ) )
& ( v3_membered @ ( empty_carrier_subset @ A ) )
& ( v4_membered @ ( empty_carrier_subset @ A ) )
& ( v5_membered @ ( empty_carrier_subset @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(265,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(2172,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)],[265]) ).
thf(102,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(1016,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)],[102]) ).
thf(131,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc11_relat_1) ).
thf(1207,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(512,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( open_subset @ ( interior @ A @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t51_tops_1) ).
thf(3723,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( open_subset @ ( interior @ A @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).
thf(346,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(2666,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(193,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(1612,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(343,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(2649,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)],[343]) ).
thf(108,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ~ ( ! [B: $i] :
~ ( ( in @ B @ A )
& ( B = empty_set ) )
& ! [B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ~ ( ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( in @ ( apply @ B @ C ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_wellord2) ).
thf(1039,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ~ ( ! [B: $i] :
~ ( ( in @ B @ A )
& ( B = empty_set ) )
& ! [B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ~ ( ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( in @ ( apply @ B @ C ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(396,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(2968,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)],[396]) ).
thf(469,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(3541,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).
thf(169,axiom,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_ordinal1) ).
thf(1451,plain,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(371,axiom,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_finset_1) ).
thf(2836,plain,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).
thf(242,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(1974,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)],[242]) ).
thf(330,axiom,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_struct_0) ).
thf(2581,plain,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(225,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_finset_1) ).
thf(1874,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(416,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(3064,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)],[416]) ).
thf(244,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(1979,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)],[244]) ).
thf(284,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(2272,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)],[284]) ).
thf(22,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(605,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)],[22]) ).
thf(36,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(647,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)],[36]) ).
thf(491,axiom,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_boole) ).
thf(3656,plain,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).
thf(389,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
& ( relation @ C )
& ( function @ C ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ( in @ ( relation_image @ C @ E ) @ B )
& ( D = F )
& ( in @ ( relation_image @ C @ F ) @ B ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ? [F: $i] :
( ( in @ F @ ( powerset @ ( relation_dom @ C ) ) )
& ( F = E )
& ( in @ ( relation_image @ C @ E ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e6_27__finset_1__1) ).
thf(2932,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
& ( relation @ C )
& ( function @ C ) )
=> ( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ( in @ ( relation_image @ C @ E ) @ B )
& ( D = F )
& ( in @ ( relation_image @ C @ F ) @ B ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ? [F: $i] :
( ( in @ F @ ( powerset @ ( relation_dom @ C ) ) )
& ( F = E )
& ( in @ ( relation_image @ C @ E ) @ B ) ) )
& ( ? [F: $i] :
( ( in @ F @ ( powerset @ ( relation_dom @ C ) ) )
& ( F = E )
& ( in @ ( relation_image @ C @ E ) @ B ) )
=> ( in @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).
thf(96,axiom,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(985,plain,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(316,axiom,
! [A: $i] :
( ( v5_membered @ A )
=> ( v4_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_membered) ).
thf(2487,plain,
! [A: $i] :
( ( v5_membered @ A )
=> ( v4_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(374,axiom,
! [A: $i] :
( ~ ( empty @ ( powerset @ A ) )
& ( cup_closed @ ( powerset @ A ) )
& ( diff_closed @ ( powerset @ A ) )
& ( preboolean @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_finsub_1) ).
thf(2845,plain,
! [A: $i] :
( ~ ( empty @ ( powerset @ A ) )
& ( cup_closed @ ( powerset @ A ) )
& ( diff_closed @ ( powerset @ A ) )
& ( preboolean @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(18,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(569,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)],[18]) ).
thf(444,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(3409,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)],[444]) ).
thf(377,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(2862,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)],[377]) ).
thf(147,axiom,
? [A: $i] : ( one_sorted_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_struct_0) ).
thf(1333,plain,
? [A: $i] : ( one_sorted_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(129,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(1202,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)],[129]) ).
thf(134,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(1241,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)],[134]) ).
thf(71,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_ordinal1) ).
thf(805,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(7,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_wellord2) ).
thf(541,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(221,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(1846,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)],[221]) ).
thf(294,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(2338,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)],[294]) ).
thf(86,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(888,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)],[86]) ).
thf(174,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(1466,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)],[174]) ).
thf(13,axiom,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_wellord2) ).
thf(558,plain,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(487,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_complement @ ( the_carrier @ A ) @ B )
= ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_pre_topc) ).
thf(3639,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_complement @ ( the_carrier @ A ) @ B )
= ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).
thf(385,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(2915,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).
thf(178,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(1515,plain,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(278,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_partfun1) ).
thf(2221,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(259,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(2121,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)],[259]) ).
thf(349,axiom,
! [A: $i,B: $i] :
( ( ( finite @ A )
& ( finite @ B ) )
=> ( finite @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc9_finset_1) ).
thf(2729,plain,
! [A: $i,B: $i] :
( ( ( finite @ A )
& ( finite @ B ) )
=> ( finite @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).
thf(506,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(3707,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)],[506]) ).
thf(1,conjecture,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_tops_2) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(513,plain,
~ ! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ~ ( ( B != empty_set )
& ( ( complements_of_subsets @ A @ B )
= empty_set ) )
& ~ ( ( ( complements_of_subsets @ A @ B )
!= empty_set )
& ( B = empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(382,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(2904,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)],[382]) ).
thf(433,axiom,
! [A: $i] :
( ( v5_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B )
& ( v4_membered @ B )
& ( v5_membered @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc20_membered) ).
thf(3289,plain,
! [A: $i] :
( ( v5_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B )
& ( v3_membered @ B )
& ( v4_membered @ B )
& ( v5_membered @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).
thf(48,axiom,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_enumset1) ).
thf(689,plain,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(501,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(3686,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)],[501]) ).
thf(378,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( subset @ B @ ( topstr_closure @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t48_pre_topc) ).
thf(2894,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( subset @ B @ ( topstr_closure @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).
thf(137,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(1251,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)],[137]) ).
thf(26,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(618,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)],[26]) ).
thf(300,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(2413,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)],[300]) ).
thf(397,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(2970,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)],[397]) ).
thf(253,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(2048,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(229,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e10_24__wellord2__2) ).
thf(1884,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) ) ) )
& ( ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ A @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ A )
& ? [I: $i] :
( ( G = I )
& ( in @ H @ I )
& ! [J: $i] :
( ( in @ J @ I )
=> ( in @ ( ordered_pair @ H @ J ) @ B ) ) ) ) )
=> ( in @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(317,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc28_membered) ).
thf(2489,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(164,axiom,
! [A: $i] :
( ( function @ A )
<=> ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ B @ D ) @ A ) )
=> ( C = D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_funct_1) ).
thf(1403,plain,
! [A: $i] :
( ( ( function @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ B @ D ) @ A ) )
=> ( C = D ) ) )
& ( ! [B: $i,C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ B @ C ) @ A )
& ( in @ ( ordered_pair @ B @ D ) @ A ) )
=> ( C = D ) )
=> ( function @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(9,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(545,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)],[9]) ).
thf(15,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(562,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)],[15]) ).
thf(87,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(904,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(194,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(1615,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(212,axiom,
! [A: $i,B: $i] :
( ( finite @ B )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc10_finset_1) ).
thf(1784,plain,
! [A: $i,B: $i] :
( ( finite @ B )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(180,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(1520,plain,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(367,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ( ( in @ E @ ( powerset @ ( relation_dom @ C ) ) )
& ( in @ ( relation_image @ C @ E ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e6_27__finset_1) ).
thf(2817,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
& ( relation @ C )
& ( function @ C ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ( ( in @ E @ ( powerset @ ( relation_dom @ C ) ) )
& ( in @ ( relation_image @ C @ E ) @ B ) ) )
& ( ( ( in @ E @ ( powerset @ ( relation_dom @ C ) ) )
& ( in @ ( relation_image @ C @ E ) @ B ) )
=> ( in @ E @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).
thf(439,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( ( topological_space @ A )
<=> ( ( in @ ( the_carrier @ A ) @ ( the_topology @ A ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( subset @ B @ ( the_topology @ A ) )
=> ( in @ ( union_of_subsets @ ( the_carrier @ A ) @ B ) @ ( the_topology @ A ) ) ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ B @ ( the_topology @ A ) )
& ( in @ C @ ( the_topology @ A ) ) )
=> ( in @ ( subset_intersection2 @ ( the_carrier @ A ) @ B @ C ) @ ( the_topology @ A ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_pre_topc) ).
thf(3340,plain,
! [A: $i] :
( ( top_str @ A )
=> ( ( ( topological_space @ A )
=> ( ( in @ ( the_carrier @ A ) @ ( the_topology @ A ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( subset @ B @ ( the_topology @ A ) )
=> ( in @ ( union_of_subsets @ ( the_carrier @ A ) @ B ) @ ( the_topology @ A ) ) ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ B @ ( the_topology @ A ) )
& ( in @ C @ ( the_topology @ A ) ) )
=> ( in @ ( subset_intersection2 @ ( the_carrier @ A ) @ B @ C ) @ ( the_topology @ A ) ) ) ) ) ) )
& ( ( ( in @ ( the_carrier @ A ) @ ( the_topology @ A ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( subset @ B @ ( the_topology @ A ) )
=> ( in @ ( union_of_subsets @ ( the_carrier @ A ) @ B ) @ ( the_topology @ A ) ) ) )
& ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ B @ ( the_topology @ A ) )
& ( in @ C @ ( the_topology @ A ) ) )
=> ( in @ ( subset_intersection2 @ ( the_carrier @ A ) @ B @ C ) @ ( the_topology @ A ) ) ) ) ) )
=> ( topological_space @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).
thf(447,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k4_lattices) ).
thf(3416,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).
thf(311,axiom,
! [A: $i] :
( ( ( empty @ A )
& ( ordinal @ A ) )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_arytm_3) ).
thf(2474,plain,
! [A: $i] :
( ( ( empty @ A )
& ( ordinal @ A ) )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(198,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(1675,plain,
! [A: $i] :
( ( set_intersection2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(409,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(3023,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)],[409]) ).
thf(269,axiom,
! [A: $i,B: $i] :
? [C: $i] :
( ( relation_of2 @ C @ A @ B )
& ( relation @ C )
& ( function @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_partfun1) ).
thf(2193,plain,
! [A: $i,B: $i] :
? [C: $i] :
( ( relation_of2 @ C @ A @ B )
& ( relation @ C )
& ( function @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(213,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(1786,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)],[213]) ).
thf(493,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( below @ A @ B @ C )
<=> ( ( join @ A @ B @ C )
= C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_lattices) ).
thf(3663,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below @ A @ B @ C )
=> ( ( join @ A @ B @ C )
= C ) )
& ( ( ( join @ A @ B @ C )
= C )
=> ( below @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).
thf(223,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(1865,plain,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(403,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(3002,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)],[403]) ).
thf(201,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(1733,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)],[201]) ).
thf(113,axiom,
! [A: $i] :
( ( epsilon_transitive @ A )
<=> ! [B: $i] :
( ( in @ B @ A )
=> ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_ordinal1) ).
thf(1090,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)],[113]) ).
thf(486,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(3619,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)],[486]) ).
thf(475,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( is_a_cover_of_carrier @ A @ B )
& ( B = empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_tops_2) ).
thf(3579,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( is_a_cover_of_carrier @ A @ B )
& ( B = empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).
thf(327,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(2573,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(510,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( join @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_lattices) ).
thf(3719,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( join @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).
thf(426,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B )
& ( C = E )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ E ) @ B ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e2_37_1_1__pre_topc__1) ).
thf(3155,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B )
& ( C = E )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ E ) @ B ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).
thf(400,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty_carrier_subset @ A )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_pre_topc) ).
thf(2987,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty_carrier_subset @ A )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).
thf(185,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(1574,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)],[185]) ).
thf(359,axiom,
( ! [A: $i] :
( ( ordinal @ A )
=> ( ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( ( in @ B @ omega )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ B ) ) )
=> ~ ( ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ! [E: $i] :
( ( ( in @ E @ C )
& ( subset @ D @ E ) )
=> ( E = D ) ) ) ) ) ) ) )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) )
=> ! [A: $i] :
( ( ordinal @ A )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s2_ordinal1__e18_27__finset_1__1) ).
thf(2774,plain,
( ! [A: $i] :
( ( ordinal @ A )
=> ( ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( ( in @ B @ omega )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ B ) ) )
=> ~ ( ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ! [E: $i] :
( ( ( in @ E @ C )
& ( subset @ D @ E ) )
=> ( E = D ) ) ) ) ) ) ) )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) )
=> ! [A: $i] :
( ( ordinal @ A )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).
thf(448,axiom,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(3419,plain,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).
thf(231,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(1933,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(191,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(1603,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(17,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(567,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(258,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
<=> ( are_equipotent @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r2_wellord2) ).
thf(2116,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)],[258]) ).
thf(507,axiom,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_boole) ).
thf(3710,plain,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).
thf(159,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(1376,plain,
! [A: $i] :
( ! [B: $i] :
( ( in @ B @ A )
=> ( ( ordinal @ B )
& ( subset @ B @ A ) ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(270,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(2197,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)],[270]) ).
thf(414,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( in @ F @ B )
& ( D
= ( set_difference @ F @ ( singleton @ A ) ) ) )
& ( C = E )
& ? [F: $i] :
( ( in @ F @ B )
& ( E
= ( set_difference @ F @ ( singleton @ A ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( powerset @ A ) )
& ( E = D )
& ? [F: $i] :
( ( in @ F @ B )
& ( D
= ( set_difference @ F @ ( singleton @ A ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e4_27_3_1__finset_1__1) ).
thf(3038,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( in @ F @ B )
& ( D
= ( set_difference @ F @ ( singleton @ A ) ) ) )
& ( C = E )
& ? [F: $i] :
( ( in @ F @ B )
& ( E
= ( set_difference @ F @ ( singleton @ A ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( powerset @ A ) )
& ( E = D )
& ? [F: $i] :
( ( in @ F @ B )
& ( D
= ( set_difference @ F @ ( singleton @ A ) ) ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( powerset @ A ) )
& ( E = D )
& ? [F: $i] :
( ( in @ F @ B )
& ( D
= ( set_difference @ F @ ( singleton @ A ) ) ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).
thf(303,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(2422,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)],[303]) ).
thf(379,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(2896,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(477,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(3586,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)],[477]) ).
thf(73,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(811,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)],[73]) ).
thf(226,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(1877,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)],[226]) ).
thf(365,axiom,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(2812,plain,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).
thf(353,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(2742,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)],[353]) ).
thf(423,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(3133,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)],[423]) ).
thf(494,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(3668,plain,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).
thf(257,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(2111,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)],[257]) ).
thf(474,axiom,
( ( empty @ empty_set )
& ( v1_membered @ empty_set )
& ( v2_membered @ empty_set )
& ( v3_membered @ empty_set )
& ( v4_membered @ empty_set )
& ( v5_membered @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc6_membered) ).
thf(3572,plain,
( ( empty @ empty_set )
& ( v1_membered @ empty_set )
& ( v2_membered @ empty_set )
& ( v3_membered @ empty_set )
& ( v4_membered @ empty_set )
& ( v5_membered @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).
thf(305,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(2446,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)],[305]) ).
thf(80,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(848,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)],[80]) ).
thf(412,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( open_subset @ ( interior @ A @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc6_tops_1) ).
thf(3034,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( open_subset @ ( interior @ A @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).
thf(476,axiom,
! [A: $i] :
( ( v2_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc11_membered) ).
thf(3583,plain,
! [A: $i] :
( ( v2_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).
thf(288,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_relat_1) ).
thf(2289,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(124,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(1177,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(383,axiom,
empty @ empty_set,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(2911,plain,
empty @ empty_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
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(2578,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(62,axiom,
? [A: $i] : ( top_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_pre_topc) ).
thf(758,plain,
? [A: $i] : ( top_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(336,axiom,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) )
& ( v4_membered @ ( set_difference @ A @ B ) )
& ( v5_membered @ ( set_difference @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc41_membered) ).
thf(2610,plain,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) )
& ( v4_membered @ ( set_difference @ A @ B ) )
& ( v5_membered @ ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(111,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(1061,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)],[111]) ).
thf(236,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(1952,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)],[236]) ).
thf(46,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_wellord1) ).
thf(685,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(468,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( closed_subset @ ( topstr_closure @ A @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_tops_1) ).
thf(3539,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( closed_subset @ ( topstr_closure @ A @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).
thf(163,axiom,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= ( singleton @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s3_funct_1__e16_22__wellord2) ).
thf(1396,plain,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_dom @ B )
= A )
& ! [C: $i] :
( ( in @ C @ A )
=> ( ( apply @ B @ C )
= ( singleton @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(275,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(2212,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)],[275]) ).
thf(23,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(608,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)],[23]) ).
thf(116,axiom,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( in @ B @ A )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ! [C: $i,D: $i] :
( ( in @ ( ordered_pair @ C @ D ) @ B )
<=> ( ( in @ C @ A )
& ( in @ C @ A )
& ( D
= ( singleton @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_funct_1__e16_22__wellord2__1) ).
thf(1113,plain,
! [A: $i] :
( ! [B: $i,C: $i,D: $i] :
( ( ( in @ B @ A )
& ( C
= ( singleton @ B ) )
& ( in @ B @ A )
& ( D
= ( singleton @ B ) ) )
=> ( C = D ) )
=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ! [C: $i,D: $i] :
( ( ( in @ ( ordered_pair @ C @ D ) @ B )
=> ( ( in @ C @ A )
& ( in @ C @ A )
& ( D
= ( singleton @ C ) ) ) )
& ( ( ( in @ C @ A )
& ( in @ C @ A )
& ( D
= ( singleton @ C ) ) )
=> ( in @ ( ordered_pair @ C @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(322,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(2535,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)],[322]) ).
thf(262,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc6_relat_1) ).
thf(2156,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(273,axiom,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(2207,plain,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(37,axiom,
! [A: $i,B: $i] :
~ ( proper_subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(653,plain,
! [A: $i] :
~ ( proper_subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(399,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( interior @ A @ B )
= ( subset_complement @ ( the_carrier @ A ) @ ( topstr_closure @ A @ ( subset_complement @ ( the_carrier @ A ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_tops_1) ).
thf(2984,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( interior @ A @ B )
= ( subset_complement @ ( the_carrier @ A ) @ ( topstr_closure @ A @ ( subset_complement @ ( the_carrier @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).
thf(39,axiom,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_relat_1) ).
thf(658,plain,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(358,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( powerset @ A ) )
& ? [E: $i] :
( ( in @ E @ B )
& ( D
= ( set_difference @ E @ ( singleton @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e4_27_3_1__finset_1) ).
thf(2765,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( powerset @ A ) )
& ? [E: $i] :
( ( in @ E @ B )
& ( D
= ( set_difference @ E @ ( singleton @ A ) ) ) ) ) )
& ( ( ( in @ D @ ( powerset @ A ) )
& ? [E: $i] :
( ( in @ E @ B )
& ( D
= ( set_difference @ E @ ( singleton @ A ) ) ) ) )
=> ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).
thf(31,axiom,
! [A: $i,B: $i] : ( equipotent @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r2_wellord2) ).
thf(636,plain,
! [A: $i] : ( equipotent @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(50,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r2_wellord2) ).
thf(695,plain,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(136,axiom,
! [A: $i] :
( ( ordinal @ A )
<=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_ordinal1) ).
thf(1245,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)],[136]) ).
thf(313,axiom,
? [A: $i] : ( latt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l3_lattices) ).
thf(2481,plain,
? [A: $i] : ( latt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(81,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc8_relat_1) ).
thf(851,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(115,axiom,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) )
& ( v4_membered @ ( set_intersection2 @ B @ A ) )
& ( v5_membered @ ( set_intersection2 @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc36_membered) ).
thf(1102,plain,
! [A: $i,B: $i] :
( ( v5_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) )
& ( v4_membered @ ( set_intersection2 @ B @ A ) )
& ( v5_membered @ ( set_intersection2 @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(58,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(722,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)],[58]) ).
thf(411,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( closed_subset @ B @ A )
<=> ( open_subset @ ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_pre_topc) ).
thf(3031,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ B @ A )
=> ( open_subset @ ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) @ A ) )
& ( ( open_subset @ ( subset_difference @ ( the_carrier @ A ) @ ( cast_as_carrier_subset @ A ) @ B ) @ A )
=> ( closed_subset @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).
thf(256,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(2096,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)],[256]) ).
thf(450,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( closed_subset @ C @ A ) ) )
=> ( closed_subset @ ( meet_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t44_pre_topc) ).
thf(3427,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( closed_subset @ C @ A ) ) )
=> ( closed_subset @ ( meet_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).
thf(207,axiom,
! [A: $i] :
( ( empty @ A )
=> ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_finset_1) ).
thf(1758,plain,
! [A: $i] :
( ( empty @ A )
=> ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(44,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_wellord2) ).
thf(680,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(150,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_pre_topc) ).
thf(1340,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(458,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = D )
& ( closed_subset @ F @ A )
& ( subset @ B @ D ) )
& ( C = E )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = E )
& ( closed_subset @ F @ A )
& ( subset @ B @ E ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = D )
& ( closed_subset @ F @ A )
& ( subset @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e1_40__pre_topc__1) ).
thf(3455,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( C = D )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = D )
& ( closed_subset @ F @ A )
& ( subset @ B @ D ) )
& ( C = E )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = E )
& ( closed_subset @ F @ A )
& ( subset @ B @ E ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = D )
& ( closed_subset @ F @ A )
& ( subset @ B @ D ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( powerset @ ( the_carrier @ A ) ) )
& ( E = D )
& ? [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
& ( F = D )
& ( closed_subset @ F @ A )
& ( subset @ B @ D ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).
thf(380,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(2899,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)],[380]) ).
thf(109,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(1049,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)],[109]) ).
thf(41,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(663,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)],[41]) ).
thf(310,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_relat_1) ).
thf(2471,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(10,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k4_relat_1) ).
thf(548,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(120,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(1145,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)],[120]) ).
thf(103,axiom,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(1018,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(65,axiom,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc39_membered) ).
thf(776,plain,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(210,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(1770,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)],[210]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t24_wellord1) ).
thf(640,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(462,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ D @ C )
<=> ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) ) ) )
& ( ( topstr_closure @ A @ B )
= ( meet_of_subsets @ ( the_carrier @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t46_pre_topc) ).
thf(3507,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ D @ C )
=> ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) ) )
& ( ( ( closed_subset @ D @ A )
& ( subset @ B @ D ) )
=> ( in @ D @ C ) ) ) )
& ( ( topstr_closure @ A @ B )
= ( meet_of_subsets @ ( the_carrier @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).
thf(345,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(2661,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)],[345]) ).
thf(192,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(1605,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)],[192]) ).
thf(492,axiom,
! [A: $i] :
( ( v3_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B )
& ( v1_rat_1 @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc12_membered) ).
thf(3659,plain,
! [A: $i] :
( ( v3_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( v1_xcmplx_0 @ B )
& ( v1_xreal_0 @ B )
& ( v1_rat_1 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).
thf(141,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(1267,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)],[141]) ).
thf(483,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( open_subset @ B @ A )
& ( closed_subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_tops_1) ).
thf(3608,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ( open_subset @ B @ A )
& ( closed_subset @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).
thf(157,axiom,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc29_membered) ).
thf(1368,plain,
! [A: $i,B: $i] :
( ( v2_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(266,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(2175,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)],[266]) ).
thf(488,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(3642,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)],[488]) ).
thf(43,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(673,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)],[43]) ).
thf(437,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(3334,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)],[437]) ).
thf(67,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_struct_0) ).
thf(791,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(348,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(2700,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)],[348]) ).
thf(277,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(2219,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_inverse_image @ B @ A ) @ ( relation_dom @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(470,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(3544,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)],[470]) ).
thf(325,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(2545,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)],[325]) ).
thf(428,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ! [B: $i,C: $i,D: $i] :
( ( ( B = C )
& ? [E: $i] :
( ( ordinal @ E )
& ( C = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) )
& ( B = D )
& ? [E: $i] :
( ( ordinal @ E )
& ( D = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( in @ C @ B )
<=> ? [D: $i] :
( ( in @ D @ ( succ @ A ) )
& ( D = C )
& ? [E: $i] :
( ( ordinal @ E )
& ( C = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e18_27__finset_1__1) ).
thf(3174,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ! [B: $i,C: $i,D: $i] :
( ( ( B = C )
& ? [E: $i] :
( ( ordinal @ E )
& ( C = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) )
& ( B = D )
& ? [E: $i] :
( ( ordinal @ E )
& ( D = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) ) )
=> ( C = D ) )
=> ? [B: $i] :
! [C: $i] :
( ( ( in @ C @ B )
=> ? [D: $i] :
( ( in @ D @ ( succ @ A ) )
& ( D = C )
& ? [E: $i] :
( ( ordinal @ E )
& ( C = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) ) ) )
& ( ? [D: $i] :
( ( in @ D @ ( succ @ A ) )
& ( D = C )
& ? [E: $i] :
( ( ordinal @ E )
& ( C = E )
& ( ( in @ E @ omega )
=> ! [F: $i] :
( ( element @ F @ ( powerset @ ( powerset @ E ) ) )
=> ~ ( ( F != empty_set )
& ! [G: $i] :
~ ( ( in @ G @ F )
& ! [H: $i] :
( ( ( in @ H @ F )
& ( subset @ G @ H ) )
=> ( H = G ) ) ) ) ) ) ) )
=> ( in @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).
thf(117,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_arytm_3) ).
thf(1130,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(419,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k2_binop_1) ).
thf(3095,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).
thf(88,axiom,
! [A: $i] :
( ( finite @ A )
<=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_rng @ B )
= A )
& ( in @ ( relation_dom @ B ) @ omega ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_finset_1) ).
thf(907,plain,
! [A: $i] :
( ( ( finite @ A )
=> ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_rng @ B )
= A )
& ( in @ ( relation_dom @ B ) @ omega ) ) )
& ( ? [B: $i] :
( ( relation @ B )
& ( function @ B )
& ( ( relation_rng @ B )
= A )
& ( in @ ( relation_dom @ B ) @ omega ) )
=> ( finite @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(299,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(2411,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)],[299]) ).
thf(183,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ A )
& ( in @ E @ A )
& ? [F: $i] :
( ( E = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e10_24__wellord2__1) ).
thf(1536,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ A )
& ( in @ E @ A )
& ? [F: $i] :
( ( E = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ A )
& ( in @ E @ A )
& ? [F: $i] :
( ( E = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(298,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ! [D: $i] :
~ ? [E: $i] :
( ( C = E )
& ( in @ D @ E )
& ! [F: $i] :
( ( in @ F @ E )
=> ( in @ ( ordered_pair @ D @ F ) @ B ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= A )
& ! [D: $i] :
( ( in @ D @ A )
=> ? [E: $i] :
( ( D = E )
& ( in @ ( apply @ C @ D ) @ E )
& ! [F: $i] :
( ( in @ F @ E )
=> ( in @ ( ordered_pair @ ( apply @ C @ D ) @ F ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s2_funct_1__e10_24__wellord2) ).
thf(2379,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ A )
& ? [F: $i] :
( ( C = F )
& ( in @ D @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ D @ G ) @ B ) ) )
& ? [F: $i] :
( ( C = F )
& ( in @ E @ F )
& ! [G: $i] :
( ( in @ G @ F )
=> ( in @ ( ordered_pair @ E @ G ) @ B ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ A )
& ! [D: $i] :
~ ? [E: $i] :
( ( C = E )
& ( in @ D @ E )
& ! [F: $i] :
( ( in @ F @ E )
=> ( in @ ( ordered_pair @ D @ F ) @ B ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= A )
& ! [D: $i] :
( ( in @ D @ A )
=> ? [E: $i] :
( ( D = E )
& ( in @ ( apply @ C @ D ) @ E )
& ! [F: $i] :
( ( in @ F @ E )
=> ( in @ ( ordered_pair @ ( apply @ C @ D ) @ F ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(142,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(1284,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)],[142]) ).
thf(241,axiom,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_funct_1) ).
thf(1972,plain,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(215,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(1797,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)],[215]) ).
thf(19,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(595,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(405,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(3007,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)],[405]) ).
thf(331,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(2584,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(500,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(3684,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)],[500]) ).
thf(98,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(991,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)],[98]) ).
thf(222,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ( ( in @ E @ ( cartesian_product2 @ A @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ A )
& ? [H: $i] :
( ( F = H )
& ( in @ G @ H )
& ! [I: $i] :
( ( in @ I @ H )
=> ( in @ ( ordered_pair @ G @ I ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e10_24__wellord2__1) ).
thf(1849,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation @ B ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ( ( in @ E @ ( cartesian_product2 @ A @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ A )
& ? [H: $i] :
( ( F = H )
& ( in @ G @ H )
& ! [I: $i] :
( ( in @ I @ H )
=> ( in @ ( ordered_pair @ G @ I ) @ B ) ) ) ) ) )
& ( ( ( in @ E @ ( cartesian_product2 @ A @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ A )
& ? [H: $i] :
( ( F = H )
& ( in @ G @ H )
& ! [I: $i] :
( ( in @ I @ H )
=> ( in @ ( ordered_pair @ G @ I ) @ B ) ) ) ) )
=> ( in @ E @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(139,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(1257,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)],[139]) ).
thf(372,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( closed_subset @ B @ A )
<=> ( open_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t29_tops_1) ).
thf(2838,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( closed_subset @ B @ A )
=> ( open_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) )
& ( ( open_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A )
=> ( closed_subset @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(53,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(703,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)],[53]) ).
thf(38,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(656,plain,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(105,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(1023,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)],[105]) ).
thf(153,axiom,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_ordinal1) ).
thf(1353,plain,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(237,axiom,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc32_membered) ).
thf(1955,plain,
! [A: $i,B: $i] :
( ( v3_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ B @ A ) )
& ( v2_membered @ ( set_intersection2 @ B @ A ) )
& ( v3_membered @ ( set_intersection2 @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(390,axiom,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(2949,plain,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(100,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(1008,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)],[100]) ).
thf(363,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(2807,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(24,axiom,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(612,plain,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(464,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B )
& ( relation @ B )
& ( function @ B )
& ( one_to_one @ B )
& ( epsilon_transitive @ B )
& ( epsilon_connected @ B )
& ( ordinal @ B )
& ( natural @ B )
& ( finite @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_finset_1) ).
thf(3520,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B )
& ( relation @ B )
& ( function @ B )
& ( one_to_one @ B )
& ( epsilon_transitive @ B )
& ( epsilon_connected @ B )
& ( ordinal @ B )
& ( natural @ B )
& ( finite @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).
thf(442,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ( ( ( B = empty_set )
=> ( A = empty_set ) )
=> ( ( quasi_total @ C @ A @ B )
<=> ( A
= ( relation_dom_as_subset @ A @ B @ C ) ) ) )
& ( ( B = empty_set )
=> ( ( A = empty_set )
| ( ( quasi_total @ C @ A @ B )
<=> ( C = empty_set ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_funct_2) ).
thf(3366,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( ( ( ( B = empty_set )
=> ( A = empty_set ) )
=> ( ( ( quasi_total @ C @ A @ B )
=> ( A
= ( relation_dom_as_subset @ A @ B @ C ) ) )
& ( ( A
= ( relation_dom_as_subset @ A @ B @ C ) )
=> ( quasi_total @ C @ A @ B ) ) ) )
& ( ( B = empty_set )
=> ( ( A = empty_set )
| ( ( ( quasi_total @ C @ A @ B )
=> ( C = empty_set ) )
& ( ( C = empty_set )
=> ( quasi_total @ C @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).
thf(312,axiom,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(2479,plain,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(125,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(1180,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)],[125]) ).
thf(219,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(1817,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)],[219]) ).
thf(179,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc7_relat_1) ).
thf(1517,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(267,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(2187,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(279,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(2226,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)],[279]) ).
thf(344,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(2658,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(431,axiom,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( interior @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_tops_1) ).
thf(3284,plain,
! [A: $i,B: $i] :
( ( ( top_str @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( element @ ( interior @ A @ B ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).
thf(79,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_relat_1) ).
thf(845,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(357,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(2760,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)],[357]) ).
thf(167,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(1421,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(387,axiom,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_relat_1) ).
thf(2924,plain,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).
thf(457,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(3452,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)],[457]) ).
thf(27,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(627,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_ordering @ B )
=> ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(143,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(1316,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)],[143]) ).
thf(197,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(1649,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)],[197]) ).
thf(252,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t15_finset_1) ).
thf(2045,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(418,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(3092,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).
thf(466,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k5_subset_1) ).
thf(3533,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_intersection2 @ A @ B @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).
thf(456,axiom,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( top_str @ B )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ B ) ) )
=> ( ( ( open_subset @ D @ B )
=> ( ( interior @ B @ D )
= D ) )
& ( ( ( interior @ A @ C )
= C )
=> ( open_subset @ C @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t55_tops_1) ).
thf(3447,plain,
! [A: $i] :
( ( ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( top_str @ B )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ B ) ) )
=> ( ( ( open_subset @ D @ B )
=> ( ( interior @ B @ D )
= D ) )
& ( ( ( interior @ A @ C )
= C )
=> ( open_subset @ C @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).
thf(196,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(1646,plain,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(145,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(1326,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)],[145]) ).
thf(52,axiom,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(699,plain,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(94,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(961,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)],[94]) ).
thf(175,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(1476,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)],[175]) ).
thf(459,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(3484,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)],[459]) ).
thf(321,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(2532,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)],[321]) ).
thf(443,axiom,
! [A: $i] :
( ( A = omega )
<=> ( ( in @ empty_set @ A )
& ( being_limit_ordinal @ A )
& ( ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( ( ( in @ empty_set @ B )
& ( being_limit_ordinal @ B ) )
=> ( subset @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_ordinal2) ).
thf(3383,plain,
! [A: $i] :
( ( ( A = omega )
=> ( ( in @ empty_set @ A )
& ( being_limit_ordinal @ A )
& ( ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( ( ( in @ empty_set @ B )
& ( being_limit_ordinal @ B ) )
=> ( subset @ A @ B ) ) ) ) )
& ( ( ( in @ empty_set @ A )
& ( being_limit_ordinal @ A )
& ( ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( ( ( in @ empty_set @ B )
& ( being_limit_ordinal @ B ) )
=> ( subset @ A @ B ) ) ) )
=> ( A = omega ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).
thf(438,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( epsilon_transitive @ B )
& ( epsilon_connected @ B )
& ( ordinal @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_arytm_3) ).
thf(3336,plain,
! [A: $i] :
( ( ordinal @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( ( epsilon_transitive @ B )
& ( epsilon_connected @ B )
& ( ordinal @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).
thf(368,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(2823,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)],[368]) ).
thf(29,axiom,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_wellord2) ).
thf(632,plain,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(430,axiom,
( ( ( ( in @ empty_set @ omega )
=> ! [A: $i] :
( ( element @ A @ ( powerset @ ( powerset @ empty_set ) ) )
=> ~ ( ( A != empty_set )
& ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i] :
( ( ( in @ C @ A )
& ( subset @ B @ C ) )
=> ( C = B ) ) ) ) ) )
& ! [A: $i] :
( ( ordinal @ A )
=> ( ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) )
=> ( ( in @ ( succ @ A ) @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) )
& ! [A: $i] :
( ( ordinal @ A )
=> ( ( ( being_limit_ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( ( in @ B @ omega )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ B ) ) )
=> ~ ( ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ! [E: $i] :
( ( ( in @ E @ C )
& ( subset @ D @ E ) )
=> ( E = D ) ) ) ) ) ) ) ) )
=> ( ( A = empty_set )
| ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ) ) )
=> ! [A: $i] :
( ( ordinal @ A )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_ordinal2__e18_27__finset_1) ).
thf(3235,plain,
( ( ( ( in @ empty_set @ omega )
=> ! [A: $i] :
( ( element @ A @ ( powerset @ ( powerset @ empty_set ) ) )
=> ~ ( ( A != empty_set )
& ! [B: $i] :
~ ( ( in @ B @ A )
& ! [C: $i] :
( ( ( in @ C @ A )
& ( subset @ B @ C ) )
=> ( C = B ) ) ) ) ) )
& ! [A: $i] :
( ( ordinal @ A )
=> ( ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) )
=> ( ( in @ ( succ @ A ) @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( succ @ A ) ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) )
& ! [A: $i] :
( ( ordinal @ A )
=> ( ( ( being_limit_ordinal @ A )
& ! [B: $i] :
( ( ordinal @ B )
=> ( ( in @ B @ A )
=> ( ( in @ B @ omega )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ B ) ) )
=> ~ ( ( C != empty_set )
& ! [D: $i] :
~ ( ( in @ D @ C )
& ! [E: $i] :
( ( ( in @ E @ C )
& ( subset @ D @ E ) )
=> ( E = D ) ) ) ) ) ) ) ) )
=> ( ( A = empty_set )
| ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ) ) )
=> ! [A: $i] :
( ( ordinal @ A )
=> ( ( in @ A @ omega )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ~ ( ( B != empty_set )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
( ( ( in @ D @ B )
& ( subset @ C @ D ) )
=> ( D = C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).
thf(181,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(1523,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)],[181]) ).
thf(261,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(2152,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)],[261]) ).
thf(59,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(731,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)],[59]) ).
thf(436,axiom,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_boole) ).
thf(3331,plain,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).
thf(247,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(2009,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)],[247]) ).
thf(393,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(2956,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)],[393]) ).
thf(384,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc5_struct_0) ).
thf(2912,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).
thf(497,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( element @ ( empty_carrier_subset @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_pre_topc) ).
thf(3677,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( element @ ( empty_carrier_subset @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).
thf(381,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(2902,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)],[381]) ).
thf(77,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(832,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)],[77]) ).
thf(489,axiom,
! [A: $i] : ( subset @ empty_set @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(3645,plain,
! [A: $i] : ( subset @ empty_set @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).
thf(352,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(2739,plain,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).
thf(70,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(800,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)],[70]) ).
thf(480,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( is_a_cover_of_carrier @ A @ B )
<=> ( ( cast_as_carrier_subset @ A )
= ( union_of_subsets @ ( the_carrier @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_pre_topc) ).
thf(3598,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( is_a_cover_of_carrier @ A @ B )
=> ( ( cast_as_carrier_subset @ A )
= ( union_of_subsets @ ( the_carrier @ A ) @ B ) ) )
& ( ( ( cast_as_carrier_subset @ A )
= ( union_of_subsets @ ( the_carrier @ A ) @ B ) )
=> ( is_a_cover_of_carrier @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).
thf(307,axiom,
! [A: $i,B: $i] : ( subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(2462,plain,
! [A: $i] : ( subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(386,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( meet_absorbing @ A )
<=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( join @ A @ ( meet @ A @ B @ C ) @ C )
= C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_lattices) ).
thf(2917,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( ( meet_absorbing @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( join @ A @ ( meet @ A @ B @ C ) @ C )
= C ) ) ) )
& ( ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( join @ A @ ( meet @ A @ B @ C ) @ C )
= C ) ) )
=> ( meet_absorbing @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(338,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(2623,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)],[338]) ).
thf(60,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(752,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(199,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(1678,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)],[199]) ).
thf(429,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( join @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_join @ A ) @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_lattices) ).
thf(3232,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( join @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_join @ A ) @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).
thf(114,axiom,
! [A: $i,B: $i] :
? [C: $i] :
( ( relation_of2 @ C @ A @ B )
& ( relation @ C )
& ( function @ C )
& ( quasi_total @ C @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_funct_2) ).
thf(1097,plain,
! [A: $i,B: $i] :
? [C: $i] :
( ( relation_of2 @ C @ A @ B )
& ( relation @ C )
& ( function @ C )
& ( quasi_total @ C @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(391,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( element @ ( cast_as_carrier_subset @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_pre_topc) ).
thf(2951,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( element @ ( cast_as_carrier_subset @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).
thf(254,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(2050,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)],[254]) ).
thf(503,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(3690,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)],[503]) ).
thf(366,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( element @ ( apply_binary_as_element @ A @ B @ C @ D @ E @ F ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_binop_1) ).
thf(2815,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( element @ ( apply_binary_as_element @ A @ B @ C @ D @ E @ F ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(351,axiom,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
& ( finite @ B ) )
=> ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t13_finset_1) ).
thf(2736,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
& ( finite @ B ) )
=> ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).
thf(82,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(854,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)],[82]) ).
thf(355,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( join_commut @ A @ B @ C )
= ( join @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k3_lattices) ).
thf(2751,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( join_commut @ A @ B @ C )
= ( join @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).
thf(435,axiom,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(3328,plain,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).
thf(271,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(2201,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)],[271]) ).
thf(296,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(2366,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)],[296]) ).
thf(76,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(828,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)],[76]) ).
thf(6,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(532,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)],[6]) ).
thf(92,axiom,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( finite @ B ) )
=> ( finite @ ( relation_image @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc13_finset_1) ).
thf(955,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( finite @ B ) )
=> ( finite @ ( relation_image @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(496,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_relat_1) ).
thf(3673,plain,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).
thf(85,axiom,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_ordinal1) ).
thf(886,plain,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(148,axiom,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(1335,plain,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(255,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(2070,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)],[255]) ).
thf(509,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( join_commut @ A @ B @ C )
= ( join_commut @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k3_lattices) ).
thf(3716,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( join_commut @ A @ B @ C )
= ( join_commut @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).
thf(123,axiom,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( being_limit_ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_ordinal2) ).
thf(1172,plain,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( being_limit_ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(64,axiom,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) )
& ( v4_membered @ ( set_intersection2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc33_membered) ).
thf(767,plain,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_intersection2 @ A @ B ) )
& ( v2_membered @ ( set_intersection2 @ A @ B ) )
& ( v3_membered @ ( set_intersection2 @ A @ B ) )
& ( v4_membered @ ( set_intersection2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(301,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc37_membered) ).
thf(2417,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(57,axiom,
? [A: $i] : ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(720,plain,
? [A: $i] : ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(295,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(2340,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)],[295]) ).
thf(347,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(2668,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)],[347]) ).
thf(121,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(1165,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)],[121]) ).
thf(133,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(1236,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)],[133]) ).
thf(243,axiom,
? [A: $i] : ( join_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l2_lattices) ).
thf(1977,plain,
? [A: $i] : ( join_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(395,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_lattices) ).
thf(2966,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).
thf(440,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( closed_subset @ B @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( open_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_tops_1) ).
thf(3360,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( closed_subset @ B @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( open_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).
thf(188,axiom,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) )
& ( v4_membered @ ( set_difference @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc40_membered) ).
thf(1582,plain,
! [A: $i,B: $i] :
( ( v4_membered @ A )
=> ( ( v1_membered @ ( set_difference @ A @ B ) )
& ( v2_membered @ ( set_difference @ A @ B ) )
& ( v3_membered @ ( set_difference @ A @ B ) )
& ( v4_membered @ ( set_difference @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(35,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(644,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)],[35]) ).
thf(408,axiom,
! [A: $i] :
( ( v2_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc17_membered) ).
thf(3020,plain,
! [A: $i] :
( ( v2_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( ( v1_membered @ B )
& ( v2_membered @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).
thf(74,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(822,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)],[74]) ).
thf(227,axiom,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l2_lattices) ).
thf(1880,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(460,axiom,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(3494,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).
thf(297,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(2372,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)],[297]) ).
thf(230,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(1930,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)],[230]) ).
thf(78,axiom,
! [A: $i] :
( ( ( ordinal @ A )
& ( natural @ A ) )
=> ( ~ ( empty @ ( succ @ A ) )
& ( epsilon_transitive @ ( succ @ A ) )
& ( epsilon_connected @ ( succ @ A ) )
& ( ordinal @ ( succ @ A ) )
& ( natural @ ( succ @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_arytm_3) ).
thf(839,plain,
! [A: $i] :
( ( ( ordinal @ A )
& ( natural @ A ) )
=> ( ~ ( empty @ ( succ @ A ) )
& ( epsilon_transitive @ ( succ @ A ) )
& ( epsilon_connected @ ( succ @ A ) )
& ( ordinal @ ( succ @ A ) )
& ( natural @ ( succ @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(441,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B )
& ( finite @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_finset_1) ).
thf(3362,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B )
& ( finite @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).
thf(350,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(2731,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)],[350]) ).
thf(445,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( join_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_lattices) ).
thf(3411,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( join_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).
thf(315,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(2485,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(161,axiom,
? [A: $i] : ( meet_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_lattices) ).
thf(1384,plain,
? [A: $i] : ( meet_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(482,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_intersection2 @ ( the_carrier @ A ) @ B @ ( cast_as_carrier_subset @ A ) )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t15_pre_topc) ).
thf(3605,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( subset_intersection2 @ ( the_carrier @ A ) @ B @ ( cast_as_carrier_subset @ A ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).
thf(356,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e2_37_1_1__pre_topc__1) ).
thf(2754,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) ) )
& ( ( ( in @ D @ ( powerset @ ( the_carrier @ A ) ) )
& ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) )
=> ( in @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(453,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(3437,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)],[453]) ).
thf(238,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(1962,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)],[238]) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_wellord1) ).
thf(543,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(61,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(755,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)],[61]) ).
thf(471,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(3553,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)],[471]) ).
thf(511,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( open_subset @ B @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( closed_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_tops_1) ).
thf(3721,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( open_subset @ B @ A )
& ( element @ B @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( closed_subset @ ( subset_complement @ ( the_carrier @ A ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).
thf(95,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(972,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)],[95]) ).
thf(173,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(1463,plain,
! [A: $i] :
( ( set_union2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(184,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(1565,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)],[184]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_wellord1) ).
thf(560,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(455,axiom,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ D @ C )
<=> ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s3_subset_1__e2_37_1_1__pre_topc) ).
thf(3443,plain,
! [A: $i,B: $i] :
( ( ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ? [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
& ! [D: $i] :
( ( element @ D @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( in @ D @ C )
=> ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B ) )
& ( ( in @ ( set_difference @ ( cast_as_carrier_subset @ A ) @ D ) @ B )
=> ( in @ D @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).
thf(4877,plain,
$false,
inference(e,[status(thm)],[1036,809,1337,629,1767,3434,3566,1950,2744,1190,2295,760,2280,3172,523,2210,1882,2644,1808,1618,2953,2248,3098,1591,3012,1205,878,3498,1793,2854,797,715,3693,1988,3004,2035,670,2575,2216,1243,709,1935,1386,565,1967,1254,3688,1746,3103,2603,1985,1760,2269,3061,3612,917,2466,1059,3617,2444,1264,1577,1455,2571,3713,2420,638,1379,2857,1013,2929,3014,2158,1773,2169,1087,1319,1704,1741,881,1489,3431,3075,1373,3295,3670,3702,1136,2291,2335,3569,2540,1843,3537,2833,634,2483,660,2809,2804,1600,3588,2205,2586,602,2964,1169,687,692,2190,3421,3036,2530,988,2286,529,793,2646,2641,2621,1868,3679,3151,2011,3531,1199,2264,957,2601,952,1487,2606,2032,3286,825,1343,1021,3603,2277,1355,1580,598,2147,3514,3682,1423,2464,1032,1365,2332,1457,1210,3028,2228,3647,3595,682,1195,551,2492,697,2990,1534,2973,1970,1006,1755,3440,1360,1350,3105,2300,2515,3413,948,2801,1804,615,2841,1328,1138,2537,783,2638,3143,1942,3000,554,2830,642,1937,2038,1744,706,1419,2460,2798,1739,1461,1414,526,1810,2172,1016,1207,3723,2666,1612,2649,1039,2968,3541,1451,2836,1974,2581,1874,3064,1979,2272,605,647,3656,2932,985,2487,2845,569,3409,2862,1333,1202,1241,805,541,1846,2338,888,1466,558,3639,2915,1515,2221,2121,2729,3707,513,2904,3289,689,3686,2894,1251,618,2413,2970,2048,1884,2489,1403,545,562,904,1615,1784,1520,2817,3340,3416,2474,1675,3023,2193,1786,3663,1865,3002,1733,1090,3619,3579,2573,3719,3155,2987,1574,2774,3419,1933,1603,567,2116,3710,1376,2197,3038,2422,2896,3586,811,1877,2812,2742,3133,3668,2111,3572,2446,848,3034,3583,2289,1177,2911,2578,758,2610,1061,1952,685,3539,1396,2212,608,1113,2535,2156,2207,653,2984,658,2765,636,695,1245,2481,851,1102,722,3031,2096,3427,1758,680,1340,3455,2899,1049,663,2471,548,1145,1018,776,1770,640,3507,2661,1605,3659,1267,3608,1368,2175,3642,673,3334,791,2700,2219,3544,2545,3174,1130,3095,907,2411,1536,2379,1284,1972,1797,595,3007,2584,3684,991,1849,1257,2838,703,656,1023,1353,1955,2949,1008,2807,612,3520,3366,2479,1180,1817,1517,2187,2226,2658,3284,845,2760,1421,2924,3452,627,1316,1649,2045,3092,3533,3447,1646,1326,699,961,1476,3484,2532,3383,3336,2823,632,3235,1523,2152,731,3331,2009,2956,2912,3677,2902,832,3645,2739,800,3598,2462,2917,2623,752,1678,3232,1097,2951,2050,3690,2815,2736,854,2751,3328,2201,2366,828,532,955,3673,886,1335,2070,3716,1172,767,2417,720,2340,2668,1165,1236,1977,2966,3360,1582,644,3020,822,1880,3494,2372,1930,839,3362,2731,3411,2485,1384,3605,2754,3437,1962,543,755,3553,3721,972,1463,1565,560,3443]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SEU326+2 : TPTP v8.1.2. Released v3.3.0.
% 0.11/0.15 % Command : run_Leo-III %s %d
% 0.15/0.35 % Computer : n001.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu May 18 13:25:35 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.84/0.83 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.51/1.06 % [INFO] Parsing done (231ms).
% 1.82/1.07 % [INFO] Running in sequential loop mode.
% 2.27/1.27 % [INFO] eprover registered as external prover.
% 2.27/1.27 % [INFO] cvc4 registered as external prover.
% 2.27/1.28 % [INFO] Scanning for conjecture ...
% 2.65/1.39 % [INFO] Found a conjecture and 538 axioms. Running axiom selection ...
% 3.25/1.51 % [INFO] Axiom selection finished. Selected 510 axioms (removed 28 axioms).
% 3.79/1.74 % [INFO] Problem is first-order (TPTP FOF).
% 4.05/1.77 % [INFO] Type checking passed.
% 4.05/1.77 % [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 ...
% 26.22/6.20 % External prover 'e' found a proof!
% 26.22/6.21 % [INFO] Killing All external provers ...
% 26.22/6.21 % Time passed: 5693ms (effective reasoning time: 5129ms)
% 26.22/6.21 % 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)>
% 26.22/6.21 % Axioms used in derivation (510): d1_lattices, t15_finset_1, d1_enumset1, redefinition_m2_relset_1, redefinition_r1_ordinal1, d1_wellord2, l3_wellord1, s3_funct_1__e16_22__wellord2, cc2_finset_1, fc1_zfmisc_1, d1_finset_1, d1_tarski, s1_xboole_0__e4_27_3_1__finset_1, t30_relat_1, t147_funct_1, t45_pre_topc, d4_funct_1, s1_tarski__e2_37_1_1__pre_topc__1, t32_wellord1, t19_xboole_1, t99_relat_1, s3_subset_1__e2_37_1_1__pre_topc, cc12_membered, t17_xboole_1, t90_relat_1, t43_subset_1, t54_wellord1, fc1_ordinal2, l82_funct_1, idempotence_k3_xboole_0, d4_ordinal1, t1_boole, t39_wellord1, fc2_pre_topc, fc37_membered, redefinition_k5_subset_1, t3_xboole_1, dt_k4_relset_1, dt_k5_relset_1, d5_relat_1, fc1_finsub_1, d6_relat_2, l1_wellord1, l2_wellord1, fc1_xboole_0, t7_mcart_1, cc10_membered, s1_tarski__e4_27_3_1__finset_1__1, dt_k2_subset_1, d5_tarski, d12_relat_1, s1_xboole_0__e18_27__finset_1__1, t44_relat_1, fc34_membered, existence_l1_lattices, cc2_arytm_3, t26_lattices, fc4_relat_1, fc4_funct_1, t1_subset, t144_relat_1, fc31_membered, t22_funct_1, existence_l3_lattices, rc1_membered, fc4_tops_1, fc3_relat_1, d8_funct_1, fc4_subset_1, dt_k1_tops_1, t117_relat_1, t2_boole, d5_ordinal2, fc6_tops_1, d13_pre_topc, d6_pre_topc, t12_pre_topc, t32_ordinal1, t47_relat_1, fc9_relat_1, l40_tops_1, l3_subset_1, t99_zfmisc_1, t47_setfam_1, fc1_funct_1, fc1_pre_topc, s1_tarski__e10_24__wellord2__2, d1_wellord1, rc2_partfun1, d9_funct_1, rc3_funct_1, d1_struct_0, t56_relat_1, t37_xboole_1, cc1_relat_1, idempotence_k5_subset_1, t55_funct_1, fc28_membered, dt_m2_relset_1, t18_finset_1, t21_ordinal1, t26_wellord2, redefinition_k6_setfam_1, dt_k1_pre_topc, commutativity_k3_xboole_0, s1_tarski__e6_22__wellord2__1, t62_funct_1, t46_pre_topc, d3_pre_topc, fc11_finset_1, d2_tarski, dt_k2_binop_1, d4_subset_1, rc1_funct_1, cc2_finsub_1, fc2_xboole_0, rc6_pre_topc, fc32_membered, fc30_membered, fc5_pre_topc, involutiveness_k3_subset_1, d1_xboole_0, l50_zfmisc_1, t23_funct_1, t46_funct_2, s1_tarski__e6_21__wellord2__1, dt_l1_lattices, t52_pre_topc, t64_relat_1, rc1_tops_1, d6_ordinal1, t118_zfmisc_1, fc10_finset_1, d3_wellord1, t14_relset_1, t26_finset_1, t16_relset_1, rc3_relat_1, t6_boole, t4_wellord2, existence_l2_lattices, dt_k6_subset_1, d1_relat_1, dt_l1_pre_topc, t34_funct_1, d13_relat_1, t46_zfmisc_1, rc2_subset_1, s1_xboole_0__e6_21__wellord2__1, fc9_finset_1, t1_xboole_1, d9_relat_2, s1_xboole_0__e2_37_1_1__pre_topc__1, t28_wellord2, t31_ordinal1, t116_relat_1, fc33_membered, cc2_ordinal1, commutativity_k2_xboole_0, fc3_subset_1, t5_subset, d4_relat_1, t4_xboole_0, d12_relat_2, cc1_arytm_3, existence_m1_relset_1, t1_zfmisc_1, dt_l3_lattices, fc2_tops_1, t17_wellord1, fc2_subset_1, d8_xboole_0, d4_wellord2, cc19_membered, commutativity_k5_subset_1, l25_zfmisc_1, t5_wellord2, d5_wellord1, d1_ordinal1, s1_relat_1__e6_21__wellord2, t3_wellord2, rc4_funct_1, dt_l2_lattices, fc2_funct_1, t44_pre_topc, t5_wellord1, t8_funct_1, fc6_membered, t166_relat_1, l32_xboole_1, t8_zfmisc_1, t23_wellord1, t2_tarski, s1_ordinal1__e8_6__wellord2, t33_xboole_1, t4_boole, d3_lattices, t146_funct_1, cc11_membered, d2_pre_topc, d13_funct_1, dt_k4_relat_1, d5_pre_topc, t4_subset, fc3_xboole_0, t178_relat_1, d2_xboole_0, dt_k2_funct_1, s3_subset_1__e1_40__pre_topc, t20_wellord1, rc1_arytm_3, t19_wellord1, fc13_relat_1, d4_tarski, t18_wellord1, d16_relat_2, t9_zfmisc_1, t174_relat_1, redefinition_k5_setfam_1, s1_tarski__e6_27__finset_1__1, involutiveness_k4_relat_1, cc1_ordinal1, d8_pre_topc, reflexivity_r2_wellord2, t30_tops_1, cc20_membered, t7_boole, t21_wellord1, cc1_funct_1, t23_ordinal1, s1_tarski__e10_24__wellord2__1, t118_relat_1, s2_ordinal1__e18_27__finset_1__1, s1_tarski__e16_22__wellord2__2, t143_relat_1, fc29_membered, t48_xboole_1, t3_ordinal1, dt_k1_wellord2, d10_relat_1, rc3_struct_0, d2_subset_1, commutativity_k4_lattices, fc1_subset_1, t33_zfmisc_1, fc5_funct_1, t13_finset_1, t24_wellord1, fc38_membered, l4_wellord1, fc3_ordinal1, rc2_funct_1, cc4_membered, d11_relat_1, connectedness_r1_ordinal1, t21_funct_2, t8_wellord1, t9_funct_2, t160_relat_1, fc10_relat_1, rc1_subset_1, cc3_ordinal1, t74_relat_1, rc1_funct_2, t2_wellord2, rc3_ordinal1, cc18_membered, s1_xboole_0__e1_40__pre_topc__1, t83_xboole_1, s1_xboole_0__e6_27__finset_1, t45_relat_1, l28_zfmisc_1, t140_relat_1, d14_relat_1, t55_tops_1, s1_funct_1__e10_24__wellord2__1, s1_xboole_0__e16_22__wellord2__1, d7_xboole_0, fc5_relat_1, cc2_funct_1, t46_setfam_1, d2_wellord1, t17_finset_1, l4_zfmisc_1, t3_subset, cc1_relset_1, t33_ordinal1, t115_relat_1, dt_u1_lattices, existence_m1_subset_1, d2_mcart_1, t22_relset_1, cc13_membered, t24_ordinal1, s1_ordinal2__e18_27__finset_1, antisymmetry_r2_xboole_0, dt_k6_setfam_1, d2_zfmisc_1, fc11_relat_1, l3_zfmisc_1, commutativity_k2_tarski, d7_relat_1, rc1_ordinal1, fc39_membered, t106_zfmisc_1, cc15_membered, t25_wellord1, t145_funct_1, fc1_finset_1, l71_subset_1, dt_k7_relat_1, t7_wellord2, l2_zfmisc_1, d8_relat_1, t42_ordinal1, t94_relat_1, t48_setfam_1, d4_wellord1, d5_subset_1, t92_zfmisc_1, t146_relat_1, t70_funct_1, t31_wellord1, d1_zfmisc_1, t69_enumset1, t41_ordinal1, d7_wellord1, dt_k5_subset_1, 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, redefinition_k2_binop_1, t17_pre_topc, s1_tarski__e16_22__wellord2__1, t6_funct_2, t51_tops_1, involutiveness_k7_setfam_1, t8_xboole_1, cc1_finsub_1, t37_relat_1, d3_tarski, fc1_struct_0, redefinition_k3_lattices, existence_l1_pre_topc, t39_zfmisc_1, t119_zfmisc_1, rc1_relat_1, t54_funct_1, rc2_finset_1, fc35_membered, dt_k5_relat_1, reflexivity_r1_tarski, t28_xboole_1, d1_relset_1, s1_xboole_0__e10_24__wellord2__1, t5_tops_2, dt_k4_lattices, t60_relat_1, t6_zfmisc_1, dt_k6_relat_1, reflexivity_r1_ordinal1, s1_tarski__e18_27__finset_1__1, t2_xboole_1, s1_xboole_0__e6_22__wellord2, fc36_membered, cc17_membered, fc6_relat_1, d1_funct_2, d12_funct_1, d3_ordinal1, d2_lattices, t46_relat_1, rc2_ordinal1, d4_xboole_0, d14_relat_2, t35_funct_1, t36_xboole_1, dt_k5_setfam_1, rc3_finset_1, t68_funct_1, t20_relat_1, dt_k1_lattices, rc1_ordinal2, t22_pre_topc, dt_k7_setfam_1, rc2_tops_1, t21_funct_1, fc2_arytm_3, d1_relat_2, fc1_relat_1, d6_wellord1, d6_relat_1, fc3_tops_1, l23_zfmisc_1, t48_pre_topc, t54_subset_1, rc1_finset_1, cc1_finset_1, s2_funct_1__e10_24__wellord2, fc12_finset_1, dt_k2_wellord1, s1_funct_1__e16_22__wellord2__1, t136_zfmisc_1, t8_boole, rc1_xboole_0, dt_u1_pre_topc, fc3_funct_1, t16_wellord1, rc5_struct_0, t12_xboole_1, fc7_relat_1, fc1_ordinal1, t49_wellord1, t86_relat_1, fc40_membered, s1_xboole_0__e8_6__wellord2__1, t10_zfmisc_1, t2_subset, cc1_membered, d3_relat_1, 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, s1_tarski__e1_40__pre_topc__1, fc8_relat_1, fc41_membered, idempotence_k2_xboole_0, d1_funct_1, l30_wellord2, t15_pre_topc, t9_tarski, t145_relat_1, cc3_arytm_3, commutativity_k3_lattices, d10_xboole_0, symmetry_r2_wellord2, t7_xboole_1, cc14_membered, cc2_membered, d8_lattices, cc3_membered, t50_subset_1, fc2_ordinal1, existence_l1_struct_0, t7_tarski, t60_xboole_1, rc1_partfun1, redefinition_k5_relset_1, t57_funct_1, t3_boole, d5_funct_1, redefinition_k6_subset_1, t12_relset_1, redefinition_k4_relset_1, dt_u2_lattices, d8_relat_2, dt_k2_lattices, d4_relat_2, t3_xboole_0, t25_relat_1, d1_tops_1, l29_wellord1, cc16_membered, fc13_finset_1, s2_funct_1__e16_22__wellord2__1, rc2_xboole_0, fc27_membered, fc4_ordinal1, t40_xboole_1, t45_xboole_1, t44_tops_1, d1_pre_topc, dt_k6_pre_topc, irreflexivity_r2_xboole_0, t53_wellord1, t65_zfmisc_1, t29_tops_1, d3_xboole_0, d2_relat_1, dt_k3_lattices, t71_relat_1, t23_relset_1, d1_mcart_1, t88_relat_1, t167_relat_1, dt_k3_subset_1, l1_zfmisc_1, t39_xboole_1, t23_lattices, t119_relat_1, antisymmetry_r2_hidden, redefinition_k4_lattices, dt_k2_pre_topc
% 26.22/6.21 % No. of inferences in proof: 1024
% 26.22/6.21 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 5693 ms resp. 5129 ms w/o parsing
% 26.72/6.43 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 26.72/6.44 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------