TSTP Solution File: SEU353+2 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SEU353+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n017.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 11:58:27 EDT 2023
% Result : Theorem 123.93s 24.58s
% Output : Refutation 124.87s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 798
% Syntax : Number of formulae : 1434 ( 103 unt; 164 typ; 0 def)
% Number of atoms : 6940 (1228 equ; 0 cnn)
% Maximal formula atoms : 49 ( 5 avg)
% Number of connectives : 21455 ( 678 ~; 34 |;2973 &;15107 @)
% ( 215 <=>;2448 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 9 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 281 ( 281 >; 0 *; 0 +; 0 <<)
% Number of symbols : 166 ( 164 usr; 3 con; 0-6 aty)
% Number of variables : 3764 ( 0 ^;3264 !; 500 ?;3764 :)
% Comments :
%------------------------------------------------------------------------------
thf(empty_carrier_type,type,
empty_carrier: $i > $o ).
thf(one_sorted_str_type,type,
one_sorted_str: $i > $o ).
thf(element_type,type,
element: $i > $i > $o ).
thf(the_carrier_type,type,
the_carrier: $i > $i ).
thf(apply_as_element_type,type,
apply_as_element: $i > $i > $i > $i > $i ).
thf(identity_on_carrier_type,type,
identity_on_carrier: $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(equipotent_type,type,
equipotent: $i > $i > $o ).
thf(are_equipotent_type,type,
are_equipotent: $i > $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(relation_of2_type,type,
relation_of2: $i > $i > $i > $o ).
thf(function_type,type,
function: $i > $o ).
thf(quasi_total_type,type,
quasi_total: $i > $i > $i > $o ).
thf(bijective_type,type,
bijective: $i > $i > $i > $o ).
thf(one_to_one_type,type,
one_to_one: $i > $o ).
thf(onto_type,type,
onto: $i > $i > $i > $o ).
thf(ordinal_type,type,
ordinal: $i > $o ).
thf(inclusion_relation_type,type,
inclusion_relation: $i > $i ).
thf(relation_empty_yielding_type,type,
relation_empty_yielding: $i > $o ).
thf(function_inverse_type,type,
function_inverse: $i > $i ).
thf(relation_inverse_type,type,
relation_inverse: $i > $i ).
thf(empty_type,type,
empty: $i > $o ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(relation_restriction_type,type,
relation_restriction: $i > $i > $i ).
thf(relation_dom_restriction_type,type,
relation_dom_restriction: $i > $i > $i ).
thf(relation_rng_restriction_type,type,
relation_rng_restriction: $i > $i > $i ).
thf(relation_dom_type,type,
relation_dom: $i > $i ).
thf(identity_relation_type,type,
identity_relation: $i > $i ).
thf(relation_rng_type,type,
relation_rng: $i > $i ).
thf(in_type,type,
in: $i > $i > $o ).
thf(relation_image_type,type,
relation_image: $i > $i > $i ).
thf(apply_type,type,
apply: $i > $i > $i ).
thf(relation_composition_type,type,
relation_composition: $i > $i > $i ).
thf(ordinal_subset_type,type,
ordinal_subset: $i > $i > $o ).
thf(well_founded_relation_type,type,
well_founded_relation: $i > $o ).
thf(connected_type,type,
connected: $i > $o ).
thf(reflexive_type,type,
reflexive: $i > $o ).
thf(transitive_type,type,
transitive: $i > $o ).
thf(fiber_type,type,
fiber: $i > $i > $i ).
thf(antisymmetric_type,type,
antisymmetric: $i > $o ).
thf(relation_inverse_image_type,type,
relation_inverse_image: $i > $i > $i ).
thf(empty_set_type,type,
empty_set: $i ).
thf(relation_isomorphism_type,type,
relation_isomorphism: $i > $i > $i > $o ).
thf(set_intersection2_type,type,
set_intersection2: $i > $i > $i ).
thf(pair_first_type,type,
pair_first: $i > $i ).
thf(pair_second_type,type,
pair_second: $i > $i ).
thf(finite_type,type,
finite: $i > $o ).
thf(set_difference_type,type,
set_difference: $i > $i > $i ).
thf(v1_partfun1_type,type,
v1_partfun1: $i > $i > $i > $o ).
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(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(powerset_type,type,
powerset: $i > $i ).
thf(succ_type,type,
succ: $i > $i ).
thf(cast_to_subset_type,type,
cast_to_subset: $i > $i ).
thf(omega_type,type,
omega: $i ).
thf(apply_binary_type,type,
apply_binary: $i > $i > $i > $i ).
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(set_meet_type,type,
set_meet: $i > $i ).
thf(strict_latt_str_type,type,
strict_latt_str: $i > $o ).
thf(latt_str_of_type,type,
latt_str_of: $i > $i > $i > $i ).
thf(latt_str_type,type,
latt_str: $i > $o ).
thf(disjoint_type,type,
disjoint: $i > $i > $o ).
thf(relation_of2_as_subset_type,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(relation_rng_as_subset_type,type,
relation_rng_as_subset: $i > $i > $i > $i ).
thf(centered_type,type,
centered: $i > $o ).
thf(rel_str_type,type,
rel_str: $i > $o ).
thf(lower_bounded_semilattstr_type,type,
lower_bounded_semilattstr: $i > $o ).
thf(boole_lattice_type,type,
boole_lattice: $i > $i ).
thf(bottom_of_semilattstr_type,type,
bottom_of_semilattstr: $i > $i ).
thf(rel_str_of_type,type,
rel_str_of: $i > $i > $i ).
thf(strict_rel_str_type,type,
strict_rel_str: $i > $o ).
thf(reflexive_relstr_type,type,
reflexive_relstr: $i > $o ).
thf(transitive_relstr_type,type,
transitive_relstr: $i > $o ).
thf(antisymmetric_relstr_type,type,
antisymmetric_relstr: $i > $o ).
thf(meet_semilatt_str_type,type,
meet_semilatt_str: $i > $o ).
thf(topological_space_type,type,
topological_space: $i > $o ).
thf(closed_subset_type,type,
closed_subset: $i > $i > $o ).
thf(cast_as_carrier_subset_type,type,
cast_as_carrier_subset: $i > $i ).
thf(symmetric_type,type,
symmetric: $i > $o ).
thf(identity_as_relation_of_type,type,
identity_as_relation_of: $i > $i ).
thf(join_semilatt_str_type,type,
join_semilatt_str: $i > $o ).
thf(unordered_triple_type,type,
unordered_triple: $i > $i > $i > $i ).
thf(meet_commutative_type,type,
meet_commutative: $i > $o ).
thf(the_L_meet_type,type,
the_L_meet: $i > $i ).
thf(v1_binop_1_type,type,
v1_binop_1: $i > $i > $o ).
thf(meet_associative_type,type,
meet_associative: $i > $o ).
thf(v2_binop_1_type,type,
v2_binop_1: $i > $i > $o ).
thf(a_2_2_lattice3_type,type,
a_2_2_lattice3: $i > $i > $i ).
thf(latt_set_smaller_type,type,
latt_set_smaller: $i > $i > $i > $o ).
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(lattice_type,type,
lattice: $i > $o ).
thf(k10_filter_1_type,type,
k10_filter_1: $i > $i > $i > $i > $i ).
thf(related_type,type,
related: $i > $i > $i > $o ).
thf(complements_of_subsets_type,type,
complements_of_subsets: $i > $i > $i ).
thf(subset_complement_type,type,
subset_complement: $i > $i > $i ).
thf(the_L_join_type,type,
the_L_join: $i > $i ).
thf(apply_binary_as_element_type,type,
apply_binary_as_element: $i > $i > $i > $i > $i > $i > $i ).
thf(k2_lattice3_type,type,
k2_lattice3: $i > $i ).
thf(relation_of_lattice_type,type,
relation_of_lattice: $i > $i ).
thf(point_neighbourhood_type,type,
point_neighbourhood: $i > $i > $i > $o ).
thf(meet_commut_type,type,
meet_commut: $i > $i > $i > $i ).
thf(poset_of_lattice_type,type,
poset_of_lattice: $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(k8_filter_1_type,type,
k8_filter_1: $i > $i > $i ).
thf(compact_top_space_type,type,
compact_top_space: $i > $o ).
thf(is_a_cover_of_carrier_type,type,
is_a_cover_of_carrier: $i > $i > $o ).
thf(open_subsets_type,type,
open_subsets: $i > $i > $o ).
thf(below_type,type,
below: $i > $i > $i > $o ).
thf(relstr_element_smaller_type,type,
relstr_element_smaller: $i > $i > $i > $o ).
thf(subset_union2_type,type,
subset_union2: $i > $i > $i > $i ).
thf(union_of_subsets_type,type,
union_of_subsets: $i > $i > $i ).
thf(interior_type,type,
interior: $i > $i > $i ).
thf(meet_absorbing_type,type,
meet_absorbing: $i > $o ).
thf(subset_difference_type,type,
subset_difference: $i > $i > $i > $i ).
thf(meet_of_latt_set_type,type,
meet_of_latt_set: $i > $i > $i ).
thf(cast_to_el_of_lattice_type,type,
cast_to_el_of_lattice: $i > $i > $i ).
thf(closed_subsets_type,type,
closed_subsets: $i > $i > $o ).
thf(empty_carrier_subset_type,type,
empty_carrier_subset: $i > $i ).
thf(relstr_set_smaller_type,type,
relstr_set_smaller: $i > $i > $i > $o ).
thf(latt_element_smaller_type,type,
latt_element_smaller: $i > $i > $i > $o ).
thf(meet_of_subsets_type,type,
meet_of_subsets: $i > $i > $i ).
thf(the_topology_type,type,
the_topology: $i > $i ).
thf(subset_intersection2_type,type,
subset_intersection2: $i > $i > $i > $i ).
thf(the_InternalRel_type,type,
the_InternalRel: $i > $i ).
thf(join_absorbing_type,type,
join_absorbing: $i > $o ).
thf(below_refl_type,type,
below_refl: $i > $i > $i > $o ).
thf(cast_to_el_of_LattPOSet_type,type,
cast_to_el_of_LattPOSet: $i > $i > $i ).
thf(complete_latt_str_type,type,
complete_latt_str: $i > $o ).
thf(a_2_3_lattice3_type,type,
a_2_3_lattice3: $i > $i > $i ).
thf(ordered_pair_as_product_element_type,type,
ordered_pair_as_product_element: $i > $i > $i > $i > $i ).
thf(related_reflexive_type,type,
related_reflexive: $i > $i > $i > $o ).
thf(join_of_latt_set_type,type,
join_of_latt_set: $i > $i > $i ).
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(a_1_0_filter_1_type,type,
a_1_0_filter_1: $i > $i ).
thf(join_associative_type,type,
join_associative: $i > $o ).
thf(4,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
<=> ( are_equipotent @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r2_wellord2) ).
thf(645,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)],[4]) ).
thf(510,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(3873,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)],[510]) ).
thf(163,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(1501,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)],[163]) ).
thf(125,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(1190,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)],[125]) ).
thf(260,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(2336,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)],[260]) ).
thf(254,axiom,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_boole) ).
thf(2263,plain,
! [A: $i] :
( ( set_difference @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(168,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(1559,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_dom_restriction @ B @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(192,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(1729,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)],[192]) ).
thf(54,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(846,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)],[54]) ).
thf(214,axiom,
? [A: $i] : ( meet_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_lattices) ).
thf(1850,plain,
? [A: $i] : ( meet_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(79,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(941,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)],[79]) ).
thf(256,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(2295,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)],[256]) ).
thf(549,axiom,
? [A: $i] :
( ( rel_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_rel_str @ A )
& ( reflexive_relstr @ A )
& ( transitive_relstr @ A )
& ( antisymmetric_relstr @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_orders_2) ).
thf(4101,plain,
? [A: $i] :
( ( rel_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_rel_str @ A )
& ( reflexive_relstr @ A )
& ( transitive_relstr @ A )
& ( antisymmetric_relstr @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).
thf(33,axiom,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_wellord2) ).
thf(760,plain,
! [A: $i] : ( transitive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(524,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ( relation_of2_as_subset @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_orders_2) ).
thf(4005,plain,
! [A: $i] :
( ( rel_str @ A )
=> ( relation_of2_as_subset @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).
thf(570,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(4290,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)],[570]) ).
thf(97,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(1031,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)],[97]) ).
thf(271,axiom,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) )
& ( reflexive @ ( identity_relation @ A ) )
& ( symmetric @ ( identity_relation @ A ) )
& ( antisymmetric @ ( identity_relation @ A ) )
& ( transitive @ ( identity_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_partfun1) ).
thf(2427,plain,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) )
& ( reflexive @ ( identity_relation @ A ) )
& ( symmetric @ ( identity_relation @ A ) )
& ( antisymmetric @ ( identity_relation @ A ) )
& ( transitive @ ( identity_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(522,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(3969,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)],[522]) ).
thf(85,axiom,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_enumset1) ).
thf(962,plain,
! [A: $i] :
( ( unordered_pair @ A @ A )
= ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(34,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(762,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( relation @ ( relation_rng_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(223,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(1924,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)],[223]) ).
thf(580,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( in @ ( ordered_pair_as_product_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ B @ C ) @ ( relation_of_lattice @ A ) )
<=> ( below_refl @ A @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t32_filter_1) ).
thf(4339,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( in @ ( ordered_pair_as_product_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ B @ C ) @ ( relation_of_lattice @ A ) )
=> ( below_refl @ A @ B @ C ) )
& ( ( below_refl @ A @ B @ C )
=> ( in @ ( ordered_pair_as_product_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ B @ C ) @ ( relation_of_lattice @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).
thf(251,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(2231,plain,
! [A: $i,B: $i] :
( ( set_difference @ A @ ( set_difference @ A @ B ) )
= ( set_intersection2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(492,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(3762,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).
thf(283,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(2495,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)],[283]) ).
thf(341,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(2908,plain,
! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( ( apply @ ( identity_relation @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(430,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(3509,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)],[430]) ).
thf(93,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(994,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)],[93]) ).
thf(600,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ B @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e4_7_2__tops_2__2) ).
thf(4479,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ B @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) ) )
& ( ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ B @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ B )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( in @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).
thf(531,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(4043,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)],[531]) ).
thf(437,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( latt_str @ B ) )
=> ( ( strict_latt_str @ ( k8_filter_1 @ A @ B ) )
& ( latt_str @ ( k8_filter_1 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k8_filter_1) ).
thf(3534,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( latt_str @ B ) )
=> ( ( strict_latt_str @ ( k8_filter_1 @ A @ B ) )
& ( latt_str @ ( k8_filter_1 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).
thf(377,axiom,
! [A: $i] : ( subset @ empty_set @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_xboole_1) ).
thf(3185,plain,
! [A: $i] : ( subset @ empty_set @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).
thf(89,axiom,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(977,plain,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(208,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(1808,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)],[208]) ).
thf(306,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(2659,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)],[306]) ).
thf(411,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_lattice2) ).
thf(3417,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).
thf(153,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(1422,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)],[153]) ).
thf(170,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(1569,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)],[170]) ).
thf(7,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(655,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)],[7]) ).
thf(268,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(2380,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)],[268]) ).
thf(592,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( lower_bounded_semilattstr @ A )
& ( latt_str @ A )
& ( ( bottom_of_semilattstr @ A )
= ( join_of_latt_set @ A @ empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t50_lattice3) ).
thf(4408,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( lower_bounded_semilattstr @ A )
& ( latt_str @ A )
& ( ( bottom_of_semilattstr @ A )
= ( join_of_latt_set @ A @ empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).
thf(12,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_wellord2) ).
thf(677,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_ordering @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(160,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_finset_1) ).
thf(1486,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(605,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_lattices) ).
thf(4532,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).
thf(188,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(1671,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)],[188]) ).
thf(178,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc11_relat_1) ).
thf(1608,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_inverse @ A ) )
& ( relation @ ( relation_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(545,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(4090,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)],[545]) ).
thf(413,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ B )
& ( latt_str @ B ) )
=> ( ( in @ A @ ( a_2_2_lattice3 @ B @ C ) )
<=> ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fraenkel_a_2_2_lattice3) ).
thf(3429,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ B )
& ( latt_str @ B ) )
=> ( ( ( in @ A @ ( a_2_2_lattice3 @ B @ C ) )
=> ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) ) )
& ( ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) )
=> ( in @ A @ ( a_2_2_lattice3 @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).
thf(356,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc37_membered) ).
thf(3049,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(301,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(2607,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( meet_semilatt_str @ A )
& ( join_semilatt_str @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(616,axiom,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_struct_0) ).
thf(4575,plain,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).
thf(352,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(3004,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)],[352]) ).
thf(119,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(1164,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)],[119]) ).
thf(275,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(2448,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)],[275]) ).
thf(626,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( relstr_set_smaller @ A @ B @ C )
<=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ D @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d9_lattice3) ).
thf(4638,plain,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( relstr_set_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ D @ C ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ D @ C ) ) )
=> ( relstr_set_smaller @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(814,plain,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(141,axiom,
? [A: $i] :
~ ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(1342,plain,
? [A: $i] :
~ ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(329,axiom,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_ordinal1) ).
thf(2844,plain,
! [A: $i,B: $i] :
( ( ordinal @ B )
=> ( ( in @ A @ B )
=> ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(477,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(3717,plain,
! [A: $i] :
( ( top_str @ A )
=> ( element @ ( the_topology @ A ) @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).
thf(158,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(1458,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(293,axiom,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_funct_1) ).
thf(2575,plain,
! [A: $i] :
( ( empty @ A )
=> ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(280,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(2476,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)],[280]) ).
thf(246,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(2168,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)],[246]) ).
thf(564,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ ( poset_of_lattice @ A ) )
& ( strict_rel_str @ ( poset_of_lattice @ A ) )
& ( reflexive_relstr @ ( poset_of_lattice @ A ) )
& ( transitive_relstr @ ( poset_of_lattice @ A ) )
& ( antisymmetric_relstr @ ( poset_of_lattice @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_lattice3) ).
thf(4247,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ ( poset_of_lattice @ A ) )
& ( strict_rel_str @ ( poset_of_lattice @ A ) )
& ( reflexive_relstr @ ( poset_of_lattice @ A ) )
& ( transitive_relstr @ ( poset_of_lattice @ A ) )
& ( antisymmetric_relstr @ ( poset_of_lattice @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).
thf(295,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(2580,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)],[295]) ).
thf(364,axiom,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_relat_1) ).
thf(3103,plain,
! [A: $i] :
( ( empty @ A )
=> ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(456,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(3629,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).
thf(95,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(1001,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)],[95]) ).
thf(317,axiom,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_zfmisc_1) ).
thf(2763,plain,
( ( powerset @ empty_set )
= ( singleton @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(427,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( k2_lattice3 @ A )
= ( relation_of_lattice @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k2_lattice3) ).
thf(3502,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( k2_lattice3 @ A )
= ( relation_of_lattice @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).
thf(233,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc7_relat_1) ).
thf(2020,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_dom @ A ) )
& ( relation @ ( relation_dom @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(227,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(1935,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)],[227]) ).
thf(481,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(3730,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)],[481]) ).
thf(145,axiom,
empty @ empty_set,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(1386,plain,
empty @ empty_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(327,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(2825,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)],[327]) ).
thf(587,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(4386,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)],[587]) ).
thf(350,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(2972,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)],[350]) ).
thf(458,axiom,
? [A: $i] :
( ( latt_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_latt_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc6_lattices) ).
thf(3634,plain,
? [A: $i] :
( ( latt_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_latt_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).
thf(389,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(3249,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)],[389]) ).
thf(541,axiom,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_subset_1) ).
thf(4079,plain,
! [A: $i] : ( element @ ( cast_to_subset @ A ) @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).
thf(130,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(1228,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)],[130]) ).
thf(445,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(3597,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)],[445]) ).
thf(583,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(4348,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)],[583]) ).
thf(237,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(2039,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)],[237]) ).
thf(539,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(4070,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)],[539]) ).
thf(593,axiom,
! [A: $i] :
( ~ ( empty_carrier @ ( boole_lattice @ A ) )
& ( strict_latt_str @ ( boole_lattice @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_lattice3) ).
thf(4415,plain,
! [A: $i] :
( ~ ( empty_carrier @ ( boole_lattice @ A ) )
& ( strict_latt_str @ ( boole_lattice @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).
thf(424,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(3495,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).
thf(359,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(3061,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)],[359]) ).
thf(451,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B )
& ( element @ C @ ( the_carrier @ A ) )
& ( element @ D @ ( the_carrier @ B ) ) )
=> ( element @ ( k10_filter_1 @ A @ B @ C @ D ) @ ( the_carrier @ ( k8_filter_1 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k10_filter_1) ).
thf(3617,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B )
& ( element @ C @ ( the_carrier @ A ) )
& ( element @ D @ ( the_carrier @ B ) ) )
=> ( element @ ( k10_filter_1 @ A @ B @ C @ D ) @ ( the_carrier @ ( k8_filter_1 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).
thf(40,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(802,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)],[40]) ).
thf(505,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(3845,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)],[505]) ).
thf(231,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(1992,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)],[231]) ).
thf(239,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(2126,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)],[239]) ).
thf(354,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(3043,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)],[354]) ).
thf(102,axiom,
! [A: $i] :
( ( being_limit_ordinal @ A )
<=> ( A
= ( union @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_ordinal1) ).
thf(1055,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)],[102]) ).
thf(382,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(3204,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)],[382]) ).
thf(324,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(2814,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)],[324]) ).
thf(595,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(4422,plain,
! [A: $i] :
( ( element @ A @ omega )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).
thf(137,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(1323,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( function @ A )
& ( finite @ B ) )
=> ( finite @ ( relation_image @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(51,axiom,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_relat_1) ).
thf(838,plain,
! [A: $i] :
( ( relation @ A )
=> ( relation @ ( relation_inverse @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(372,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(3148,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( disjoint @ B @ C ) )
=> ( disjoint @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(465,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(3653,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)],[465]) ).
thf(135,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(1313,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)],[135]) ).
thf(133,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ( ( strict_latt_str @ ( latt_str_of @ A @ B @ C ) )
& ( latt_str @ ( latt_str_of @ A @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_g3_lattices) ).
thf(1279,plain,
! [A: $i,B: $i,C: $i] :
( ( ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ( ( strict_latt_str @ ( latt_str_of @ A @ B @ C ) )
& ( latt_str @ ( latt_str_of @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(76,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(934,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)],[76]) ).
thf(485,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ( ( antisymmetric_relstr @ A )
<=> ( is_antisymmetric_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_orders_2) ).
thf(3744,plain,
! [A: $i] :
( ( rel_str @ A )
=> ( ( ( antisymmetric_relstr @ A )
=> ( is_antisymmetric_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) ) )
& ( ( is_antisymmetric_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) )
=> ( antisymmetric_relstr @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).
thf(45,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_wellord2) ).
thf(817,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( well_founded_relation @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(362,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_relset_1) ).
thf(3099,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(334,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_partfun1) ).
thf(2857,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A )
& ( empty @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(459,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( meet_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k16_lattice3) ).
thf(3638,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( meet_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).
thf(288,axiom,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ( ( ( function @ B )
& ( v1_partfun1 @ B @ A @ A )
& ( reflexive @ B )
& ( quasi_total @ B @ A @ A ) )
=> ( ( function @ B )
& ( one_to_one @ B )
& ( quasi_total @ B @ A @ A )
& ( onto @ B @ A @ A )
& ( bijective @ B @ A @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc4_funct_2) ).
thf(2552,plain,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ( ( ( function @ B )
& ( v1_partfun1 @ B @ A @ A )
& ( reflexive @ B )
& ( quasi_total @ B @ A @ A ) )
=> ( ( function @ B )
& ( one_to_one @ B )
& ( quasi_total @ B @ A @ A )
& ( onto @ B @ A @ A )
& ( bijective @ B @ A @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(220,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(1890,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(267,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(2373,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)],[267]) ).
thf(244,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(2158,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)],[244]) ).
thf(502,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( below_refl @ A @ B @ C )
<=> ( related_reflexive @ ( poset_of_lattice @ A ) @ ( cast_to_el_of_LattPOSet @ A @ B ) @ ( cast_to_el_of_LattPOSet @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_lattice3) ).
thf(3835,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below_refl @ A @ B @ C )
=> ( related_reflexive @ ( poset_of_lattice @ A ) @ ( cast_to_el_of_LattPOSet @ A @ B ) @ ( cast_to_el_of_LattPOSet @ A @ C ) ) )
& ( ( related_reflexive @ ( poset_of_lattice @ A ) @ ( cast_to_el_of_LattPOSet @ A @ B ) @ ( cast_to_el_of_LattPOSet @ A @ C ) )
=> ( below_refl @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).
thf(257,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(2301,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(588,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(4390,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)],[588]) ).
thf(218,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(1875,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)],[218]) ).
thf(565,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( relation @ ( relation_of_lattice @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k9_filter_1) ).
thf(4253,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( relation @ ( relation_of_lattice @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).
thf(213,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(1847,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(221,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(1892,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)],[221]) ).
thf(403,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(3385,plain,
! [A: $i,B: $i] :
( ( ( finite @ A )
& ( finite @ B ) )
=> ( finite @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).
thf(399,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(3312,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)],[399]) ).
thf(609,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_complement @ A @ ( union_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t11_tops_2) ).
thf(4551,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( meet_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_complement @ A @ ( union_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).
thf(143,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( subset @ A @ B )
& ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d10_xboole_0) ).
thf(1362,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)],[143]) ).
thf(228,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(1963,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)],[228]) ).
thf(161,axiom,
! [A: $i] :
( ( epsilon_transitive @ A )
<=> ! [B: $i] :
( ( in @ B @ A )
=> ( subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_ordinal1) ).
thf(1489,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)],[161]) ).
thf(434,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(3524,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)],[434]) ).
thf(187,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(1664,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)],[187]) ).
thf(515,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_orders_2) ).
thf(3934,plain,
! [A: $i] :
( ( rel_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).
thf(617,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ C )
= ( set_union2 @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k4_subset_1) ).
thf(4578,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ C )
= ( set_union2 @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).
thf(144,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(1373,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)],[144]) ).
thf(321,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(2782,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)],[321]) ).
thf(468,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty_carrier_subset @ A )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_pre_topc) ).
thf(3670,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( empty_carrier_subset @ A )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).
thf(435,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(3527,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)],[435]) ).
thf(49,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(834,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( well_ordering @ B )
=> ( well_ordering @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(552,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(4115,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)],[552]) ).
thf(507,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(3851,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)],[507]) ).
thf(69,axiom,
! [A: $i,B: $i] :
~ ( proper_subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',irreflexivity_r2_xboole_0) ).
thf(905,plain,
! [A: $i] :
~ ( proper_subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(414,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(3438,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)],[414]) ).
thf(217,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(1864,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)],[217]) ).
thf(117,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(1151,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)],[117]) ).
thf(77,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(937,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( relation_composition @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(90,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(981,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)],[90]) ).
thf(438,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(3537,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).
thf(535,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(4055,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)],[535]) ).
thf(528,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(4033,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)],[528]) ).
thf(428,axiom,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_finset_1) ).
thf(3505,plain,
! [A: $i] :
( ( finite @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( finite @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).
thf(266,axiom,
! [A: $i,B: $i] :
( ( finite @ B )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc10_finset_1) ).
thf(2371,plain,
! [A: $i,B: $i] :
( ( finite @ B )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(374,axiom,
! [A: $i] :
( ( v5_membered @ A )
=> ( v4_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_membered) ).
thf(3157,plain,
! [A: $i] :
( ( v5_membered @ A )
=> ( v4_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(9,axiom,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_xboole_1) ).
thf(660,plain,
! [A: $i,B: $i] :
~ ( ( subset @ A @ B )
& ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(490,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( finite @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
<=> ( finite @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t13_tops_2) ).
thf(3756,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( finite @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
=> ( finite @ B ) )
& ( ( finite @ B )
=> ( finite @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).
thf(314,axiom,
! [A: $i] :
? [B: $i] :
( ( relation_of2 @ B @ A @ A )
& ( relation @ B )
& ( reflexive @ B )
& ( symmetric @ B )
& ( antisymmetric @ B )
& ( transitive @ B )
& ( v1_partfun1 @ B @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_partfun1) ).
thf(2750,plain,
! [A: $i] :
? [B: $i] :
( ( relation_of2 @ B @ A @ A )
& ( relation @ B )
& ( reflexive @ B )
& ( symmetric @ B )
& ( antisymmetric @ B )
& ( transitive @ B )
& ( v1_partfun1 @ B @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(627,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_pre_topc) ).
thf(4643,plain,
! [A: $i] :
( ( top_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).
thf(176,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(1600,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)],[176]) ).
thf(621,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_associative @ A )
& ( join_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_join @ A ) )
& ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_join @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_lattice2) ).
thf(4588,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_associative @ A )
& ( join_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_join @ A ) )
& ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_join @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).
thf(423,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(3492,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)],[423]) ).
thf(337,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(2884,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)],[337]) ).
thf(16,axiom,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(687,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(631,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(4660,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)],[631]) ).
thf(142,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(1345,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)],[142]) ).
thf(92,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(988,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)],[92]) ).
thf(441,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( relstr_element_smaller @ A @ B @ C )
<=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ C @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_lattice3) ).
thf(3559,plain,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( relstr_element_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ C @ D ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ B )
=> ( related @ A @ C @ D ) ) )
=> ( relstr_element_smaller @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).
thf(55,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(849,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)],[55]) ).
thf(353,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(3011,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)],[353]) ).
thf(417,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(3450,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)],[417]) ).
thf(396,axiom,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_ordinal1) ).
thf(3303,plain,
! [A: $i] :
( ( succ @ A )
= ( set_union2 @ A @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).
thf(156,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(1438,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)],[156]) ).
thf(37,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(793,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)],[37]) ).
thf(43,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(810,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)],[43]) ).
thf(611,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(4557,plain,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).
thf(358,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(3054,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)],[358]) ).
thf(194,axiom,
! [A: $i] :
? [B: $i] :
( ( relation_of2 @ B @ A @ A )
& ( relation @ B )
& ( function @ B )
& ( one_to_one @ B )
& ( quasi_total @ B @ A @ A )
& ( onto @ B @ A @ A )
& ( bijective @ B @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_funct_2) ).
thf(1736,plain,
! [A: $i] :
? [B: $i] :
( ( relation_of2 @ B @ A @ A )
& ( relation @ B )
& ( function @ B )
& ( one_to_one @ B )
& ( quasi_total @ B @ A @ A )
& ( onto @ B @ A @ A )
& ( bijective @ B @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(440,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( latt_set_smaller @ A @ B @ C )
<=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d16_lattice3) ).
thf(3550,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( ( latt_set_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ B @ D ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ B @ D ) ) )
=> ( latt_set_smaller @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).
thf(497,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(3795,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)],[497]) ).
thf(99,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(1047,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(28,axiom,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_wellord1) ).
thf(746,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(447,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(3603,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).
thf(122,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(1174,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)],[122]) ).
thf(197,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(1751,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)],[197]) ).
thf(450,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(3610,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)],[450]) ).
thf(18,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(693,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)],[18]) ).
thf(146,axiom,
! [A: $i] :
( ( v4_membered @ A )
=> ( v3_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_membered) ).
thf(1387,plain,
! [A: $i] :
( ( v4_membered @ A )
=> ( v3_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(71,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(910,plain,
! [A: $i,B: $i] :
( ( proper_subset @ A @ B )
=> ~ ( proper_subset @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(83,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(957,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)],[83]) ).
thf(65,axiom,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_ordinal1) ).
thf(894,plain,
! [A: $i,B: $i] :
( ( ( ordinal @ A )
& ( ordinal @ B ) )
=> ( ordinal_subset @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(475,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(3711,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)],[475]) ).
thf(48,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(825,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)],[48]) ).
thf(127,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(1217,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)],[127]) ).
thf(242,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(2147,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)],[242]) ).
thf(488,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(3752,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( element @ ( subset_complement @ A @ B ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).
thf(601,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(4521,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)],[601]) ).
thf(222,axiom,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_ordinal1) ).
thf(1920,plain,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(534,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(4053,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)],[534]) ).
thf(56,axiom,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1_zfmisc_1) ).
thf(852,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(487,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(3750,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)],[487]) ).
thf(249,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( symmetric @ A )
& ( transitive @ A ) )
=> ( ( relation @ A )
& ( reflexive @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_partfun1) ).
thf(2200,plain,
! [A: $i] :
( ( ( relation @ A )
& ( symmetric @ A )
& ( transitive @ A ) )
=> ( ( relation @ A )
& ( reflexive @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(64,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(889,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)],[64]) ).
thf(206,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(1787,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)],[206]) ).
thf(14,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(682,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)],[14]) ).
thf(112,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_binop_1) ).
thf(1128,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(463,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( poset_of_lattice @ B ) ) )
=> ( ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ C )
<=> ( latt_set_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t29_lattice3) ).
thf(3647,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( poset_of_lattice @ B ) ) )
=> ( ( ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ C )
=> ( latt_set_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A ) )
& ( ( latt_set_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A )
=> ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).
thf(514,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( below_refl @ A @ B @ C )
<=> ( below @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r3_lattices) ).
thf(3931,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( ( below_refl @ A @ B @ C )
=> ( below @ A @ B @ C ) )
& ( ( below @ A @ B @ C )
=> ( below_refl @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).
thf(171,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(1573,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)],[171]) ).
thf(319,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc6_relat_1) ).
thf(2769,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(519,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(3958,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)],[519]) ).
thf(274,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(2445,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)],[274]) ).
thf(572,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( closed_subsets @ B @ A )
<=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( closed_subset @ C @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_tops_2) ).
thf(4295,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( closed_subsets @ B @ A )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( closed_subset @ C @ A ) ) ) )
& ( ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( closed_subset @ C @ A ) ) )
=> ( closed_subsets @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).
thf(276,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(2464,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)],[276]) ).
thf(315,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t24_wellord1) ).
thf(2758,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( transitive @ B )
=> ( transitive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(290,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).
thf(2565,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)],[290]) ).
thf(380,axiom,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_boole) ).
thf(3198,plain,
! [A: $i] :
( ( set_difference @ empty_set @ A )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).
thf(32,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(757,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)],[32]) ).
thf(261,axiom,
! [A: $i] :
( ( empty @ A )
=> ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_finset_1) ).
thf(2345,plain,
! [A: $i] :
( ( empty @ A )
=> ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(148,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_xboole_0) ).
thf(1392,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(467,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(3667,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)],[467]) ).
thf(554,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ ( poset_of_lattice @ A ) ) ) )
=> ( element @ ( cast_to_el_of_lattice @ A @ B ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_lattice3) ).
thf(4123,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ ( poset_of_lattice @ A ) ) ) )
=> ( element @ ( cast_to_el_of_lattice @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).
thf(365,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(3105,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)],[365]) ).
thf(420,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(3482,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).
thf(340,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(2895,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)],[340]) ).
thf(328,axiom,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ( ( strict_rel_str @ ( rel_str_of @ A @ B ) )
& ( rel_str @ ( rel_str_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_g1_orders_2) ).
thf(2841,plain,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ( ( strict_rel_str @ ( rel_str_of @ A @ B ) )
& ( rel_str @ ( rel_str_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(550,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(4108,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)],[550]) ).
thf(294,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(2577,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)],[294]) ).
thf(607,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(4544,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)],[607]) ).
thf(625,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(4636,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)],[625]) ).
thf(472,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(3704,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)],[472]) ).
thf(521,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(3963,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)],[521]) ).
thf(115,axiom,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
thf(1138,plain,
! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(457,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( point_neighbourhood @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_m1_connsp_2) ).
thf(3632,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( point_neighbourhood @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( the_carrier @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).
thf(464,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( closed_subsets @ B @ A )
<=> ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t16_tops_2) ).
thf(3650,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( closed_subsets @ B @ A )
=> ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) )
& ( ( open_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A )
=> ( closed_subsets @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).
thf(562,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(4219,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)],[562]) ).
thf(633,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(4668,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)],[633]) ).
thf(5,axiom,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r2_wellord2) ).
thf(650,plain,
! [A: $i,B: $i] :
( ( equipotent @ A @ B )
=> ( equipotent @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(80,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(948,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)],[80]) ).
thf(234,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(2023,plain,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ B )
= ( set_intersection2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(453,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(3622,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)],[453]) ).
thf(415,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(3441,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)],[415]) ).
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(642,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(212,axiom,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_funct_1) ).
thf(1842,plain,
! [A: $i] :
( ( relation @ ( identity_relation @ A ) )
& ( function @ ( identity_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(559,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( B
= ( meet_of_latt_set @ A @ C ) )
<=> ( ( latt_set_smaller @ A @ B @ C )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_set_smaller @ A @ D @ C )
=> ( below_refl @ A @ D @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t34_lattice3) ).
thf(4191,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( ( B
= ( meet_of_latt_set @ A @ C ) )
=> ( ( latt_set_smaller @ A @ B @ C )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_set_smaller @ A @ D @ C )
=> ( below_refl @ A @ D @ B ) ) ) ) )
& ( ( ( latt_set_smaller @ A @ B @ C )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_set_smaller @ A @ D @ C )
=> ( below_refl @ A @ D @ B ) ) ) )
=> ( B
= ( meet_of_latt_set @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).
thf(126,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(1214,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)],[126]) ).
thf(461,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(3643,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)],[461]) ).
thf(195,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_ordinal1) ).
thf(1744,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(533,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(4049,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B )
& ( finite @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).
thf(62,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(884,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)],[62]) ).
thf(408,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(3407,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)],[408]) ).
thf(30,axiom,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_wellord2) ).
thf(753,plain,
! [A: $i] : ( relation @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(476,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_complement @ A @ ( meet_of_subsets @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_tops_2) ).
thf(3714,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ A ) ) )
=> ( ( B != empty_set )
=> ( ( union_of_subsets @ A @ ( complements_of_subsets @ A @ B ) )
= ( subset_complement @ A @ ( meet_of_subsets @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).
thf(480,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( open_subset @ B @ A )
& ( in @ C @ B ) )
=> ( point_neighbourhood @ B @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_connsp_2) ).
thf(3728,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( open_subset @ B @ A )
& ( in @ C @ B ) )
=> ( point_neighbourhood @ B @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).
thf(278,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_wellord1) ).
thf(2472,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( antisymmetric @ B )
=> ( antisymmetric @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(13,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_relat_1) ).
thf(679,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(370,axiom,
? [A: $i] : ( latt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l3_lattices) ).
thf(3126,plain,
? [A: $i] : ( latt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).
thf(282,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(2487,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)],[282]) ).
thf(209,axiom,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t99_zfmisc_1) ).
thf(1831,plain,
! [A: $i] :
( ( union @ ( powerset @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(113,axiom,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_xboole_1) ).
thf(1131,plain,
! [A: $i] :
( ( subset @ A @ empty_set )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(495,axiom,
! [A: $i,B: $i] :
( ( ( strict_latt_str @ B )
& ( latt_str @ B ) )
=> ( ( B
= ( boole_lattice @ A ) )
<=> ( ( ( the_carrier @ B )
= ( powerset @ A ) )
& ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( apply_binary @ ( the_L_join @ B ) @ C @ D )
= ( subset_union2 @ A @ C @ D ) )
& ( ( apply_binary @ ( the_L_meet @ B ) @ C @ D )
= ( subset_intersection2 @ A @ C @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_lattice3) ).
thf(3770,plain,
! [A: $i,B: $i] :
( ( ( strict_latt_str @ B )
& ( latt_str @ B ) )
=> ( ( ( B
= ( boole_lattice @ A ) )
=> ( ( ( the_carrier @ B )
= ( powerset @ A ) )
& ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( apply_binary @ ( the_L_join @ B ) @ C @ D )
= ( subset_union2 @ A @ C @ D ) )
& ( ( apply_binary @ ( the_L_meet @ B ) @ C @ D )
= ( subset_intersection2 @ A @ C @ D ) ) ) ) ) ) )
& ( ( ( ( the_carrier @ B )
= ( powerset @ A ) )
& ! [C: $i] :
( ( element @ C @ ( powerset @ A ) )
=> ! [D: $i] :
( ( element @ D @ ( powerset @ A ) )
=> ( ( ( apply_binary @ ( the_L_join @ B ) @ C @ D )
= ( subset_union2 @ A @ C @ D ) )
& ( ( apply_binary @ ( the_L_meet @ B ) @ C @ D )
= ( subset_intersection2 @ A @ C @ D ) ) ) ) ) )
=> ( B
= ( boole_lattice @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).
thf(332,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(2851,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)],[332]) ).
thf(226,axiom,
! [A: $i,B: $i] :
( ( set_union2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).
thf(1932,plain,
! [A: $i] :
( ( set_union2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(100,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(1050,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)],[100]) ).
thf(91,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ B )
=> ! [C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B ) )
=> ( ( function @ C )
& ( v1_partfun1 @ C @ A @ B )
& ( quasi_total @ C @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc5_funct_2) ).
thf(984,plain,
! [A: $i,B: $i] :
( ~ ( empty @ B )
=> ! [C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B ) )
=> ( ( function @ C )
& ( v1_partfun1 @ C @ A @ B )
& ( quasi_total @ C @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(553,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(4120,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)],[553]) ).
thf(215,axiom,
! [A: $i] :
( ( strict_latt_str @ ( boole_lattice @ A ) )
& ( latt_str @ ( boole_lattice @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_lattice3) ).
thf(1852,plain,
! [A: $i] :
( ( strict_latt_str @ ( boole_lattice @ A ) )
& ( latt_str @ ( boole_lattice @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(281,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(2482,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)],[281]) ).
thf(203,axiom,
! [A: $i,B: $i] :
( ( ( reflexive @ B )
& ( antisymmetric @ B )
& ( transitive @ B )
& ( v1_partfun1 @ B @ A @ A )
& ( relation_of2 @ B @ A @ A ) )
=> ( ( strict_rel_str @ ( rel_str_of @ A @ B ) )
& ( reflexive_relstr @ ( rel_str_of @ A @ B ) )
& ( transitive_relstr @ ( rel_str_of @ A @ B ) )
& ( antisymmetric_relstr @ ( rel_str_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_orders_2) ).
thf(1776,plain,
! [A: $i,B: $i] :
( ( ( reflexive @ B )
& ( antisymmetric @ B )
& ( transitive @ B )
& ( v1_partfun1 @ B @ A @ A )
& ( relation_of2 @ B @ A @ A ) )
=> ( ( strict_rel_str @ ( rel_str_of @ A @ B ) )
& ( reflexive_relstr @ ( rel_str_of @ A @ B ) )
& ( transitive_relstr @ ( rel_str_of @ A @ B ) )
& ( antisymmetric_relstr @ ( rel_str_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(466,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(3656,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)],[466]) ).
thf(284,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(2497,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)],[284]) ).
thf(606,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(4540,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)],[606]) ).
thf(439,axiom,
! [A: $i] :
( ( top_str @ A )
=> ( ( compact_top_space @ A )
<=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( is_a_cover_of_carrier @ A @ B )
& ( open_subsets @ B @ A )
& ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( subset @ C @ B )
& ( is_a_cover_of_carrier @ A @ C )
& ( finite @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_compts_1) ).
thf(3540,plain,
! [A: $i] :
( ( top_str @ A )
=> ( ( ( compact_top_space @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( is_a_cover_of_carrier @ A @ B )
& ( open_subsets @ B @ A )
& ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( subset @ C @ B )
& ( is_a_cover_of_carrier @ A @ C )
& ( finite @ C ) ) ) ) ) )
& ( ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( is_a_cover_of_carrier @ A @ B )
& ( open_subsets @ B @ A )
& ! [C: $i] :
( ( element @ C @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( subset @ C @ B )
& ( is_a_cover_of_carrier @ A @ C )
& ( finite @ C ) ) ) ) )
=> ( compact_top_space @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).
thf(409,axiom,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_boole) ).
thf(3409,plain,
! [A: $i] :
( ( set_intersection2 @ A @ empty_set )
= empty_set ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).
thf(335,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(2862,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)],[335]) ).
thf(556,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(4152,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)],[556]) ).
thf(190,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(1720,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)],[190]) ).
thf(567,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ B )
& ( in @ E @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e4_7_2__tops_2__1) ).
thf(4259,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ B )
& ( in @ E @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ B )
& ( in @ E @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).
thf(360,axiom,
! [A: $i] :
( ( v3_membered @ A )
=> ( v2_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_membered) ).
thf(3083,plain,
! [A: $i] :
( ( v3_membered @ A )
=> ( v2_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(376,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(3162,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)],[376]) ).
thf(381,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(3201,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)],[381]) ).
thf(200,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ B )
<=> ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t37_zfmisc_1) ).
thf(1759,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)],[200]) ).
thf(345,axiom,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_funct_1) ).
thf(2926,plain,
? [A: $i] :
( ( relation @ A )
& ( empty @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).
thf(250,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(2203,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)],[250]) ).
thf(106,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(1086,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)],[106]) ).
thf(397,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(3306,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)],[397]) ).
thf(121,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(1170,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)],[121]) ).
thf(529,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ C )
= ( subset_union2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k4_subset_1) ).
thf(4036,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ C )
= ( subset_union2 @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).
thf(120,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(1167,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)],[120]) ).
thf(576,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ( element @ ( cast_to_el_of_LattPOSet @ A @ B ) @ ( the_carrier @ ( poset_of_lattice @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_lattice3) ).
thf(4308,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ( element @ ( cast_to_el_of_LattPOSet @ A @ B ) @ ( the_carrier @ ( poset_of_lattice @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).
thf(615,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ B ) )
=> ( ( latt_set_smaller @ B @ C @ A )
<=> ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_lattice3) ).
thf(4572,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ B ) )
=> ( ( ( latt_set_smaller @ B @ C @ A )
=> ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) ) )
& ( ( relstr_element_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) )
=> ( latt_set_smaller @ B @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).
thf(216,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(1857,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)],[216]) ).
thf(558,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(4184,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)],[558]) ).
thf(41,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(805,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)],[41]) ).
thf(542,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ( ( transitive_relstr @ A )
<=> ( is_transitive_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_orders_2) ).
thf(4081,plain,
! [A: $i] :
( ( rel_str @ A )
=> ( ( ( transitive_relstr @ A )
=> ( is_transitive_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) ) )
& ( ( is_transitive_in @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) )
=> ( transitive_relstr @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).
thf(400,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(3321,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_field @ A )
= ( set_union2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).
thf(265,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(2368,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)],[265]) ).
thf(124,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc8_relat_1) ).
thf(1187,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( empty @ ( relation_rng @ A ) )
& ( relation @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(272,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(2440,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(201,axiom,
! [A: $i] :
( ~ ( empty @ ( singleton @ A ) )
& ( finite @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_finset_1) ).
thf(1764,plain,
! [A: $i] :
( ~ ( empty @ ( singleton @ A ) )
& ( finite @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(198,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(1754,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( B
= ( set_union2 @ A @ ( set_difference @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(509,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(3871,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)],[509]) ).
thf(297,axiom,
? [A: $i] :
( ( rel_str @ A )
& ( strict_rel_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_orders_2) ).
thf(2585,plain,
? [A: $i] :
( ( rel_str @ A )
& ( strict_rel_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(291,axiom,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_ordinal1) ).
thf(2570,plain,
! [A: $i] :
~ ( empty @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(589,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(4393,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)],[589]) ).
thf(484,axiom,
! [A: $i] :
( ( ( reflexive_relstr @ A )
& ( transitive_relstr @ A )
& ( antisymmetric_relstr @ A )
& ( rel_str @ A ) )
=> ( ( relation @ ( the_InternalRel @ A ) )
& ( reflexive @ ( the_InternalRel @ A ) )
& ( antisymmetric @ ( the_InternalRel @ A ) )
& ( transitive @ ( the_InternalRel @ A ) )
& ( v1_partfun1 @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_orders_2) ).
thf(3738,plain,
! [A: $i] :
( ( ( reflexive_relstr @ A )
& ( transitive_relstr @ A )
& ( antisymmetric_relstr @ A )
& ( rel_str @ A ) )
=> ( ( relation @ ( the_InternalRel @ A ) )
& ( reflexive @ ( the_InternalRel @ A ) )
& ( antisymmetric @ ( the_InternalRel @ A ) )
& ( transitive @ ( the_InternalRel @ A ) )
& ( v1_partfun1 @ ( the_InternalRel @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).
thf(586,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [D: $i] :
~ ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [D: $i] :
( ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = D )
=> ( ( apply @ C @ D )
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s2_funct_1__e4_7_1__tops_2) ).
thf(4359,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [D: $i] :
~ ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [D: $i] :
( ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = D )
=> ( ( apply @ C @ D )
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).
thf(24,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( one_to_one @ C )
& ( quasi_total @ C @ A @ B )
& ( onto @ C @ A @ B ) )
=> ( ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( bijective @ C @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_funct_2) ).
thf(733,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( one_to_one @ C )
& ( quasi_total @ C @ A @ B )
& ( onto @ C @ A @ B ) )
=> ( ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( bijective @ C @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(253,axiom,
! [A: $i,B: $i] :
( ( set_intersection2 @ A @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
thf(2260,plain,
! [A: $i] :
( ( set_intersection2 @ A @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(86,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc11_finset_1) ).
thf(965,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(191,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t22_wellord1) ).
thf(1727,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( reflexive @ B )
=> ( reflexive @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(622,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ( ( in @ E @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e4_7_1__tops_2__1) ).
thf(4594,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ( ( in @ E @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) ) )
& ( ( ( in @ E @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) )
=> ( in @ E @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).
thf(449,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( point_neighbourhood @ C @ A @ B )
<=> ( in @ B @ ( interior @ A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_connsp_2) ).
thf(3607,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( ( point_neighbourhood @ C @ A @ B )
=> ( in @ B @ ( interior @ A @ C ) ) )
& ( ( in @ B @ ( interior @ A @ C ) )
=> ( point_neighbourhood @ C @ A @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).
thf(57,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(856,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)],[57]) ).
thf(613,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ( ( compact_top_space @ A )
<=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( centered @ B )
& ( closed_subsets @ B @ A )
& ( ( meet_of_subsets @ ( the_carrier @ A ) @ B )
= empty_set ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t13_compts_1) ).
thf(4562,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ( ( ( compact_top_space @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( centered @ B )
& ( closed_subsets @ B @ A )
& ( ( meet_of_subsets @ ( the_carrier @ A ) @ B )
= empty_set ) ) ) )
& ( ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ~ ( ( centered @ B )
& ( closed_subsets @ B @ A )
& ( ( meet_of_subsets @ ( the_carrier @ A ) @ B )
= empty_set ) ) )
=> ( compact_top_space @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).
thf(46,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(819,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)],[46]) ).
thf(349,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(2970,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)],[349]) ).
thf(494,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(3768,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)],[494]) ).
thf(512,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( strict_latt_str @ A )
=> ( A
= ( latt_str_of @ ( the_carrier @ A ) @ ( the_L_join @ A ) @ ( the_L_meet @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',abstractness_v3_lattices) ).
thf(3925,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( strict_latt_str @ A )
=> ( A
= ( latt_str_of @ ( the_carrier @ A ) @ ( the_L_join @ A ) @ ( the_L_meet @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).
thf(386,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_relat_1) ).
thf(3240,plain,
( ( empty @ empty_set )
& ( relation @ empty_set )
& ( relation_empty_yielding @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(483,axiom,
? [A: $i] : ( one_sorted_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_struct_0) ).
thf(3736,plain,
? [A: $i] : ( one_sorted_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).
thf(27,axiom,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',involutiveness_k4_relat_1) ).
thf(743,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_inverse @ ( relation_inverse @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(419,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
<=> ? [E: $i] :
( ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e4_7_1__tops_2__1) ).
thf(3458,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
! [D: $i] :
( ( ( in @ D @ C )
=> ? [E: $i] :
( ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) )
& ( ? [E: $i] :
( ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ E @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = E )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( in @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).
thf(129,axiom,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_ordinal1) ).
thf(1226,plain,
! [A: $i] : ( in @ A @ ( succ @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(446,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(3599,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)],[446]) ).
thf(88,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(975,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( finite @ ( relation_dom @ A ) )
=> ( finite @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(404,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(3387,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)],[404]) ).
thf(378,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(3187,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)],[378]) ).
thf(568,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(4283,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)],[568]) ).
thf(205,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(1784,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)],[205]) ).
thf(147,axiom,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_relat_1) ).
thf(1389,plain,
? [A: $i] :
( ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(391,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(3255,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)],[391]) ).
thf(53,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(843,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)],[53]) ).
thf(111,axiom,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_relat_1) ).
thf(1125,plain,
( ( empty @ empty_set )
& ( relation @ empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(305,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(2657,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)],[305]) ).
thf(331,axiom,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_xboole_1) ).
thf(2849,plain,
! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(508,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( poset_of_lattice @ A )
= ( rel_str_of @ ( the_carrier @ A ) @ ( k2_lattice3 @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_lattice3) ).
thf(3868,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( poset_of_lattice @ A )
= ( rel_str_of @ ( the_carrier @ A ) @ ( k2_lattice3 @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).
thf(623,axiom,
! [A: $i] :
( ~ ( empty_carrier @ ( boole_lattice @ A ) )
& ( strict_latt_str @ ( boole_lattice @ A ) )
& ( join_commutative @ ( boole_lattice @ A ) )
& ( join_associative @ ( boole_lattice @ A ) )
& ( meet_commutative @ ( boole_lattice @ A ) )
& ( meet_associative @ ( boole_lattice @ A ) )
& ( meet_absorbing @ ( boole_lattice @ A ) )
& ( join_absorbing @ ( boole_lattice @ A ) )
& ( lattice @ ( boole_lattice @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_lattice3) ).
thf(4612,plain,
! [A: $i] :
( ~ ( empty_carrier @ ( boole_lattice @ A ) )
& ( strict_latt_str @ ( boole_lattice @ A ) )
& ( join_commutative @ ( boole_lattice @ A ) )
& ( join_associative @ ( boole_lattice @ A ) )
& ( meet_commutative @ ( boole_lattice @ A ) )
& ( meet_associative @ ( boole_lattice @ A ) )
& ( meet_absorbing @ ( boole_lattice @ A ) )
& ( join_absorbing @ ( boole_lattice @ A ) )
& ( lattice @ ( boole_lattice @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).
thf(325,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(2817,plain,
! [A: $i,B: $i] :
? [C: $i] :
( ( relation_of2 @ C @ A @ B )
& ( relation @ C )
& ( function @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(603,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(4526,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)],[603]) ).
thf(368,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(3118,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)],[368]) ).
thf(547,axiom,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_lattices) ).
thf(4096,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).
thf(628,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( B
= ( bottom_of_semilattstr @ A ) )
<=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d16_lattices) ).
thf(4645,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( ( B
= ( bottom_of_semilattstr @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) )
& ( ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) )
=> ( B
= ( bottom_of_semilattstr @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).
thf(347,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(2935,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)],[347]) ).
thf(273,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(2442,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)],[273]) ).
thf(537,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(4060,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)],[537]) ).
thf(460,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(3641,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)],[460]) ).
thf(279,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(2474,plain,
! [A: $i,B: $i,C: $i] :
( ( ( subset @ A @ B )
& ( subset @ B @ C ) )
=> ( subset @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(443,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation_of2 @ B @ A @ A ) )
=> ( ~ ( empty_carrier @ ( rel_str_of @ A @ B ) )
& ( strict_rel_str @ ( rel_str_of @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_orders_2) ).
thf(3568,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ( relation_of2 @ B @ A @ A ) )
=> ( ~ ( empty_carrier @ ( rel_str_of @ A @ B ) )
& ( strict_rel_str @ ( rel_str_of @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).
thf(107,axiom,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_subset_1) ).
thf(1090,plain,
! [A: $i] :
( ( cast_to_subset @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(452,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(3619,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)],[452]) ).
thf(598,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(4433,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)],[598]) ).
thf(292,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(2573,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)],[292]) ).
thf(478,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(3719,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)],[478]) ).
thf(436,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(3532,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)],[436]) ).
thf(11,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( bijective @ C @ A @ B ) )
=> ( ( function @ C )
& ( one_to_one @ C )
& ( quasi_total @ C @ A @ B )
& ( onto @ C @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_funct_2) ).
thf(672,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( bijective @ C @ A @ B ) )
=> ( ( function @ C )
& ( one_to_one @ C )
& ( quasi_total @ C @ A @ B )
& ( onto @ C @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(35,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(764,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)],[35]) ).
thf(363,axiom,
! [A: $i,B: $i] : ( subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(3101,plain,
! [A: $i] : ( subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(312,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(2721,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)],[312]) ).
thf(67,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(900,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)],[67]) ).
thf(134,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(1282,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)],[134]) ).
thf(499,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(3815,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)],[499]) ).
thf(109,axiom,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_boole) ).
thf(1117,plain,
! [A: $i] :
( ( set_union2 @ A @ empty_set )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(432,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( strict_rel_str @ ( poset_of_lattice @ A ) )
& ( reflexive_relstr @ ( poset_of_lattice @ A ) )
& ( transitive_relstr @ ( poset_of_lattice @ A ) )
& ( antisymmetric_relstr @ ( poset_of_lattice @ A ) )
& ( rel_str @ ( poset_of_lattice @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k3_lattice3) ).
thf(3515,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( strict_rel_str @ ( poset_of_lattice @ A ) )
& ( reflexive_relstr @ ( poset_of_lattice @ A ) )
& ( transitive_relstr @ ( poset_of_lattice @ A ) )
& ( antisymmetric_relstr @ ( poset_of_lattice @ A ) )
& ( rel_str @ ( poset_of_lattice @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).
thf(177,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ! [C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B ) )
=> ( ( function @ C )
& ~ ( empty @ C )
& ( v1_partfun1 @ C @ A @ B )
& ( quasi_total @ C @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc6_funct_2) ).
thf(1603,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ! [C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( quasi_total @ C @ A @ B ) )
=> ( ( function @ C )
& ~ ( empty @ C )
& ( v1_partfun1 @ C @ A @ B )
& ( quasi_total @ C @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(620,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(4586,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)],[620]) ).
thf(25,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(737,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)],[25]) ).
thf(426,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(3500,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)],[426]) ).
thf(573,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( relation_of_lattice @ A )
= ( a_1_0_filter_1 @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d8_filter_1) ).
thf(4300,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( relation_of_lattice @ A )
= ( a_1_0_filter_1 @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).
thf(248,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t28_xboole_1) ).
thf(2197,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_intersection2 @ A @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(59,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(863,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)],[59]) ).
thf(238,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(2097,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)],[238]) ).
thf(392,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(3270,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ B )
= ( set_union2 @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(561,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(4213,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)],[561]) ).
thf(61,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(880,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)],[61]) ).
thf(540,axiom,
! [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(4073,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)],[540]) ).
thf(597,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(4430,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)],[597]) ).
thf(486,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( below_refl @ A @ B @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r3_lattices) ).
thf(3747,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( below_refl @ A @ B @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).
thf(303,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(2625,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)],[303]) ).
thf(193,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(1731,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)],[193]) ).
thf(138,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(1325,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)],[138]) ).
thf(322,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(2785,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)],[322]) ).
thf(159,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(1460,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)],[159]) ).
thf(189,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(1688,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)],[189]) ).
thf(302,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(2610,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)],[302]) ).
thf(140,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(1339,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)],[140]) ).
thf(594,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(4420,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)],[594]) ).
thf(58,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(858,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)],[58]) ).
thf(526,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(4009,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)],[526]) ).
thf(39,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(799,plain,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(22,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(726,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)],[22]) ).
thf(560,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ( ( ( join @ ( boole_lattice @ A ) @ B @ C )
= ( set_union2 @ B @ C ) )
& ( ( meet @ ( boole_lattice @ A ) @ B @ C )
= ( set_intersection2 @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_lattice3) ).
thf(4208,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ( ( ( join @ ( boole_lattice @ A ) @ B @ C )
= ( set_union2 @ B @ C ) )
& ( ( meet @ ( boole_lattice @ A ) @ B @ C )
= ( set_intersection2 @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).
thf(557,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k15_lattice3) ).
thf(4181,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).
thf(75,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(932,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)],[75]) ).
thf(373,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ! [D: $i,E: $i,F: $i] :
( ( ( latt_str_of @ A @ B @ C )
= ( latt_str_of @ D @ E @ F ) )
=> ( ( A = D )
& ( B = E )
& ( C = F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',free_g3_lattices) ).
thf(3150,plain,
! [A: $i,B: $i,C: $i] :
( ( ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ! [D: $i,E: $i,F: $i] :
( ( ( latt_str_of @ A @ B @ C )
= ( latt_str_of @ D @ E @ F ) )
=> ( ( A = D )
& ( B = E )
& ( C = F ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).
thf(259,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(2329,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)],[259]) ).
thf(342,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(2911,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)],[342]) ).
thf(610,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ( ( strict_rel_str @ A )
=> ( A
= ( rel_str_of @ ( the_carrier @ A ) @ ( the_InternalRel @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',abstractness_v1_orders_2) ).
thf(4554,plain,
! [A: $i] :
( ( rel_str @ A )
=> ( ( strict_rel_str @ A )
=> ( A
= ( rel_str_of @ ( the_carrier @ A ) @ ( the_InternalRel @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).
thf(224,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(1926,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ ( union @ A ) )
& ( epsilon_connected @ ( union @ A ) )
& ( ordinal @ ( union @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(455,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(3627,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)],[455]) ).
thf(60,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(868,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)],[60]) ).
thf(366,axiom,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_relat_1) ).
thf(3110,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(375,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc28_membered) ).
thf(3159,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(155,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(1435,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)],[155]) ).
thf(473,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( poset_of_lattice @ B ) ) )
=> ( ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ C )
<=> ( latt_element_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t31_lattice3) ).
thf(3706,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( poset_of_lattice @ B ) ) )
=> ( ( ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ C )
=> ( latt_element_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A ) )
& ( ( latt_element_smaller @ B @ ( cast_to_el_of_lattice @ B @ C ) @ A )
=> ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).
thf(348,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(2967,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)],[348]) ).
thf(431,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(3512,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)],[431]) ).
thf(243,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(2156,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)],[243]) ).
thf(538,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty @ A )
& ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ( ~ ( empty_carrier @ ( latt_str_of @ A @ B @ C ) )
& ( strict_latt_str @ ( latt_str_of @ A @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_lattices) ).
thf(4067,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty @ A )
& ( function @ B )
& ( quasi_total @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ B @ ( cartesian_product2 @ A @ A ) @ A )
& ( function @ C )
& ( quasi_total @ C @ ( cartesian_product2 @ A @ A ) @ A )
& ( relation_of2 @ C @ ( cartesian_product2 @ A @ A ) @ A ) )
=> ( ~ ( empty_carrier @ ( latt_str_of @ A @ B @ C ) )
& ( strict_latt_str @ ( latt_str_of @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).
thf(511,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(3876,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)],[511]) ).
thf(229,axiom,
! [A: $i] :
( ( A = empty_set )
<=> ! [B: $i] :
~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
thf(1973,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)],[229]) ).
thf(8,axiom,
! [A: $i,B: $i] : ( equipotent @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r2_wellord2) ).
thf(658,plain,
! [A: $i] : ( equipotent @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(72,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(912,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)],[72]) ).
thf(15,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(685,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)],[15]) ).
thf(270,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(2401,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)],[270]) ).
thf(175,axiom,
! [A: $i] :
( ( lower_bounded_semilattstr @ ( boole_lattice @ A ) )
& ( ( bottom_of_semilattstr @ ( boole_lattice @ A ) )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_lattice3) ).
thf(1594,plain,
! [A: $i] :
( ( lower_bounded_semilattstr @ ( boole_lattice @ A ) )
& ( ( bottom_of_semilattstr @ ( boole_lattice @ A ) )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(491,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( cast_to_el_of_LattPOSet @ A @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_lattice3) ).
thf(3759,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( cast_to_el_of_LattPOSet @ A @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).
thf(503,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(3838,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)],[503]) ).
thf(318,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(2765,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)],[318]) ).
thf(412,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_associative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_lattice2) ).
thf(3423,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_associative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).
thf(1,conjecture,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( apply_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( identity_on_carrier @ A ) @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t91_tmap_1) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( apply_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( identity_on_carrier @ A ) @ B )
= B ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(636,plain,
~ ! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( apply_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( identity_on_carrier @ A ) @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(162,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(1496,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)],[162]) ).
thf(258,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(2303,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)],[258]) ).
thf(422,axiom,
! [A: $i] :
( ( ( transitive_relstr @ A )
& ( rel_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( ( related @ A @ B @ C )
& ( related @ A @ C @ D ) )
=> ( related @ A @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_orders_2) ).
thf(3490,plain,
! [A: $i] :
( ( ( transitive_relstr @ A )
& ( rel_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( ( related @ A @ B @ C )
& ( related @ A @ C @ D ) )
=> ( related @ A @ B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).
thf(543,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(4084,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)],[543]) ).
thf(172,axiom,
? [A: $i] : ( rel_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_orders_2) ).
thf(1583,plain,
? [A: $i] : ( rel_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(527,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(4013,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)],[527]) ).
thf(516,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( function @ ( identity_on_carrier @ A ) )
& ( quasi_total @ ( identity_on_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( identity_on_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k7_grcat_1) ).
thf(3936,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( function @ ( identity_on_carrier @ A ) )
& ( quasi_total @ ( identity_on_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( identity_on_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).
thf(10,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(663,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)],[10]) ).
thf(286,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(2503,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)],[286]) ).
thf(252,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(2234,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)],[252]) ).
thf(114,axiom,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc3_ordinal1) ).
thf(1134,plain,
! [A: $i] :
( ( empty @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(571,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(4292,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).
thf(29,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(749,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)],[29]) ).
thf(78,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(939,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)],[78]) ).
thf(418,axiom,
! [A: $i] :
( ( ( antisymmetric_relstr @ A )
& ( rel_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( related @ A @ B @ C )
& ( related @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t25_orders_2) ).
thf(3455,plain,
! [A: $i] :
( ( ( antisymmetric_relstr @ A )
& ( rel_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( related @ A @ B @ C )
& ( related @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).
thf(330,axiom,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l25_zfmisc_1) ).
thf(2846,plain,
! [A: $i,B: $i] :
~ ( ( disjoint @ ( singleton @ A ) @ B )
& ( in @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(50,axiom,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_wellord2) ).
thf(836,plain,
! [A: $i] : ( antisymmetric @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(444,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_funct_1__e4_7_1__tops_2__1) ).
thf(3571,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) )
& ( ( ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ( in @ D @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).
thf(506,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(3849,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( cast_as_carrier_subset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).
thf(38,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(795,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(406,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).
thf(3395,plain,
! [A: $i,B: $i] :
~ ( empty @ ( ordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).
thf(383,axiom,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).
thf(3206,plain,
! [A: $i,B: $i] :
( ~ ( empty @ A )
=> ~ ( empty @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
thf(73,axiom,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_relat_1) ).
thf(927,plain,
! [A: $i] : ( relation @ ( identity_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(604,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(4529,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)],[604]) ).
thf(236,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(2037,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)],[236]) ).
thf(300,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(2597,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)],[300]) ).
thf(618,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(4581,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)],[618]) ).
thf(87,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(968,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)],[87]) ).
thf(123,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(1181,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)],[123]) ).
thf(566,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B )
& ( finite @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc4_finset_1) ).
thf(4255,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B )
& ( finite @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).
thf(207,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(1798,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)],[207]) ).
thf(17,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(690,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)],[17]) ).
thf(179,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(1611,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)],[179]) ).
thf(385,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(3214,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)],[385]) ).
thf(183,axiom,
! [A: $i] :
( ( ordinal @ A )
<=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_ordinal1) ).
thf(1647,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)],[183]) ).
thf(496,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( complete_latt_str @ B )
& ( latt_str @ B ) )
=> ( ( in @ A @ ( a_2_3_lattice3 @ B @ C ) )
<=> ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fraenkel_a_2_3_lattice3) ).
thf(3786,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( complete_latt_str @ B )
& ( latt_str @ B ) )
=> ( ( ( in @ A @ ( a_2_3_lattice3 @ B @ C ) )
=> ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) ) )
& ( ? [D: $i] :
( ( element @ D @ ( the_carrier @ B ) )
& ( A = D )
& ( latt_set_smaller @ B @ D @ C ) )
=> ( in @ A @ ( a_2_3_lattice3 @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).
thf(308,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
<=> ( ( set_difference @ A @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t83_xboole_1) ).
thf(2689,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)],[308]) ).
thf(532,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(4045,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)],[532]) ).
thf(630,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(4658,plain,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ A ) )
=> ( v1_membered @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).
thf(269,axiom,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc4_funct_1) ).
thf(2397,plain,
? [A: $i] :
( ( relation @ A )
& ( relation_empty_yielding @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(429,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ? [C: $i] : ( point_neighbourhood @ C @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_connsp_2) ).
thf(3507,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A )
& ( element @ B @ ( the_carrier @ A ) ) )
=> ? [C: $i] : ( point_neighbourhood @ C @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).
thf(298,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(2588,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)],[298]) ).
thf(489,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_lattices) ).
thf(3754,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).
thf(569,axiom,
! [A: $i,B: $i] :
( ( element @ B @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ( ( below @ ( boole_lattice @ A ) @ B @ C )
<=> ( subset @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_lattice3) ).
thf(4287,plain,
! [A: $i,B: $i] :
( ( element @ B @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ ( boole_lattice @ A ) ) )
=> ( ( ( below @ ( boole_lattice @ A ) @ B @ C )
=> ( subset @ B @ C ) )
& ( ( subset @ B @ C )
=> ( below @ ( boole_lattice @ A ) @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).
thf(326,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(2821,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)],[326]) ).
thf(384,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(3209,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)],[384]) ).
thf(591,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ( ( in @ A @ ( a_1_0_filter_1 @ B ) )
<=> ? [C: $i,D: $i] :
( ( element @ C @ ( the_carrier @ B ) )
& ( element @ D @ ( the_carrier @ B ) )
& ( A
= ( ordered_pair_as_product_element @ ( the_carrier @ B ) @ ( the_carrier @ B ) @ C @ D ) )
& ( below_refl @ B @ C @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fraenkel_a_1_0_filter_1) ).
thf(4398,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ( ( ( in @ A @ ( a_1_0_filter_1 @ B ) )
=> ? [C: $i,D: $i] :
( ( element @ C @ ( the_carrier @ B ) )
& ( element @ D @ ( the_carrier @ B ) )
& ( A
= ( ordered_pair_as_product_element @ ( the_carrier @ B ) @ ( the_carrier @ B ) @ C @ D ) )
& ( below_refl @ B @ C @ D ) ) )
& ( ? [C: $i,D: $i] :
( ( element @ C @ ( the_carrier @ B ) )
& ( element @ D @ ( the_carrier @ B ) )
& ( A
= ( ordered_pair_as_product_element @ ( the_carrier @ B ) @ ( the_carrier @ B ) @ C @ D ) )
& ( below_refl @ B @ C @ D ) )
=> ( in @ A @ ( a_1_0_filter_1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).
thf(517,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( C
= ( join_of_latt_set @ A @ B ) )
<=> ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d21_lattice3) ).
thf(3940,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( C
= ( join_of_latt_set @ A @ B ) )
=> ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) ) )
& ( ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) )
=> ( C
= ( join_of_latt_set @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).
thf(98,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(1038,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)],[98]) ).
thf(608,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( element @ C @ A )
& ( element @ D @ B ) )
=> ( element @ ( ordered_pair_as_product_element @ A @ B @ C @ D ) @ ( cartesian_product2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_domain_1) ).
thf(4549,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( element @ C @ A )
& ( element @ D @ B ) )
=> ( element @ ( ordered_pair_as_product_element @ A @ B @ C @ D ) @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).
thf(52,axiom,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_zfmisc_1) ).
thf(840,plain,
! [A: $i,B: $i] :
( ( subset @ ( singleton @ A ) @ ( singleton @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(584,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(4351,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)],[584]) ).
thf(185,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(1658,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)],[185]) ).
thf(263,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(2354,plain,
! [A: $i,B: $i] :
( ( set_difference @ ( set_union2 @ A @ B ) @ B )
= ( set_difference @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(313,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( v1_partfun1 @ C @ A @ B ) )
=> ( ( function @ C )
& ( quasi_total @ C @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_funct_2) ).
thf(2747,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( ( ( function @ C )
& ( v1_partfun1 @ C @ A @ B ) )
=> ( ( function @ C )
& ( quasi_total @ C @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(101,axiom,
? [A: $i] : ( top_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_pre_topc) ).
thf(1053,plain,
? [A: $i] : ( top_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(388,axiom,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc27_membered) ).
thf(3246,plain,
! [A: $i,B: $i] :
( ( v1_membered @ A )
=> ( v1_membered @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(599,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_tarski__e4_7_1__tops_2__2) ).
thf(4437,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
( ! [D: $i,E: $i,F: $i] :
( ( ( D = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) )
& ( D = F )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= F )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( E = F ) )
=> ? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) ) )
& ( ? [F: $i] :
( ( in @ F @ ( cartesian_product2 @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ C ) )
& ( F = E )
& ? [G: $i,H: $i] :
( ( ( ordered_pair @ G @ H )
= E )
& ( in @ G @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) )
& ! [I: $i] :
( ( element @ I @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( I = G )
=> ( H
= ( subset_complement @ ( the_carrier @ A ) @ I ) ) ) ) ) )
=> ( in @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).
thf(245,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(2165,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( ( set_union2 @ ( singleton @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(500,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
<=> ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d13_lattices) ).
thf(3818,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( ( lower_bounded_semilattstr @ A )
=> ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) )
& ( ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) )
=> ( lower_bounded_semilattstr @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).
thf(470,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(3691,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( cast_as_carrier_subset @ A )
= ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).
thf(180,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(1637,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(393,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(3273,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)],[393]) ).
thf(311,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(2701,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)],[311]) ).
thf(42,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(808,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng @ ( relation_rng_restriction @ A @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(410,axiom,
! [A: $i] :
( ( v1_partfun1 @ ( identity_as_relation_of @ A ) @ A @ A )
& ( relation_of2_as_subset @ ( identity_as_relation_of @ A ) @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k6_partfun1) ).
thf(3412,plain,
! [A: $i] :
( ( v1_partfun1 @ ( identity_as_relation_of @ A ) @ A @ A )
& ( relation_of2_as_subset @ ( identity_as_relation_of @ A ) @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).
thf(442,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k4_subset_1) ).
thf(3564,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( ( subset_union2 @ A @ B @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).
thf(285,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_difference @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc12_finset_1) ).
thf(2500,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_difference @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(536,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(4057,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)],[536]) ).
thf(402,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(3356,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)],[402]) ).
thf(128,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(1221,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)],[128]) ).
thf(264,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(2357,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)],[264]) ).
thf(196,axiom,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t36_xboole_1) ).
thf(1749,plain,
! [A: $i,B: $i] : ( subset @ ( set_difference @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(546,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(4093,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)],[546]) ).
thf(581,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_lattices) ).
thf(4342,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).
thf(339,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(2892,plain,
! [A: $i,B: $i] :
( ( set_union2 @ A @ ( set_difference @ B @ A ) )
= ( set_union2 @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(255,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(2266,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)],[255]) ).
thf(551,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(4111,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)],[551]) ).
thf(395,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(3288,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)],[395]) ).
thf(289,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(2558,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)],[289]) ).
thf(20,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(703,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)],[20]) ).
thf(501,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( reflexive @ ( k2_lattice3 @ A ) )
& ( antisymmetric @ ( k2_lattice3 @ A ) )
& ( transitive @ ( k2_lattice3 @ A ) )
& ( v1_partfun1 @ ( k2_lattice3 @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( k2_lattice3 @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_lattice3) ).
thf(3829,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ( ( reflexive @ ( k2_lattice3 @ A ) )
& ( antisymmetric @ ( k2_lattice3 @ A ) )
& ( transitive @ ( k2_lattice3 @ A ) )
& ( v1_partfun1 @ ( k2_lattice3 @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( k2_lattice3 @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).
thf(369,axiom,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l28_zfmisc_1) ).
thf(3124,plain,
! [A: $i,B: $i] :
( ~ ( in @ A @ B )
=> ( disjoint @ ( singleton @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(166,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(1536,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)],[166]) ).
thf(230,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(1981,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)],[230]) ).
thf(173,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(1585,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)],[173]) ).
thf(116,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(1140,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)],[116]) ).
thf(634,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(4671,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)],[634]) ).
thf(407,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(3398,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)],[407]) ).
thf(316,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(2760,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)],[316]) ).
thf(344,axiom,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_relat_1) ).
thf(2924,plain,
! [A: $i] :
( ( ~ ( empty @ A )
& ( relation @ A ) )
=> ~ ( empty @ ( relation_dom @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(211,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(1839,plain,
! [A: $i] :
( ! [B: $i] :
( ( in @ B @ A )
=> ( ( ordinal @ B )
& ( subset @ B @ A ) ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(136,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(1316,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)],[136]) ).
thf(416,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B )
& ( element @ C @ ( the_carrier @ A ) )
& ( element @ D @ ( the_carrier @ B ) ) )
=> ( ( k10_filter_1 @ A @ B @ C @ D )
= ( ordered_pair @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k10_filter_1) ).
thf(3447,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A )
& ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B )
& ( element @ C @ ( the_carrier @ A ) )
& ( element @ D @ ( the_carrier @ B ) ) )
=> ( ( k10_filter_1 @ A @ B @ C @ D )
= ( ordered_pair @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).
thf(68,axiom,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_wellord2) ).
thf(903,plain,
! [A: $i] : ( reflexive @ ( inclusion_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(247,axiom,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(2194,plain,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(513,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ ( poset_of_lattice @ A ) ) )
=> ( ( cast_to_el_of_lattice @ A @ B )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d4_lattice3) ).
thf(3928,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ ( poset_of_lattice @ A ) ) )
=> ( ( cast_to_el_of_lattice @ A @ B )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).
thf(204,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(1781,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)],[204]) ).
thf(157,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(1448,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)],[157]) ).
thf(96,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(1010,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)],[96]) ).
thf(577,axiom,
? [A: $i] :
( ( latt_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_latt_str @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( lattice @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc9_lattices) ).
thf(4310,plain,
? [A: $i] :
( ( latt_str @ A )
& ~ ( empty_carrier @ A )
& ( strict_latt_str @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( lattice @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).
thf(309,axiom,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t15_finset_1) ).
thf(2696,plain,
! [A: $i,B: $i] :
( ( finite @ A )
=> ( finite @ ( set_intersection2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(167,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(1553,plain,
? [A: $i] :
( ~ ( empty @ A )
& ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( natural @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(118,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(1157,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)],[118]) ).
thf(210,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(1834,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)],[210]) ).
thf(629,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( relation_of2 @ C @ A @ B )
& ( element @ D @ A ) )
=> ( element @ ( apply_as_element @ A @ B @ C @ D ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k8_funct_2) ).
thf(4656,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( relation_of2 @ C @ A @ B )
& ( element @ D @ A ) )
=> ( element @ ( apply_as_element @ A @ B @ C @ D ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).
thf(262,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(2347,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)],[262]) ).
thf(151,axiom,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t60_relat_1) ).
thf(1412,plain,
( ( ( relation_dom @ empty_set )
= empty_set )
& ( ( relation_rng @ empty_set )
= empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(612,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(4559,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)],[612]) ).
thf(165,axiom,
! [A: $i] :
( ( centered @ A )
<=> ( ( A != empty_set )
& ! [B: $i] :
~ ( ( B != empty_set )
& ( subset @ B @ A )
& ( finite @ B )
& ( ( set_meet @ B )
= empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_compts_1) ).
thf(1517,plain,
! [A: $i] :
( ( ( centered @ A )
=> ( ( A != empty_set )
& ! [B: $i] :
~ ( ( B != empty_set )
& ( subset @ B @ A )
& ( finite @ B )
& ( ( set_meet @ B )
= empty_set ) ) ) )
& ( ( ( A != empty_set )
& ! [B: $i] :
~ ( ( B != empty_set )
& ( subset @ B @ A )
& ( finite @ B )
& ( ( set_meet @ B )
= empty_set ) ) )
=> ( centered @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(454,axiom,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(3625,plain,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).
thf(479,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(3725,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)],[479]) ).
thf(299,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(2594,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ( ( relation @ ( function_inverse @ A ) )
& ( function @ ( function_inverse @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(421,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_join @ A ) )
& ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_join @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_lattice2) ).
thf(3484,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_join @ A ) )
& ( function @ ( the_L_join @ A ) )
& ( quasi_total @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_join @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_join @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).
thf(240,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(2135,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)],[240]) ).
thf(103,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(1062,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)],[103]) ).
thf(6,axiom,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( well_orders @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_wellord2) ).
thf(652,plain,
! [A: $i] :
? [B: $i] :
( ( relation @ B )
& ( well_orders @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(323,axiom,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
thf(2811,plain,
! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( set_union2 @ A @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(235,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(2026,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)],[235]) ).
thf(131,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(1244,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)],[131]) ).
thf(23,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(731,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)],[23]) ).
thf(390,axiom,
? [A: $i] :
( ( latt_str @ A )
& ( strict_latt_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_lattices) ).
thf(3252,plain,
? [A: $i] :
( ( latt_str @ A )
& ( strict_latt_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(361,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(3085,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)],[361]) ).
thf(132,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(1276,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(174,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(1590,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)],[174]) ).
thf(333,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(2855,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_inverse_image @ B @ A ) @ ( relation_dom @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(474,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(3709,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)],[474]) ).
thf(181,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(1639,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( in @ B @ C )
& ( in @ C @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(471,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(3694,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)],[471]) ).
thf(401,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(3324,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)],[401]) ).
thf(578,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( latt_element_smaller @ A @ B @ C )
<=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d17_lattice3) ).
thf(4321,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( ( latt_element_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) )
=> ( latt_element_smaller @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).
thf(425,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(3497,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)],[425]) ).
thf(74,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(929,plain,
! [A: $i,B: $i,C: $i] :
( ( ( singleton @ A )
= ( unordered_pair @ B @ C ) )
=> ( B = C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(105,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(1077,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)],[105]) ).
thf(307,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(2680,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)],[307]) ).
thf(186,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_ordinal1) ).
thf(1661,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(462,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(3645,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)],[462]) ).
thf(343,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(2917,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)],[343]) ).
thf(575,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(4306,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)],[575]) ).
thf(548,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( reflexive_relstr @ A )
& ( rel_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( related_reflexive @ A @ B @ C )
<=> ( related @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r3_orders_2) ).
thf(4098,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( reflexive_relstr @ A )
& ( rel_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( ( related_reflexive @ A @ B @ C )
=> ( related @ A @ B @ C ) )
& ( ( related @ A @ B @ C )
=> ( related_reflexive @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).
thf(614,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(4570,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( element @ ( empty_carrier_subset @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).
thf(530,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B )
& ( closed_subset @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc7_pre_topc) ).
thf(4039,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( topological_space @ A )
& ( top_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B )
& ( closed_subset @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).
thf(94,axiom,
? [A: $i] : ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(999,plain,
? [A: $i] : ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(182,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(1642,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)],[182]) ).
thf(164,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(1512,plain,
? [A: $i] :
( ( epsilon_transitive @ A )
& ( epsilon_connected @ A )
& ( ordinal @ A )
& ( being_limit_ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(585,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( meet_of_latt_set @ A @ B )
= ( join_of_latt_set @ A @ ( a_2_2_lattice3 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d22_lattice3) ).
thf(4356,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( meet_of_latt_set @ A @ B )
= ( join_of_latt_set @ A @ ( a_2_2_lattice3 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).
thf(149,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(1395,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)],[149]) ).
thf(602,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( relation_of2 @ C @ A @ B )
& ( element @ D @ A ) )
=> ( ( apply_as_element @ A @ B @ C @ D )
= ( apply @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k8_funct_2) ).
thf(4523,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ( function @ C )
& ( quasi_total @ C @ A @ B )
& ( relation_of2 @ C @ A @ B )
& ( element @ D @ A ) )
=> ( ( apply_as_element @ A @ B @ C @ D )
= ( apply @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).
thf(433,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(3521,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)],[433]) ).
thf(36,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(790,plain,
! [A: $i,B: $i] :
( ( relation @ A )
=> ( relation @ ( relation_dom_restriction @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(346,axiom,
( ( epsilon_transitive @ omega )
& ( epsilon_connected @ omega )
& ( ordinal @ omega )
& ~ ( empty @ omega ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_ordinal2) ).
thf(2930,plain,
( ( epsilon_transitive @ omega )
& ( epsilon_connected @ omega )
& ( ordinal @ omega )
& ~ ( empty @ omega ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(555,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= B )
& ! [D: $i] :
( ( in @ D @ B )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = D )
=> ( ( apply @ C @ D )
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s2_funct_1__e4_7_2__tops_2) ).
thf(4125,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
& ! [C: $i] :
~ ( ( in @ C @ B )
& ! [D: $i] :
~ ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ( ( relation_dom @ C )
= B )
& ! [D: $i] :
( ( in @ D @ B )
=> ! [E: $i] :
( ( element @ E @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( E = D )
=> ( ( apply @ C @ D )
= ( subset_complement @ ( the_carrier @ A ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).
thf(81,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(950,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)],[81]) ).
thf(482,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(3733,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)],[482]) ).
thf(199,axiom,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_ordinal1) ).
thf(1757,plain,
! [A: $i] :
( ( ( epsilon_transitive @ A )
& ( epsilon_connected @ A ) )
=> ( ordinal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(632,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(4663,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)],[632]) ).
thf(518,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(3956,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)],[518]) ).
thf(574,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( open_subsets @ B @ A )
<=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_tops_2) ).
thf(4303,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( open_subsets @ B @ A )
=> ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A ) )
& ( ( closed_subsets @ ( complements_of_subsets @ ( the_carrier @ A ) @ B ) @ A )
=> ( open_subsets @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).
thf(394,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(3277,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)],[394]) ).
thf(624,axiom,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( open_subsets @ B @ A )
<=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( open_subset @ C @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_tops_2) ).
thf(4631,plain,
! [A: $i] :
( ( top_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) )
=> ( ( ( open_subsets @ B @ A )
=> ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( open_subset @ C @ A ) ) ) )
& ( ! [C: $i] :
( ( element @ C @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( in @ C @ B )
=> ( open_subset @ C @ A ) ) )
=> ( open_subsets @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).
thf(596,axiom,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( related @ A @ B @ C )
<=> ( in @ ( ordered_pair @ B @ C ) @ ( the_InternalRel @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d9_orders_2) ).
thf(4427,plain,
! [A: $i] :
( ( rel_str @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( related @ A @ B @ C )
=> ( in @ ( ordered_pair @ B @ C ) @ ( the_InternalRel @ A ) ) )
& ( ( in @ ( ordered_pair @ B @ C ) @ ( the_InternalRel @ A ) )
=> ( related @ A @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).
thf(525,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(4007,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)],[525]) ).
thf(139,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(1335,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)],[139]) ).
thf(169,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(1561,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)],[169]) ).
thf(379,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t92_zfmisc_1) ).
thf(3196,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(493,axiom,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( identity_on_carrier @ A )
= ( identity_as_relation_of @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d11_grcat_1) ).
thf(3765,plain,
! [A: $i] :
( ( one_sorted_str @ A )
=> ( ( identity_on_carrier @ A )
= ( identity_as_relation_of @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).
thf(84,axiom,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_wellord2) ).
thf(960,plain,
! [A: $i] :
( ( ordinal @ A )
=> ( connected @ ( inclusion_relation @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(19,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(696,plain,
! [A: $i] :
( ( ( relation_dom @ ( identity_relation @ A ) )
= A )
& ( ( relation_rng @ ( identity_relation @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(469,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( in @ E @ D )
<=> ( ( in @ E @ ( cartesian_product2 @ B @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ B )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_xboole_0__e4_7_2__tops_2__1) ).
thf(3673,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ! [C: $i] :
? [D: $i] :
! [E: $i] :
( ( ( in @ E @ D )
=> ( ( in @ E @ ( cartesian_product2 @ B @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ B )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) ) )
& ( ( ( in @ E @ ( cartesian_product2 @ B @ C ) )
& ? [F: $i,G: $i] :
( ( ( ordered_pair @ F @ G )
= E )
& ( in @ F @ B )
& ! [H: $i] :
( ( element @ H @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( H = F )
=> ( G
= ( subset_complement @ ( the_carrier @ A ) @ H ) ) ) ) ) )
=> ( in @ E @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).
thf(398,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_funct_1) ).
thf(3309,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).
thf(304,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(2636,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)],[304]) ).
thf(66,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(897,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)],[66]) ).
thf(150,axiom,
! [A: $i] :
( ( v2_membered @ A )
=> ( v1_membered @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc4_membered) ).
thf(1410,plain,
! [A: $i] :
( ( v2_membered @ A )
=> ( v1_membered @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(355,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(3045,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)],[355]) ).
thf(504,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(3843,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)],[504]) ).
thf(82,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(955,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_image @ B @ A ) @ ( relation_rng @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(63,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(886,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)],[63]) ).
thf(287,axiom,
! [A: $i] :
( ( identity_as_relation_of @ A )
= ( identity_relation @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k6_partfun1) ).
thf(2549,plain,
! [A: $i] :
( ( identity_as_relation_of @ A )
= ( identity_relation @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(371,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(3128,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)],[371]) ).
thf(448,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_union2 @ A @ B @ C ) @ ( powerset @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_subset_1) ).
thf(3605,plain,
! [A: $i,B: $i,C: $i] :
( ( ( element @ B @ ( powerset @ A ) )
& ( element @ C @ ( powerset @ A ) ) )
=> ( element @ ( subset_union2 @ A @ B @ C ) @ ( powerset @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).
thf(296,axiom,
? [A: $i] : ( join_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l2_lattices) ).
thf(2583,plain,
? [A: $i] : ( join_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(70,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(908,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)],[70]) ).
thf(21,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(723,plain,
! [A: $i] :
( ( relation @ A )
=> ( ( relation_image @ A @ ( relation_dom @ A ) )
= ( relation_rng @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(47,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(822,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)],[47]) ).
thf(405,axiom,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
& ( finite @ B ) )
=> ( finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t13_finset_1) ).
thf(3392,plain,
! [A: $i,B: $i] :
( ( ( subset @ A @ B )
& ( finite @ B ) )
=> ( finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).
thf(590,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(4395,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)],[590]) ).
thf(579,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(4330,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)],[579]) ).
thf(367,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(3113,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)],[367]) ).
thf(582,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ B ) )
=> ( ( latt_element_smaller @ B @ C @ A )
<=> ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t30_lattice3) ).
thf(4345,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ B )
& ( lattice @ B )
& ( latt_str @ B ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ B ) )
=> ( ( ( latt_element_smaller @ B @ C @ A )
=> ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) ) )
& ( ( relstr_set_smaller @ ( poset_of_lattice @ B ) @ A @ ( cast_to_el_of_LattPOSet @ B @ C ) )
=> ( latt_element_smaller @ B @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).
thf(544,axiom,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l2_lattices) ).
thf(4088,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).
thf(110,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(1120,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)],[110]) ).
thf(563,axiom,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( in @ ( ordered_pair @ D @ E ) @ C )
<=> ( ( in @ D @ B )
& ( in @ D @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s1_funct_1__e4_7_2__tops_2__1) ).
thf(4221,plain,
! [A: $i,B: $i] :
( ( ( one_sorted_str @ A )
& ( element @ B @ ( powerset @ ( powerset @ ( the_carrier @ A ) ) ) ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( D
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) )
& ( in @ C @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = C )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( D = E ) )
=> ? [C: $i] :
( ( relation @ C )
& ( function @ C )
& ! [D: $i,E: $i] :
( ( ( in @ ( ordered_pair @ D @ E ) @ C )
=> ( ( in @ D @ B )
& ( in @ D @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) ) )
& ( ( ( in @ D @ B )
& ( in @ D @ B )
& ! [F: $i] :
( ( element @ F @ ( powerset @ ( the_carrier @ A ) ) )
=> ( ( F = D )
=> ( E
= ( subset_complement @ ( the_carrier @ A ) @ F ) ) ) ) )
=> ( in @ ( ordered_pair @ D @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).
thf(108,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(1093,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)],[108]) ).
thf(225,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(1930,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( subset @ ( relation_rng_restriction @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(387,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l50_zfmisc_1) ).
thf(3244,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( subset @ A @ ( union @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).
thf(26,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(740,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)],[26]) ).
thf(336,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(2864,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)],[336]) ).
thf(219,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(1880,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)],[219]) ).
thf(154,axiom,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_funct_1) ).
thf(1431,plain,
? [A: $i] :
( ( relation @ A )
& ( function @ A )
& ( one_to_one @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(310,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(2699,plain,
! [A: $i,B: $i] :
( ( ( relation @ A )
& ( relation @ B ) )
=> ( relation @ ( set_union2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(498,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( element @ C @ A )
& ( element @ D @ B ) )
=> ( ( ordered_pair_as_product_element @ A @ B @ C @ D )
= ( ordered_pair @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k1_domain_1) ).
thf(3812,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( element @ C @ A )
& ( element @ D @ B ) )
=> ( ( ordered_pair_as_product_element @ A @ B @ C @ D )
= ( ordered_pair @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).
thf(241,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(2138,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)],[241]) ).
thf(635,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(4674,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)],[635]) ).
thf(520,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(3960,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)],[520]) ).
thf(184,axiom,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ! [C: $i,D: $i] :
( ( ( rel_str_of @ A @ B )
= ( rel_str_of @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',free_g1_orders_2) ).
thf(1653,plain,
! [A: $i,B: $i] :
( ( relation_of2 @ B @ A @ A )
=> ! [C: $i,D: $i] :
( ( ( rel_str_of @ A @ B )
= ( rel_str_of @ C @ D ) )
=> ( ( A = C )
& ( B = D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(619,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(4584,plain,
! [A: $i] :
( ( v1_membered @ A )
=> ! [B: $i] :
( ( element @ B @ A )
=> ( v1_xcmplx_0 @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).
thf(338,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(2887,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)],[338]) ).
thf(351,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(2998,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)],[351]) ).
thf(232,axiom,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry_r1_xboole_0) ).
thf(2018,plain,
! [A: $i,B: $i] :
( ( disjoint @ A @ B )
=> ( disjoint @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(357,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(3052,plain,
! [A: $i,B: $i] :
( ( ( relation @ B )
& ( function @ B ) )
=> ( ( finite @ A )
=> ( finite @ ( relation_image @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).
thf(277,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(2470,plain,
! [A: $i] :
( ( relation @ A )
=> ( subset @ A @ ( cartesian_product2 @ ( relation_dom @ A ) @ ( relation_rng @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(104,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(1069,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)],[104]) ).
thf(523,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( reflexive_relstr @ A )
& ( rel_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( related_reflexive @ A @ B @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r3_orders_2) ).
thf(4002,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( reflexive_relstr @ A )
& ( rel_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( related_reflexive @ A @ B @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).
thf(202,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(1769,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)],[202]) ).
thf(31,axiom,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_wellord1) ).
thf(755,plain,
! [A: $i,B: $i] :
( ( relation @ B )
=> ( ( connected @ B )
=> ( connected @ ( relation_restriction @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(152,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(1417,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)],[152]) ).
thf(320,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(2771,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)],[320]) ).
thf(58830,plain,
$false,
inference(cvc4,[status(thm)],[645,3873,1501,1190,2336,2263,1559,1729,846,1850,941,2295,4101,760,4005,4290,1031,2427,3969,962,762,1924,4339,2231,3762,2495,2908,3509,994,4479,4043,3534,3185,977,1808,2659,3417,1422,1569,655,2380,4408,677,1486,4532,1671,1608,4090,3429,3049,2607,4575,3004,1164,2448,4638,814,1342,2844,3717,1458,2575,2476,2168,4247,2580,3103,3629,1001,2763,3502,2020,1935,3730,1386,2825,4386,2972,3634,3249,4079,1228,3597,4348,2039,4070,4415,3495,3061,3617,802,3845,1992,2126,3043,1055,3204,2814,4422,1323,838,3148,3653,1313,1279,934,3744,817,3099,2857,3638,2552,1890,2373,2158,3835,2301,4390,1875,4253,1847,1892,3385,3312,4551,1362,1963,1489,3524,1664,3934,4578,1373,2782,3670,3527,834,4115,3851,905,3438,1864,1151,937,981,3537,4055,4033,3505,2371,3157,660,3756,2750,4643,1600,4588,3492,2884,687,4660,1345,988,3559,849,3011,3450,3303,1438,793,810,4557,3054,1736,3550,3795,1047,746,3603,1174,1751,3610,693,1387,910,957,894,3711,825,1217,2147,3752,4521,1920,4053,852,3750,2200,889,1787,682,1128,3647,3931,1573,2769,3958,2445,4295,2464,2758,2565,3198,757,2345,1392,3667,4123,3105,3482,2895,2841,4108,2577,4544,4636,3704,3963,1138,3632,3650,4219,4668,650,948,2023,3622,3441,642,1842,4191,1214,3643,1744,4049,884,3407,753,3714,3728,2472,679,3126,2487,1831,1131,3770,2851,1932,1050,984,4120,1852,2482,1776,3656,2497,4540,3540,3409,2862,4152,1720,4259,3083,3162,3201,1759,2926,2203,1086,3306,1170,4036,1167,4308,4572,1857,4184,805,4081,3321,2368,1187,2440,1764,1754,3871,2585,2570,4393,3738,4359,733,2260,965,1727,4594,3607,856,4562,819,2970,3768,3925,3240,3736,743,3458,1226,3599,975,3387,3187,4283,1784,1389,3255,843,1125,2657,2849,3868,4612,2817,4526,3118,4096,4645,2935,2442,4060,3641,2474,3568,1090,3619,4433,2573,3719,3532,672,764,3101,2721,900,1282,3815,1117,3515,1603,4586,737,3500,4300,2197,863,2097,3270,4213,880,4073,4430,3747,2625,1731,1325,2785,1460,1688,2610,1339,4420,858,4009,799,726,4208,4181,932,3150,2329,2911,4554,1926,3627,868,3110,3159,1435,3706,2967,3512,2156,4067,3876,1973,658,912,685,2401,1594,3759,3838,2765,3423,636,1496,2303,3490,4084,1583,4013,3936,663,2503,2234,1134,4292,749,939,3455,2846,836,3571,3849,795,3395,3206,927,4529,2037,2597,4581,968,1181,4255,1798,690,1611,3214,1647,3786,2689,4045,4658,2397,3507,2588,3754,4287,2821,3209,4398,3940,1038,4549,840,4351,1658,2354,2747,1053,3246,4437,2165,3818,3691,1637,3273,2701,808,3412,3564,2500,4057,3356,1221,2357,1749,4093,4342,2892,2266,4111,3288,2558,703,3829,3124,1536,1981,1585,1140,4671,3398,2760,2924,1839,1316,3447,903,2194,3928,1781,1448,1010,4310,2696,1553,1157,1834,4656,2347,1412,4559,1517,3625,3725,2594,3484,2135,1062,652,2811,2026,1244,731,3252,3085,1276,1590,2855,3709,1639,3694,3324,4321,3497,929,1077,2680,1661,3645,2917,4306,4098,4570,4039,999,1642,1512,4356,1395,4523,3521,790,2930,4125,950,3733,1757,4663,3956,4303,3277,4631,4427,4007,1335,1561,3196,3765,960,696,3673,3309,2636,897,1410,3045,3843,955,886,2549,3128,3605,2583,908,723,822,3392,4395,4330,3113,4345,4088,1120,4221,1093,1930,3244,740,2864,1880,1431,2699,3812,2138,4674,3960,1653,4584,2887,2998,2018,3052,2470,1069,4002,1769,755,1417,2771]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU353+2 : TPTP v8.1.2. Released v3.3.0.
% 0.03/0.15 % Command : run_Leo-III %s %d
% 0.14/0.35 % Computer : n017.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu May 18 12:29:16 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.96/0.83 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.81/1.09 % [INFO] Parsing done (259ms).
% 1.81/1.10 % [INFO] Running in sequential loop mode.
% 2.38/1.30 % [INFO] eprover registered as external prover.
% 2.38/1.30 % [INFO] cvc4 registered as external prover.
% 2.38/1.31 % [INFO] Scanning for conjecture ...
% 2.72/1.42 % [INFO] Found a conjecture and 664 axioms. Running axiom selection ...
% 3.56/1.58 % [INFO] Axiom selection finished. Selected 633 axioms (removed 31 axioms).
% 4.43/1.87 % [INFO] Problem is first-order (TPTP FOF).
% 4.43/1.90 % [INFO] Type checking passed.
% 4.43/1.90 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 123.93/24.57 % External prover 'cvc4' found a proof!
% 123.93/24.57 % [INFO] Killing All external provers ...
% 123.93/24.57 % Time passed: 24057ms (effective reasoning time: 23469ms)
% 123.93/24.57 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 123.93/24.57 % Axioms used in derivation (633): d1_lattices, t15_finset_1, d1_enumset1, redefinition_m2_relset_1, dt_k7_grcat_1, d16_lattice3, 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, fc2_partfun1, s3_subset_1__e2_37_1_1__pre_topc, cc12_membered, t17_xboole_1, t90_relat_1, t43_subset_1, t54_wellord1, fc1_ordinal2, d6_relat_2, l82_funct_1, fraenkel_a_1_0_filter_1, idempotence_k3_xboole_0, t1_boole, t39_wellord1, cc4_funct_2, fc2_pre_topc, fc37_membered, redefinition_k5_subset_1, d13_lattices, t3_xboole_1, dt_k4_relset_1, dt_k5_relset_1, d5_relat_1, d4_ordinal1, rc4_finset_1, l1_wellord1, l2_wellord1, fc1_xboole_0, t10_tops_2, t7_mcart_1, d16_lattices, t17_tops_2, 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, t29_lattice3, t44_relat_1, existence_l1_lattices, cc2_arytm_3, t26_lattices, fc4_relat_1, fc4_funct_1, t1_subset, t28_lattice3, fc34_membered, fc4_lattice3, t144_relat_1, fc31_membered, t22_funct_1, existence_l3_lattices, rc1_membered, dt_k5_lattices, fc4_tops_1, fc3_relat_1, d8_funct_1, fc4_subset_1, dt_k1_tops_1, fc4_lattice2, d6_orders_2, t117_relat_1, t2_boole, d5_ordinal2, d13_pre_topc, s1_tarski__e4_7_2__tops_2__1, d6_pre_topc, t12_pre_topc, t32_ordinal1, t47_relat_1, fc9_relat_1, dt_k4_subset_1, fc6_tops_1, l40_tops_1, t99_zfmisc_1, d1_tops_2, rc2_orders_2, t47_setfam_1, fc1_funct_1, t34_lattice3, s1_tarski__e10_24__wellord2__2, l3_subset_1, d2_lattice3, d1_wellord1, rc2_partfun1, d9_funct_1, rc3_funct_1, dt_k6_partfun1, fc1_pre_topc, d1_struct_0, t56_relat_1, t37_xboole_1, d3_compts_1, fc2_lattice2, cc1_relat_1, d22_lattice3, idempotence_k5_subset_1, t55_funct_1, dt_k10_filter_1, 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, fc28_membered, fc3_lattice2, t62_funct_1, t46_pre_topc, d3_pre_topc, fc11_finset_1, d2_tarski, dt_k4_lattice3, dt_k2_binop_1, d4_subset_1, fc2_xboole_0, dt_u1_orders_2, rc6_pre_topc, fc32_membered, fc30_membered, fc5_pre_topc, involutiveness_k3_subset_1, d2_tops_2, rc1_funct_1, reflexivity_r3_lattices, d1_xboole_0, d9_orders_2, l50_zfmisc_1, dt_k5_lattice3, rc3_lattices, 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, cc2_funct_2, 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, redefinition_r3_orders_2, rc3_partfun1, d9_relat_2, s1_xboole_0__e2_37_1_1__pre_topc__1, t28_wellord2, t31_ordinal1, t116_relat_1, fc33_membered, t31_lattice3, cc2_ordinal1, commutativity_k2_xboole_0, t2_lattice3, fc3_subset_1, t5_subset, t30_lattice3, d4_relat_1, d12_relat_2, redefinition_k10_filter_1, cc1_arytm_3, existence_m1_relset_1, t1_zfmisc_1, dt_l3_lattices, fc2_tops_1, t17_wellord1, fc2_subset_1, dt_k8_funct_2, t4_xboole_0, s1_xboole_0__e4_7_1__tops_2__1, d4_wellord2, cc19_membered, commutativity_k5_subset_1, l25_zfmisc_1, fraenkel_a_2_3_lattice3, d8_xboole_0, fc2_lattice3, t5_wellord2, d5_wellord1, d1_ordinal1, t3_wellord2, t50_lattice3, dt_l2_lattices, cc3_funct_2, t13_compts_1, fc2_funct_1, t5_wellord1, t8_funct_1, fc6_membered, dt_k3_lattice3, t166_relat_1, l32_xboole_1, t8_zfmisc_1, t11_tops_2, fc1_lattice3, s1_relat_1__e6_21__wellord2, s1_tarski__e4_7_2__tops_2__2, rc4_funct_1, d4_lattice3, t44_pre_topc, t23_wellord1, t2_tarski, s1_ordinal1__e8_6__wellord2, t33_xboole_1, d3_lattices, t146_funct_1, 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, rc1_arytm_3, cc1_partfun1, t19_wellord1, fc13_relat_1, dt_k9_filter_1, t4_boole, d21_lattice3, cc11_membered, t20_wellord1, cc1_funct_2, d4_tarski, t13_tops_2, t18_wellord1, d16_relat_2, t9_zfmisc_1, t174_relat_1, t12_tops_2, d3_lattice3, redefinition_k5_setfam_1, s1_tarski__e6_27__finset_1__1, involutiveness_k4_relat_1, fc2_orders_2, cc1_ordinal1, d8_pre_topc, reflexivity_r2_wellord2, t30_tops_1, s1_funct_1__e4_7_2__tops_2__1, dt_k1_domain_1, cc20_membered, t7_boole, t21_wellord1, existence_l1_orders_2, cc1_funct_1, t23_ordinal1, d17_lattice3, s1_tarski__e10_24__wellord2__1, t118_relat_1, s1_tarski__e16_22__wellord2__2, t143_relat_1, reflexivity_r3_orders_2, t48_xboole_1, t3_ordinal1, dt_k1_wellord2, rc1_orders_2, s2_ordinal1__e18_27__finset_1__1, 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, fc29_membered, redefinition_k6_partfun1, fc38_membered, l4_wellord1, fc5_lattice2, fc3_ordinal1, dt_g1_orders_2, rc2_funct_1, t3_lattice3, cc4_membered, d11_relat_1, t21_funct_2, d2_compts_1, dt_k8_filter_1, t9_funct_2, t160_relat_1, fc10_relat_1, rc1_subset_1, cc3_ordinal1, t74_relat_1, redefinition_k1_domain_1, connectedness_r1_ordinal1, t8_wellord1, cc1_lattices, rc1_funct_2, t2_wellord2, rc3_ordinal1, cc18_membered, s1_xboole_0__e1_40__pre_topc__1, s1_xboole_0__e6_27__finset_1, t45_relat_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, abstractness_v1_orders_2, t83_xboole_1, redefinition_r3_lattices, l28_zfmisc_1, cc2_funct_1, d2_wellord1, l4_zfmisc_1, t3_subset, cc1_relset_1, t33_ordinal1, rc9_lattices, t115_relat_1, dt_u1_lattices, existence_m1_subset_1, d2_mcart_1, dt_k16_lattice3, t46_setfam_1, dt_k15_lattice3, s1_tarski__e4_7_1__tops_2__2, t17_finset_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, s1_xboole_0__e4_7_2__tops_2__1, fc39_membered, t106_zfmisc_1, cc15_membered, t145_funct_1, fc1_finset_1, l71_subset_1, existence_m1_connsp_2, dt_k7_relat_1, redefinition_k4_subset_1, t7_wellord2, l2_zfmisc_1, t25_wellord1, s2_funct_1__e4_7_2__tops_2, d8_relat_1, t42_ordinal1, t94_relat_1, t48_setfam_1, d4_wellord1, d5_subset_1, t92_zfmisc_1, t146_relat_1, d5_orders_2, t70_funct_1, t31_wellord1, t69_enumset1, t41_ordinal1, commutativity_k4_subset_1, d7_wellord1, dt_k5_subset_1, existence_m2_relset_1, t16_tops_2, t38_zfmisc_1, l55_zfmisc_1, s1_tarski__e8_6__wellord2__1, t10_ordinal1, d1_zfmisc_1, fc3_orders_2, 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, dt_l1_orders_2, fraenkel_a_2_2_lattice3, fc1_orders_2, t51_tops_1, involutiveness_k7_setfam_1, t6_funct_2, dt_m1_connsp_2, t8_xboole_1, d3_tarski, fc1_struct_0, cc6_funct_2, 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, free_g1_orders_2, t37_relat_1, t28_xboole_1, d1_relset_1, d8_filter_1, redefinition_k3_lattices, s1_xboole_0__e10_24__wellord2__1, t5_tops_2, dt_k4_lattices, t60_relat_1, t6_zfmisc_1, reflexivity_r1_ordinal1, s1_tarski__e18_27__finset_1__1, t2_xboole_1, s1_xboole_0__e6_22__wellord2, fc36_membered, cc17_membered, d11_grcat_1, rc6_lattices, d1_funct_2, dt_k6_relat_1, dt_k2_lattice3, t7_lattice3, d1_connsp_2, d12_funct_1, d3_ordinal1, d2_lattices, dt_g3_lattices, fc6_relat_1, rc7_pre_topc, free_g3_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, d1_relat_2, fc1_relat_1, d6_wellord1, d6_relat_1, redefinition_k2_lattice3, l23_zfmisc_1, t48_pre_topc, t54_subset_1, rc1_finset_1, dt_k1_lattice3, cc1_finset_1, s2_funct_1__e10_24__wellord2, fc12_finset_1, dt_k2_wellord1, abstractness_v3_lattices, fc2_arytm_3, s1_funct_1__e16_22__wellord2__1, t136_zfmisc_1, t8_boole, rc1_xboole_0, redefinition_k8_funct_2, fc3_tops_1, dt_u1_pre_topc, fc3_funct_1, t16_wellord1, rc5_struct_0, t12_xboole_1, fc7_relat_1, fc1_ordinal1, d8_lattice3, t49_wellord1, t86_relat_1, t26_orders_2, d1_lattice3, 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, fc40_membered, s1_tarski__e4_7_1__tops_2__1, s1_tarski__e1_40__pre_topc__1, fc8_relat_1, fc41_membered, idempotence_k2_xboole_0, s1_funct_1__e4_7_1__tops_2__1, d1_funct_1, l30_wellord2, t15_pre_topc, t32_filter_1, t9_tarski, t145_relat_1, cc3_arytm_3, commutativity_k3_lattices, d10_xboole_0, t1_lattice3, 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, dt_u2_lattices, d8_relat_2, dt_k2_lattices, d4_relat_2, t3_xboole_0, t5_connsp_2, t12_relset_1, redefinition_k4_relset_1, cc5_funct_2, t25_relat_1, d1_tops_1, l29_wellord1, cc16_membered, fc13_finset_1, d9_lattice3, t25_orders_2, idempotence_k4_subset_1, s2_funct_1__e16_22__wellord2__1, d1_binop_1, rc2_xboole_0, fc27_membered, s2_funct_1__e4_7_1__tops_2, fc4_ordinal1, t40_xboole_1, t45_xboole_1, t44_tops_1, d1_pre_topc, dt_k6_pre_topc, fc3_lattices, 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, cc2_lattices, t167_relat_1, dt_k3_subset_1, l1_zfmisc_1, t39_xboole_1, t23_lattices, t119_relat_1, antisymmetry_r2_hidden, rc2_funct_2, redefinition_k4_lattices, dt_k2_pre_topc
% 123.93/24.57 % No. of inferences in proof: 1270
% 123.93/24.58 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 24057 ms resp. 23469 ms w/o parsing
% 124.87/24.91 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 124.87/24.92 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------