TPTP Problem File: SEU370+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU370+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t4_yellow_1
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-t4_yellow_1 [Urb07]
% Status : Theorem
% Rating : 0.97 v8.2.0, 1.00 v6.2.0, 0.96 v6.1.0, 1.00 v3.3.0
% Syntax : Number of formulae : 154 ( 44 unt; 0 def)
% Number of atoms : 653 ( 42 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 591 ( 92 ~; 1 |; 385 &)
% ( 10 <=>; 103 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 50 ( 48 usr; 1 prp; 0-3 aty)
% Number of functors : 30 ( 30 usr; 1 con; 0-3 aty)
% Number of variables : 236 ( 208 !; 28 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(abstractness_v1_orders_2,axiom,
! [A] :
( rel_str(A)
=> ( strict_rel_str(A)
=> A = rel_str_of(the_carrier(A),the_InternalRel(A)) ) ) ).
fof(abstractness_v3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( strict_latt_str(A)
=> A = latt_str_of(the_carrier(A),the_L_join(A),the_L_meet(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_knaster,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A) )
=> ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A)
& lattice(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A)
& bounded_lattstr(A) ) ) ) ).
fof(cc1_lattice3,axiom,
! [A] :
( rel_str(A)
=> ( with_suprema_relstr(A)
=> ~ empty_carrier(A) ) ) ).
fof(cc1_lattices,axiom,
! [A] :
( 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) ) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(cc1_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& complete_relstr(A) )
=> ( ~ empty_carrier(A)
& with_suprema_relstr(A)
& with_infima_relstr(A) ) ) ) ).
fof(cc2_lattice3,axiom,
! [A] :
( rel_str(A)
=> ( with_infima_relstr(A)
=> ~ empty_carrier(A) ) ) ).
fof(cc2_lattices,axiom,
! [A] :
( 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) ) ) ) ).
fof(cc2_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& trivial_carrier(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& complete_relstr(A) ) ) ) ).
fof(cc3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A) )
=> ( ~ empty_carrier(A)
& bounded_lattstr(A) ) ) ) ).
fof(cc3_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& complete_relstr(A) )
=> ( ~ empty_carrier(A)
& bounded_relstr(A) ) ) ) ).
fof(cc4_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& bounded_lattstr(A) )
=> ( ~ empty_carrier(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A) ) ) ) ).
fof(cc4_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( bounded_relstr(A)
=> ( lower_bounded_relstr(A)
& upper_bounded_relstr(A) ) ) ) ).
fof(cc5_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& boolean_lattstr(A) )
=> ( ~ empty_carrier(A)
& distributive_lattstr(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A)
& bounded_lattstr(A)
& complemented_lattstr(A) ) ) ) ).
fof(cc5_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( lower_bounded_relstr(A)
& upper_bounded_relstr(A) )
=> bounded_relstr(A) ) ) ).
fof(cc6_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& distributive_lattstr(A)
& bounded_lattstr(A)
& complemented_lattstr(A) )
=> ( ~ empty_carrier(A)
& boolean_lattstr(A) ) ) ) ).
fof(cc7_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& lattice(A)
& distributive_lattstr(A) )
=> ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A)
& lattice(A)
& modular_lattstr(A) ) ) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(commutativity_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,C) = subset_union2(A,C,B) ) ).
fof(commutativity_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = subset_intersection2(A,C,B) ) ).
fof(d1_lattice3,axiom,
! [A,B] :
( ( strict_latt_str(B)
& latt_str(B) )
=> ( B = boole_lattice(A)
<=> ( the_carrier(B) = powerset(A)
& ! [C] :
( element(C,powerset(A))
=> ! [D] :
( 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) ) ) ) ) ) ) ).
fof(d1_wellord2,axiom,
! [A,B] :
( relation(B)
=> ( B = inclusion_relation(A)
<=> ( relation_field(B) = A
& ! [C,D] :
( ( in(C,A)
& in(D,A) )
=> ( in(ordered_pair(C,D),B)
<=> subset(C,D) ) ) ) ) ) ).
fof(d1_yellow_1,axiom,
! [A] : incl_POSet(A) = rel_str_of(A,inclusion_order(A)) ).
fof(d2_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> poset_of_lattice(A) = rel_str_of(the_carrier(A),k2_lattice3(A)) ) ).
fof(d2_yellow_1,axiom,
! [A] : boole_POSet(A) = poset_of_lattice(boole_lattice(A)) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d6_relat_1,axiom,
! [A] :
( relation(A)
=> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)) ) ).
fof(d9_orders_2,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( related(A,B,C)
<=> in(ordered_pair(B,C),the_InternalRel(A)) ) ) ) ) ).
fof(dt_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ( strict_rel_str(rel_str_of(A,B))
& rel_str(rel_str_of(A,B)) ) ) ).
fof(dt_g3_lattices,axiom,
! [A,B,C] :
( ( 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)) ) ) ).
fof(dt_k1_binop_1,axiom,
$true ).
fof(dt_k1_lattice3,axiom,
! [A] :
( strict_latt_str(boole_lattice(A))
& latt_str(boole_lattice(A)) ) ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_wellord2,axiom,
! [A] : relation(inclusion_relation(A)) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_yellow_1,axiom,
! [A] :
( reflexive(inclusion_order(A))
& antisymmetric(inclusion_order(A))
& transitive(inclusion_order(A))
& v1_partfun1(inclusion_order(A),A,A)
& relation_of2_as_subset(inclusion_order(A),A,A) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_lattice3,axiom,
! [A] :
( ( ~ 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)) ) ) ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k2_yellow_1,axiom,
! [A] :
( strict_rel_str(incl_POSet(A))
& rel_str(incl_POSet(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_lattice3,axiom,
! [A] :
( ( ~ 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)) ) ) ).
fof(dt_k3_relat_1,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k3_yellow_1,axiom,
! [A] :
( strict_rel_str(boole_POSet(A))
& rel_str(boole_POSet(A)) ) ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_dom_as_subset(A,B,C),powerset(A)) ) ).
fof(dt_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_union2(A,B,C),powerset(A)) ) ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_rng_as_subset(A,B,C),powerset(B)) ) ).
fof(dt_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_intersection2(A,B,C),powerset(A)) ) ).
fof(dt_k9_filter_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> relation(relation_of_lattice(A)) ) ).
fof(dt_l1_lattices,axiom,
! [A] :
( meet_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( join_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( meet_semilatt_str(A)
& join_semilatt_str(A) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_u1_lattices,axiom,
! [A] :
( 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)) ) ) ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_lattices,axiom,
! [A] :
( 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)) ) ) ).
fof(existence_l1_lattices,axiom,
? [A] : meet_semilatt_str(A) ).
fof(existence_l1_orders_2,axiom,
? [A] : rel_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_l2_lattices,axiom,
? [A] : join_semilatt_str(A) ).
fof(existence_l3_lattices,axiom,
? [A] : latt_str(A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(fc1_knaster,axiom,
! [A] :
( ~ 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))
& distributive_lattstr(boole_lattice(A))
& modular_lattstr(boole_lattice(A))
& lower_bounded_semilattstr(boole_lattice(A))
& upper_bounded_semilattstr(boole_lattice(A))
& bounded_lattstr(boole_lattice(A))
& complemented_lattstr(boole_lattice(A))
& boolean_lattstr(boole_lattice(A))
& complete_latt_str(boole_lattice(A)) ) ).
fof(fc1_lattice3,axiom,
! [A] :
( ~ empty_carrier(boole_lattice(A))
& strict_latt_str(boole_lattice(A)) ) ).
fof(fc1_orders_2,axiom,
! [A,B] :
( ( ~ empty(A)
& relation_of2(B,A,A) )
=> ( ~ empty_carrier(rel_str_of(A,B))
& strict_rel_str(rel_str_of(A,B)) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_yellow_0,axiom,
! [A,B] :
( relation_of2(B,singleton(A),singleton(A))
=> ( ~ empty_carrier(rel_str_of(singleton(A),B))
& strict_rel_str(rel_str_of(singleton(A),B))
& trivial_carrier(rel_str_of(singleton(A),B)) ) ) ).
fof(fc1_yellow_1,axiom,
! [A] :
( ( ~ 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))
& with_suprema_relstr(poset_of_lattice(A))
& with_infima_relstr(poset_of_lattice(A)) ) ) ).
fof(fc2_lattice3,axiom,
! [A] :
( ~ 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)) ) ).
fof(fc2_orders_2,axiom,
! [A] :
( ( 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)) ) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc2_yellow_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& upper_bounded_semilattstr(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))
& upper_bounded_relstr(poset_of_lattice(A))
& with_suprema_relstr(poset_of_lattice(A))
& with_infima_relstr(poset_of_lattice(A)) ) ) ).
fof(fc3_lattice3,axiom,
! [A] :
( ~ 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))
& distributive_lattstr(boole_lattice(A))
& modular_lattstr(boole_lattice(A))
& lower_bounded_semilattstr(boole_lattice(A))
& upper_bounded_semilattstr(boole_lattice(A))
& bounded_lattstr(boole_lattice(A))
& complemented_lattstr(boole_lattice(A))
& boolean_lattstr(boole_lattice(A)) ) ).
fof(fc3_lattices,axiom,
! [A,B,C] :
( ( ~ 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)) ) ) ).
fof(fc3_orders_2,axiom,
! [A,B] :
( ( 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)) ) ) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(fc3_yellow_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& lower_bounded_semilattstr(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))
& lower_bounded_relstr(poset_of_lattice(A))
& with_suprema_relstr(poset_of_lattice(A))
& with_infima_relstr(poset_of_lattice(A)) ) ) ).
fof(fc4_lattice3,axiom,
! [A] :
( ( ~ 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)) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc4_yellow_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(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))
& lower_bounded_relstr(poset_of_lattice(A))
& upper_bounded_relstr(poset_of_lattice(A))
& bounded_relstr(poset_of_lattice(A))
& with_suprema_relstr(poset_of_lattice(A))
& with_infima_relstr(poset_of_lattice(A))
& complete_relstr(poset_of_lattice(A)) ) ) ).
fof(fc5_yellow_1,axiom,
! [A] :
( strict_rel_str(incl_POSet(A))
& reflexive_relstr(incl_POSet(A))
& transitive_relstr(incl_POSet(A))
& antisymmetric_relstr(incl_POSet(A)) ) ).
fof(fc6_yellow_1,axiom,
! [A] :
( ~ empty(A)
=> ( ~ empty_carrier(incl_POSet(A))
& strict_rel_str(incl_POSet(A))
& reflexive_relstr(incl_POSet(A))
& transitive_relstr(incl_POSet(A))
& antisymmetric_relstr(incl_POSet(A)) ) ) ).
fof(fc7_yellow_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A)) ) ).
fof(fc8_yellow_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A))
& lower_bounded_relstr(boole_POSet(A))
& upper_bounded_relstr(boole_POSet(A))
& bounded_relstr(boole_POSet(A))
& with_suprema_relstr(boole_POSet(A))
& with_infima_relstr(boole_POSet(A))
& complete_relstr(boole_POSet(A)) ) ).
fof(free_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ! [C,D] :
( rel_str_of(A,B) = rel_str_of(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(free_g3_lattices,axiom,
! [A,B,C] :
( ( 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,E,F] :
( latt_str_of(A,B,C) = latt_str_of(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(idempotence_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,B) = B ) ).
fof(idempotence_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,B) = B ) ).
fof(rc10_lattices,axiom,
? [A] :
( 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)
& distributive_lattstr(A)
& modular_lattstr(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A) ) ).
fof(rc11_lattices,axiom,
? [A] :
( 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)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A)
& bounded_lattstr(A) ) ).
fof(rc12_lattices,axiom,
? [A] :
( 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)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A)
& bounded_lattstr(A)
& complemented_lattstr(A) ) ).
fof(rc13_lattices,axiom,
? [A] :
( 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)
& distributive_lattstr(A)
& lower_bounded_semilattstr(A)
& upper_bounded_semilattstr(A)
& bounded_lattstr(A)
& complemented_lattstr(A)
& boolean_lattstr(A) ) ).
fof(rc1_lattice3,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& complete_relstr(A) ) ).
fof(rc1_orders_2,axiom,
? [A] :
( rel_str(A)
& strict_rel_str(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc1_yellow_0,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A)
& trivial_carrier(A) ) ).
fof(rc2_lattice3,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A) ) ).
fof(rc2_orders_2,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A) ) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc2_yellow_0,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A) ) ).
fof(rc3_lattices,axiom,
? [A] :
( latt_str(A)
& strict_latt_str(A) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(rc6_lattices,axiom,
? [A] :
( latt_str(A)
& ~ empty_carrier(A)
& strict_latt_str(A) ) ).
fof(rc9_lattices,axiom,
? [A] :
( 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) ) ).
fof(redefinition_k1_yellow_1,axiom,
! [A] : inclusion_order(A) = inclusion_relation(A) ).
fof(redefinition_k2_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> k2_lattice3(A) = relation_of_lattice(A) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_dom_as_subset(A,B,C) = relation_dom(C) ) ).
fof(redefinition_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,C) = set_union2(B,C) ) ).
fof(redefinition_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_rng_as_subset(A,B,C) = relation_rng(C) ) ).
fof(redefinition_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = set_intersection2(B,C) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(redefinition_r3_orders_2,axiom,
! [A,B,C] :
( ( ~ 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) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(reflexivity_r3_orders_2,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& rel_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> related_reflexive(A,B,B) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t22_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,B,A)
=> ( ! [D] :
~ ( in(D,B)
& ! [E] : ~ in(ordered_pair(D,E),C) )
<=> relation_dom_as_subset(B,A,C) = B ) ) ).
fof(t23_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> ( ! [D] :
~ ( in(D,B)
& ! [E] : ~ in(ordered_pair(E,D),C) )
<=> relation_rng_as_subset(A,B,C) = B ) ) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_yellow_1,axiom,
! [A,B] :
( element(B,the_carrier(boole_POSet(A)))
=> ! [C] :
( element(C,the_carrier(boole_POSet(A)))
=> ( related_reflexive(boole_POSet(A),B,C)
<=> subset(B,C) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t4_yellow_1,conjecture,
! [A] : boole_POSet(A) = incl_POSet(powerset(A)) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
%------------------------------------------------------------------------------