TPTP Problem File: SEU345+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU345+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t3_lattice3
% 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-t3_lattice3 [Urb07]
% Status : Theorem
% Rating : 0.61 v8.2.0, 0.58 v8.1.0, 0.67 v7.5.0, 0.69 v7.4.0, 0.53 v7.3.0, 0.59 v7.1.0, 0.52 v7.0.0, 0.60 v6.4.0, 0.62 v6.3.0, 0.54 v6.2.0, 0.64 v6.1.0, 0.70 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.79 v5.3.0, 0.81 v5.2.0, 0.70 v5.1.0, 0.71 v5.0.0, 0.75 v4.1.0, 0.74 v4.0.1, 0.78 v4.0.0, 0.79 v3.7.0, 0.80 v3.5.0, 0.79 v3.3.0
% Syntax : Number of formulae : 102 ( 37 unt; 0 def)
% Number of atoms : 324 ( 39 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 273 ( 51 ~; 1 |; 150 &)
% ( 5 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 22 ( 22 usr; 1 con; 0-6 aty)
% Number of variables : 182 ( 165 !; 17 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
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_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(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(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_lattices,axiom,
! [A,B,C] :
( ( ~ 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) ) ).
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(d13_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ( lower_bounded_semilattstr(A)
<=> ? [B] :
( element(B,the_carrier(A))
& ! [C] :
( element(C,the_carrier(A))
=> ( meet(A,B,C) = B
& meet(A,C,B) = B ) ) ) ) ) ).
fof(d16_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ( lower_bounded_semilattstr(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ( B = bottom_of_semilattstr(A)
<=> ! [C] :
( element(C,the_carrier(A))
=> ( meet(A,B,C) = B
& meet(A,C,B) = B ) ) ) ) ) ) ).
fof(d1_binop_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] : apply_binary(A,B,C) = apply(A,ordered_pair(B,C)) ) ).
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_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( 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) ) ) ) ).
fof(d2_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( 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) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
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_funct_1,axiom,
$true ).
fof(dt_k1_lattice3,axiom,
! [A] :
( strict_latt_str(boole_lattice(A))
& latt_str(boole_lattice(A)) ) ).
fof(dt_k1_lattices,axiom,
! [A,B,C] :
( ( ~ 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)) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ 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) ) ).
fof(dt_k2_lattices,axiom,
! [A,B,C] :
( ( ~ 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)) ) ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_lattices,axiom,
! [A,B,C] :
( ( ~ 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)) ) ).
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_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> element(bottom_of_semilattstr(A),the_carrier(A)) ) ).
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_l1_lattices,axiom,
! [A] :
( meet_semilatt_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_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_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_lattice3,axiom,
! [A] :
( ~ empty_carrier(boole_lattice(A))
& strict_latt_str(boole_lattice(A)) ) ).
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(fc2_lattice2,axiom,
! [A] :
( ( ~ 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)) ) ) ).
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_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_lattice2,axiom,
! [A] :
( ( ~ 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)) ) ) ).
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_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(fc4_lattice2,axiom,
! [A] :
( ( ~ 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)) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc5_lattice2,axiom,
! [A] :
( ( ~ 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)) ) ) ).
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(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(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_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ 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) ) ).
fof(redefinition_k4_lattices,axiom,
! [A,B,C] :
( ( ~ 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) ) ).
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_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(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_lattice3,axiom,
! [A,B] :
( element(B,the_carrier(boole_lattice(A)))
=> ! [C] :
( 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) ) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,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_xboole_1,axiom,
! [A] : subset(empty_set,A) ).
fof(t3_lattice3,conjecture,
! [A] :
( lower_bounded_semilattstr(boole_lattice(A))
& bottom_of_semilattstr(boole_lattice(A)) = empty_set ) ).
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(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) ) ).
%------------------------------------------------------------------------------