TPTP Problem File: SEU342+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU342+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t32_filter_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-t32_filter_1 [Urb07]
% Status : Theorem
% Rating : 0.56 v8.1.0, 0.64 v7.5.0, 0.62 v7.4.0, 0.47 v7.3.0, 0.52 v7.1.0, 0.43 v7.0.0, 0.53 v6.4.0, 0.54 v6.3.0, 0.50 v6.2.0, 0.56 v6.1.0, 0.60 v6.0.0, 0.57 v5.5.0, 0.59 v5.4.0, 0.64 v5.3.0, 0.70 v5.2.0, 0.65 v5.1.0, 0.67 v4.1.0, 0.65 v4.0.1, 0.74 v4.0.0, 0.75 v3.7.0, 0.70 v3.5.0, 0.68 v3.4.0, 0.74 v3.3.0
% Syntax : Number of formulae : 44 ( 20 unt; 0 def)
% Number of atoms : 125 ( 11 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 108 ( 27 ~; 1 |; 51 &)
% ( 4 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-4 aty)
% Number of variables : 65 ( 55 !; 10 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
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(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(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d8_filter_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> relation_of_lattice(A) = a_1_0_filter_1(A) ) ).
fof(dt_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty(B)
& element(C,A)
& element(D,B) )
=> element(ordered_pair_as_product_element(A,B,C,D),cartesian_product2(A,B)) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
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_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_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
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_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fraenkel_a_1_0_filter_1,axiom,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ( in(A,a_1_0_filter_1(B))
<=> ? [C,D] :
( 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) ) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(redefinition_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty(B)
& element(C,A)
& element(D,B) )
=> ordered_pair_as_product_element(A,B,C,D) = ordered_pair(C,D) ) ).
fof(redefinition_r3_lattices,axiom,
! [A,B,C] :
( ( ~ 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) ) ) ).
fof(reflexivity_r3_lattices,axiom,
! [A,B,C] :
( ( ~ 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) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) ).
fof(t32_filter_1,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( 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) ) ) ) ) ).
fof(t33_zfmisc_1,axiom,
! [A,B,C,D] :
( ordered_pair(A,B) = ordered_pair(C,D)
=> ( A = C
& B = D ) ) ).
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) ) ).
%------------------------------------------------------------------------------