TPTP Problem File: LAT310+1.p
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%------------------------------------------------------------------------------
% File : LAT310+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Ideals T39
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Ban96] Bancerek (1996), Ideals
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t39_filter_2 [Urb08]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.31 v8.1.0, 0.28 v7.5.0, 0.34 v7.4.0, 0.23 v7.3.0, 0.28 v7.2.0, 0.24 v7.1.0, 0.30 v7.0.0, 0.23 v6.4.0, 0.31 v6.3.0, 0.25 v6.2.0, 0.28 v6.1.0, 0.37 v6.0.0, 0.26 v5.5.0, 0.44 v5.4.0, 0.46 v5.3.0, 0.56 v5.2.0, 0.35 v5.1.0, 0.33 v5.0.0, 0.38 v4.1.0, 0.43 v4.0.0, 0.46 v3.7.0, 0.45 v3.5.0, 0.47 v3.4.0
% Syntax : Number of formulae : 44 ( 15 unt; 0 def)
% Number of atoms : 136 ( 3 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 125 ( 33 ~; 1 |; 54 &)
% ( 2 <=>; 35 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 1 con; 0-2 aty)
% Number of variables : 67 ( 53 !; 14 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t39_filter_2,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( r1_tarski(C,D)
=> r1_tarski(k19_filter_2(A,C),k19_filter_2(A,D)) )
& r1_tarski(k19_filter_2(A,k19_filter_2(A,B)),k19_filter_2(A,B)) ) ) ) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(cc1_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A) ) ) ) ).
fof(cc2_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A) )
=> ( ~ v3_struct_0(A)
& v10_lattices(A) ) ) ) ).
fof(d11_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_filter_2(C,A)
=> ( C = k19_filter_2(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( m2_filter_2(D,A)
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ) ).
fof(dt_k19_filter_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m2_filter_2(k19_filter_2(A,B),A) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_l1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> l1_struct_0(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( l1_lattices(A)
& l2_lattices(A) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_filter_2(B,A)
=> ( ~ v1_xboole_0(B)
& m2_lattice4(B,A) ) ) ) ).
fof(dt_m2_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_lattice4(B,A)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_lattices,axiom,
? [A] : l1_lattices(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l2_lattices,axiom,
? [A] : l2_lattices(A) ).
fof(existence_l3_lattices,axiom,
? [A] : l3_lattices(A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m2_filter_2(B,A) ) ).
fof(existence_m2_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m2_lattice4(B,A) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(rc1_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] :
( m2_lattice4(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t1_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C) )
=> r1_tarski(A,C) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
%------------------------------------------------------------------------------