TPTP Problem File: LAT302+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LAT302+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Ideals T26
% 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 : t26_filter_2 [Urb08]
% Status : Theorem
% Rating : 0.39 v8.2.0, 0.36 v8.1.0, 0.33 v7.5.0, 0.38 v7.4.0, 0.27 v7.3.0, 0.34 v7.1.0, 0.30 v7.0.0, 0.37 v6.4.0, 0.38 v6.3.0, 0.29 v6.2.0, 0.40 v6.1.0, 0.53 v6.0.0, 0.43 v5.5.0, 0.63 v5.4.0, 0.61 v5.3.0, 0.67 v5.2.0, 0.55 v5.1.0, 0.57 v5.0.0, 0.58 v4.1.0, 0.61 v4.0.1, 0.57 v4.0.0, 0.58 v3.7.0, 0.50 v3.5.0, 0.58 v3.4.0
% Syntax : Number of formulae : 92 ( 21 unt; 0 def)
% Number of atoms : 338 ( 10 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 315 ( 69 ~; 1 |; 174 &)
% ( 5 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 37 ( 35 usr; 1 prp; 0-3 aty)
% Number of functors : 12 ( 12 usr; 1 con; 0-3 aty)
% Number of variables : 131 ( 104 !; 27 ?)
% 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(t26_filter_2,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( v13_lattices(A)
=> m2_filter_2(k6_domain_1(u1_struct_0(A),k5_lattices(A)),A) ) ) ).
fof(abstractness_v3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( v3_lattices(A)
=> A = g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) ) ) ).
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(cc1_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ~ v1_xboole_0(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(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(cc3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v13_lattices(A)
& v14_lattices(A) )
=> ( ~ v3_struct_0(A)
& v15_lattices(A) ) ) ) ).
fof(cc4_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v15_lattices(A) )
=> ( ~ v3_struct_0(A)
& v13_lattices(A)
& v14_lattices(A) ) ) ) ).
fof(d2_lattice2,axiom,
! [A] :
( l3_lattices(A)
=> k1_lattice2(A) = g3_lattices(u1_struct_0(A),u1_lattices(A),u2_lattices(A)) ) ).
fof(dt_g3_lattices,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v3_lattices(g3_lattices(A,B,C))
& l3_lattices(g3_lattices(A,B,C)) ) ) ).
fof(dt_k1_lattice2,axiom,
! [A] :
( l3_lattices(A)
=> ( v3_lattices(k1_lattice2(A))
& l3_lattices(k1_lattice2(A)) ) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k5_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> m1_subset_1(k5_lattices(A),u1_struct_0(A)) ) ).
fof(dt_k6_domain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k6_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> m1_subset_1(k6_lattices(A),u1_struct_0(A)) ) ).
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_filter_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(dt_m1_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_2(B,A)
=> ( ~ v1_xboole_0(B)
& m2_lattice4(B,A) ) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
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_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_u1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> ( v1_funct_1(u1_lattices(A))
& v1_funct_2(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> ( v1_funct_1(u2_lattices(A))
& v1_funct_2(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
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_filter_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m1_filter_0(B,A) ) ).
fof(existence_m1_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m1_filter_2(B,A) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
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(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(fc1_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(k1_lattice2(A))
& v3_lattices(k1_lattice2(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(fc2_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& l2_lattices(A) )
=> ( v1_relat_1(u2_lattices(A))
& v1_funct_1(u2_lattices(A))
& v1_funct_2(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& v1_binop_1(u2_lattices(A),u1_struct_0(A))
& v1_partfun1(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc3_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_lattices(A)
& l2_lattices(A) )
=> ( v1_relat_1(u2_lattices(A))
& v1_funct_1(u2_lattices(A))
& v1_funct_2(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& v2_binop_1(u2_lattices(A),u1_struct_0(A))
& v1_partfun1(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(fc3_lattices,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ~ v3_struct_0(g3_lattices(A,B,C))
& v3_lattices(g3_lattices(A,B,C)) ) ) ).
fof(fc3_realset1,axiom,
! [A] :
( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A))
& v1_realset1(k1_tarski(A)) ) ).
fof(fc4_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& l1_lattices(A) )
=> ( v1_relat_1(u1_lattices(A))
& v1_funct_1(u1_lattices(A))
& v1_funct_2(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& v1_binop_1(u1_lattices(A),u1_struct_0(A))
& v1_partfun1(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc5_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_lattices(A)
& l1_lattices(A) )
=> ( v1_relat_1(u1_lattices(A))
& v1_funct_1(u1_lattices(A))
& v1_funct_2(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& v2_binop_1(u1_lattices(A),u1_struct_0(A))
& v1_partfun1(u1_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(fc6_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( ~ v3_struct_0(k1_lattice2(A))
& v3_lattices(k1_lattice2(A))
& v4_lattices(k1_lattice2(A))
& v5_lattices(k1_lattice2(A))
& v6_lattices(k1_lattice2(A))
& v7_lattices(k1_lattice2(A))
& v8_lattices(k1_lattice2(A))
& v9_lattices(k1_lattice2(A))
& v10_lattices(k1_lattice2(A)) ) ) ).
fof(free_g3_lattices,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D,E,F] :
( g3_lattices(A,B,C) = g3_lattices(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(rc11_lattices,axiom,
? [A] :
( l3_lattices(A)
& ~ v3_struct_0(A)
& v3_lattices(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
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_realset1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_realset1(A) ) ).
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_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
fof(rc2_realset1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& ~ v1_realset1(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_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(rc3_lattices,axiom,
? [A] :
( l3_lattices(A)
& v3_lattices(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(rc6_lattices,axiom,
? [A] :
( l3_lattices(A)
& ~ v3_struct_0(A)
& v3_lattices(A) ) ).
fof(rc9_lattices,axiom,
? [A] :
( l3_lattices(A)
& ~ v3_struct_0(A)
& v3_lattices(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A) ) ).
fof(redefinition_k6_domain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k6_domain_1(A,B) = k1_tarski(B) ) ).
fof(redefinition_m1_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_2(B,A)
<=> m1_filter_0(B,A) ) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t13_filter_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> m1_filter_0(k6_domain_1(u1_struct_0(A),k6_lattices(A)),A) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t21_filter_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_filter_2(B,A)
<=> m1_filter_2(B,k1_lattice2(A)) ) ) ).
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(t63_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( v13_lattices(A)
<=> v14_lattices(k1_lattice2(A)) ) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t78_lattice2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> k5_lattices(A) = k6_lattices(k1_lattice2(A)) ) ).
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) ) ).
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