SET007 Axioms: SET007+205.ax
%------------------------------------------------------------------------------
% File : SET007+205 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Lattice Theory
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : lattices [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 127 ( 30 unt; 0 def)
% Number of atoms : 680 ( 65 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 644 ( 91 ~; 0 |; 335 &)
% ( 23 <=>; 195 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 0 con; 1-6 aty)
% Number of variables : 231 ( 212 !; 19 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_lattices,axiom,
? [A] :
( l1_lattices(A)
& v1_lattices(A) ) ).
fof(rc2_lattices,axiom,
? [A] :
( l2_lattices(A)
& v2_lattices(A) ) ).
fof(rc3_lattices,axiom,
? [A] :
( l3_lattices(A)
& v3_lattices(A) ) ).
fof(fc1_lattices,axiom,
! [A,B] :
( ( ~ 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) )
=> ( ~ v3_struct_0(g2_lattices(A,B))
& v2_lattices(g2_lattices(A,B)) ) ) ).
fof(fc2_lattices,axiom,
! [A,B] :
( ( ~ 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) )
=> ( ~ v3_struct_0(g1_lattices(A,B))
& v1_lattices(g1_lattices(A,B)) ) ) ).
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(rc4_lattices,axiom,
? [A] :
( l2_lattices(A)
& ~ v3_struct_0(A)
& v2_lattices(A) ) ).
fof(rc5_lattices,axiom,
? [A] :
( l1_lattices(A)
& ~ v3_struct_0(A)
& v1_lattices(A) ) ).
fof(rc6_lattices,axiom,
? [A] :
( l3_lattices(A)
& ~ v3_struct_0(A)
& v3_lattices(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(rc7_lattices,axiom,
? [A] :
( l2_lattices(A)
& ~ v3_struct_0(A)
& v2_lattices(A)
& v4_lattices(A)
& v5_lattices(A) ) ).
fof(rc8_lattices,axiom,
? [A] :
( l1_lattices(A)
& ~ v3_struct_0(A)
& v1_lattices(A)
& v6_lattices(A)
& v7_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(rc10_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)
& v11_lattices(A)
& v12_lattices(A)
& v13_lattices(A)
& v14_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(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(rc12_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)
& v16_lattices(A) ) ).
fof(cc5_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v17_lattices(A) )
=> ( ~ v3_struct_0(A)
& v11_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A) ) ) ) ).
fof(cc6_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v11_lattices(A)
& v15_lattices(A)
& v16_lattices(A) )
=> ( ~ v3_struct_0(A)
& v17_lattices(A) ) ) ) ).
fof(rc13_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)
& v11_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& v17_lattices(A) ) ).
fof(cc7_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v12_lattices(A) ) ) ) ).
fof(d1_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_lattices(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u2_lattices(A),B,C) ) ) ) ).
fof(d2_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_lattices(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_lattices(A),B,C) ) ) ) ).
fof(d3_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_lattices(A,B,C)
<=> k1_lattices(A,B,C) = C ) ) ) ) ).
fof(d4_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ( v4_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_lattices(A,B,C) = k1_lattices(A,C,B) ) ) ) ) ).
fof(d5_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ( v5_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_lattices(A,B,k1_lattices(A,C,D)) = k1_lattices(A,k1_lattices(A,B,C),D) ) ) ) ) ) ).
fof(d6_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ( v6_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_lattices(A,B,C) = k2_lattices(A,C,B) ) ) ) ) ).
fof(d7_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ( v7_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_lattices(A,B,k2_lattices(A,C,D)) = k2_lattices(A,k2_lattices(A,B,C),D) ) ) ) ) ) ).
fof(d8_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v8_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_lattices(A,k2_lattices(A,B,C),C) = C ) ) ) ) ).
fof(d9_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v9_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_lattices(A,B,k1_lattices(A,B,C)) = B ) ) ) ) ).
fof(d10_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v10_lattices(A)
<=> ( v4_lattices(A)
& v5_lattices(A)
& v8_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v9_lattices(A) ) ) ) ).
fof(d11_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v11_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_lattices(A,B,k1_lattices(A,C,D)) = k1_lattices(A,k2_lattices(A,B,C),k2_lattices(A,B,D)) ) ) ) ) ) ).
fof(d12_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v12_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lattices(A,B,D)
=> k1_lattices(A,B,k2_lattices(A,C,D)) = k2_lattices(A,k1_lattices(A,B,C),D) ) ) ) ) ) ) ).
fof(d13_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ( v13_lattices(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_lattices(A,B,C) = B
& k2_lattices(A,C,B) = B ) ) ) ) ) ).
fof(d14_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ( v14_lattices(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_lattices(A,B,C) = B
& k1_lattices(A,C,B) = B ) ) ) ) ) ).
fof(d15_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v15_lattices(A)
<=> ( v13_lattices(A)
& v14_lattices(A) ) ) ) ).
fof(d16_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ( v13_lattices(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k5_lattices(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_lattices(A,B,C) = B
& k2_lattices(A,C,B) = B ) ) ) ) ) ) ).
fof(d17_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> ( v14_lattices(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k6_lattices(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_lattices(A,B,C) = B
& k1_lattices(A,C,B) = B ) ) ) ) ) ) ).
fof(d18_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_lattices(A,B,C)
<=> ( k1_lattices(A,B,C) = k6_lattices(A)
& k1_lattices(A,C,B) = k6_lattices(A)
& k2_lattices(A,B,C) = k5_lattices(A)
& k2_lattices(A,C,B) = k5_lattices(A) ) ) ) ) ) ).
fof(d19_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v16_lattices(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r2_lattices(A,C,B) ) ) ) ) ).
fof(d20_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v17_lattices(A)
<=> ( v15_lattices(A)
& v16_lattices(A)
& v11_lattices(A) ) ) ) ).
fof(t1_lattices,axiom,
$true ).
fof(t2_lattices,axiom,
$true ).
fof(t3_lattices,axiom,
$true ).
fof(t4_lattices,axiom,
$true ).
fof(t5_lattices,axiom,
$true ).
fof(t6_lattices,axiom,
$true ).
fof(t7_lattices,axiom,
$true ).
fof(t8_lattices,axiom,
$true ).
fof(t9_lattices,axiom,
$true ).
fof(t10_lattices,axiom,
$true ).
fof(t11_lattices,axiom,
$true ).
fof(t12_lattices,axiom,
$true ).
fof(t13_lattices,axiom,
$true ).
fof(t14_lattices,axiom,
$true ).
fof(t15_lattices,axiom,
$true ).
fof(t16_lattices,axiom,
$true ).
fof(t17_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_lattices(A,B,B) = B ) ) ).
fof(t18_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,B,B) = B ) ) ).
fof(t19_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_lattices(A,B,k3_lattices(A,C,D)) = k3_lattices(A,k4_lattices(A,B,C),k4_lattices(A,B,D)) ) ) )
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k3_lattices(A,B,k4_lattices(A,C,D)) = k4_lattices(A,k3_lattices(A,B,C),k3_lattices(A,B,D)) ) ) ) ) ) ).
fof(t20_lattices,axiom,
$true ).
fof(t21_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_lattices(A,B,C)
<=> k2_lattices(A,B,C) = B ) ) ) ) ).
fof(t22_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_lattices(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_lattices(A,B,k1_lattices(A,B,C)) ) ) ) ).
fof(t23_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& v8_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_lattices(A,k4_lattices(A,B,C),B) ) ) ) ).
fof(t24_lattices,axiom,
$true ).
fof(t25_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_lattices(A)
& l2_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_lattices(A,B,C)
& r1_lattices(A,C,D) )
=> r1_lattices(A,B,D) ) ) ) ) ) ).
fof(t26_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& l2_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( r1_lattices(A,B,C)
& r1_lattices(A,C,B) )
=> B = C ) ) ) ) ).
fof(t27_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lattices(A,B,C)
=> r1_lattices(A,k2_lattices(A,B,D),k2_lattices(A,C,D)) ) ) ) ) ) ).
fof(t28_lattices,axiom,
$true ).
fof(t29_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k3_lattices(A,k3_lattices(A,k4_lattices(A,B,C),k4_lattices(A,C,D)),k4_lattices(A,D,B)) = k4_lattices(A,k4_lattices(A,k3_lattices(A,B,C),k3_lattices(A,C,D)),k3_lattices(A,D,B)) ) ) )
=> v11_lattices(A) ) ) ).
fof(t30_lattices,axiom,
$true ).
fof(t31_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k3_lattices(A,B,k4_lattices(A,C,D)) = k4_lattices(A,k3_lattices(A,B,C),k3_lattices(A,B,D)) ) ) ) ) ).
fof(t32_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( k4_lattices(A,B,C) = k4_lattices(A,B,D)
& k3_lattices(A,B,C) = k3_lattices(A,B,D) )
=> C = D ) ) ) ) ) ).
fof(t33_lattices,axiom,
$true ).
fof(t34_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_lattices(A,k4_lattices(A,k3_lattices(A,B,C),k3_lattices(A,C,D)),k3_lattices(A,D,B)) = k3_lattices(A,k3_lattices(A,k4_lattices(A,B,C),k4_lattices(A,C,D)),k4_lattices(A,D,B)) ) ) ) ) ).
fof(t35_lattices,axiom,
$true ).
fof(t36_lattices,axiom,
$true ).
fof(t37_lattices,axiom,
$true ).
fof(t38_lattices,axiom,
$true ).
fof(t39_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_lattices(A,k5_lattices(A),B) = B ) ) ).
fof(t40_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,k5_lattices(A),B) = k5_lattices(A) ) ) ).
fof(t41_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r3_lattices(A,k5_lattices(A),B) ) ) ).
fof(t42_lattices,axiom,
$true ).
fof(t43_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,k6_lattices(A),B) = B ) ) ).
fof(t44_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_lattices(A,k6_lattices(A),B) = k6_lattices(A) ) ) ).
fof(t45_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r3_lattices(A,B,k6_lattices(A)) ) ) ).
fof(d21_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& v16_lattices(A)
& l3_lattices(A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k7_lattices(A,B)
<=> r2_lattices(A,C,B) ) ) ) ) ) ).
fof(t46_lattices,axiom,
$true ).
fof(t47_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,k7_lattices(A,B),B) = k5_lattices(A) ) ) ).
fof(t48_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_lattices(A,k7_lattices(A,B),B) = k6_lattices(A) ) ) ).
fof(t49_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k7_lattices(A,k7_lattices(A,B)) = B ) ) ).
fof(t50_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k7_lattices(A,k4_lattices(A,B,C)) = k3_lattices(A,k7_lattices(A,B),k7_lattices(A,C)) ) ) ) ).
fof(t51_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k7_lattices(A,k3_lattices(A,B,C)) = k4_lattices(A,k7_lattices(A,B),k7_lattices(A,C)) ) ) ) ).
fof(t52_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k4_lattices(A,B,C) = k5_lattices(A)
<=> r3_lattices(A,B,k7_lattices(A,C)) ) ) ) ) ).
fof(t53_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,B,C)
=> r3_lattices(A,k7_lattices(A,C),k7_lattices(A,B)) ) ) ) ) ).
fof(dt_l1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_lattices,axiom,
? [A] : l1_lattices(A) ).
fof(dt_l2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> l1_struct_0(A) ) ).
fof(existence_l2_lattices,axiom,
? [A] : l2_lattices(A) ).
fof(dt_l3_lattices,axiom,
! [A] :
( l3_lattices(A)
=> ( l1_lattices(A)
& l2_lattices(A) ) ) ).
fof(existence_l3_lattices,axiom,
? [A] : l3_lattices(A) ).
fof(abstractness_v1_lattices,axiom,
! [A] :
( l1_lattices(A)
=> ( v1_lattices(A)
=> A = g1_lattices(u1_struct_0(A),u1_lattices(A)) ) ) ).
fof(abstractness_v2_lattices,axiom,
! [A] :
( l2_lattices(A)
=> ( v2_lattices(A)
=> A = g2_lattices(u1_struct_0(A),u2_lattices(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(reflexivity_r3_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r3_lattices(A,B,B) ) ).
fof(redefinition_r3_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r3_lattices(A,B,C)
<=> r1_lattices(A,B,C) ) ) ).
fof(dt_k1_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k1_lattices(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k2_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_lattices(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k3_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& l2_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_lattices(A,B,C),u1_struct_0(A)) ) ).
fof(commutativity_k3_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& l2_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_lattices(A,B,C) = k3_lattices(A,C,B) ) ).
fof(redefinition_k3_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& l2_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_lattices(A,B,C) = k1_lattices(A,B,C) ) ).
fof(dt_k4_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& l1_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_lattices(A,B,C),u1_struct_0(A)) ) ).
fof(commutativity_k4_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& l1_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k4_lattices(A,B,C) = k4_lattices(A,C,B) ) ).
fof(redefinition_k4_lattices,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_lattices(A)
& l1_lattices(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k4_lattices(A,B,C) = k2_lattices(A,B,C) ) ).
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_lattices,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_lattices(A) )
=> m1_subset_1(k6_lattices(A),u1_struct_0(A)) ) ).
fof(dt_k7_lattices,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k7_lattices(A,B),u1_struct_0(A)) ) ).
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_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(dt_g1_lattices,axiom,
! [A,B] :
( ( 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_lattices(g1_lattices(A,B))
& l1_lattices(g1_lattices(A,B)) ) ) ).
fof(free_g1_lattices,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g1_lattices(A,B) = g1_lattices(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(dt_g2_lattices,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v2_lattices(g2_lattices(A,B))
& l2_lattices(g2_lattices(A,B)) ) ) ).
fof(free_g2_lattices,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g2_lattices(A,B) = g2_lattices(C,D)
=> ( A = C
& B = D ) ) ) ).
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(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 ) ) ) ).
%------------------------------------------------------------------------------