SET007 Axioms: SET007+375.ax
%------------------------------------------------------------------------------
% File : SET007+375 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homomorphisms of Lattices, Finite Join and Finite Meet
% 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 : lattice4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 82 ( 5 unt; 0 def)
% Number of atoms : 659 ( 43 equ)
% Maximal formula atoms : 16 ( 8 avg)
% Number of connectives : 701 ( 124 ~; 0 |; 339 &)
% ( 18 <=>; 220 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 35 ( 35 usr; 1 con; 0-5 aty)
% Number of variables : 230 ( 217 !; 13 ?)
% 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_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] :
( m2_lattice4(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(fc1_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k4_lattice4(A,B)) ) ).
fof(t1_lattice4,axiom,
$true ).
fof(t2_lattice4,axiom,
$true ).
fof(t3_lattice4,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( B != k1_xboole_0
& r1_tarski(B,A)
& v6_ordinal1(B)
& ! [C] :
~ ( r2_hidden(C,A)
& ! [D] :
( r2_hidden(D,B)
=> r1_tarski(D,C) ) ) )
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& C != B
& r1_tarski(B,C) ) ) ) ).
fof(t4_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> v2_filter_0(k1_filter_0(A),A) ) ).
fof(t5_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_filter_0(C,A)
=> ( r1_tarski(B,k3_filter_0(A,k1_finsub_1(k1_zfmisc_1(u1_struct_0(A)),B,C)))
& r1_tarski(C,k3_filter_0(A,k1_finsub_1(k1_zfmisc_1(u1_struct_0(A)),B,C))) ) ) ) ) ).
fof(t6_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,k3_filter_0(A,k1_finsub_1(k1_zfmisc_1(u1_struct_0(A)),k2_filter_0(A,D),B)))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,B)
& r3_lattices(A,k4_lattices(A,D,E),C) ) ) ) ) ) ) ) ).
fof(d1_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( m1_lattice4(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k3_lattices(A,D,E)) = k3_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
& k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k4_lattices(A,D,E)) = k4_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ).
fof(t7_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_lattice4(E,A,B)
=> ( r3_lattices(A,C,D)
=> r3_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C),k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,D)) ) ) ) ) ) ) ).
fof(d2_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_lattice4(C,A,B)
<=> v2_funct_1(C) ) ) ) ) ).
fof(d3_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_lattice4(C,A,B)
<=> k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ).
fof(t8_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_lattice4(E,A,B)
=> ( v1_lattice4(E,A,B)
=> ( r3_lattices(A,C,D)
<=> r3_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C),k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,D)) ) ) ) ) ) ) ) ).
fof(t9_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_lattice4(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& D = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) ).
fof(d4_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( v3_lattice4(C,A,B)
<=> ( v1_lattice4(C,A,B)
& v2_lattice4(C,A,B) ) ) ) ) ) ).
fof(d5_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( r1_filter_1(A,B)
<=> ? [C] :
( m1_lattice4(C,A,B)
& v3_lattice4(C,A,B) ) ) ) ) ).
fof(d6_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( r1_lattice4(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k4_filter_0(A,D,E)) = k4_filter_0(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(d7_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( r2_lattice4(A,B,C)
<=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k6_lattices(A)) = k6_lattices(B) ) ) ) ) ).
fof(d8_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( r3_lattice4(A,B,C)
<=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k5_lattices(A)) = k5_lattices(B) ) ) ) ) ).
fof(d9_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( r4_lattice4(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k7_lattices(A,D)) = k7_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) ) ) ) ) ) ).
fof(d10_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( m2_lattice4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k4_lattices(A,C,D),B)
& r2_hidden(k3_lattices(A,C,D),B) ) ) ) ) ) ) ) ).
fof(t10_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> m2_lattice4(u1_struct_0(A),A) ) ).
fof(t11_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> m2_lattice4(B,A) ) ) ).
fof(d11_lattice4,axiom,
$true ).
fof(d12_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k1_lattice4(A,B) = k2_lattice2(u1_struct_0(A),A,B,k7_grcat_1(A)) ) ) ).
fof(d13_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k2_lattice4(A,B) = k3_lattice2(u1_struct_0(A),A,B,k7_grcat_1(A)) ) ) ).
fof(t12_lattice4,axiom,
$true ).
fof(t13_lattice4,axiom,
$true ).
fof(t14_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k2_lattice4(A,B) = k7_setwiseo(u1_struct_0(A),u1_struct_0(A),u1_lattices(A),B,k7_grcat_1(A)) ) ) ).
fof(t15_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k1_lattice4(A,B) = k7_setwiseo(u1_struct_0(A),u1_struct_0(A),u2_lattices(A),B,k7_grcat_1(A)) ) ) ).
fof(t16_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_lattice4(A,k2_setwiseo(u1_struct_0(A),B)) = B ) ) ).
fof(t17_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_lattice4(A,k2_setwiseo(u1_struct_0(A),B)) = B ) ) ).
fof(t18_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v11_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ( v2_lattice4(C,B,A)
=> v11_lattices(A) ) ) ) ) ).
fof(t19_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v13_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ( v2_lattice4(C,B,A)
=> ( v13_lattices(A)
& r3_lattice4(B,A,C) ) ) ) ) ) ).
fof(t20_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> k2_lattice2(u1_struct_0(A),A,k5_setwiseo(u1_struct_0(A),B,k2_setwiseo(u1_struct_0(A),C)),D) = k3_lattices(A,k2_lattice2(u1_struct_0(A),A,B,D),k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,C)) ) ) ) ) ).
fof(t21_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_lattice4(A,k5_setwiseo(u1_struct_0(A),B,k2_setwiseo(u1_struct_0(A),C))) = k3_lattices(A,k1_lattice4(A,B),C) ) ) ) ).
fof(t22_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(A)))
=> k3_lattices(A,k1_lattice4(A,B),k1_lattice4(A,C)) = k1_lattice4(A,k5_setwiseo(u1_struct_0(A),B,C)) ) ) ) ).
fof(t23_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> k1_lattice4(A,k1_setwiseo(u1_struct_0(A))) = k5_lattices(A) ) ).
fof(t24_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_lattice4(B,A)
=> ( r2_hidden(k5_lattices(A),B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(A)))
=> ( r1_tarski(C,B)
=> r2_hidden(k1_lattice4(A,C),B) ) ) ) ) ) ).
fof(t25_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v14_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ( v2_lattice4(C,B,A)
=> ( v14_lattices(A)
& r2_lattice4(B,A,C) ) ) ) ) ) ).
fof(t26_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> k2_lattice4(A,k1_setwiseo(u1_struct_0(A))) = k6_lattices(A) ) ).
fof(t27_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> k3_lattice2(u1_struct_0(A),A,k5_setwiseo(u1_struct_0(A),B,k2_setwiseo(u1_struct_0(A),C)),D) = k4_lattices(A,k3_lattice2(u1_struct_0(A),A,B,D),k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,C)) ) ) ) ) ).
fof(t28_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_lattice4(A,k5_setwiseo(u1_struct_0(A),B,k2_setwiseo(u1_struct_0(A),C))) = k4_lattices(A,k2_lattice4(A,B),C) ) ) ) ).
fof(t29_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> k3_lattice2(u1_struct_0(A),A,k8_setwiseo(u1_struct_0(A),u1_struct_0(A),C,B),D) = k3_lattice2(u1_struct_0(A),A,B,k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,D)) ) ) ) ) ).
fof(t30_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(A)))
=> k4_lattices(A,k2_lattice4(A,B),k2_lattice4(A,C)) = k2_lattice4(A,k5_setwiseo(u1_struct_0(A),B,C)) ) ) ) ).
fof(t31_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m2_lattice4(B,A)
=> ( r2_hidden(k6_lattices(A),B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(A)))
=> ( r1_tarski(C,B)
=> r2_hidden(k2_lattice4(A,C),B) ) ) ) ) ) ).
fof(t32_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_lattices(A,k2_lattice4(A,B),C) = k2_lattice4(A,k8_setwiseo(u1_struct_0(A),u1_struct_0(A),k7_funcop_1(u1_struct_0(A),u1_struct_0(A),u2_lattices(A),k7_grcat_1(A),C),B)) ) ) ) ).
fof(t33_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v3_filter_0(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> r3_lattices(A,k4_lattices(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,D),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,k4_filter_0(B,D,E))),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E)) ) ) ) ) ) ).
fof(t34_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v3_filter_0(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( v1_lattice4(C,B,A)
& r3_lattices(A,k4_lattices(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,D),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E)),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,F)) )
=> r3_lattices(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E),k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,k4_filter_0(B,D,F))) ) ) ) ) ) ) ) ).
fof(t35_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v3_filter_0(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ( v3_lattice4(C,B,A)
=> ( v3_filter_0(A)
& r1_lattice4(B,A,C) ) ) ) ) ) ).
fof(t36_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> k7_lattices(A,k6_lattices(A)) = k5_lattices(A) ) ).
fof(t37_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> k7_lattices(A,k5_lattices(A)) = k6_lattices(A) ) ).
fof(t38_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v17_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,B,A)
=> ( v2_lattice4(C,B,A)
=> ( v17_lattices(A)
& r4_lattice4(B,A,C) ) ) ) ) ) ).
fof(d14_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m3_lattice4(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k4_lattices(A,C,D),B)
& r2_hidden(k7_lattices(A,C),B) ) ) ) ) ) ) ) ).
fof(t39_lattice4,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))
=> ! [D] :
( m3_lattice4(D,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> r2_hidden(k3_lattices(A,B,C),D) ) ) ) ) ) ).
fof(t40_lattice4,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))
=> ! [D] :
( m3_lattice4(D,A)
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> r2_hidden(k4_filter_0(A,B,C),D) ) ) ) ) ) ).
fof(t41_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> m3_lattice4(u1_struct_0(A),A) ) ).
fof(t42_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m3_lattice4(B,A)
=> m2_lattice4(B,A) ) ) ).
fof(d15_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_lattice4(C,A)
=> ( C = k3_lattice4(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( m3_lattice4(D,A)
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ) ).
fof(t43_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v17_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,k4_lattice4(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k4_filter_0(B,D,E)
& r2_hidden(D,C)
& r2_hidden(E,C) ) ) ) ) ) ).
fof(t44_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k4_lattice4(A,B))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& C = k3_lattices(A,k7_lattices(A,D),E)
& r2_hidden(D,B)
& r2_hidden(E,B) ) ) ) ) ) ) ).
fof(d17_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( B = k5_lattice4(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k7_lattices(A,C) ) ) ) ) ).
fof(t45_lattice4,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,k5_finsub_1(u1_struct_0(A)))
=> k2_lattice2(u1_struct_0(A),A,k5_setwiseo(u1_struct_0(A),C,k2_setwiseo(u1_struct_0(A),B)),k5_lattice4(A)) = k3_lattices(A,k2_lattice2(u1_struct_0(A),A,C,k5_lattice4(A)),k7_lattices(A,B)) ) ) ) ).
fof(t46_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k7_lattices(A,k1_lattice4(A,B)) = k3_lattice2(u1_struct_0(A),A,B,k5_lattice4(A)) ) ) ).
fof(t47_lattice4,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,k5_finsub_1(u1_struct_0(A)))
=> k3_lattice2(u1_struct_0(A),A,k5_setwiseo(u1_struct_0(A),C,k2_setwiseo(u1_struct_0(A),B)),k5_lattice4(A)) = k4_lattices(A,k3_lattice2(u1_struct_0(A),A,C,k5_lattice4(A)),k7_lattices(A,B)) ) ) ) ).
fof(t48_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(u1_struct_0(A)))
=> k7_lattices(A,k2_lattice4(A,B)) = k2_lattice2(u1_struct_0(A),A,B,k5_lattice4(A)) ) ) ).
fof(t49_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_lattice4(B,A) )
=> ( ( r2_hidden(k5_lattices(A),B)
& r2_hidden(k6_lattices(A),B) )
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(u1_struct_0(A)))
=> ~ ( r1_tarski(C,k4_lattice4(A,B))
& ! [D] :
( m1_subset_1(D,k5_finsub_1(u1_struct_0(A)))
=> ~ ( r1_tarski(D,k4_lattice4(A,B))
& k2_lattice2(u1_struct_0(A),A,C,k5_lattice4(A)) = k2_lattice4(A,D) ) ) ) ) ) ) ) ).
fof(dt_m1_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_lattice4(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ) ).
fof(existence_m1_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ? [C] : m1_lattice4(C,A,B) ) ).
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(existence_m2_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] : m2_lattice4(B,A) ) ).
fof(dt_m3_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m3_lattice4(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m3_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ? [B] : m3_lattice4(B,A) ) ).
fof(dt_k1_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k5_finsub_1(u1_struct_0(A))) )
=> m1_subset_1(k1_lattice4(A,B),u1_struct_0(A)) ) ).
fof(dt_k2_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_subset_1(B,k5_finsub_1(u1_struct_0(A))) )
=> m1_subset_1(k2_lattice4(A,B),u1_struct_0(A)) ) ).
fof(dt_k3_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m3_lattice4(k3_lattice4(A,B),A) ) ).
fof(dt_k4_lattice4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k4_lattice4(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k5_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ( v1_funct_1(k5_lattice4(A))
& v1_funct_2(k5_lattice4(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k5_lattice4(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(d16_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k4_lattice4(A,B) = a_2_0_lattice4(A,B) ) ) ).
fof(t50_lattice4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_lattice4(B,A) )
=> ( ( r2_hidden(k5_lattices(A),B)
& r2_hidden(k6_lattices(A),B) )
=> a_2_1_lattice4(A,B) = k3_lattice4(A,B) ) ) ) ).
fof(fraenkel_a_2_0_lattice4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v17_lattices(B)
& l3_lattices(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_lattice4(B,C))
<=> ? [D,E] :
( m1_subset_1(D,u1_struct_0(B))
& m1_subset_1(E,u1_struct_0(B))
& A = k4_filter_0(B,D,E)
& r2_hidden(D,C)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_2_1_lattice4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v17_lattices(B)
& l3_lattices(B)
& ~ v1_xboole_0(C)
& m2_lattice4(C,B) )
=> ( r2_hidden(A,a_2_1_lattice4(B,C))
<=> ? [D] :
( m1_subset_1(D,k5_finsub_1(u1_struct_0(B)))
& A = k2_lattice4(B,D)
& r1_tarski(D,k4_lattice4(B,C)) ) ) ) ).
%------------------------------------------------------------------------------