SET007 Axioms: SET007+640.ax
%------------------------------------------------------------------------------
% File : SET007+640 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Representation Theorem for Finite Distributive Lattices
% 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 : lattice7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 41 ( 2 unt; 0 def)
% Number of atoms : 394 ( 22 equ)
% Maximal formula atoms : 19 ( 9 avg)
% Number of connectives : 383 ( 30 ~; 0 |; 242 &)
% ( 15 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-4 aty)
% Number of functors : 22 ( 22 usr; 4 con; 0-4 aty)
% Number of variables : 108 ( 97 !; 11 ?)
% 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_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v5_orders_2(B,A) ) ) ).
fof(rc2_lattice7,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v1_yellow_0(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v2_waybel_1(A)
& v6_group_1(A) ) ).
fof(rc3_lattice7,axiom,
? [A] :
( m2_lattice7(A)
& ~ v1_xboole_0(A) ) ).
fof(fc1_lattice7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_lattice7(A) )
=> ( ~ v3_struct_0(k2_yellow_1(A))
& v1_orders_2(k2_yellow_1(A))
& v2_orders_2(k2_yellow_1(A))
& v3_orders_2(k2_yellow_1(A))
& v4_orders_2(k2_yellow_1(A))
& v1_lattice3(k2_yellow_1(A))
& v2_lattice3(k2_yellow_1(A))
& v2_waybel_1(k2_yellow_1(A)) ) ) ).
fof(d1_lattice7,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_lattice7(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r2_hidden(D,C) ) ) ) ) ) ) ).
fof(d2_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_orders_2(A,B,C)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v5_orders_2(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m1_lattice7(D,A,B,C)
<=> ( r2_hidden(B,D)
& r2_hidden(C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,D)
=> ( r3_orders_2(A,B,E)
& r3_orders_2(A,E,C) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_orders_2(A,B,C)
=> m1_lattice7(k2_struct_0(A,B,C),A,B,C) ) ) ) ) ).
fof(d3_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k1_lattice7(A,B)
<=> ( ? [D] :
( m1_lattice7(D,A,k3_yellow_0(A),B)
& C = k4_card_1(D) )
& ! [D] :
( m1_lattice7(D,A,k3_yellow_0(A),B)
=> r1_xreal_0(k4_card_1(D),C) ) ) ) ) ) ) ).
fof(t2_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_orders_2(A,B,C)
& r1_xreal_0(k1_lattice7(A,C),k1_lattice7(A,B)) ) ) ) ) ).
fof(t3_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v5_orders_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(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_orders_2(A,C,D)
<=> ~ r1_xreal_0(k1_lattice7(A,D),k1_lattice7(A,C)) ) ) ) ) ) ) ).
fof(t4_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v5_orders_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(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) )
=> ( C = D
<=> k1_lattice7(A,C) = k1_lattice7(A,D) ) ) ) ) ) ) ).
fof(t5_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v5_orders_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(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) )
=> ( r3_orders_2(A,C,D)
<=> r1_xreal_0(k1_lattice7(A,C),k1_lattice7(A,D)) ) ) ) ) ) ) ).
fof(t6_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_lattice7(A,B) = np__1
<=> B = k3_yellow_0(A) ) ) ) ).
fof(t7_lattice7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_xreal_0(np__1,k1_lattice7(A,B)) ) ) ).
fof(d4_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_lattice7(A,B,C)
<=> ( r2_orders_2(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_orders_2(A,B,D)
& r2_orders_2(A,D,C) ) ) ) ) ) ) ) ).
fof(t8_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(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,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,B)
& r2_orders_2(A,C,D) ) ) ) ) ) ).
fof(d5_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v5_orders_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k2_lattice7(A,B)
<=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r3_orders_2(A,D,C) ) )
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t9_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k3_yellow_0(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ r2_lattice7(A,C,B) ) ) ) ) ).
fof(t10_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( r2_hidden(B,k3_lattice7(A))
<=> ( B != k3_yellow_0(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( B = k13_lattice3(A,C,D)
& B != C
& B != D ) ) ) ) ) ) ) ).
fof(t11_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_1(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( r2_hidden(B,k3_lattice7(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_orders_2(A,C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_orders_2(A,D,B)
=> r3_orders_2(A,D,C) ) ) ) ) ) ) ) ).
fof(t12_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_1(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_yellow_0(A,k5_subset_1(u1_struct_0(A),k6_waybel_0(A,B),k3_lattice7(A))) = B ) ) ).
fof(t13_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_1(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))))
& v23_waybel_0(B,A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_waybel_0(A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))),B,C) = k5_subset_1(u1_struct_0(A),k6_waybel_0(A,C),k3_lattice7(A)) ) ) ) ).
fof(t14_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_1(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> r5_waybel_1(A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))) ) ).
fof(d8_lattice7,axiom,
! [A] :
( m2_lattice7(A)
<=> r1_cohsp_1(A,A) ) ).
fof(t15_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> m2_lattice7(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))) ) ).
fof(t16_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A) )
=> ( v2_waybel_1(A)
<=> ? [B] :
( ~ v1_xboole_0(B)
& m2_lattice7(B)
& r5_waybel_1(A,k2_yellow_1(B)) ) ) ) ).
fof(s1_lattice7,axiom,
( ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_lattice7))
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(f1_s1_lattice7))
=> ( r2_orders_2(f1_s1_lattice7,B,A)
=> p1_s1_lattice7(B) ) )
=> p1_s1_lattice7(A) ) )
=> ! [A] :
( m1_subset_1(A,u1_struct_0(f1_s1_lattice7))
=> p1_s1_lattice7(A) ) ) ).
fof(dt_m1_lattice7,axiom,
! [A,B,C] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ! [D] :
( m1_lattice7(D,A,B,C)
=> ( ~ v1_xboole_0(D)
& v5_orders_2(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m1_lattice7,axiom,
! [A,B,C] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ? [D] : m1_lattice7(D,A,B,C) ) ).
fof(dt_m2_lattice7,axiom,
$true ).
fof(existence_m2_lattice7,axiom,
? [A] : m2_lattice7(A) ).
fof(reflexivity_r1_lattice7,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_lattice7(A,B,B) ) ).
fof(redefinition_r1_lattice7,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_lattice7(A,B,C)
<=> r1_tarski(B,C) ) ) ).
fof(dt_k1_lattice7,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m2_subset_1(k1_lattice7(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k2_lattice7,axiom,
! [A,B] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(A)
& l1_orders_2(A)
& ~ v1_xboole_0(B)
& v5_orders_2(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k2_lattice7(A,B),u1_struct_0(A)) ) ).
fof(dt_k3_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> m1_subset_1(k3_lattice7(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_lattice7,axiom,
! [A] :
( l1_orders_2(A)
=> ~ v1_xboole_0(k4_lattice7(A)) ) ).
fof(d6_lattice7,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> k3_lattice7(A) = a_1_0_lattice7(A) ) ).
fof(d7_lattice7,axiom,
! [A] :
( l1_orders_2(A)
=> k4_lattice7(A) = a_1_1_lattice7(A) ) ).
fof(fraenkel_a_1_0_lattice7,axiom,
! [A,B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& l1_orders_2(B) )
=> ( r2_hidden(A,a_1_0_lattice7(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& A = C
& C != k3_yellow_0(B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ~ ( C = k13_lattice3(B,D,E)
& C != D
& C != E ) ) ) ) ) ) ).
fof(fraenkel_a_1_1_lattice7,axiom,
! [A,B] :
( l1_orders_2(B)
=> ( r2_hidden(A,a_1_1_lattice7(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v12_waybel_0(C,B) ) ) ) ).
%------------------------------------------------------------------------------