SET007 Axioms: SET007+544.ax
%------------------------------------------------------------------------------
% File : SET007+544 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Characterization of Modular and 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 : yellow11 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 8 unt; 0 def)
% Number of atoms : 363 ( 94 equ)
% Maximal formula atoms : 70 ( 10 avg)
% Number of connectives : 395 ( 67 ~; 0 |; 251 &)
% ( 7 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 29 ( 28 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 6 con; 0-5 aty)
% Number of variables : 84 ( 69 !; 15 ?)
% 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(fc1_yellow11,axiom,
( v1_orders_2(k1_yellow11)
& v2_orders_2(k1_yellow11)
& v3_orders_2(k1_yellow11)
& v4_orders_2(k1_yellow11) ) ).
fof(fc2_yellow11,axiom,
( ~ v3_struct_0(k1_yellow11)
& v1_lattice3(k1_yellow11)
& v2_lattice3(k1_yellow11) ) ).
fof(fc3_yellow11,axiom,
( v1_orders_2(k2_yellow11)
& v2_orders_2(k2_yellow11)
& v3_orders_2(k2_yellow11)
& v4_orders_2(k2_yellow11) ) ).
fof(fc4_yellow11,axiom,
( ~ v3_struct_0(k2_yellow11)
& v1_lattice3(k2_yellow11)
& v2_lattice3(k2_yellow11) ) ).
fof(cc1_yellow11,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& v2_waybel_1(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& v1_yellow11(A) ) ) ) ).
fof(cc2_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v1_xboole_0(B)
& v2_yellow11(B,A) )
=> v1_waybel_0(B,A) ) ) ) ).
fof(cc3_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v1_xboole_0(B)
& v2_yellow11(B,A) )
=> v2_waybel_0(B,A) ) ) ) ).
fof(fc5_yellow11,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> v2_yellow11(k3_yellow11(A,B,C),A) ) ).
fof(fc6_yellow11,axiom,
! [A,B,C] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v2_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v3_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v4_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v4_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A)
& v5_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A) ) ) ).
fof(fc7_yellow11,axiom,
! [A,B,C] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v2_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v3_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v4_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
& v4_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A)
& v6_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A) ) ) ).
fof(rc1_yellow11,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)
& v6_group_1(A) ) ).
fof(cc4_yellow11,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& v6_group_1(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_lattice3(A)
& v1_yellow_0(A) ) ) ) ).
fof(cc5_yellow11,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v6_group_1(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) ) ) ) ).
fof(t1_yellow11,axiom,
np__3 = k1_enumset1(np__0,np__1,np__2) ).
fof(t2_yellow11,axiom,
k4_xboole_0(np__2,np__1) = k1_tarski(np__1) ).
fof(t3_yellow11,axiom,
k4_xboole_0(np__3,np__1) = k2_tarski(np__1,np__2) ).
fof(t4_yellow11,axiom,
k4_xboole_0(np__3,np__2) = k1_tarski(np__2) ).
fof(t5_yellow11,axiom,
! [A] :
( ( v2_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))
=> ( k12_lattice3(A,B,C) = C
<=> k13_lattice3(A,B,C) = B ) ) ) ) ).
fof(t6_yellow11,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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r3_orders_2(A,k13_lattice3(A,k12_lattice3(A,B,C),k12_lattice3(A,B,D)),k12_lattice3(A,B,k13_lattice3(A,C,D))) ) ) ) ) ).
fof(t7_yellow11,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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r3_orders_2(A,k13_lattice3(A,B,k12_lattice3(A,C,D)),k12_lattice3(A,k13_lattice3(A,B,C),k13_lattice3(A,B,D))) ) ) ) ) ).
fof(t8_yellow11,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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_orders_2(A,B,D)
=> r3_orders_2(A,k13_lattice3(A,B,k12_lattice3(A,C,D)),k12_lattice3(A,k13_lattice3(A,B,C),D)) ) ) ) ) ) ).
fof(d1_yellow11,axiom,
k1_yellow11 = k2_yellow_1(k3_enumset1(np__0,k4_xboole_0(np__3,np__1),np__2,k4_xboole_0(np__3,np__2),np__3)) ).
fof(d2_yellow11,axiom,
k2_yellow11 = k2_yellow_1(k3_enumset1(np__0,np__1,k4_xboole_0(np__2,np__1),k4_xboole_0(np__3,np__2),np__3)) ).
fof(t9_yellow11,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] :
( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A)
& r5_waybel_1(k1_yellow11,B) )
& ! [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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( B != C
& B != D
& B != E
& B != F
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& k12_lattice3(A,B,C) = B
& k12_lattice3(A,B,D) = B
& k12_lattice3(A,D,F) = D
& k12_lattice3(A,E,F) = E
& k12_lattice3(A,C,D) = B
& k12_lattice3(A,C,E) = C
& k12_lattice3(A,D,E) = B
& k13_lattice3(A,C,D) = F
& k13_lattice3(A,D,E) = F ) ) ) ) ) ) )
& ~ ( ? [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))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& B != C
& B != D
& B != E
& B != F
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& k12_lattice3(A,B,C) = B
& k12_lattice3(A,B,D) = B
& k12_lattice3(A,D,F) = D
& k12_lattice3(A,E,F) = E
& k12_lattice3(A,C,D) = B
& k12_lattice3(A,C,E) = C
& k12_lattice3(A,D,E) = B
& k13_lattice3(A,C,D) = F
& k13_lattice3(A,D,E) = F ) ) ) ) )
& ! [B] :
( ( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A) )
=> ~ r5_waybel_1(k1_yellow11,B) ) ) ) ) ).
fof(t10_yellow11,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] :
( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A)
& r5_waybel_1(k2_yellow11,B) )
& ! [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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( B != C
& B != D
& B != E
& B != F
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& k12_lattice3(A,B,C) = B
& k12_lattice3(A,B,D) = B
& k12_lattice3(A,B,E) = B
& k12_lattice3(A,C,F) = C
& k12_lattice3(A,D,F) = D
& k12_lattice3(A,E,F) = E
& k12_lattice3(A,C,D) = B
& k12_lattice3(A,C,E) = B
& k12_lattice3(A,D,E) = B
& k13_lattice3(A,C,D) = F
& k13_lattice3(A,C,E) = F
& k13_lattice3(A,D,E) = F ) ) ) ) ) ) )
& ~ ( ? [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))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& B != C
& B != D
& B != E
& B != F
& C != D
& C != E
& C != F
& D != E
& D != F
& E != F
& k12_lattice3(A,B,C) = B
& k12_lattice3(A,B,D) = B
& k12_lattice3(A,B,E) = B
& k12_lattice3(A,C,F) = C
& k12_lattice3(A,D,F) = D
& k12_lattice3(A,E,F) = E
& k12_lattice3(A,C,D) = B
& k12_lattice3(A,C,E) = B
& k12_lattice3(A,D,E) = B
& k13_lattice3(A,C,D) = F
& k13_lattice3(A,C,E) = F
& k13_lattice3(A,D,E) = F ) ) ) ) )
& ! [B] :
( ( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A) )
=> ~ r5_waybel_1(k2_yellow11,B) ) ) ) ) ).
fof(d3_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_yellow11(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_orders_2(A,B,D)
=> k10_lattice3(A,B,k11_lattice3(A,C,D)) = k11_lattice3(A,k10_lattice3(A,B,C),D) ) ) ) ) ) ) ).
fof(t11_yellow11,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( v1_yellow11(A)
<=> ! [B] :
( ( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A) )
=> ~ r5_waybel_1(k1_yellow11,B) ) ) ) ).
fof(t12_yellow11,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( v1_yellow11(A)
=> ( v2_waybel_1(A)
<=> ! [B] :
( ( v4_yellow_0(B,A)
& v5_yellow_0(B,A)
& v6_yellow_0(B,A)
& m1_yellow_0(B,A) )
=> ~ r5_waybel_1(k2_yellow11,B) ) ) ) ) ).
fof(d4_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k3_yellow11(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,D)
<=> ( r1_orders_2(A,B,E)
& r1_orders_2(A,E,C) ) ) ) ) ) ) ) ) ).
fof(d5_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_yellow11(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& B = k3_yellow11(A,C,D) ) ) ) ) ) ).
fof(t13_yellow11,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_yellow11(A,B,C) = k5_subset_1(u1_struct_0(A),k7_waybel_0(A,B),k6_waybel_0(A,C)) ) ) ) ).
fof(t14_yellow11,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))
=> ( v1_yellow11(A)
=> r5_waybel_1(k5_yellow_0(A,k3_yellow11(A,C,k13_lattice3(A,B,C))),k5_yellow_0(A,k3_yellow11(A,k12_lattice3(A,B,C),B))) ) ) ) ) ).
fof(dt_k1_yellow11,axiom,
l1_orders_2(k1_yellow11) ).
fof(dt_k2_yellow11,axiom,
l1_orders_2(k2_yellow11) ).
fof(dt_k3_yellow11,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_yellow11(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
%------------------------------------------------------------------------------