SET007 Axioms: SET007+662.ax
%------------------------------------------------------------------------------
% File : SET007+662 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Isomorphism between Finite Chains
% 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 : orders_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 19 ( 1 unt; 0 def)
% Number of atoms : 137 ( 13 equ)
% Maximal formula atoms : 23 ( 7 avg)
% Number of connectives : 136 ( 18 ~; 1 |; 70 &)
% ( 3 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 36 ( 35 usr; 0 prp; 1-3 aty)
% Number of functors : 13 ( 13 usr; 3 con; 0-2 aty)
% Number of variables : 39 ( 35 !; 4 ?)
% 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(cc1_orders_4,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_struct_0(A)
=> ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A) ) ) ) ).
fof(cc2_orders_4,axiom,
! [A] :
( m1_orders_4(A)
=> ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A) ) ) ).
fof(rc1_orders_4,axiom,
? [A] :
( m1_orders_4(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A) ) ).
fof(cc3_orders_4,axiom,
! [A] :
( m1_orders_4(A)
=> ( ~ v3_struct_0(A)
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v1_lattice3(A)
& v2_lattice3(A) ) ) ) ).
fof(rc2_orders_4,axiom,
? [A] :
( m1_orders_4(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v16_waybel_0(A)
& v24_waybel_0(A)
& v1_waybel_8(A)
& v2_waybel_8(A)
& v3_waybel_8(A)
& ~ v1_yellow_3(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_3(A)
& v3_waybel_3(A)
& v6_group_1(A) ) ).
fof(rc3_orders_4,axiom,
? [A] :
( m1_orders_4(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_orders_4(A) ) ).
fof(cc4_orders_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v16_waybel_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_yellow_0(B,A)
=> ( ( ~ v3_struct_0(B)
& v4_yellow_0(B,A) )
=> ( ~ v3_struct_0(B)
& v16_waybel_0(B) ) ) ) ) ).
fof(cc5_orders_4,axiom,
! [A] :
( ( v6_group_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_yellow_0(B,A)
=> v6_group_1(B) ) ) ).
fof(d1_orders_4,axiom,
! [A] :
( l1_orders_2(A)
=> ( m1_orders_4(A)
<=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& l1_orders_2(A) )
| v3_struct_0(A) ) ) ) ).
fof(d2_orders_4,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_orders_4(A)
<=> v1_card_4(u1_struct_0(A)) ) ) ).
fof(t1_orders_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> ( v4_yellow_0(k2_yellow_1(A),k2_yellow_1(B))
& m1_yellow_0(k2_yellow_1(A),k2_yellow_1(B)) ) ) ) ) ).
fof(d3_orders_4,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B,C] :
( r1_orders_4(A,B,C)
<=> ( k2_xboole_0(B,C) = u1_struct_0(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(D,B)
& r2_hidden(E,C) )
=> r2_orders_2(A,D,E) ) ) ) ) ) ) ).
fof(t2_orders_4,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B,C] :
( r1_orders_4(A,B,C)
=> r1_xboole_0(B,C) ) ) ).
fof(t3_orders_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& l1_orders_2(A) )
=> r1_orders_4(A,k4_xboole_0(u1_struct_0(A),k1_struct_0(A,k4_yellow_0(A))),k1_struct_0(A,k4_yellow_0(A))) ) ).
fof(t4_orders_4,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(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)) )
=> ( v23_waybel_0(C,A,B)
=> ( ~ ( u1_struct_0(A) != k1_xboole_0
& u1_struct_0(B) = k1_xboole_0 )
& ~ ( u1_struct_0(B) != k1_xboole_0
& u1_struct_0(A) = k1_xboole_0 )
& ~ ( u1_struct_0(B) = k1_xboole_0
& u1_struct_0(A) != k1_xboole_0 )
& ( u1_struct_0(A) = k1_xboole_0
=> u1_struct_0(B) = k1_xboole_0 )
& ( u1_struct_0(B) = k1_xboole_0
=> u1_struct_0(A) = k1_xboole_0 ) ) ) ) ) ) ).
fof(t5_orders_4,axiom,
! [A] :
( ( v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v4_orders_2(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_orders_4(A,C,D)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ( r1_orders_4(B,E,F)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k5_yellow_0(A,C)),u1_struct_0(k5_yellow_0(B,E)))
& m2_relset_1(G,u1_struct_0(k5_yellow_0(A,C)),u1_struct_0(k5_yellow_0(B,E))) )
=> ( v23_waybel_0(G,k5_yellow_0(A,C),k5_yellow_0(B,E))
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,u1_struct_0(k5_yellow_0(A,D)),u1_struct_0(k5_yellow_0(B,F)))
& m2_relset_1(H,u1_struct_0(k5_yellow_0(A,D)),u1_struct_0(k5_yellow_0(B,F))) )
=> ~ ( v23_waybel_0(H,k5_yellow_0(A,D),k5_yellow_0(B,F))
& ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(I,u1_struct_0(A),u1_struct_0(B)) )
=> ~ ( I = k1_funct_4(G,H)
& v23_waybel_0(I,A,B) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_orders_4,axiom,
! [A] :
( ( v6_group_1(A)
& m1_orders_4(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k1_card_1(u1_struct_0(A)) = B
=> r5_waybel_1(A,k2_yellow_1(B)) ) ) ) ).
fof(dt_m1_orders_4,axiom,
! [A] :
( m1_orders_4(A)
=> l1_orders_2(A) ) ).
fof(existence_m1_orders_4,axiom,
? [A] : m1_orders_4(A) ).
%------------------------------------------------------------------------------