SET007 Axioms: SET007+284.ax
%------------------------------------------------------------------------------
% File : SET007+284 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Metric-Affine Configurations in Metric Affine Planes - Part I
% 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 : conaffm [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 13 ( 0 unt; 0 def)
% Number of atoms : 187 ( 27 equ)
% Maximal formula atoms : 27 ( 14 avg)
% Number of connectives : 190 ( 16 ~; 36 |; 60 &)
% ( 7 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 16 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-5 aty)
% Number of functors : 1 ( 1 usr; 0 con; 1-1 aty)
% Number of variables : 59 ( 59 !; 0 ?)
% 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(d1_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v1_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r6_analmetr(A,B,C,D)
& r6_analmetr(A,B,E,F)
& r6_analmetr(A,B,G,H)
& r4_analmetr(A,C,E,D,F)
& r4_analmetr(A,C,G,D,H) )
=> ( B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| r6_analmetr(A,E,F,C)
| r6_analmetr(A,C,D,G)
| r4_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v2_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,C,E,D,F)
& r4_analmetr(A,B,C,E,G)
& r5_analmetr(A,C,G,D,H) )
=> ( r4_analmetr(A,B,G,B,C)
| r4_analmetr(A,B,C,B,E)
| r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_conaffm(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))
=> ~ ( ~ r6_analmetr(A,B,C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r5_analmetr(A,E,B,C,D)
& r5_analmetr(A,E,C,B,D)
& r5_analmetr(A,E,D,B,C) ) ) ) ) ) ) ) ) ).
fof(d4_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v4_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,C,E,D,F)
& r5_analmetr(A,C,G,D,H) )
=> ( r4_analmetr(A,B,G,B,C)
| r4_analmetr(A,B,C,B,E)
| r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r6_analmetr(A,B,C,E)
& r6_analmetr(A,B,D,F)
& r5_analmetr(A,C,G,D,H)
& r5_analmetr(A,E,G,F,H) )
=> ( B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| C = E
| r6_analmetr(A,B,G,C)
| r4_analmetr(A,C,D,E,F) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v6_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r6_analmetr(A,B,C,E)
& r6_analmetr(A,B,D,F)
& r5_analmetr(A,C,G,D,H)
& r4_analmetr(A,C,D,E,F) )
=> ( B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| C = E
| r6_analmetr(A,B,G,C)
| r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v7_conaffm(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r4_analmetr(A,C,D,E,F)
& r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r6_analmetr(A,B,C,E)
& r6_analmetr(A,B,D,F)
& r5_analmetr(A,C,G,D,H)
& r5_analmetr(A,E,G,F,H) )
=> ( B = C
| B = D
| B = E
| B = F
| B = G
| B = H
| C = E
| r6_analmetr(A,B,G,C)
| r5_analmetr(A,B,G,B,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v4_conaffm(A)
=> v1_conaffm(A) ) ) ).
fof(t2_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v4_conaffm(A)
=> v2_conaffm(A) ) ) ).
fof(t3_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conaffm(A)
=> v6_conaffm(A) ) ) ).
fof(t4_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v6_conaffm(A)
=> v7_conaffm(A) ) ) ).
fof(t5_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conaffm(A)
=> v4_conaffm(A) ) ) ).
fof(t6_conaffm,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conaffm(A)
=> v3_conaffm(A) ) ) ).
%------------------------------------------------------------------------------