SET007 Axioms: SET007+272.ax
%------------------------------------------------------------------------------
% File : SET007+272 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Ordered Rings - Part II
% 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 : o_ring_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 10 ( 0 unt; 0 def)
% Number of atoms : 148 ( 0 equ)
% Maximal formula atoms : 19 ( 14 avg)
% Number of connectives : 201 ( 63 ~; 2 |; 96 &)
% ( 0 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 15 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 10 ( 10 usr; 0 prp; 1-2 aty)
% Number of functors : 2 ( 2 usr; 0 con; 1-3 aty)
% Number of variables : 30 ( 30 !; 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(t1_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( ( v1_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
| ( v3_o_ring_1(B,A)
& v1_o_ring_1(C,A) ) )
=> v3_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t2_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( ( v7_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
| ( v7_o_ring_1(B,A)
& v5_o_ring_1(C,A) ) )
=> v7_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t3_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v9_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v9_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v9_o_ring_1(C,A) ) )
=> v11_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t4_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v1_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v1_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v1_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v1_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v1_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v1_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t5_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v3_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v3_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v3_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v3_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v3_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v3_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t6_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v5_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v5_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t7_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v7_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v7_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v7_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v7_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v7_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t8_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v9_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
& ~ ( v9_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v9_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v9_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v9_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t9_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v11_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v11_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
fof(t10_o_ring_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ ( ~ ( v13_o_ring_1(B,A)
& v1_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v3_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v5_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v7_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v9_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v11_o_ring_1(C,A) )
& ~ ( v13_o_ring_1(B,A)
& v13_o_ring_1(C,A) ) )
=> v13_o_ring_1(k2_rlvect_1(A,B,C),A) ) ) ) ) ).
%------------------------------------------------------------------------------