SET007 Axioms: SET007+272.ax


%------------------------------------------------------------------------------
% File     : SET007+272 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  148 (   0 equality)
%            Maximal formula depth :   17 (  14 average)
%            Number of connectives :  201 (  63 ~  ;   2  |;  96  &)
%                                         (   0 <=>;  40 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   10 (   0 propositional; 1-2 arity)
%            Number of functors    :    2 (   0 constant; 1-3 arity)
%            Number of variables   :   30 (   0 singleton;  30 !;   0 ?)
%            Maximal term depth    :    2 (   1 average)
% 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) ) ) ) ) )).
%------------------------------------------------------------------------------