SET007 Axioms: SET007+352.ax


%------------------------------------------------------------------------------
% File     : SET007+352 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Go-Board Theorem
% 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    : goboard4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    9 (   2 unit)
%            Number of atoms       :  113 (   2 equality)
%            Maximal formula depth :   22 (  13 average)
%            Number of connectives :  115 (  11 ~  ;   0  |;  73  &)
%                                         (   1 <=>;  30 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (   4 constant; 0-4 arity)
%            Number of variables   :   27 (   0 singleton;  27 !;   0 ?)
%            Maximal term depth    :    5 (   2 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_goboard4,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
                  & r1_xreal_0(np__1,k3_finseq_1(C))
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
                  & r1_xboole_0(k2_relat_1(B),k2_relat_1(C)) ) ) ) ) )).

fof(t2_goboard4,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
                  & r1_xreal_0(np__2,k3_finseq_1(C))
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
                  & r1_xboole_0(k5_topreal1(np__2,B),k5_topreal1(np__2,C)) ) ) ) ) )).

fof(t3_goboard4,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
                  & r1_xreal_0(np__2,k3_finseq_1(C))
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
                  & r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
                  & r1_xboole_0(k5_topreal1(np__2,B),k5_topreal1(np__2,C)) ) ) ) ) )).

fof(d1_goboard4,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_goboard4(A,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r2_hidden(D,k4_finseq_1(A))
                     => ( r1_xreal_0(B,k1_goboard1(A,D))
                        & r1_xreal_0(k1_goboard1(A,D),C) ) ) ) ) ) ) ) )).

fof(t4_goboard4,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( r1_xreal_0(np__2,k3_finseq_1(A))
              & r1_xreal_0(np__2,k3_finseq_1(B))
              & v1_topreal1(A)
              & v1_topreal1(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r2_hidden(C,k4_finseq_1(A))
                      & r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A))
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(C,np__1)) ) )
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r2_hidden(C,k4_finseq_1(B))
                      & r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(B))
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1)) ) )
              & r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
              & r1_goboard4(k2_goboard1(B),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
              & r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
              & r1_goboard4(k3_goboard1(B),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
              & r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) )).

fof(t5_goboard4,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v2_funct_1(A)
              & v1_topreal1(A)
              & r1_xreal_0(np__2,k3_finseq_1(A))
              & v2_funct_1(B)
              & v1_topreal1(B)
              & r1_xreal_0(np__2,k3_finseq_1(B))
              & r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
              & r1_goboard4(k2_goboard1(B),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
              & r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
              & r1_goboard4(k3_goboard1(B),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
              & r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) )).

fof(t6_goboard4,axiom,(
    $true )).

fof(t7_goboard4,axiom,(
    $true )).

fof(t8_goboard4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ~ ( r1_topreal4(A,C,E)
                              & r1_topreal4(B,D,F)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B))
                                   => ( r1_xreal_0(k21_euclid(C),k21_euclid(G))
                                      & r1_xreal_0(k21_euclid(G),k21_euclid(E)) ) ) )
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B))
                                   => ( r1_xreal_0(k22_euclid(D),k22_euclid(G))
                                      & r1_xreal_0(k22_euclid(G),k22_euclid(F)) ) ) )
                              & r1_xboole_0(A,B) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------