SET007 Axioms: SET007+706.ax


%------------------------------------------------------------------------------
% File     : SET007+706 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Remarks on Finite Sequences on Go-Boards
% 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    : jordan1f [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   13 (   0 unit)
%            Number of atoms       :  165 (  19 equality)
%            Maximal formula depth :   22 (  13 average)
%            Number of connectives :  189 (  37 ~  ;   2  |;  97  &)
%                                         (   0 <=>;  53 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   0 propositional; 1-3 arity)
%            Number of functors    :   42 (   6 constant; 0-4 arity)
%            Number of variables   :   51 (   0 singleton;  51 !;   0 ?)
%            Maximal term depth    :    8 (   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_jordan1f,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( ( ~ v3_relat_1(E)
                        & v1_matrix_1(E)
                        & v3_goboard1(E)
                        & v4_goboard1(E)
                        & v5_goboard1(E)
                        & v6_goboard1(E)
                        & m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
                          & ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D))
                          & r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
                          & r2_hidden(k4_tarski(A,C),k2_matrix_1(E))
                          & r1_xreal_0(B,C)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r1_xreal_0(B,F)
                                  & r1_xreal_0(F,C)
                                  & k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,F)) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) )).

fof(t2_jordan1f,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( ( ~ v3_relat_1(E)
                        & v1_matrix_1(E)
                        & v3_goboard1(E)
                        & v4_goboard1(E)
                        & v5_goboard1(E)
                        & v6_goboard1(E)
                        & m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
                          & ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D))
                          & r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
                          & r2_hidden(k4_tarski(A,C),k2_matrix_1(E))
                          & r1_xreal_0(B,C)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r1_xreal_0(B,F)
                                  & r1_xreal_0(F,C)
                                  & k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,F)) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,C)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) )).

fof(t3_jordan1f,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( ( ~ v3_relat_1(E)
                        & v1_matrix_1(E)
                        & v3_goboard1(E)
                        & v4_goboard1(E)
                        & v5_goboard1(E)
                        & v6_goboard1(E)
                        & m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
                          & ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D))
                          & r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
                          & r2_hidden(k4_tarski(C,B),k2_matrix_1(E))
                          & r1_xreal_0(A,C)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r1_xreal_0(A,F)
                                  & r1_xreal_0(F,C)
                                  & k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,F,B)) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) )).

fof(t4_jordan1f,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( ( ~ v3_relat_1(E)
                        & v1_matrix_1(E)
                        & v3_goboard1(E)
                        & v4_goboard1(E)
                        & v5_goboard1(E)
                        & v6_goboard1(E)
                        & m2_finseq_1(E,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,E)
                          & ~ r1_xboole_0(k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D))
                          & r2_hidden(k4_tarski(A,B),k2_matrix_1(E))
                          & r2_hidden(k4_tarski(C,B),k2_matrix_1(E))
                          & r1_xreal_0(A,C)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r1_xreal_0(A,F)
                                  & r1_xreal_0(F,C)
                                  & k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,F,B)) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),E,C,B)),k5_topreal1(np__2,D)))) ) ) ) ) ) ) ) ) )).

fof(t5_jordan1f,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),np__1) = k30_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) )).

fof(t6_jordan1f,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),np__1) = k34_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) )).

fof(t7_jordan1f,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),k3_finseq_1(k1_jordan1e(A,B))) = k34_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) )).

fof(t8_jordan1f,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),k3_finseq_1(k2_jordan1e(A,B))) = k30_pscomp_1(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) )).

fof(t9_jordan1f,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( k5_topreal1(np__2,k1_jordan1e(A,B)) = k8_jordan6(k5_topreal1(np__2,k1_jordan9(A,B)))
              & k5_topreal1(np__2,k2_jordan1e(A,B)) = k9_jordan6(k5_topreal1(np__2,k1_jordan9(A,B))) )
            | ( k5_topreal1(np__2,k1_jordan1e(A,B)) = k9_jordan6(k5_topreal1(np__2,k1_jordan9(A,B)))
              & k5_topreal1(np__2,k2_jordan1e(A,B)) = k8_jordan6(k5_topreal1(np__2,k1_jordan9(A,B))) ) ) ) ) )).

fof(t10_jordan1f,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_goboard1(u1_struct_0(k15_euclid(np__2)),k1_jordan1e(A,B),k1_jordan8(A,B)) ) ) )).

fof(t11_jordan1f,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] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
                                      & r2_hidden(k4_tarski(F,G),k2_matrix_1(A))
                                      & B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E)
                                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,F,G) )
                                   => k2_xcmplx_0(k18_complex1(k6_xcmplx_0(F,D)),k18_complex1(k6_xcmplx_0(G,E))) = np__1 ) ) ) ) ) )
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ~ ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
                              & B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E) ) ) )
                  | r1_goboard1(u1_struct_0(k15_euclid(np__2)),k8_finseq_1(u1_struct_0(k15_euclid(np__2)),k12_finseq_1(u1_struct_0(k15_euclid(np__2)),B),C),A) ) ) ) ) ) )).

fof(t12_jordan1f,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_goboard1(u1_struct_0(k15_euclid(np__2)),k2_jordan1e(A,B),k1_jordan8(A,B)) ) ) )).

fof(t13_jordan1f,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_topreal2(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(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)))
             => ~ ( k21_euclid(C) = k7_xcmplx_0(k2_xcmplx_0(k18_pscomp_1(B),k20_pscomp_1(B)),np__2)
                  & k22_euclid(C) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,np__1),k1_jordan1a(k1_jordan8(B,np__1)),np__1),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,np__1),k1_jordan1a(k1_jordan8(B,np__1)),k1_matrix_1(k1_jordan8(B,np__1)))),k8_jordan6(k5_topreal1(np__2,k1_jordan9(B,k1_nat_1(A,np__1)))))))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k1_matrix_1(k1_jordan8(B,k1_nat_1(A,np__1))))
                          & C = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,k1_nat_1(A,np__1)),k1_jordan1a(k1_jordan8(B,k1_nat_1(A,np__1))),D) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------