SET007 Axioms: SET007+470.ax


%------------------------------------------------------------------------------
% File     : SET007+470 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Examples of Category Structures
% 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    : msinst_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   2 unit)
%            Number of atoms       :  200 (  22 equality)
%            Maximal formula depth :   26 (  12 average)
%            Number of connectives :  236 (  54 ~  ;   0  |; 107  &)
%                                         (   6 <=>;  69 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-4 arity)
%            Number of functors    :   25 (   2 constant; 0-6 arity)
%            Number of variables   :   76 (   0 singleton;  71 !;   5 ?)
%            Maximal term depth    :    4 (   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(fc1_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k1_msinst_1(A))
        & v2_altcat_1(k1_msinst_1(A))
        & v6_altcat_1(k1_msinst_1(A))
        & v11_altcat_1(k1_msinst_1(A))
        & v12_altcat_1(k1_msinst_1(A)) ) ) )).

fof(rc1_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ? [B] :
          ( l3_msualg_1(B,A)
          & v4_msualg_1(B,A)
          & v1_msualg_6(B,A) ) ) )).

fof(fc2_msinst_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_msinst_1(A,B)) ) )).

fof(fc3_msinst_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & ~ v1_xboole_0(B) )
     => ( ~ v3_struct_0(k4_msinst_1(A,B))
        & v2_altcat_1(k4_msinst_1(A,B))
        & v6_altcat_1(k4_msinst_1(A,B))
        & v11_altcat_1(k4_msinst_1(A,B))
        & v12_altcat_1(k4_msinst_1(A,B)) ) ) )).

fof(d1_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_altcat_1(B)
            & l2_altcat_1(B) )
         => ( B = k1_msinst_1(A)
          <=> ( u1_struct_0(B) = k5_msalimit(A)
              & ! [C] :
                  ( m3_msalimit(C,k5_msalimit(A))
                 => ! [D] :
                      ( m3_msalimit(D,k5_msalimit(A))
                     => k1_binop_1(u1_altcat_1(B),C,D) = k6_msalimit(C,D) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(B))
                         => ( ( r2_hidden(C,k5_msalimit(A))
                              & r2_hidden(D,k5_msalimit(A))
                              & r2_hidden(E,k5_msalimit(A)) )
                           => ! [F] :
                                ( ( v1_relat_1(F)
                                  & v1_funct_1(F) )
                               => ! [G] :
                                    ( ( v1_relat_1(G)
                                      & v1_funct_1(G) )
                                   => ! [H] :
                                        ( ( v1_relat_1(H)
                                          & v1_funct_1(H) )
                                       => ! [I] :
                                            ( ( v1_relat_1(I)
                                              & v1_funct_1(I) )
                                           => ( ( r2_hidden(k4_tarski(F,G),k1_binop_1(u1_altcat_1(B),C,D))
                                                & r2_hidden(k4_tarski(H,I),k1_binop_1(u1_altcat_1(B),D,E)) )
                                             => k1_binop_1(k4_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B),C,D,E),k4_tarski(H,I),k4_tarski(F,G)) = k4_tarski(k5_relat_1(F,H),k5_relat_1(G,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v11_altcat_1(B)
            & v12_altcat_1(B)
            & l2_altcat_1(B) )
         => ( B = k1_msinst_1(A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(B))
               => ( ~ v3_struct_0(C)
                  & ~ v2_msualg_1(C)
                  & l1_msualg_1(C) ) ) ) ) ) )).

fof(d2_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( C = k2_msinst_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ? [E] :
                      ( v4_msualg_1(E,A)
                      & v1_msualg_6(E,A)
                      & l3_msualg_1(E,A)
                      & D = E
                      & ! [F] :
                          ( m1_subset_1(F,k2_relat_1(u4_msualg_1(A,E)))
                         => r1_tarski(F,B) ) ) ) ) ) ) )).

fof(t2_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( l3_msualg_1(C,B)
             => ( r2_hidden(C,k2_msinst_1(B,A))
               => ( r2_hidden(u4_msualg_1(B,C),k1_funct_2(u1_struct_0(B),k1_zfmisc_1(A)))
                  & r2_hidden(u5_msualg_1(B,C),k1_funct_2(u1_msualg_1(B),k4_partfun1(k4_partfun1(k5_numbers,A),A))) ) ) ) ) ) )).

fof(t3_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_msualg_1(A))
         => ! [C] :
              ( l3_msualg_1(C,A)
             => ! [D] :
                  ( l3_msualg_1(D,A)
                 => ~ ( r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                      & k3_msualg_1(A,B,C) != k1_xboole_0
                      & k3_msualg_1(A,B,D) = k1_xboole_0 ) ) ) ) ) )).

fof(t4_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_msualg_1(A))
         => ! [C] :
              ( ( v1_msualg_6(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( ( v1_msualg_6(D,A)
                    & l3_msualg_1(D,A) )
                 => ! [E] :
                      ( ( v1_msualg_6(E,A)
                        & l3_msualg_1(E,A) )
                     => ! [F] :
                          ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                         => ! [G] :
                              ( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,E))
                             => ! [H] :
                                  ( m1_subset_1(H,k3_msualg_1(A,B,C))
                                 => ~ ( k3_msualg_1(A,B,C) != k1_xboole_0
                                      & r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                                      & r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,E))
                                      & ! [I] :
                                          ( m3_pboole(I,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,E))
                                         => ~ ( I = k13_pboole(F,G)
                                              & k5_msualg_3(A,C,E,B,I,H) = k5_msualg_3(A,D,E,B,G,k5_msualg_3(A,C,D,B,F,H)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t5_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v1_msualg_6(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v1_msualg_6(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( ( v1_msualg_6(D,A)
                    & l3_msualg_1(D,A) )
                 => ! [E] :
                      ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
                     => ! [F] :
                          ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                         => ~ ( r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
                              & r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                              & r1_msualg_3(A,B,C,E)
                              & r1_msualg_3(A,C,D,F)
                              & ! [G] :
                                  ( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,D))
                                 => ~ ( G = k13_pboole(E,F)
                                      & r1_msualg_3(A,B,D,G) ) ) ) ) ) ) ) ) ) )).

fof(d3_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C,D] :
              ( ( r2_hidden(C,k2_msinst_1(A,B))
                & r2_hidden(D,k2_msinst_1(A,B)) )
             => ! [E] :
                  ( E = k3_msinst_1(A,B,C,D)
                <=> ! [F] :
                      ( r2_hidden(F,E)
                    <=> ? [G] :
                          ( v4_msualg_1(G,A)
                          & v1_msualg_6(G,A)
                          & l3_msualg_1(G,A)
                          & ? [H] :
                              ( v4_msualg_1(H,A)
                              & v1_msualg_6(H,A)
                              & l3_msualg_1(H,A)
                              & ? [I] :
                                  ( m3_pboole(I,u1_struct_0(A),u4_msualg_1(A,G),u4_msualg_1(A,H))
                                  & G = C
                                  & H = D
                                  & I = F
                                  & r1_pzfmisc1(u1_struct_0(A),u4_msualg_1(A,G),u4_msualg_1(A,H))
                                  & r1_msualg_3(A,G,H,I) ) ) ) ) ) ) ) ) )).

fof(d4_msinst_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_altcat_1(C)
                & l2_altcat_1(C) )
             => ( C = k4_msinst_1(A,B)
              <=> ( u1_struct_0(C) = k2_msinst_1(A,B)
                  & ! [D] :
                      ( m1_subset_1(D,k2_msinst_1(A,B))
                     => ! [E] :
                          ( m1_subset_1(E,k2_msinst_1(A,B))
                         => k1_binop_1(u1_altcat_1(C),D,E) = k3_msinst_1(A,B,D,E) ) )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(C))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(C))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(C))
                             => ! [G] :
                                  ( ( v1_relat_1(G)
                                    & v1_funct_1(G)
                                    & v1_funcop_1(G) )
                                 => ! [H] :
                                      ( ( v1_relat_1(H)
                                        & v1_funct_1(H)
                                        & v1_funcop_1(H) )
                                     => ( ( r2_hidden(G,k1_binop_1(u1_altcat_1(C),D,E))
                                          & r2_hidden(H,k1_binop_1(u1_altcat_1(C),E,F)) )
                                       => k1_binop_1(k4_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C),D,E,F),H,G) = k13_pboole(G,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t6_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_altcat_1(C)
                & v11_altcat_1(C)
                & v12_altcat_1(C)
                & l2_altcat_1(C) )
             => ( C = k4_msinst_1(B,A)
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(C))
                   => ( v4_msualg_1(D,B)
                      & v1_msualg_6(D,B)
                      & l3_msualg_1(D,B) ) ) ) ) ) ) )).

fof(dt_k1_msinst_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k1_msinst_1(A))
        & v6_altcat_1(k1_msinst_1(A))
        & l2_altcat_1(k1_msinst_1(A)) ) ) )).

fof(dt_k2_msinst_1,axiom,(
    $true )).

fof(dt_k3_msinst_1,axiom,(
    $true )).

fof(dt_k4_msinst_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & ~ v1_xboole_0(B) )
     => ( ~ v3_struct_0(k4_msinst_1(A,B))
        & v6_altcat_1(k4_msinst_1(A,B))
        & l2_altcat_1(k4_msinst_1(A,B)) ) ) )).
%------------------------------------------------------------------------------