SET007 Axioms: SET007+91.ax


%------------------------------------------------------------------------------
% File     : SET007+91 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Three-Argument Operations and Four-Argument Operations
% 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    : multop_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   4 unit)
%            Number of atoms       :  229 (  20 equality)
%            Maximal formula depth :   24 (  14 average)
%            Number of connectives :  243 (  38 ~  ;   0  |;  99  &)
%                                         (   0 <=>; 106 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-5 arity)
%            Number of functors    :   41 (  25 constant; 0-10 arity)
%            Number of variables   :  159 (   0 singleton; 147 !;  12 ?)
%            Maximal term depth    :    3 (   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(d1_multop_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : k1_multop_1(A,B,C,D) = k1_funct_1(A,k3_mcart_1(B,C,D)) ) )).

fof(t1_multop_1,axiom,(
    $true )).

fof(t2_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E] :
          ( ( v1_funct_1(E)
            & v1_funct_2(E,k3_zfmisc_1(B,C,D),A)
            & m2_relset_1(E,k3_zfmisc_1(B,C,D),A) )
         => ! [F] :
              ( ( v1_funct_1(F)
                & v1_funct_2(F,k3_zfmisc_1(B,C,D),A)
                & m2_relset_1(F,k3_zfmisc_1(B,C,D),A) )
             => ( ! [G,H,I] :
                    ( ( r2_hidden(G,B)
                      & r2_hidden(H,C)
                      & r2_hidden(I,D) )
                   => k1_funct_1(E,k3_mcart_1(G,H,I)) = k1_funct_1(F,k3_mcart_1(G,H,I)) )
               => E = F ) ) ) ) )).

fof(t3_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
                        & m2_relset_1(E,k3_zfmisc_1(A,B,C),D) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k3_zfmisc_1(A,B,C),D)
                            & m2_relset_1(F,k3_zfmisc_1(A,B,C),D) )
                         => ( ! [G] :
                                ( m1_subset_1(G,A)
                               => ! [H] :
                                    ( m1_subset_1(H,B)
                                   => ! [I] :
                                        ( m1_subset_1(I,C)
                                       => k8_funct_2(k3_zfmisc_1(A,B,C),D,E,k4_domain_1(A,B,C,G,H,I)) = k8_funct_2(k3_zfmisc_1(A,B,C),D,F,k4_domain_1(A,B,C,G,H,I)) ) ) )
                           => E = F ) ) ) ) ) ) ) )).

fof(t4_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
                        & m2_relset_1(E,k3_zfmisc_1(A,B,C),D) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k3_zfmisc_1(A,B,C),D)
                            & m2_relset_1(F,k3_zfmisc_1(A,B,C),D) )
                         => ( ! [G] :
                                ( m1_subset_1(G,A)
                               => ! [H] :
                                    ( m1_subset_1(H,B)
                                   => ! [I] :
                                        ( m1_subset_1(I,C)
                                       => k2_multop_1(A,B,C,D,E,G,H,I) = k2_multop_1(A,B,C,D,F,G,H,I) ) ) )
                           => E = F ) ) ) ) ) ) ) )).

fof(d2_multop_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D,E] : k3_multop_1(A,B,C,D,E) = k1_funct_1(A,k4_mcart_1(B,C,D,E)) ) )).

fof(t5_multop_1,axiom,(
    $true )).

fof(t6_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E,F] :
          ( ( v1_funct_1(F)
            & v1_funct_2(F,k4_zfmisc_1(B,C,D,E),A)
            & m2_relset_1(F,k4_zfmisc_1(B,C,D,E),A) )
         => ! [G] :
              ( ( v1_funct_1(G)
                & v1_funct_2(G,k4_zfmisc_1(B,C,D,E),A)
                & m2_relset_1(G,k4_zfmisc_1(B,C,D,E),A) )
             => ( ! [H,I,J,K] :
                    ( ( r2_hidden(H,B)
                      & r2_hidden(I,C)
                      & r2_hidden(J,D)
                      & r2_hidden(K,E) )
                   => k1_funct_1(F,k4_mcart_1(H,I,J,K)) = k1_funct_1(G,k4_mcart_1(H,I,J,K)) )
               => F = G ) ) ) ) )).

fof(t7_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
                            & m2_relset_1(F,k4_zfmisc_1(A,B,C,D),E) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,k4_zfmisc_1(A,B,C,D),E)
                                & m2_relset_1(G,k4_zfmisc_1(A,B,C,D),E) )
                             => ( ! [H] :
                                    ( m1_subset_1(H,A)
                                   => ! [I] :
                                        ( m1_subset_1(I,B)
                                       => ! [J] :
                                            ( m1_subset_1(J,C)
                                           => ! [K] :
                                                ( m1_subset_1(K,D)
                                               => k8_funct_2(k4_zfmisc_1(A,B,C,D),E,F,k5_domain_1(A,B,C,D,H,I,J,K)) = k8_funct_2(k4_zfmisc_1(A,B,C,D),E,G,k5_domain_1(A,B,C,D,H,I,J,K)) ) ) ) )
                               => F = G ) ) ) ) ) ) ) ) )).

fof(t8_multop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
                            & m2_relset_1(F,k4_zfmisc_1(A,B,C,D),E) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,k4_zfmisc_1(A,B,C,D),E)
                                & m2_relset_1(G,k4_zfmisc_1(A,B,C,D),E) )
                             => ( ! [H] :
                                    ( m1_subset_1(H,A)
                                   => ! [I] :
                                        ( m1_subset_1(I,B)
                                       => ! [J] :
                                            ( m1_subset_1(J,C)
                                           => ! [K] :
                                                ( m1_subset_1(K,D)
                                               => k4_multop_1(A,B,C,D,E,F,H,I,J,K) = k4_multop_1(A,B,C,D,E,G,H,I,J,K) ) ) ) )
                               => F = G ) ) ) ) ) ) ) ) )).

fof(s1_multop_1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_multop_1)
       => ! [B] :
            ( m1_subset_1(B,f2_s1_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f3_s1_multop_1)
               => ? [D] :
                    ( m1_subset_1(D,f4_s1_multop_1)
                    & p1_s1_multop_1(A,B,C,D) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1)
        & m2_relset_1(A,k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1)
        & ! [B] :
            ( m1_subset_1(B,f1_s1_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f2_s1_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f3_s1_multop_1)
                   => p1_s1_multop_1(B,C,D,k8_funct_2(k3_zfmisc_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1),f4_s1_multop_1,A,k4_domain_1(f1_s1_multop_1,f2_s1_multop_1,f3_s1_multop_1,B,C,D))) ) ) ) ) )).

fof(s2_multop_1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s2_multop_1)
       => ! [B] :
            ( m1_subset_1(B,f1_s2_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f1_s2_multop_1)
               => ? [D] :
                    ( m1_subset_1(D,f1_s2_multop_1)
                    & p1_s2_multop_1(A,B,C,D) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k3_zfmisc_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1),f1_s2_multop_1)
        & m2_relset_1(A,k3_zfmisc_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1),f1_s2_multop_1)
        & ! [B] :
            ( m1_subset_1(B,f1_s2_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f1_s2_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f1_s2_multop_1)
                   => p1_s2_multop_1(B,C,D,k2_multop_1(f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1,f1_s2_multop_1,A,B,C,D)) ) ) ) ) )).

fof(s3_multop_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1)
      & m2_relset_1(A,k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s3_multop_1)
         => ! [C] :
              ( m1_subset_1(C,f2_s3_multop_1)
             => ! [D] :
                  ( m1_subset_1(D,f3_s3_multop_1)
                 => k8_funct_2(k3_zfmisc_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1),f4_s3_multop_1,A,k4_domain_1(f1_s3_multop_1,f2_s3_multop_1,f3_s3_multop_1,B,C,D)) = f5_s3_multop_1(B,C,D) ) ) ) ) )).

fof(s4_multop_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k3_zfmisc_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1),f4_s4_multop_1)
      & m2_relset_1(A,k3_zfmisc_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1),f4_s4_multop_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s4_multop_1)
         => ! [C] :
              ( m1_subset_1(C,f2_s4_multop_1)
             => ! [D] :
                  ( m1_subset_1(D,f3_s4_multop_1)
                 => k2_multop_1(f1_s4_multop_1,f2_s4_multop_1,f3_s4_multop_1,f4_s4_multop_1,A,B,C,D) = f5_s4_multop_1(B,C,D) ) ) ) ) )).

fof(s5_multop_1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s5_multop_1)
       => ! [B] :
            ( m1_subset_1(B,f2_s5_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f3_s5_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f4_s5_multop_1)
                   => ? [E] :
                        ( m1_subset_1(E,f5_s5_multop_1)
                        & p1_s5_multop_1(A,B,C,D,E) ) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1)
        & m2_relset_1(A,k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1)
        & ! [B] :
            ( m1_subset_1(B,f1_s5_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f2_s5_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f3_s5_multop_1)
                   => ! [E] :
                        ( m1_subset_1(E,f4_s5_multop_1)
                       => p1_s5_multop_1(B,C,D,E,k8_funct_2(k4_zfmisc_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1),f5_s5_multop_1,A,k5_domain_1(f1_s5_multop_1,f2_s5_multop_1,f3_s5_multop_1,f4_s5_multop_1,B,C,D,E))) ) ) ) ) ) )).

fof(s6_multop_1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s6_multop_1)
       => ! [B] :
            ( m1_subset_1(B,f1_s6_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f1_s6_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f1_s6_multop_1)
                   => ? [E] :
                        ( m1_subset_1(E,f1_s6_multop_1)
                        & p1_s6_multop_1(A,B,C,D,E) ) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k4_zfmisc_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1),f1_s6_multop_1)
        & m2_relset_1(A,k4_zfmisc_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1),f1_s6_multop_1)
        & ! [B] :
            ( m1_subset_1(B,f1_s6_multop_1)
           => ! [C] :
                ( m1_subset_1(C,f1_s6_multop_1)
               => ! [D] :
                    ( m1_subset_1(D,f1_s6_multop_1)
                   => ! [E] :
                        ( m1_subset_1(E,f1_s6_multop_1)
                       => p1_s6_multop_1(B,C,D,E,k4_multop_1(f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,f1_s6_multop_1,A,B,C,D,E)) ) ) ) ) ) )).

fof(s7_multop_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1)
      & m2_relset_1(A,k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s7_multop_1)
         => ! [C] :
              ( m1_subset_1(C,f2_s7_multop_1)
             => ! [D] :
                  ( m1_subset_1(D,f3_s7_multop_1)
                 => ! [E] :
                      ( m1_subset_1(E,f4_s7_multop_1)
                     => k8_funct_2(k4_zfmisc_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1),f5_s7_multop_1,A,k5_domain_1(f1_s7_multop_1,f2_s7_multop_1,f3_s7_multop_1,f4_s7_multop_1,B,C,D,E)) = f6_s7_multop_1(B,C,D,E) ) ) ) ) ) )).

fof(s8_multop_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k4_zfmisc_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1),f1_s8_multop_1)
      & m2_relset_1(A,k4_zfmisc_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1),f1_s8_multop_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s8_multop_1)
         => ! [C] :
              ( m1_subset_1(C,f1_s8_multop_1)
             => ! [D] :
                  ( m1_subset_1(D,f1_s8_multop_1)
                 => ! [E] :
                      ( m1_subset_1(E,f1_s8_multop_1)
                     => k4_multop_1(f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,f1_s8_multop_1,A,B,C,D,E) = f2_s8_multop_1(B,C,D,E) ) ) ) ) ) )).

fof(dt_k1_multop_1,axiom,(
    $true )).

fof(dt_k2_multop_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
        & m1_relset_1(E,k3_zfmisc_1(A,B,C),D)
        & m1_subset_1(F,A)
        & m1_subset_1(G,B)
        & m1_subset_1(H,C) )
     => m1_subset_1(k2_multop_1(A,B,C,D,E,F,G,H),D) ) )).

fof(redefinition_k2_multop_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k3_zfmisc_1(A,B,C),D)
        & m1_relset_1(E,k3_zfmisc_1(A,B,C),D)
        & m1_subset_1(F,A)
        & m1_subset_1(G,B)
        & m1_subset_1(H,C) )
     => k2_multop_1(A,B,C,D,E,F,G,H) = k1_multop_1(E,F,G,H) ) )).

fof(dt_k3_multop_1,axiom,(
    $true )).

fof(dt_k4_multop_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E)
        & v1_funct_1(F)
        & v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
        & m1_relset_1(F,k4_zfmisc_1(A,B,C,D),E)
        & m1_subset_1(G,A)
        & m1_subset_1(H,B)
        & m1_subset_1(I,C)
        & m1_subset_1(J,D) )
     => m1_subset_1(k4_multop_1(A,B,C,D,E,F,G,H,I,J),E) ) )).

fof(redefinition_k4_multop_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E)
        & v1_funct_1(F)
        & v1_funct_2(F,k4_zfmisc_1(A,B,C,D),E)
        & m1_relset_1(F,k4_zfmisc_1(A,B,C,D),E)
        & m1_subset_1(G,A)
        & m1_subset_1(H,B)
        & m1_subset_1(I,C)
        & m1_subset_1(J,D) )
     => k4_multop_1(A,B,C,D,E,F,G,H,I,J) = k3_multop_1(F,G,H,I,J) ) )).
%------------------------------------------------------------------------------