SET007 Axioms: SET007+464.ax


%------------------------------------------------------------------------------
% File     : SET007+464 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Functor 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    : functor1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   27 (   2 unit)
%            Number of atoms       :  311 (  16 equality)
%            Maximal formula depth :   21 (  11 average)
%            Number of connectives :  349 (  65 ~  ;   0  |; 169  &)
%                                         (   3 <=>; 112 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   1 propositional; 0-3 arity)
%            Number of functors    :   19 (   0 constant; 1-6 arity)
%            Number of variables   :  105 (   0 singleton; 103 !;   2 ?)
%            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(rc1_functor1,axiom,(
    ? [A] :
      ( l2_altcat_1(A)
      & ~ v3_struct_0(A)
      & v2_altcat_1(A)
      & v12_altcat_1(A)
      & v1_altcat_2(A) ) )).

fof(rc2_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ? [B] :
          ( m1_altcat_2(B,A)
          & ~ v3_struct_0(B)
          & v1_altcat_2(B) ) ) )).

fof(fc1_functor1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A)
        & ~ v3_struct_0(B)
        & v1_altcat_2(B)
        & l2_altcat_1(B)
        & v8_functor0(C,A,B)
        & l2_functor0(C,A,B)
        & ~ v3_struct_0(D)
        & v1_altcat_2(D)
        & m1_altcat_2(D,A) )
     => v8_functor0(k14_functor0(A,B,D,C),D,B) ) )).

fof(t1_functor1,axiom,(
    ! [A] : v2_funct_2(k6_partfun1(A),A,A) )).

fof(t2_functor1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(A)) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(A)) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(B)) )
                 => ( C = D
                   => k6_funct_3(A,C) = k7_funct_2(D,B,A,k6_funct_3(B,D),k6_funct_3(A,B)) ) ) ) ) ) )).

fof(t3_functor1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( v3_funct_2(C,A,B)
       => ( v1_funct_1(k2_funct_1(C))
          & v1_funct_2(k2_funct_1(C),B,A)
          & m2_relset_1(k2_funct_1(C),B,A) ) ) ) )).

fof(t4_functor1,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ! [E] :
              ( ( v1_funct_1(E)
                & v1_funct_2(E,B,C)
                & m2_relset_1(E,B,C) )
             => ~ ( v3_funct_2(D,A,B)
                  & v3_funct_2(E,B,C)
                  & ! [F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,A,C)
                        & m2_relset_1(F,A,C) )
                     => ~ ( F = k1_partfun1(A,B,B,C,D,E)
                          & v3_funct_2(F,A,C) ) ) ) ) ) ) )).

fof(t5_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_altcat_2(C,A) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & m1_altcat_2(D,B) )
                 => ( C = D
                   => k10_functor0(A,C) = k13_functor0(D,B,A,k10_functor0(B,D),k10_functor0(A,B)) ) ) ) ) ) )).

fof(t6_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( l2_functor0(C,A,B)
             => ( v21_functor0(C,A,B)
               => ( v3_funct_2(u1_functor0(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)))
                  & v1_msualg_3(u2_functor0(A,B,C)) ) ) ) ) ) )).

fof(t7_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v4_functor0(D,A,B)
                          & v4_functor0(E,B,C) )
                       => v4_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t8_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v17_functor0(D,A,B)
                          & v17_functor0(E,B,C) )
                       => v17_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t9_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v5_functor0(D,A,B)
                          & v5_functor0(E,B,C) )
                       => v5_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t10_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v18_functor0(D,A,B)
                          & v18_functor0(E,B,C) )
                       => v18_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t11_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v19_functor0(D,A,B)
                          & v19_functor0(E,B,C) )
                       => v19_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t12_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v20_functor0(D,A,B)
                          & v20_functor0(E,B,C) )
                       => v20_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t13_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l1_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l1_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ( ( v21_functor0(D,A,B)
                          & v21_functor0(E,B,C) )
                       => v21_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) )).

fof(t14_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & m1_altcat_2(C,A) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & m1_altcat_2(D,A) )
                 => ! [E] :
                      ( ( ~ v3_struct_0(E)
                        & m1_altcat_2(E,C) )
                     => ( D = E
                       => ! [F] :
                            ( l2_functor0(F,A,B)
                           => k14_functor0(A,B,D,F) = k14_functor0(C,B,E,k14_functor0(A,B,C,F)) ) ) ) ) ) ) ) )).

fof(t15_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_altcat_2(C)
                & l2_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( l2_functor0(E,B,C)
                     => ! [F] :
                          ( ( ~ v3_struct_0(F)
                            & v1_altcat_2(F)
                            & m1_altcat_2(F,A) )
                         => k14_functor0(A,C,F,k13_functor0(A,B,C,D,E)) = k13_functor0(F,B,C,k14_functor0(A,B,F,D),E) ) ) ) ) ) ) )).

fof(t16_functor1,axiom,(
    $true )).

fof(t17_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_altcat_2(B,A) )
         => ( v2_altcat_2(B,A)
          <=> v18_functor0(k10_functor0(A,B),B,A) ) ) ) )).

fof(t18_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( v10_functor0(C,A,B)
                & l2_functor0(C,A,B) )
             => ( v18_functor0(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => v2_funct_2(k5_functor0(A,B,C,D,E),k1_altcat_1(A,D,E),k1_altcat_1(B,k3_functor0(A,B,C,D),k3_functor0(A,B,C,E))) ) ) ) ) ) ) )).

fof(t19_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( v10_functor0(C,A,B)
                & l2_functor0(C,A,B) )
             => ( v17_functor0(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => v2_funct_1(k5_functor0(A,B,C,D,E)) ) ) ) ) ) ) )).

fof(t20_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => k15_functor0(A,A,k12_functor0(A)) = k12_functor0(A) ) )).

fof(t21_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v12_altcat_1(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( v8_functor0(C,A,B)
                & l2_functor0(C,A,B) )
             => ( v21_functor0(C,A,B)
               => ! [D] :
                    ( ( v8_functor0(D,B,A)
                      & l2_functor0(D,B,A) )
                   => ( g2_functor0(B,A,u1_functor0(B,A,D),u2_functor0(B,A,D)) = k15_functor0(A,B,C)
                     => k13_functor0(B,A,B,D,C) = k12_functor0(B) ) ) ) ) ) ) )).

fof(t22_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v12_altcat_1(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( v8_functor0(C,A,B)
                & l2_functor0(C,A,B) )
             => ( v21_functor0(C,A,B)
               => k13_functor0(A,B,A,C,k15_functor0(A,B,C)) = k12_functor0(A) ) ) ) ) )).

fof(t23_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v12_altcat_1(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( v8_functor0(C,A,B)
                & l2_functor0(C,A,B) )
             => ( v21_functor0(C,A,B)
               => k15_functor0(B,A,k15_functor0(A,B,C)) = g2_functor0(A,B,u1_functor0(A,B,C),u2_functor0(A,B,C)) ) ) ) ) )).

fof(t24_functor1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & v1_altcat_2(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v12_altcat_1(B)
            & v1_altcat_2(B)
            & l2_altcat_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_altcat_1(C)
                & v12_altcat_1(C)
                & v1_altcat_2(C)
                & l2_altcat_1(C) )
             => ! [D] :
                  ( ( v8_functor0(D,A,B)
                    & l2_functor0(D,A,B) )
                 => ! [E] :
                      ( ( v8_functor0(E,B,C)
                        & l2_functor0(E,B,C) )
                     => ! [F] :
                          ( ( v8_functor0(F,B,A)
                            & l2_functor0(F,B,A) )
                         => ! [G] :
                              ( ( v8_functor0(G,C,B)
                                & l2_functor0(G,C,B) )
                             => ( ( v21_functor0(E,B,C)
                                  & v21_functor0(D,A,B)
                                  & v21_functor0(G,C,B)
                                  & v21_functor0(F,B,A)
                                  & g2_functor0(B,A,u1_functor0(B,A,F),u2_functor0(B,A,F)) = k15_functor0(A,B,D)
                                  & g2_functor0(C,B,u1_functor0(C,B,G),u2_functor0(C,B,G)) = k15_functor0(B,C,E) )
                               => k15_functor0(A,C,k13_functor0(A,B,C,D,E)) = k13_functor0(C,B,A,G,F) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------