SET007 Axioms: SET007+511.ax


%------------------------------------------------------------------------------
% File     : SET007+511 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lattice of Substitutions
% 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    : substlat [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   50 (   7 unit)
%            Number of atoms       :  196 (  34 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  155 (   9 ~  ;   0  |;  65  &)
%                                         (   5 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   0 propositional; 1-3 arity)
%            Number of functors    :   21 (   1 constant; 0-4 arity)
%            Number of variables   :  188 (   0 singleton; 182 !;   6 ?)
%            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(fc1_substlat,axiom,(
    ! [A,B] : ~ v1_xboole_0(k1_substlat(A,B)) )).

fof(rc1_substlat,axiom,(
    ! [A,B] :
    ? [C] :
      ( m1_subset_1(C,k1_substlat(A,B))
      & ~ v1_xboole_0(C) ) )).

fof(cc1_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_substlat(A,B))
     => v1_finset_1(C) ) )).

fof(cc2_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_substlat(A,B))
     => ( v1_finset_1(C)
        & v1_fraenkel(C) ) ) )).

fof(rc2_substlat,axiom,(
    ! [A,B] :
    ? [C] :
      ( m1_subset_1(C,k4_partfun1(A,B))
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_finset_1(C) ) )).

fof(fc2_substlat,axiom,(
    ! [A,B] :
      ( ~ v3_struct_0(k5_substlat(A,B))
      & v3_lattices(k5_substlat(A,B)) ) )).

fof(fc3_substlat,axiom,(
    ! [A,B] :
      ( ~ v3_struct_0(k5_substlat(A,B))
      & v3_lattices(k5_substlat(A,B))
      & v4_lattices(k5_substlat(A,B))
      & v5_lattices(k5_substlat(A,B))
      & v6_lattices(k5_substlat(A,B))
      & v7_lattices(k5_substlat(A,B))
      & v8_lattices(k5_substlat(A,B))
      & v9_lattices(k5_substlat(A,B))
      & v10_lattices(k5_substlat(A,B)) ) )).

fof(fc4_substlat,axiom,(
    ! [A,B] :
      ( ~ v3_struct_0(k5_substlat(A,B))
      & v3_lattices(k5_substlat(A,B))
      & v4_lattices(k5_substlat(A,B))
      & v5_lattices(k5_substlat(A,B))
      & v6_lattices(k5_substlat(A,B))
      & v7_lattices(k5_substlat(A,B))
      & v8_lattices(k5_substlat(A,B))
      & v9_lattices(k5_substlat(A,B))
      & v10_lattices(k5_substlat(A,B))
      & v11_lattices(k5_substlat(A,B))
      & v12_lattices(k5_substlat(A,B))
      & v13_lattices(k5_substlat(A,B))
      & v14_lattices(k5_substlat(A,B))
      & v15_lattices(k5_substlat(A,B)) ) )).

fof(t1_substlat,axiom,(
    ! [A,B] : r2_hidden(k1_xboole_0,k1_substlat(A,B)) )).

fof(t2_substlat,axiom,(
    ! [A,B] : r2_hidden(k1_tarski(k1_xboole_0),k1_substlat(A,B)) )).

fof(t3_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k4_substlat(A,B,C,D) = k4_substlat(A,B,D,C) ) ) )).

fof(t4_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ( C = k1_tarski(k1_xboole_0)
           => k4_substlat(A,B,D,C) = D ) ) ) )).

fof(t5_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D,E] :
          ( ( r2_hidden(C,k1_substlat(A,B))
            & r2_hidden(D,C)
            & r2_hidden(E,C)
            & r1_tarski(D,E) )
         => D = E ) ) )).

fof(t6_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( r2_hidden(D,k3_substlat(A,B,C))
         => ( r2_hidden(D,C)
            & ! [E] :
                ( ( r2_hidden(E,C)
                  & r1_tarski(E,D) )
               => E = D ) ) ) ) )).

fof(t7_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( v1_finset_1(D)
         => ( ( r2_hidden(D,C)
              & ! [E] :
                  ( v1_finset_1(E)
                 => ( ( r2_hidden(E,C)
                      & r1_tarski(E,D) )
                   => E = D ) ) )
           => r2_hidden(D,k3_substlat(A,B,C)) ) ) ) )).

fof(t8_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => r1_tarski(k3_substlat(A,B,C),C) ) )).

fof(t9_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ( C = k1_xboole_0
       => k3_substlat(A,B,C) = k1_xboole_0 ) ) )).

fof(t10_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( v1_finset_1(D)
         => ~ ( r2_hidden(D,C)
              & ! [E] :
                  ~ ( r1_tarski(E,D)
                    & r2_hidden(E,k3_substlat(A,B,C)) ) ) ) ) )).

fof(t11_substlat,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => k3_substlat(A,B,C) = C ) )).

fof(t12_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => r1_tarski(k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),C,D)),k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k3_substlat(A,B,C),D)) ) ) )).

fof(t13_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k3_substlat(A,B,C),D)) = k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),C,D)) ) ) )).

fof(t14_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
             => ( r1_tarski(C,D)
               => r1_tarski(k4_substlat(A,B,C,E),k4_substlat(A,B,D,E)) ) ) ) ) )).

fof(t15_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ~ ( r2_hidden(E,k4_substlat(A,B,C,D))
                & ! [F,G] :
                    ~ ( r2_hidden(F,C)
                      & r2_hidden(G,D)
                      & E = k2_xboole_0(F,G) ) ) ) ) )).

fof(t16_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k4_partfun1(A,B))
             => ! [F] :
                  ( m1_subset_1(F,k4_partfun1(A,B))
                 => ( ( r2_hidden(E,C)
                      & r2_hidden(F,D)
                      & r1_partfun1(E,F) )
                   => r2_hidden(k2_xboole_0(E,F),k4_substlat(A,B,C,D)) ) ) ) ) ) )).

fof(t17_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => r1_tarski(k3_substlat(A,B,k4_substlat(A,B,C,D)),k4_substlat(A,B,k3_substlat(A,B,C),D)) ) ) )).

fof(t18_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
             => ( r1_tarski(C,D)
               => r1_tarski(k4_substlat(A,B,E,C),k4_substlat(A,B,E,D)) ) ) ) ) )).

fof(t19_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k3_substlat(A,B,k4_substlat(A,B,k3_substlat(A,B,C),D)) = k3_substlat(A,B,k4_substlat(A,B,C,D)) ) ) )).

fof(t20_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k3_substlat(A,B,k4_substlat(A,B,C,k3_substlat(A,B,D))) = k3_substlat(A,B,k4_substlat(A,B,C,D)) ) ) )).

fof(t21_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
             => k4_substlat(A,B,C,k4_substlat(A,B,D,E)) = k4_substlat(A,B,k4_substlat(A,B,C,D),E) ) ) ) )).

fof(t22_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => ! [E] :
              ( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,B)))
             => k4_substlat(A,B,C,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),D,E)) = k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k4_substlat(A,B,C,D),k4_substlat(A,B,C,E)) ) ) ) )).

fof(t23_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => r1_tarski(C,k4_substlat(A,B,C,C)) ) )).

fof(t24_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => k3_substlat(A,B,k4_substlat(A,B,C,C)) = k3_substlat(A,B,C) ) )).

fof(t25_substlat,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
     => k3_substlat(A,B,k4_substlat(A,B,C,C)) = C ) )).

fof(d4_substlat,axiom,(
    ! [A,B,C] :
      ( ( v3_lattices(C)
        & l3_lattices(C) )
     => ( C = k5_substlat(A,B)
      <=> ( u1_struct_0(C) = k1_substlat(A,B)
          & ! [D] :
              ( m2_subset_1(D,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
             => ! [E] :
                  ( m2_subset_1(E,k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B))
                 => ( k1_binop_1(u2_lattices(C),D,E) = k3_substlat(A,B,k2_substlat(A,B,D,E))
                    & k1_binop_1(u1_lattices(C),D,E) = k3_substlat(A,B,k4_substlat(A,B,D,E)) ) ) ) ) ) ) )).

fof(t26_substlat,axiom,(
    ! [A,B] : k5_lattices(k5_substlat(A,B)) = k1_xboole_0 )).

fof(t27_substlat,axiom,(
    ! [A,B] : k6_lattices(k5_substlat(A,B)) = k1_tarski(k1_xboole_0) )).

fof(dt_k1_substlat,axiom,(
    ! [A,B] : m1_subset_1(k1_substlat(A,B),k1_zfmisc_1(k5_finsub_1(k4_partfun1(A,B)))) )).

fof(dt_k2_substlat,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_substlat(A,B))
        & m1_subset_1(D,k1_substlat(A,B)) )
     => m1_subset_1(k2_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) )).

fof(commutativity_k2_substlat,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_substlat(A,B))
        & m1_subset_1(D,k1_substlat(A,B)) )
     => k2_substlat(A,B,C,D) = k2_substlat(A,B,D,C) ) )).

fof(idempotence_k2_substlat,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_substlat(A,B))
        & m1_subset_1(D,k1_substlat(A,B)) )
     => k2_substlat(A,B,C,C) = C ) )).

fof(redefinition_k2_substlat,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k1_substlat(A,B))
        & m1_subset_1(D,k1_substlat(A,B)) )
     => k2_substlat(A,B,C,D) = k2_xboole_0(C,D) ) )).

fof(dt_k3_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => m2_subset_1(k3_substlat(A,B,C),k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B)) ) )).

fof(dt_k4_substlat,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
     => m1_subset_1(k4_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) )).

fof(dt_k5_substlat,axiom,(
    ! [A,B] :
      ( v3_lattices(k5_substlat(A,B))
      & l3_lattices(k5_substlat(A,B)) ) )).

fof(d1_substlat,axiom,(
    ! [A,B] : k1_substlat(A,B) = a_2_0_substlat(A,B) )).

fof(d2_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => k3_substlat(A,B,C) = a_3_0_substlat(A,B,C) ) )).

fof(d3_substlat,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k4_substlat(A,B,C,D) = a_4_0_substlat(A,B,C,D) ) ) )).

fof(fraenkel_a_2_0_substlat,axiom,(
    ! [A,B,C] :
      ( r2_hidden(A,a_2_0_substlat(B,C))
    <=> ? [D] :
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
          & A = D
          & ! [E] :
              ( r2_hidden(E,D)
             => v1_finset_1(E) )
          & ! [E] :
              ( m1_subset_1(E,k4_partfun1(B,C))
             => ! [F] :
                  ( m1_subset_1(F,k4_partfun1(B,C))
                 => ( ( r2_hidden(E,D)
                      & r2_hidden(F,D)
                      & r1_tarski(E,F) )
                   => E = F ) ) ) ) ) )).

fof(fraenkel_a_3_0_substlat,axiom,(
    ! [A,B,C,D] :
      ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
     => ( r2_hidden(A,a_3_0_substlat(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k4_partfun1(B,C))
            & A = E
            & v1_finset_1(E)
            & ! [F] :
                ( m1_subset_1(F,k4_partfun1(B,C))
               => ( ( r2_hidden(F,D)
                    & r1_tarski(F,E) )
                <=> F = E ) ) ) ) ) )).

fof(fraenkel_a_4_0_substlat,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C))) )
     => ( r2_hidden(A,a_4_0_substlat(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,k4_partfun1(B,C))
            & m1_subset_1(G,k4_partfun1(B,C))
            & A = k2_xboole_0(F,G)
            & r2_hidden(F,D)
            & r2_hidden(G,E)
            & r1_partfun1(F,G) ) ) ) )).
%------------------------------------------------------------------------------