SET007 Axioms: SET007+640.ax


%------------------------------------------------------------------------------
% File     : SET007+640 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Representation Theorem for Finite Distributive Lattices
% 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    : lattice7 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   41 (   2 unit)
%            Number of atoms       :  394 (  22 equality)
%            Maximal formula depth :   17 (  10 average)
%            Number of connectives :  383 (  30 ~  ;   0  |; 242  &)
%                                         (  15 <=>;  96 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-4 arity)
%            Number of functors    :   22 (   4 constant; 0-4 arity)
%            Number of variables   :  108 (   0 singleton;  97 !;  11 ?)
%            Maximal term depth    :    6 (   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_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v5_orders_2(B,A) ) ) )).

fof(rc2_lattice7,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_lattice3(A)
      & v2_lattice3(A)
      & v3_lattice3(A)
      & v1_yellow_0(A)
      & v2_yellow_0(A)
      & v3_yellow_0(A)
      & v2_waybel_1(A)
      & v6_group_1(A) ) )).

fof(rc3_lattice7,axiom,(
    ? [A] :
      ( m2_lattice7(A)
      & ~ v1_xboole_0(A) ) )).

fof(fc1_lattice7,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_lattice7(A) )
     => ( ~ v3_struct_0(k2_yellow_1(A))
        & v1_orders_2(k2_yellow_1(A))
        & v2_orders_2(k2_yellow_1(A))
        & v3_orders_2(k2_yellow_1(A))
        & v4_orders_2(k2_yellow_1(A))
        & v1_lattice3(k2_yellow_1(A))
        & v2_lattice3(k2_yellow_1(A))
        & v2_waybel_1(k2_yellow_1(A)) ) ) )).

fof(d1_lattice7,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r1_lattice7(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,B)
                     => r2_hidden(D,C) ) ) ) ) ) ) )).

fof(d2_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r3_orders_2(A,B,C)
               => ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & v5_orders_2(D,A)
                      & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                   => ( m1_lattice7(D,A,B,C)
                    <=> ( r2_hidden(B,D)
                        & r2_hidden(C,D)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ( r2_hidden(E,D)
                             => ( r3_orders_2(A,B,E)
                                & r3_orders_2(A,E,C) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r3_orders_2(A,B,C)
               => m1_lattice7(k2_struct_0(A,B,C),A,B,C) ) ) ) ) )).

fof(d3_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k1_lattice7(A,B)
              <=> ( ? [D] :
                      ( m1_lattice7(D,A,k3_yellow_0(A),B)
                      & C = k4_card_1(D) )
                  & ! [D] :
                      ( m1_lattice7(D,A,k3_yellow_0(A),B)
                     => r1_xreal_0(k4_card_1(D),C) ) ) ) ) ) ) )).

fof(t2_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( r2_orders_2(A,B,C)
                  & r1_xreal_0(k1_lattice7(A,C),k1_lattice7(A,B)) ) ) ) ) )).

fof(t3_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v5_orders_2(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => ( r2_orders_2(A,C,D)
                    <=> ~ r1_xreal_0(k1_lattice7(A,D),k1_lattice7(A,C)) ) ) ) ) ) ) )).

fof(t4_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v5_orders_2(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => ( C = D
                    <=> k1_lattice7(A,C) = k1_lattice7(A,D) ) ) ) ) ) ) )).

fof(t5_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v5_orders_2(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => ( r3_orders_2(A,C,D)
                    <=> r1_xreal_0(k1_lattice7(A,C),k1_lattice7(A,D)) ) ) ) ) ) ) )).

fof(t6_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k1_lattice7(A,B) = np__1
          <=> B = k3_yellow_0(A) ) ) ) )).

fof(t7_lattice7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_xreal_0(np__1,k1_lattice7(A,B)) ) ) )).

fof(d4_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_lattice7(A,B,C)
              <=> ( r2_orders_2(A,B,C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_orders_2(A,B,D)
                          & r2_orders_2(A,D,C) ) ) ) ) ) ) ) )).

fof(t8_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & r2_hidden(C,B)
              & ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( r2_hidden(D,B)
                      & r2_orders_2(A,C,D) ) ) ) ) ) )).

fof(d5_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v5_orders_2(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = k2_lattice7(A,B)
              <=> ( ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( r2_hidden(D,B)
                       => r3_orders_2(A,D,C) ) )
                  & r2_hidden(C,B) ) ) ) ) ) )).

fof(t9_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ~ ( B != k3_yellow_0(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ r2_lattice7(A,C,B) ) ) ) ) )).

fof(t10_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r2_hidden(B,k3_lattice7(A))
          <=> ( B != k3_yellow_0(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( B = k13_lattice3(A,C,D)
                          & B != C
                          & B != D ) ) ) ) ) ) ) )).

fof(t11_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v2_waybel_1(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ~ ( r2_hidden(B,k3_lattice7(A))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( r2_orders_2(A,C,B)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( r2_orders_2(A,D,B)
                           => r3_orders_2(A,D,C) ) ) ) ) ) ) ) )).

fof(t12_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v2_waybel_1(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_yellow_0(A,k5_subset_1(u1_struct_0(A),k6_waybel_0(A,B),k3_lattice7(A))) = B ) ) )).

fof(t13_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v2_waybel_1(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))))
          & m2_relset_1(B,u1_struct_0(A),u1_struct_0(k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))))
          & v23_waybel_0(B,A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))))
          & ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_waybel_0(A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))),B,C) = k5_subset_1(u1_struct_0(A),k6_waybel_0(A,C),k3_lattice7(A)) ) ) ) )).

fof(t14_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v2_waybel_1(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => r5_waybel_1(A,k2_yellow_1(k4_lattice7(k5_yellow_0(A,k3_lattice7(A))))) ) )).

fof(d8_lattice7,axiom,(
    ! [A] :
      ( m2_lattice7(A)
    <=> r1_cohsp_1(A,A) ) )).

fof(t15_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => m2_lattice7(k4_lattice7(k5_yellow_0(A,k3_lattice7(A)))) ) )).

fof(t16_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A) )
     => ( v2_waybel_1(A)
      <=> ? [B] :
            ( ~ v1_xboole_0(B)
            & m2_lattice7(B)
            & r5_waybel_1(A,k2_yellow_1(B)) ) ) ) )).

fof(s1_lattice7,axiom,
    ( ! [A] :
        ( m1_subset_1(A,u1_struct_0(f1_s1_lattice7))
       => ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(f1_s1_lattice7))
             => ( r2_orders_2(f1_s1_lattice7,B,A)
               => p1_s1_lattice7(B) ) )
         => p1_s1_lattice7(A) ) )
   => ! [A] :
        ( m1_subset_1(A,u1_struct_0(f1_s1_lattice7))
       => p1_s1_lattice7(A) ) )).

fof(dt_m1_lattice7,axiom,(
    ! [A,B,C] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ! [D] :
          ( m1_lattice7(D,A,B,C)
         => ( ~ v1_xboole_0(D)
            & v5_orders_2(D,A)
            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) ) ) ) )).

fof(existence_m1_lattice7,axiom,(
    ! [A,B,C] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ? [D] : m1_lattice7(D,A,B,C) ) )).

fof(dt_m2_lattice7,axiom,(
    $true )).

fof(existence_m2_lattice7,axiom,(
    ? [A] : m2_lattice7(A) )).

fof(reflexivity_r1_lattice7,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => r1_lattice7(A,B,B) ) )).

fof(redefinition_r1_lattice7,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( r1_lattice7(A,B,C)
      <=> r1_tarski(B,C) ) ) )).

fof(dt_k1_lattice7,axiom,(
    ! [A,B] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m2_subset_1(k1_lattice7(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k2_lattice7,axiom,(
    ! [A,B] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v6_group_1(A)
        & l1_orders_2(A)
        & ~ v1_xboole_0(B)
        & v5_orders_2(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k2_lattice7(A,B),u1_struct_0(A)) ) )).

fof(dt_k3_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => m1_subset_1(k3_lattice7(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k4_lattice7,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ~ v1_xboole_0(k4_lattice7(A)) ) )).

fof(d6_lattice7,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => k3_lattice7(A) = a_1_0_lattice7(A) ) )).

fof(d7_lattice7,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => k4_lattice7(A) = a_1_1_lattice7(A) ) )).

fof(fraenkel_a_1_0_lattice7,axiom,(
    ! [A,B] :
      ( ( v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v1_lattice3(B)
        & v2_lattice3(B)
        & l1_orders_2(B) )
     => ( r2_hidden(A,a_1_0_lattice7(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = C
            & C != k3_yellow_0(B)
            & ! [D] :
                ( m1_subset_1(D,u1_struct_0(B))
               => ! [E] :
                    ( m1_subset_1(E,u1_struct_0(B))
                   => ~ ( C = k13_lattice3(B,D,E)
                        & C != D
                        & C != E ) ) ) ) ) ) )).

fof(fraenkel_a_1_1_lattice7,axiom,(
    ! [A,B] :
      ( l1_orders_2(B)
     => ( r2_hidden(A,a_1_1_lattice7(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
            & A = C
            & v12_waybel_0(C,B) ) ) ) )).
%------------------------------------------------------------------------------