SET007 Axioms: SET007+544.ax


%------------------------------------------------------------------------------
% File     : SET007+544 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Characterization of Modular and 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    : yellow11 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   8 unit)
%            Number of atoms       :  363 (  94 equality)
%            Maximal formula depth :   38 (   8 average)
%            Number of connectives :  395 (  67 ~  ;   0  |; 251  &)
%                                         (   7 <=>;  70 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   0 propositional; 1-3 arity)
%            Number of functors    :   23 (   6 constant; 0-5 arity)
%            Number of variables   :   84 (   0 singleton;  69 !;  15 ?)
%            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_yellow11,axiom,
    ( v1_orders_2(k1_yellow11)
    & v2_orders_2(k1_yellow11)
    & v3_orders_2(k1_yellow11)
    & v4_orders_2(k1_yellow11) )).

fof(fc2_yellow11,axiom,
    ( ~ v3_struct_0(k1_yellow11)
    & v1_lattice3(k1_yellow11)
    & v2_lattice3(k1_yellow11) )).

fof(fc3_yellow11,axiom,
    ( v1_orders_2(k2_yellow11)
    & v2_orders_2(k2_yellow11)
    & v3_orders_2(k2_yellow11)
    & v4_orders_2(k2_yellow11) )).

fof(fc4_yellow11,axiom,
    ( ~ v3_struct_0(k2_yellow11)
    & v1_lattice3(k2_yellow11)
    & v2_lattice3(k2_yellow11) )).

fof(cc1_yellow11,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v4_orders_2(A)
          & v2_lattice3(A)
          & v2_waybel_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v4_orders_2(A)
          & v2_lattice3(A)
          & v1_yellow11(A) ) ) ) )).

fof(cc2_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( ~ v1_xboole_0(B)
              & v2_yellow11(B,A) )
           => v1_waybel_0(B,A) ) ) ) )).

fof(cc3_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( ~ v1_xboole_0(B)
              & v2_yellow11(B,A) )
           => v2_waybel_0(B,A) ) ) ) )).

fof(fc5_yellow11,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => v2_yellow11(k3_yellow11(A,B,C),A) ) )).

fof(fc6_yellow11,axiom,(
    ! [A,B,C] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v2_lattice3(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v2_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v3_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v4_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v4_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A)
        & v5_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A) ) ) )).

fof(fc7_yellow11,axiom,(
    ! [A,B,C] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v2_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v3_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v4_orders_2(k5_yellow_0(A,k3_yellow11(A,B,C)))
        & v4_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A)
        & v6_yellow_0(k5_yellow_0(A,k3_yellow11(A,B,C)),A) ) ) )).

fof(rc1_yellow11,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)
      & v6_group_1(A) ) )).

fof(cc4_yellow11,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v2_lattice3(A)
          & v6_group_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v2_lattice3(A)
          & v1_yellow_0(A) ) ) ) )).

fof(cc5_yellow11,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v6_group_1(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) ) ) ) )).

fof(t1_yellow11,axiom,(
    np__3 = k1_enumset1(np__0,np__1,np__2) )).

fof(t2_yellow11,axiom,(
    k4_xboole_0(np__2,np__1) = k1_tarski(np__1) )).

fof(t3_yellow11,axiom,(
    k4_xboole_0(np__3,np__1) = k2_tarski(np__1,np__2) )).

fof(t4_yellow11,axiom,(
    k4_xboole_0(np__3,np__2) = k1_tarski(np__2) )).

fof(t5_yellow11,axiom,(
    ! [A] :
      ( ( v2_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))
             => ( k12_lattice3(A,B,C) = C
              <=> k13_lattice3(A,B,C) = B ) ) ) ) )).

fof(t6_yellow11,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))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r3_orders_2(A,k13_lattice3(A,k12_lattice3(A,B,C),k12_lattice3(A,B,D)),k12_lattice3(A,B,k13_lattice3(A,C,D))) ) ) ) ) )).

fof(t7_yellow11,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))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r3_orders_2(A,k13_lattice3(A,B,k12_lattice3(A,C,D)),k12_lattice3(A,k13_lattice3(A,B,C),k13_lattice3(A,B,D))) ) ) ) ) )).

fof(t8_yellow11,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))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r3_orders_2(A,B,D)
                   => r3_orders_2(A,k13_lattice3(A,B,k12_lattice3(A,C,D)),k12_lattice3(A,k13_lattice3(A,B,C),D)) ) ) ) ) ) )).

fof(d1_yellow11,axiom,(
    k1_yellow11 = k2_yellow_1(k3_enumset1(np__0,k4_xboole_0(np__3,np__1),np__2,k4_xboole_0(np__3,np__2),np__3)) )).

fof(d2_yellow11,axiom,(
    k2_yellow11 = k2_yellow_1(k3_enumset1(np__0,np__1,k4_xboole_0(np__2,np__1),k4_xboole_0(np__3,np__2),np__3)) )).

fof(t9_yellow11,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] :
                ( v4_yellow_0(B,A)
                & v5_yellow_0(B,A)
                & v6_yellow_0(B,A)
                & m1_yellow_0(B,A)
                & r5_waybel_1(k1_yellow11,B) )
            & ! [B] :
                ( m1_subset_1(B,u1_struct_0(A))
               => ! [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ~ ( B != C
                                    & B != D
                                    & B != E
                                    & B != F
                                    & C != D
                                    & C != E
                                    & C != F
                                    & D != E
                                    & D != F
                                    & E != F
                                    & k12_lattice3(A,B,C) = B
                                    & k12_lattice3(A,B,D) = B
                                    & k12_lattice3(A,D,F) = D
                                    & k12_lattice3(A,E,F) = E
                                    & k12_lattice3(A,C,D) = B
                                    & k12_lattice3(A,C,E) = C
                                    & k12_lattice3(A,D,E) = B
                                    & k13_lattice3(A,C,D) = F
                                    & k13_lattice3(A,D,E) = F ) ) ) ) ) ) )
        & ~ ( ? [B] :
                ( m1_subset_1(B,u1_struct_0(A))
                & ? [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                    & ? [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                        & ? [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                            & ? [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                                & B != C
                                & B != D
                                & B != E
                                & B != F
                                & C != D
                                & C != E
                                & C != F
                                & D != E
                                & D != F
                                & E != F
                                & k12_lattice3(A,B,C) = B
                                & k12_lattice3(A,B,D) = B
                                & k12_lattice3(A,D,F) = D
                                & k12_lattice3(A,E,F) = E
                                & k12_lattice3(A,C,D) = B
                                & k12_lattice3(A,C,E) = C
                                & k12_lattice3(A,D,E) = B
                                & k13_lattice3(A,C,D) = F
                                & k13_lattice3(A,D,E) = F ) ) ) ) )
            & ! [B] :
                ( ( v4_yellow_0(B,A)
                  & v5_yellow_0(B,A)
                  & v6_yellow_0(B,A)
                  & m1_yellow_0(B,A) )
               => ~ r5_waybel_1(k1_yellow11,B) ) ) ) ) )).

fof(t10_yellow11,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] :
                ( v4_yellow_0(B,A)
                & v5_yellow_0(B,A)
                & v6_yellow_0(B,A)
                & m1_yellow_0(B,A)
                & r5_waybel_1(k2_yellow11,B) )
            & ! [B] :
                ( m1_subset_1(B,u1_struct_0(A))
               => ! [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ~ ( B != C
                                    & B != D
                                    & B != E
                                    & B != F
                                    & C != D
                                    & C != E
                                    & C != F
                                    & D != E
                                    & D != F
                                    & E != F
                                    & k12_lattice3(A,B,C) = B
                                    & k12_lattice3(A,B,D) = B
                                    & k12_lattice3(A,B,E) = B
                                    & k12_lattice3(A,C,F) = C
                                    & k12_lattice3(A,D,F) = D
                                    & k12_lattice3(A,E,F) = E
                                    & k12_lattice3(A,C,D) = B
                                    & k12_lattice3(A,C,E) = B
                                    & k12_lattice3(A,D,E) = B
                                    & k13_lattice3(A,C,D) = F
                                    & k13_lattice3(A,C,E) = F
                                    & k13_lattice3(A,D,E) = F ) ) ) ) ) ) )
        & ~ ( ? [B] :
                ( m1_subset_1(B,u1_struct_0(A))
                & ? [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                    & ? [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                        & ? [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                            & ? [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                                & B != C
                                & B != D
                                & B != E
                                & B != F
                                & C != D
                                & C != E
                                & C != F
                                & D != E
                                & D != F
                                & E != F
                                & k12_lattice3(A,B,C) = B
                                & k12_lattice3(A,B,D) = B
                                & k12_lattice3(A,B,E) = B
                                & k12_lattice3(A,C,F) = C
                                & k12_lattice3(A,D,F) = D
                                & k12_lattice3(A,E,F) = E
                                & k12_lattice3(A,C,D) = B
                                & k12_lattice3(A,C,E) = B
                                & k12_lattice3(A,D,E) = B
                                & k13_lattice3(A,C,D) = F
                                & k13_lattice3(A,C,E) = F
                                & k13_lattice3(A,D,E) = F ) ) ) ) )
            & ! [B] :
                ( ( v4_yellow_0(B,A)
                  & v5_yellow_0(B,A)
                  & v6_yellow_0(B,A)
                  & m1_yellow_0(B,A) )
               => ~ r5_waybel_1(k2_yellow11,B) ) ) ) ) )).

fof(d3_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_yellow11(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r1_orders_2(A,B,D)
                     => k10_lattice3(A,B,k11_lattice3(A,C,D)) = k11_lattice3(A,k10_lattice3(A,B,C),D) ) ) ) ) ) ) )).

fof(t11_yellow11,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ( v1_yellow11(A)
      <=> ! [B] :
            ( ( v4_yellow_0(B,A)
              & v5_yellow_0(B,A)
              & v6_yellow_0(B,A)
              & m1_yellow_0(B,A) )
           => ~ r5_waybel_1(k1_yellow11,B) ) ) ) )).

fof(t12_yellow11,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ( v1_yellow11(A)
       => ( v2_waybel_1(A)
        <=> ! [B] :
              ( ( v4_yellow_0(B,A)
                & v5_yellow_0(B,A)
                & v6_yellow_0(B,A)
                & m1_yellow_0(B,A) )
             => ~ r5_waybel_1(k2_yellow11,B) ) ) ) ) )).

fof(d4_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( D = k3_yellow11(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( r2_hidden(E,D)
                        <=> ( r1_orders_2(A,B,E)
                            & r1_orders_2(A,E,C) ) ) ) ) ) ) ) ) )).

fof(d5_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_yellow11(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & B = k3_yellow11(A,C,D) ) ) ) ) ) )).

fof(t13_yellow11,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k3_yellow11(A,B,C) = k5_subset_1(u1_struct_0(A),k7_waybel_0(A,B),k6_waybel_0(A,C)) ) ) ) )).

fof(t14_yellow11,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))
             => ( v1_yellow11(A)
               => r5_waybel_1(k5_yellow_0(A,k3_yellow11(A,C,k13_lattice3(A,B,C))),k5_yellow_0(A,k3_yellow11(A,k12_lattice3(A,B,C),B))) ) ) ) ) )).

fof(dt_k1_yellow11,axiom,(
    l1_orders_2(k1_yellow11) )).

fof(dt_k2_yellow11,axiom,(
    l1_orders_2(k2_yellow11) )).

fof(dt_k3_yellow11,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k3_yellow11(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).
%------------------------------------------------------------------------------