%------------------------------------------------------------------------------
% File     : SEV000^5 : TPTP v8.2.0. Released v4.0.0.
% Domain   : Set Theory (Relations)
% Problem  : TPS problem MODULAR-EQUIV-THM
% Version  : Especial.
% English  : The equivalence of two definitions of modularity.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0562 [Bro09]
%          : MODULAR-EQUIV-THM [TPS]

% Status   : Theorem
% Rating   : 0.10 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v6.1.0, 0.29 v5.5.0, 0.50 v5.4.0, 0.60 v5.1.0, 0.80 v5.0.0, 1.00 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0
% Syntax   : Number of formulae    :    2 (   0 unt;   1 typ;   0 def)
%            Number of atoms       :   11 (  11 equ;   0 cnn)
%            Maximal formula atoms :   11 (  11 avg)
%            Number of connectives :   68 (   0   ~;   0   |;   7   &;  58   @)
%                                         (   1 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (  13 avg)
%            Number of types       :    1 (   1 usr)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    1 (   0 usr;   0 con; 2-2 aty)
%            Number of variables   :   24 (   0   ^;  24   !;   0   ?;  24   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%            Mellon University. Distributed under the Creative Commons copyleft
%            license: http://creativecommons.org/licenses/by-sa/3.0/
%          : Polymorphic definitions expanded.
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thf(a_type,type,
    a: $tType ).

thf(cMODULAR_EQUIV_THM_pme,conjecture,
    ! [JOIN: a > a > a,MEET: a > a > a] :
      ( ( ! [Xx: a] :
            ( ( JOIN @ Xx @ Xx )
            = Xx )
        & ! [Xx: a] :
            ( ( MEET @ Xx @ Xx )
            = Xx )
        & ! [Xx: a,Xy: a,Xz: a] :
            ( ( JOIN @ ( JOIN @ Xx @ Xy ) @ Xz )
            = ( JOIN @ Xx @ ( JOIN @ Xy @ Xz ) ) )
        & ! [Xx: a,Xy: a,Xz: a] :
            ( ( MEET @ ( MEET @ Xx @ Xy ) @ Xz )
            = ( MEET @ Xx @ ( MEET @ Xy @ Xz ) ) )
        & ! [Xx: a,Xy: a] :
            ( ( JOIN @ Xx @ Xy )
            = ( JOIN @ Xy @ Xx ) )
        & ! [Xx: a,Xy: a] :
            ( ( MEET @ Xx @ Xy )
            = ( MEET @ Xy @ Xx ) )
        & ! [Xx: a,Xy: a] :
            ( ( JOIN @ ( MEET @ Xx @ Xy ) @ Xy )
            = Xy )
        & ! [Xx: a,Xy: a] :
            ( ( MEET @ ( JOIN @ Xx @ Xy ) @ Xy )
            = Xy ) )
     => ( ! [Xx: a,Xy: a,Xz: a] :
            ( ( ( JOIN @ Xx @ Xz )
              = Xz )
           => ( ( JOIN @ Xx @ ( MEET @ Xy @ Xz ) )
              = ( MEET @ ( JOIN @ Xx @ Xy ) @ Xz ) ) )
      <=> ! [Xx: a,Xy: a,Xz: a] :
            ( ( JOIN @ Xx @ ( MEET @ Xy @ ( JOIN @ Xx @ Xz ) ) )
            = ( MEET @ ( JOIN @ Xx @ Xy ) @ ( JOIN @ Xx @ Xz ) ) ) ) ) ).

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