## TPTP Problem File: SEV000^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV000^5 : TPTP v7.5.0. 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   : 0.00 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 unit;   1 type;   0 defn)
%            Number of atoms       :   91 (  11 equality;  80 variable)
%            Maximal formula depth :   18 (  10 average)
%            Number of connectives :   68 (   0   ~;   0   |;   7   &;  58   @)
%                                         (   1 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    2 (   1   :;   0   =)
%            Number of variables   :   24 (   0 sgn;  24   !;   0   ?;   0   ^)
%                                         (  24   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% 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
%          : Polymorphic definitions expanded.
%------------------------------------------------------------------------------
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 ) ) ) ) ) )).

%------------------------------------------------------------------------------
```