## TPTP Problem File: SEV004^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV004^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory (Relations)
% Problem  : TPS problem from LATTICES-THMS
% Version  : Especial.
% English  :

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_1226 [Bro09]

% Status   : Theorem
% Rating   : 0.82 v7.5.0, 1.00 v7.4.0, 0.78 v7.2.0, 0.75 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.71 v6.1.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :    2 (   0 unit;   1 type;   0 defn)
%            Number of atoms       :  205 (  39 equality; 166 variable)
%            Maximal formula depth :   43 (  22 average)
%            Number of connectives :  137 (  11   ~;   0   |;  36   &;  88   @)
%                                         (   1 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    2 (   1   :;   0   =)
%            Number of variables   :   26 (   0 sgn;  21   !;   5   ?;   0   ^)
%                                         (  26   :;   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
%------------------------------------------------------------------------------
thf(a_type,type,(
a: \$tType )).

thf(cMODULAR_THM2_DEF2_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 @ ( MEET @ Xy @ ( JOIN @ Xx @ Xz ) ) )
= ( MEET @ ( JOIN @ Xx @ Xy ) @ ( JOIN @ Xx @ Xz ) ) )
<=> ~ ( ? [Xx: a,Xy: a,Xa: a,Xb: a,Xc: a] :
( ( Xa != Xb )
& ( Xa != Xc )
& ( Xa != Xx )
& ( Xa != Xy )
& ( Xb != Xc )
& ( Xb != Xx )
& ( Xb != Xy )
& ( Xc != Xx )
& ( Xc != Xy )
& ( Xx != Xy )
& ( ( MEET @ Xx @ Xy )
= Xy )
& ( ( JOIN @ Xx @ Xy )
= Xx )
& ( ( MEET @ Xx @ Xa )
= Xa )
& ( ( JOIN @ Xx @ Xa )
= Xx )
& ( ( MEET @ Xx @ Xb )
= Xb )
& ( ( JOIN @ Xx @ Xb )
= Xx )
& ( ( MEET @ Xx @ Xc )
= Xc )
& ( ( JOIN @ Xx @ Xc )
= Xx )
& ( ( MEET @ Xa @ Xb )
= Xy )
& ( ( JOIN @ Xa @ Xb )
= Xx )
& ( ( MEET @ Xa @ Xc )
= Xa )
& ( ( JOIN @ Xa @ Xc )
= Xc )
& ( ( MEET @ Xa @ Xy )
= Xy )
& ( ( JOIN @ Xa @ Xy )
= Xa )
& ( ( MEET @ Xb @ Xc )
= Xy )
& ( ( JOIN @ Xb @ Xc )
= Xx )
& ( ( MEET @ Xb @ Xy )
= Xy )
& ( ( JOIN @ Xb @ Xy )
= Xb )
& ( ( MEET @ Xc @ Xy )
= Xy )
& ( ( JOIN @ Xc @ Xy )
= Xc ) ) ) ) ) )).

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