## TPTP Problem File: SEV190^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV190^5 : TPTP v7.5.0. Bugfixed v5.3.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem THM580
% Version  : Especial.
% English  : The join (in the initial pairing algebra \$) of x and x is x.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0471 [Bro09]
%          : THM580 [TPS]

% Status   : Theorem
% Rating   : 0.64 v7.5.0, 0.57 v7.4.0, 0.67 v7.2.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.83 v5.4.0, 0.60 v5.3.0
% Syntax   : Number of formulae    :    7 (   0 unit;   5 type;   1 defn)
%            Number of atoms       :   66 (   5 equality;  49 variable)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives :   55 (   1   ~;   0   |;   9   &;  40   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   19 (  19   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5   :;   0   =)
%            Number of variables   :   22 (   0 sgn;  19   !;   0   ?;   3   ^)
%                                         (  22   :;   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
%            Mellon University. Distributed under the Creative Commons copyleft
%          : Polymorphic definitions expanded.
% Bugfixes : v5.2.0 - Added missing type declarations.
%          : v5.3.0 - Fixed tType to \$tType from last bugfixes.
%------------------------------------------------------------------------------
thf(iS_type,type,(
iS: \$tType )).

thf(c0_type,type,(
c0: iS )).

thf(cJOIN_type,type,(
cJOIN: iS > iS > iS > \$o )).

thf(cP_type,type,(
cP: iS > iS > iS )).

thf(cS_JOIN_CLOS_type,type,(
cS_JOIN_CLOS: iS > ( iS > iS > iS ) > ( iS > iS > iS > \$o ) > \$o )).

thf(cS_JOIN_CLOS_def,definition,
( cS_JOIN_CLOS
= ( ^ [X0: iS,P: iS > iS > iS,JOIN: iS > iS > iS > \$o] :
( ! [Xx: iS] :
( JOIN @ Xx @ X0 @ Xx )
& ! [Xy: iS] :
( JOIN @ X0 @ Xy @ Xy )
& ! [Xx: iS,Xy: iS,Xz: iS,Xu: iS,Xv: iS,Xw: iS] :
( ( ( JOIN @ Xx @ Xy @ Xz )
& ( JOIN @ Xu @ Xv @ Xw ) )
=> ( JOIN @ ( P @ Xx @ Xu ) @ ( P @ Xy @ Xv ) @ ( P @ Xz @ Xw ) ) ) ) ) )).

thf(cTHM580_pme,conjecture,
( ( ! [Xx: iS,Xy: iS] :
( ( cP @ Xx @ Xy )
!= c0 )
& ! [Xx: iS,Xy: iS,Xu: iS,Xv: iS] :
( ( ( cP @ Xx @ Xu )
= ( cP @ Xy @ Xv ) )
=> ( ( Xx = Xy )
& ( Xu = Xv ) ) )
& ! [X: iS > \$o] :
( ( ( X @ c0 )
& ! [Xx: iS,Xy: iS] :
( ( ( X @ Xx )
& ( X @ Xy ) )
=> ( X @ ( cP @ Xx @ Xy ) ) ) )
=> ! [Xx: iS] :
( X @ Xx ) )
& ( cS_JOIN_CLOS @ c0 @ cP @ cJOIN ) )
=> ! [Xx: iS] :
( cJOIN @ Xx @ Xx @ Xx ) )).

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