## TPTP Problem File: SEV071^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV071^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory (Relations)
% Problem  : TPS problem THM576
% Version  : Especial.
% English  : Existence of symmetric closure of a relation.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0477 [Bro09]
%          : THM576 [TPS]

% Status   : Theorem
% Rating   : 0.67 v7.4.0, 0.56 v7.3.0, 0.60 v7.2.0, 0.62 v7.1.0, 0.57 v7.0.0, 0.50 v6.4.0, 0.57 v6.3.0, 0.50 v6.2.0, 0.67 v6.0.0, 0.50 v5.5.0, 0.40 v5.4.0, 0.50 v4.1.0, 0.33 v4.0.0
% Syntax   : Number of formulae    :    2 (   0 unit;   1 type;   0 defn)
%            Number of atoms       :   30 (   0 equality;  30 variable)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :   29 (   0   ~;   0   |;   3   &;  20   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    3 (   1   :;   0   =)
%            Number of variables   :   13 (   0 sgn;  12   !;   1   ?;   0   ^)
%                                         (  13   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_NEQ_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(cTHM576_pme,conjecture,(
! [Xr: a > a > \$o] :
? [Xs: a > a > \$o] :
( ! [Xa: a,Xb: a] :
( ( Xr @ Xa @ Xb )
=> ( Xs @ Xa @ Xb ) )
& ! [Xx: a,Xy: a] :
( ( Xs @ Xx @ Xy )
=> ( Xs @ Xy @ Xx ) )
& ! [Xt: a > a > \$o] :
( ( ! [Xa: a,Xb: a] :
( ( Xr @ Xa @ Xb )
=> ( Xt @ Xa @ Xb ) )
& ! [Xx: a,Xy: a] :
( ( Xt @ Xx @ Xy )
=> ( Xt @ Xy @ Xx ) ) )
=> ! [Xa: a,Xb: a] :
( ( Xs @ Xa @ Xb )
=> ( Xt @ Xa @ Xb ) ) ) ) )).

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