%------------------------------------------------------------------------------
% File     : GRP001^5 : TPTP v8.2.0. Released v4.0.0.
% Domain   : Group Theory
% Problem  : TPS problem GRP-COMM2
% Version  : Especial.
% English  : Group is Abelian iff every element has order 2.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0516 [Bro09]
%          : GRP-COMM2 [TPS]

% Status   : Theorem
% Rating   : 0.10 v8.2.0, 0.08 v8.1.0, 0.09 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.00 v4.0.0
% Syntax   : Number of formulae    :    3 (   0 unt;   2 typ;   0 def)
%            Number of atoms       :    5 (   5 equ;   0 cnn)
%            Maximal formula atoms :    5 (   5 avg)
%            Number of connectives :   22 (   0   ~;   0   |;   3   &;  18   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   8 avg)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    2 (   2   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    3 (   2 usr;   1 con; 0-2 aty)
%            Number of variables   :    8 (   0   ^;   8   !;   0   ?;   8   :)
% 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/
%          : 
%------------------------------------------------------------------------------
thf(cP,type,
    cP: $i > $i > $i ).

thf(e,type,
    e: $i ).

thf(cGRP_COMM2,conjecture,
    ( ( ! [Xx: $i] :
          ( ( cP @ e @ Xx )
          = Xx )
      & ! [Xy: $i] :
          ( ( cP @ Xy @ e )
          = Xy )
      & ! [Xz: $i] :
          ( ( cP @ Xz @ Xz )
          = e )
      & ! [Xx: $i,Xy: $i,Xz: $i] :
          ( ( cP @ ( cP @ Xx @ Xy ) @ Xz )
          = ( cP @ Xx @ ( cP @ Xy @ Xz ) ) ) )
   => ! [Xa: $i,Xb: $i] :
        ( ( cP @ Xa @ Xb )
        = ( cP @ Xb @ Xa ) ) ) ).

%------------------------------------------------------------------------------