## TPTP Problem File: ALG275^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : ALG275^5 : TPTP v7.0.0. Bugfixed v5.3.0.
% Domain   : General Algebra
% Problem  : TPS problem from GRP-THMS
% Version  : Especial.
% English  :

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0640 [Bro09]

% Status   : CounterSatisfiable
% Rating   : 0.33 v6.4.0, 0.67 v6.3.0, 0.33 v5.3.0
% Syntax   : Number of formulae    :   15 (   0 unit;   8 type;   6 defn)
%            Number of atoms       :   60 (  11 equality;  35 variable)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :   31 (   0   ~;   0   |;   4   &;  26   @)
%                                         (   1 <=>;   0  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   37 (  37   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   21 (   0 sgn;   8   !;   4   ?;   9   ^)
%                                         (  21   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_CSA_EQU_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
% Bugfixes : v5.2.0 - Added missing type declarations.
%          : v5.3.0 - Fixed tType to \$tType from last bugfixes.
%------------------------------------------------------------------------------
thf(g_type,type,(
g: \$tType )).

thf(f_type,type,(
f: g > g > g )).

thf(cGROUP2_type,type,(
cGROUP2: ( g > g > g ) > g > \$o )).

thf(cGROUP4_type,type,(
cGROUP4: ( g > g > g ) > \$o )).

thf(cGRP_ASSOC_type,type,(
cGRP_ASSOC: ( g > g > g ) > \$o )).

thf(cGRP_DIVISORS_type,type,(
cGRP_DIVISORS: ( g > g > g ) > \$o )).

thf(cGRP_LEFT_INVERSE_type,type,(
cGRP_LEFT_INVERSE: ( g > g > g ) > g > \$o )).

thf(cGRP_LEFT_UNIT_type,type,(
cGRP_LEFT_UNIT: ( g > g > g ) > g > \$o )).

thf(cGRP_ASSOC_def,definition,
( cGRP_ASSOC
= ( ^ [Xf: g > g > g] :
! [Xa: g,Xb: g,Xc: g] :
( ( Xf @ ( Xf @ Xa @ Xb ) @ Xc )
= ( Xf @ Xa @ ( Xf @ Xb @ Xc ) ) ) ) )).

thf(cGRP_DIVISORS_def,definition,
( cGRP_DIVISORS
= ( ^ [Xf: g > g > g] :
! [Xa: g,Xb: g] :
( ? [Xx: g] :
( ( Xf @ Xa @ Xx )
= Xb )
& ? [Xy: g] :
( ( Xf @ Xy @ Xa )
= Xb ) ) ) )).

thf(cGRP_LEFT_INVERSE_def,definition,
( cGRP_LEFT_INVERSE
= ( ^ [Xf: g > g > g,Xe: g] :
! [Xa: g] :
? [Xb: g] :
( ( Xf @ Xb @ Xa )
= Xe ) ) )).

thf(cGRP_LEFT_UNIT_def,definition,
( cGRP_LEFT_UNIT
= ( ^ [Xf: g > g > g,Xe: g] :
! [Xa: g] :
( ( Xf @ Xe @ Xa )
= Xa ) ) )).

thf(cGROUP2_def,definition,
( cGROUP2
= ( ^ [Xf: g > g > g,Xe: g] :
( ( cGRP_ASSOC @ Xf )
& ( cGRP_LEFT_UNIT @ Xf @ Xe )
& ( cGRP_LEFT_INVERSE @ Xf @ Xe ) ) ) )).

thf(cGROUP4_def,definition,
( cGROUP4
= ( ^ [Xf: g > g > g] :
( ( cGRP_ASSOC @ Xf )
& ( cGRP_DIVISORS @ Xf ) ) ) )).

thf(cEQUIV_02_04,conjecture,
( ! [Xf0: g > g > g] :
? [Xe: g] :
( cGROUP2 @ Xf0 @ Xe )
<=> ( cGROUP4 @ f ) )).

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