## TPTP Problem File: ALG276^5.p

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```%------------------------------------------------------------------------------
% File     : ALG276^5 : TPTP v8.1.2. 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_0641 [Bro09]

% Status   : Theorem
% Rating   : 0.54 v8.1.0, 0.36 v7.5.0, 0.43 v7.4.0, 0.33 v7.3.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0, 0.86 v5.5.0, 1.00 v5.3.0
% Syntax   : Number of formulae    :   14 (   6 unt;   7 typ;   6 def)
%            Number of atoms       :   26 (  13 equ;   0 cnn)
%            Maximal formula atoms :    2 (   3 avg)
%            Number of connectives :   37 (   0   ~;   0   |;   6   &;  30   @)
%                                         (   1 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   2 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   35 (  35   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   6 usr;   0 con; 1-2 aty)
%            Number of variables   :   21 (   9   ^   8   !;   4   ?;  21   :)
% 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
% 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(cGROUP1_type,type,
cGROUP1: ( 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_INVERSE_type,type,
cGRP_INVERSE: ( g > g > g ) > g > \$o ).

thf(cGRP_UNIT_type,type,
cGRP_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_INVERSE_def,definition,
( cGRP_INVERSE
= ( ^ [Xf: g > g > g,Xe: g] :
! [Xa: g] :
? [Xb: g] :
( ( ( Xf @ Xa @ Xb )
= Xe )
& ( ( Xf @ Xb @ Xa )
= Xe ) ) ) ) ).

thf(cGRP_UNIT_def,definition,
( cGRP_UNIT
= ( ^ [Xf: g > g > g,Xe: g] :
! [Xa: g] :
( ( ( Xf @ Xe @ Xa )
= Xa )
& ( ( Xf @ Xa @ Xe )
= Xa ) ) ) ) ).

thf(cGROUP1_def,definition,
( cGROUP1
= ( ^ [Xf: g > g > g,Xe: g] :
( ( cGRP_ASSOC @ Xf )
& ( cGRP_UNIT @ Xf @ Xe )
& ( cGRP_INVERSE @ Xf @ Xe ) ) ) ) ).

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

thf(cEQUIV_01_04,conjecture,
! [Xf: g > g > g] :
( ? [Xe: g] : ( cGROUP1 @ Xf @ Xe )
<=> ( cGROUP4 @ Xf ) ) ).

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