## TPTP Axioms File: GRP004+0.ax

```%--------------------------------------------------------------------------
% File     : GRP004+0 : TPTP v7.5.0. Released v1.0.0.
% Domain   : Group Theory
% Axioms   : Group theory (equality) axioms
% Version  : [MOW76] (equality) axioms :
%            Reduced > Complete.
% English  :

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
%          : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr
% Source   : [TPTP]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    3 (   3 unit)
%            Number of atoms       :    3 (   3 equality)
%            Maximal formula depth :    4 (   3 average)
%            Number of connectives :    0 (   0 ~  ;   0  |;   0  &)
%                                         (   0 <=>;   0 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    1 (   0 propositional; 2-2 arity)
%            Number of functors    :    3 (   1 constant; 0-2 arity)
%            Number of variables   :    5 (   0 singleton;   5 !;   0 ?)
%            Maximal term depth    :    3 (   2 average)
% SPC      :

% Comments : Reverse engineered from GRP004-0.ax.
%--------------------------------------------------------------------------
%----There exists an identity element
fof(left_identity,axiom,
( ! [X] : multiply(identity,X) = X )).

%----For any x in the group, there exists an element y such that x*y = y*x
%----= identity.
fof(left_inverse,axiom,
( ! [X] : multiply(inverse(X),X) = identity )).

%----The operation '*' is associative
fof(associativity,axiom,
( ! [X,Y,Z] : multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )).

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