%--------------------------------------------------------------------------
% File     : GRP004-0 : TPTP v9.0.0. 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   : [ANL]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :    3 (   3 unt;   0 nHn;   0 RR)
%            Number of literals    :    3 (   3 equ;   0 neg)
%            Maximal clause size   :    1 (   1 avg)
%            Maximal term depth    :    3 (   2 avg)
%            Number of predicates  :    1 (   0 usr;   0 prp; 2-2 aty)
%            Number of functors    :    3 (   3 usr;   1 con; 0-2 aty)
%            Number of variables   :    5 (   0 sgn)
% SPC      : 

% Comments : [MOW76] also contains redundant right_identity and
%            right_inverse axioms.
%          : These axioms are also used in [Wos88] p.186, also with
%            right_identity and right_inverse.
%--------------------------------------------------------------------------
%----For any x and y in the group x*y is also in the group. No clause
%----is needed here since this is an instance of reflexivity

%----There exists an identity element
cnf(left_identity,axiom,
    multiply(identity,X) = X ).

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

%----The operation '*' is associative
cnf(associativity,axiom,
    multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ).

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