%--------------------------------------------------------------------------
% File     : GRP003+0 : TPTP v9.0.0. Released v2.5.0.
% Domain   : Group Theory
% Axioms   : Group theory axioms
% Version  : [MOW76] axioms.
% English  :

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
%          : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr
%          : [Ver93] Veroff (1993), Email to G. Sutcliffe
% Source   : TPTP
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   5 unt;   0 def)
%            Number of atoms       :   16 (   1 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :    8 (   0   ~;   0   |;   5   &)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   5 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 2-3 aty)
%            Number of functors    :    3 (   3 usr;   1 con; 0-2 aty)
%            Number of variables   :   22 (  22   !;   0   ?)
% SPC      : 

% Comments :
%--------------------------------------------------------------------------
fof(left_identity,axiom,
    ! [X] : product(identity,X,X) ).

fof(right_identity,axiom,
    ! [X] : product(X,identity,X) ).

fof(left_inverse,axiom,
    ! [X] : product(inverse(X),X,identity) ).

fof(right_inverse,axiom,
    ! [X] : product(X,inverse(X),identity) ).

%----This axiom is called closure or totality in some axiomatisations
fof(total_function1,axiom,
    ! [X,Y] : product(X,Y,multiply(X,Y)) ).

%----This axiom is called well_definedness in some axiomatisations
fof(total_function2,axiom,
    ! [W,X,Y,Z] :
      ( ( product(X,Y,Z)
        & product(X,Y,W) )
     => Z = W ) ).

fof(associativity1,axiom,
    ! [X,Y,Z,U,V,W] :
      ( ( product(X,Y,U)
        & product(Y,Z,V)
        & product(U,Z,W) )
     => product(X,V,W) ) ).

fof(associativity2,axiom,
    ! [X,Y,Z,U,V,W] :
      ( ( product(X,Y,U)
        & product(Y,Z,V)
        & product(X,V,W) )
     => product(U,Z,W) ) ).

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