TPTP Axioms File: KLE002+0.ax

```%------------------------------------------------------------------------------
% File     : KLE002+0 : TPTP v8.1.0. Released v3.6.0.
% Domain   : Kleene algebra
% Axioms   : Kleene algebra
% Version  : [Hoe08] axioms.
% English  :

% Refs     : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   16 (  13 unt;   0 def)
%            Number of atoms       :   19 (  12 equ)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :    3 (   0   ~;   0   |;   0   &)
%                                         (   1 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   3 avg)
%            Maximal term depth    :    4 (   2 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    5 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :   30 (  30   !;   0   ?)
% SPC      :

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

! [A] : addition(A,zero) = A ).

! [A] : addition(A,A) = A ).

%----Multiplicative and commutative monoid
fof(multiplicative_associativity,axiom,
! [A,B,C] : multiplication(A,multiplication(B,C)) = multiplication(multiplication(A,B),C) ).

fof(multiplicative_right_identity,axiom,
! [A] : multiplication(A,one) = A ).

fof(multiplicative_left_identity,axiom,
! [A] : multiplication(one,A) = A ).

%----Distributivity laws
fof(right_distributivity,axiom,

fof(left_distributivity,axiom,

%----Annihilation
fof(right_annihilation,axiom,
! [A] : multiplication(A,zero) = zero ).

fof(left_annihilation,axiom,
! [A] : multiplication(zero,A) = zero ).

%----Order
fof(order,axiom,
! [A,B] :
( leq(A,B)
<=> addition(A,B) = B ) ).

%----Finite iteration (star)

%----Unfold laws
fof(star_unfold_right,axiom,

fof(star_unfold_left,axiom,

%----Induction laws
fof(star_induction_left,axiom,
! [A,B,C] :