## TPTP Axioms File: KLE003+0.ax

```%------------------------------------------------------------------------------
% File     : KLE003+0 : TPTP v7.5.0. Released v3.6.0.
% Domain   : Kleene Algebra
% Axioms   : Omega algebra
% Version  : [Hoe08] axioms.
% English  :

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

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (  14 unit)
%            Number of atoms       :   22 (  13 equality)
%            Maximal formula depth :    5 (   3 average)
%            Number of connectives :    4 (   0 ~  ;   0  |;   0  &)
%                                         (   1 <=>;   3 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    2 (   0 propositional; 2-2 arity)
%            Number of functors    :    6 (   2 constant; 0-2 arity)
%            Number of variables   :   34 (   0 singleton;  34 !;   0 ?)
%            Maximal term depth    :    4 (   2 average)
% 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] :
=> leq(multiplication(star(A),C),B) ) )).

fof(star_induction_right,axiom,(
! [A,B,C] :
=> leq(multiplication(C,star(B)),A) ) )).

%----Infinite iteration (omega)

%----Unfold law
fof(omega_unfold,axiom,(
! [A] : multiplication(A,omega(A)) = omega(A) )).

%----Co-Induction law
fof(omega_co_induction,axiom,(
! [A,B,C] :