## TPTP Axioms File: KLE001+5.ax

```%------------------------------------------------------------------------------
% File     : KLE001+5 : TPTP v7.5.0. Released v3.6.0.
% Domain   : Kleene Algebra
% Axioms   : Domain (not Boolean domain!)
% Version  : [Hoe08] axioms.
% English  :

% Refs     : [DS08]  Desharnais & Struth (2008), Modal Semirings Revisited
%          : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    5 (   5 unit)
%            Number of atoms       :    5 (   5 equality)
%            Maximal formula depth :    3 (   2 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    :    5 (   2 constant; 0-2 arity)
%            Number of variables   :    6 (   0 singleton;   6 !;   0 ?)
%            Maximal term depth    :    4 (   3 average)
% SPC      :

% Comments : The domain algebra is not necessarily Boolean
%          : Requires KLE001+0.ax, KLE002+0.ax or KLE003+0.ax
%          : Combined with KLE001+0 generates Idempotent semirings with tests
%            Combined with KLE002+0 generates Kleene Algebra with tests
%            Combined with KLE003+0 generates Omega Algebra with tests
%------------------------------------------------------------------------------
%----Domain axioms (a la Desharnais & Struth)
fof(domain1,axiom,(
! [X0] : addition(X0,multiplication(domain(X0),X0)) = multiplication(domain(X0),X0) )).

fof(domain2,axiom,(
! [X0,X1] : domain(multiplication(X0,X1)) = domain(multiplication(X0,domain(X1))) )).

fof(domain3,axiom,(
! [X0] : addition(domain(X0),one) = one )).

fof(domain4,axiom,(
domain(zero) = zero )).

fof(domain5,axiom,(