%------------------------------------------------------------------------------
% File     : KLE001+6 : TPTP v9.0.0. Released v3.6.0.
% Domain   : Kleene Algebra
% Axioms   : Modal operators
% Version  : [Hoe08] axioms.
% English  :

% Refs     : [DMS06] Desharnais et al. (2006), Kleene Algebra with Domain
%          : [MS06]  Moeller & Struth (2006), Algebras of Modal Operators a
%          : [DS08]  Desharnais & Struth (2008), Modal Semirings Revisited
%          : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    6 (   6 unt;   0 def)
%            Number of atoms       :    6 (   6 equ)
%            Maximal formula atoms :    1 (   1 avg)
%            Number of connectives :    0 (   0   ~;   0   |;   0   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    3 (   3 avg)
%            Maximal term depth    :    4 (   2 avg)
%            Number of predicates  :    1 (   0 usr;   0 prp; 2-2 aty)
%            Number of functors    :   10 (  10 usr;   0 con; 1-2 aty)
%            Number of variables   :   11 (  11   !;   0   ?)
% SPC      : 

% Comments : Requires KLE001+4.ax
%          : With KLE001+0 and KLE001+4.ax generates modal semirings
%            With KLE002+0 and KLE001+4.ax generates modal Kleene Algebra
%            With KLE003+0 and KLE001+4.ax generates modal Omega Algebra
%          : Defines forward/backward box and diamond (and domain).
%------------------------------------------------------------------------------
%----Standard axioms for forward/backward box and diamond
fof(complement,axiom,
    ! [X0] : c(X0) = antidomain(domain(X0)) ).

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

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

fof(backward_diamond,axiom,
    ! [X0,X1] : backward_diamond(X0,X1) = codomain(multiplication(codomain(X1),X0)) ).

fof(forward_box,axiom,
    ! [X0,X1] : forward_box(X0,X1) = c(forward_diamond(X0,c(X1))) ).

fof(backward_box,axiom,
    ! [X0,X1] : backward_box(X0,X1) = c(backward_diamond(X0,c(X1))) ).

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