%------------------------------------------------------------------------------
% File     : KLE026+2 : TPTP v9.0.0. Released v4.0.0.
% Domain   : Kleene Algebra (Idempotent Test Semirings)
% Problem  : Two ways of expressing the Hoare triple {p}x{q}
% Version  : [Hoe08] axioms : Augmented.
% English  :

% Refs     : [Koz00] Kozen (2000), On Hoare Logic and Kleene Algebra with T
%          : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Theorem
% Rating   : 0.33 v9.0.0, 0.36 v8.2.0, 0.33 v7.5.0, 0.44 v7.4.0, 0.23 v7.3.0, 0.34 v7.2.0, 0.38 v7.1.0, 0.35 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.43 v6.0.0, 0.48 v5.5.0, 0.52 v5.4.0, 0.54 v5.3.0, 0.56 v5.2.0, 0.40 v5.1.0, 0.43 v5.0.0, 0.33 v4.1.0, 0.35 v4.0.1, 0.39 v4.0.0
% Syntax   : Number of formulae    :   19 (  11 unt;   0 def)
%            Number of atoms       :   34 (  20 equ)
%            Maximal formula atoms :    4 (   1 avg)
%            Number of connectives :   16 (   1   ~;   0   |;   5   &)
%                                         (   4 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   4 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    5 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :   36 (  35   !;   1   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Equational encoding : additionally the de Morgan laws
%------------------------------------------------------------------------------
%---Include axioms for idempotent test semiring
include('Axioms/KLE001+0.ax').
%---Include test axioms
include('Axioms/KLE001+1.ax').
%---Include additionally axioms
include('Axioms/KLE001+2.ax').
%------------------------------------------------------------------------------
fof(goals,conjecture,
    ! [X0,X1,X2] :
      ( ( test(X1)
        & test(X2) )
     => ( multiplication(X1,X0) = multiplication(multiplication(X1,X0),X2)
       => leq(multiplication(X1,X0),multiplication(X0,X2)) ) ) ).

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