## TPTP Problem File: KLE101+1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : KLE101+1 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Kleene Algebra (Modal Semirings)
% Problem  : Forward and backward diamonds are conjugates
% Version  : [Hoe08] axioms.
% English  :

% Refs     : [JT51]  Jonsson & Tarski (1951), Boolean Algebras with Operator
%          : [DMS04] Desharnais et al. (2004), Termination in Modal Kleene
%          : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Theorem
% Rating   : 0.78 v7.5.0, 0.75 v7.4.0, 0.73 v7.3.0, 0.72 v7.2.0, 0.76 v7.1.0, 0.74 v7.0.0, 0.83 v6.4.0, 0.81 v6.3.0, 0.79 v6.2.0, 0.84 v6.1.0, 0.87 v6.0.0, 0.83 v5.5.0, 0.89 v5.4.0, 0.93 v5.2.0, 0.85 v5.1.0, 0.86 v5.0.0, 0.92 v4.1.0, 0.87 v4.0.1, 0.83 v4.0.0
% Syntax   : Number of formulae    :   27 (  25 unit)
%            Number of atoms       :   29 (  28 equality)
%            Maximal formula depth :    5 (   3 average)
%            Number of connectives :    2 (   0   ~;   0   |;   0   &)
%                                         (   1 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of predicates  :    2 (   0 propositional; 2-2 arity)
%            Number of functors    :   14 (   2 constant; 0-2 arity)
%            Number of variables   :   46 (   0 sgn;  46   !;   0   ?)
%            Maximal term depth    :    6 (   2 average)
% SPC      : FOF_THM_RFO_SEQ

%------------------------------------------------------------------------------
%---Include axioms for modal semiring
include('Axioms/KLE001+0.ax').
%---Include axioms for Boolean domain/codomain
include('Axioms/KLE001+4.ax').
%---Include axioms for diamond and boxes
include('Axioms/KLE001+6.ax').
%------------------------------------------------------------------------------
fof(goals,conjecture,(
! [X0,X1,X2] :
( multiplication(forward_diamond(X0,domain(X1)),domain(X2)) = zero
=> multiplication(domain(X1),backward_diamond(X0,domain(X2))) = zero ) )).

%------------------------------------------------------------------------------
```