## TPTP Problem File: LCL874^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : LCL874^1 : TPTP v7.0.0. Bugfixed v5.0.0.
% Domain   : Logical Calculi (Modal logic)
% Problem  : Inclusion statement in a 2-D logic of knowledge and belief
% Version  : [Ben10] axioms.
% English  : Suppose we want to work with a 2-dimensional logic combining a
%            modality box_k of knowledge with a modality box_b of belief.
%            Moreover, suppose we model box_k as an S5 modality and box_b as
%            an D45 modality and let us furthermore add two axioms
%            characterizing their relationship. We may then want to check
%            whether or not box_k and box_b coincide, i.e., whether box_k
%            includes box_b

% Refs     : [Ben10a] Benzmueller (2010), Email to Geoff Sutcliffe
%          : [Ben10b] Benzmueller (2010), Simple Type Theory as a Framework
% Source   : [Ben10a]
% Names    : Problem 36 [Ben10b]

% Status   : Theorem
% Rating   : 0.50 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.60 v6.2.0, 0.71 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.60 v5.0.0
% Syntax   : Number of formulae    :   74 (   0 unit;  34 type;  31 defn)
%            Number of atoms       :  263 (  36 equality; 136 variable)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :  155 (   4   ~;   4   |;   8   &; 131   @)
%                                         (   0 <=>;   8  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  175 ( 175   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   38 (  34   :;   0   =)
%            Number of variables   :   87 (   3 sgn;  29   !;   6   ?;  52   ^)
%                                         (  87   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Bugfixes : v5.0.0 - Bugfix to LCL013^0.ax
%------------------------------------------------------------------------------
%----Include the definitions for quantified multimodal logic
include('Axioms/LCL013^0.ax').
%------------------------------------------------------------------------------
thf(rk,type,(
rk: \$i > \$i > \$o )).

thf(rb,type,(
rb: \$i > \$i > \$o )).

thf(ax1,axiom,
( mreflexive @ rk )).

thf(ax2,axiom,
( mserial @ rb )).

thf(ax3,axiom,
( mtransitive @ rk )).

thf(ax4,axiom,
( mtransitive @ rb )).

thf(ax5,axiom,
( meuclidean @ rk )).

thf(ax6,axiom,
( meuclidean @ rb )).

thf(ax7,axiom,
( mvalid
@ ( mforall_prop
@ ^ [A: \$i > \$o] :
( mimplies @ ( mbox @ rk @ A ) @ ( mbox @ rb @ A ) ) ) )).

thf(ax8,axiom,
( mvalid
@ ( mforall_prop
@ ^ [A: \$i > \$o] :
( mimplies @ ( mbox @ rb @ A ) @ ( mbox @ rb @ ( mbox @ rk @ A ) ) ) ) )).

thf(conj,conjecture,
( mvalid
@ ( mforall_prop
@ ^ [A: \$i > \$o] :
( mimplies @ ( mbox @ rb @ A ) @ ( mbox @ rk @ A ) ) ) )).

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