%--------------------------------------------------------------------------
% File     : SYN550+1 : TPTP v8.2.0. Released v2.2.0.
% Domain   : Syntactic (Translated)
% Problem  : dia box p <=> dia box dia box p
% Version  : Especial.
% English  :

% Refs     : [Wei94] Weidenbach (1994), Email to G. Sutcliffe
%          : [Che80] Chellas (1980), Modal Logic
% Source   : [Wei94]
% Names    :

% Status   : Theorem
% Rating   : 0.06 v8.2.0, 0.13 v8.1.0, 0.07 v7.5.0, 0.05 v7.4.0, 0.12 v7.3.0, 0.00 v6.4.0, 0.07 v6.3.0, 0.15 v6.2.0, 0.09 v6.1.0, 0.12 v6.0.0, 0.00 v5.5.0, 0.17 v5.3.0, 0.26 v5.2.0, 0.21 v5.1.0, 0.29 v5.0.0, 0.25 v4.1.0, 0.28 v4.0.1, 0.26 v4.0.0, 0.25 v3.7.0, 0.33 v3.5.0, 0.12 v3.4.0, 0.25 v3.3.0, 0.22 v3.2.0, 0.44 v3.1.0, 0.67 v2.7.0, 0.33 v2.6.0, 0.00 v2.5.0, 0.33 v2.4.0, 0.33 v2.2.1
% Syntax   : Number of formulae    :    3 (   1 unt;   0 def)
%            Number of atoms       :   12 (   0 equ)
%            Maximal formula atoms :    8 (   4 avg)
%            Number of connectives :    9 (   0   ~;   0   |;   4   &)
%                                         (   1 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   6 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :    2 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :    1 (   1 usr;   1 con; 0-0 aty)
%            Number of variables   :   10 (   7   !;   3   ?)
% SPC      : FOF_THM_RFO_NEQ

% Comments : These are modal logic problems translated to FOF. The problems
%            can be easily solved by theorem proving methods for modal logic,
%            but their (naive) translation into first-order logic results in
%            hard first-order problems. The first-order translation of the
%            box, diamond operators is:
%                [box P]x  :=  (all y R(x,y) -> [P]y)
%                [dia P]x  :=  (exists R(x,y) & [P]y)
%                [P]x  :=  P(x)  if P is an atom
%            where we start with some initial context 'o' (the initial world),
%            i.e. we translate a formula P into [P]o. [Wei94]
%--------------------------------------------------------------------------
fof(reflexivity_of_reachable,axiom,
    ! [X] : reachable(X,X) ).

fof(transitivity_of_reachable,axiom,
    ! [X,Y,Z] :
      ( ( reachable(X,Y)
        & reachable(Y,Z) )
     => reachable(X,Z) ) ).

%---- ((exists x (R(o,x) & (all y (R(x,y) -> P(y))))) <->
%---- (exists x (R(o,x) & (all y (R(x,y) ->
%---- (exists z (R(y,z) & (all w (R(z,w) -> P(w)))))))))).
fof(prove_this,conjecture,
    ( ? [X] :
        ( reachable(initial_world,X)
        & ! [Y] :
            ( reachable(X,Y)
           => p(Y) ) )
  <=> ? [X] :
        ( reachable(initial_world,X)
        & ! [Y] :
            ( reachable(X,Y)
           => ? [Z] :
                ( reachable(Y,Z)
                & ! [W] :
                    ( reachable(Z,W)
                   => p(W) ) ) ) ) ) ).

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