%------------------------------------------------------------------------------
% File     : LCL633^1 : TPTP v4.1.0. Released v3.6.0.
% Domain   : Logical Calculi
% Problem  : Goedel's ontological argument on the existence of God
% Version  : [Ben08] axioms.
% English  :

% Refs     : [Fit00] Fitting (2000), Higher-Order Modal Logic - A Sketch
%          : [Ben08] Benzmueller (2008), Email to G. Sutcliffe
% Source   : [Ben08]
% Names    : Fitting-HOLML-Ex-God-alternative-a [Ben08]

% Status   : CounterSatisfiable
% Rating   : 0.33 v4.1.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :   44 (   3 unit;  25 type;  17 defn)
%            Number of atoms       :  290 (  17 equality;  58 variable)
%            Maximal formula depth :   15 (   5 average)
%            Number of connectives :   66 (   3   ~;   1   |;   2   &;  59   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :  100 ( 100   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   31 (  25   :;   0  :=)
%            Number of variables   :   49 (   2 sgn;   9   !;   4   ?;  36   ^)
%                                         (  49   :;   0  :=;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : THF_CSA_EQU

% Comments : THF0 syntax.
%          : This problem is known to be bugged. It is intended to be a theorem
%            but there is a flaw in the encoding. Benzmueller says "... makes
%            really no sense, I have used normal quantification in there 
%            instead of modal quantification. It is unclear what this does."
%------------------------------------------------------------------------------
%----Include simple maths definitions and axioms
include('Axioms/LCL008^0.ax').
%------------------------------------------------------------------------------
%----Signature
thf(a,type,(
    a: $tType )).

thf(p,type,(
    p: ( a > $i > $o ) > $i > $o )).

thf(g,type,(
    g: a > $i > $o )).

thf(e,type,(
    e: ( a > $i > $o ) > a > $i > $o )).

thf(r,type,(
    r: $i > $i > $o )).

%----Axioms
thf(positiveness,axiom,(
    ! [X: a > $i > $o] :
      ( mvalid
      @ ( mimpl @ ( mnot @ ( p @ X ) )
        @ ( p
          @ ^ [Z: a] :
              ( mnot @ ( X @ Z ) ) ) ) ) )).

thf(g,definition,
    ( g
    = ( ^ [Z: a,W: $i] :
        ! [X: a > $i > $o] :
          ( mimpl @ ( p @ X ) @ ( X @ Z ) @ W ) ) )).

thf(e,definition,
    ( e
    = ( ^ [X: a > $i > $o,Z: a,P: $i] :
        ! [Y: a > $i > $o] :
          ( mimpl @ ( Y @ Z )
          @ ( mbox @ r
            @ ^ [Q: $i] :
              ! [W: a] :
                ( mimpl @ ( X @ W ) @ ( Y @ W ) @ Q ) )
          @ P ) ) )).

%----Conjecture
thf(thm,conjecture,
    ( mvalid
    @ ^ [W: $i] :
      ! [Z: a] :
        ( mimpl @ ( g @ Z ) @ ( e @ g @ Z ) @ W ) )).

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