%--------------------------------------------------------------------------
% File     : GEO004+2 : TPTP v9.0.0. Released v2.4.0.
% Domain   : Geometry (Oriented curves)
% Axioms   : Oriented curves
% Version  : [EHK99] axioms.
% English  :

% Refs     : [KE99]  Kulik & Eschenbach (1999), A Geometry of Oriented Curv
%          : [EHK99] Eschenbach et al. (1999), Representing Simple Trajecto
% Source   : [EHK99]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   10 (   1 unt;   0 def)
%            Number of atoms       :   39 (   5 equ)
%            Maximal formula atoms :    7 (   3 avg)
%            Number of connectives :   32 (   3   ~;   1   |;  13   &)
%                                         (  10 <=>;   5  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 1-4 aty)
%            Number of functors    :    1 (   1 usr;   0 con; 1-1 aty)
%            Number of variables   :   36 (  32   !;   4   ?)
% SPC      : 

% Comments : Requires GEO004+0.ax GEO004+1.ax
%--------------------------------------------------------------------------
fof(between_o_defn,axiom,
    ! [O,P,Q,R] :
      ( between_o(O,P,Q,R)
    <=> ( ( ordered_by(O,P,Q)
          & ordered_by(O,Q,R) )
        | ( ordered_by(O,R,Q)
          & ordered_by(O,Q,P) ) ) ) ).

fof(start_point_defn,axiom,
    ! [P,O] :
      ( start_point(P,O)
    <=> ( incident_o(P,O)
        & ! [Q] :
            ( ( P != Q
              & incident_o(Q,O) )
           => ordered_by(O,P,Q) ) ) ) ).

fof(finish_point_defn,axiom,
    ! [P,O] :
      ( finish_point(P,O)
    <=> ( incident_o(P,O)
        & ! [Q] :
            ( ( P != Q
              & incident_o(Q,O) )
           => ordered_by(O,Q,P) ) ) ) ).

fof(o1,axiom,
    ! [O,P,Q] :
      ( ordered_by(O,P,Q)
     => ( incident_o(P,O)
        & incident_o(Q,O) ) ) ).

fof(o2,axiom,
    ! [O] :
    ? [C] :
      ( open(C)
      & ! [P] :
          ( incident_o(P,O)
        <=> incident_c(P,C) ) ) ).

fof(o3,axiom,
    ! [P,Q,R,O] :
      ( between_o(O,P,Q,R)
    <=> ? [C] :
          ( ! [P] :
              ( incident_o(P,O)
            <=> incident_c(P,C) )
          & between_c(C,P,Q,R) ) ) ).

fof(o4,axiom,
    ! [O] :
    ? [P] : start_point(P,O) ).

fof(o5,axiom,
    ! [P,Q,C] :
      ( ( open(C)
        & P != Q
        & incident_c(P,C)
        & incident_c(Q,C) )
     => ? [O] :
          ( ! [R] :
              ( incident_o(R,O)
            <=> incident_c(R,C) )
          & ordered_by(O,P,Q) ) ) ).

fof(o6,axiom,
    ! [O1,O2] :
      ( ! [P,Q] :
          ( ordered_by(O1,P,Q)
        <=> ordered_by(O2,P,Q) )
     => O1 = O2 ) ).

fof(underlying_curve_defn,axiom,
    ! [C,O] :
      ( C = underlying_curve(O)
    <=> ! [P] :
          ( incident_o(P,O)
        <=> incident_c(P,C) ) ) ).

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