%--------------------------------------------------------------------------
% File     : GEO004+3 : TPTP v8.2.0. Released v2.4.0.
% Domain   : Geometry (Oriented curves)
% Axioms   : Trajectories
% 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    :    9 (   1 unt;   0 def)
%            Number of atoms       :   20 (   1 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :   12 (   1   ~;   0   |;   3   &)
%                                         (   4 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    5 (   4 usr;   0 prp; 1-3 aty)
%            Number of functors    :    3 (   3 usr;   0 con; 1-2 aty)
%            Number of variables   :   24 (  22   !;   2   ?)
% SPC      : 

% Comments : Requires GEO004+0.ax GEO004+1.ax GEO004+2.ax
%--------------------------------------------------------------------------
fof(connect_defn,axiom,
    ! [X,Y,P] :
      ( connect(X,Y,P)
    <=> once(at_the_same_time(at(X,P),at(Y,P))) ) ).

fof(symmetry_of_at_the_same_time,axiom,
    ! [A,B] :
      ( once(at_the_same_time(A,B))
    <=> once(at_the_same_time(B,A)) ) ).

fof(assciativity_of_at_the_same_time,axiom,
    ! [A,B,C] :
      ( once(at_the_same_time(at_the_same_time(A,B),C))
    <=> once(at_the_same_time(A,at_the_same_time(B,C))) ) ).

fof(idempotence_of_at_the_same_time,axiom,
    ! [A] :
      ( once(A)
     => once(at_the_same_time(A,A)) ) ).

fof(conjunction_at_the_same_time,axiom,
    ! [A,B] :
      ( once(at_the_same_time(A,B))
     => ( once(A)
        & once(B) ) ) ).

fof(at_on_trajectory,axiom,
    ! [X,P] :
      ( once(at(X,P))
    <=> incident_o(P,trajectory_of(X)) ) ).

fof(trajectories_are_oriented_curves,axiom,
    ! [X] :
    ? [O] : trajectory_of(X) = O ).

fof(homogeneous_behaviour,axiom,
    ! [P1,P2,Q1,Q2,X,Y] :
      ( ( once(at_the_same_time(at(X,P1),at(Y,P2)))
        & once(at_the_same_time(at(X,Q1),at(Y,Q2))) )
     => ~ ( ordered_by(trajectory_of(X),P1,Q1)
          & ordered_by(trajectory_of(Y),Q2,P2) ) ) ).

fof(localization,axiom,
    ! [A] :
      ( once(A)
     => ! [X] :
        ? [P] : once(at_the_same_time(A,at(X,P))) ) ).

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