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Datei: gs_only.pvs Sprache: PVS

Original von: PVS^©

%------------------------------------------------------------------------------
% gs_only.pvs
% ACCoRD v.1.0
%
% Line solutions. Solve the following equation on k and l:
%   k*v = l*vo-vi.
%
% Circle solutions. Solve the following equation on l:
%   sq(s+t*(l*vo-vi)) = sq(D).
%
%------------------------------------------------------------------------------

gs_only[D:posreal] : THEORY
BEGIN

  IMPORTING reals@min_max,
            vectors@vectors_2D,
     vectors@det_2D,
            reals@quadratic_2b,
            horizontal[D]

  j,k      : VAR real
  vo,vi,
  s,v,nvo,
  v1,v2    : VAR Vect2
  t        : VAR posreal
  dir,irt  : VAR Sign
  u,
  vnzo     : VAR Nz_vect2

  %---------------%
  % Line Solution %
  %---------------%

  k_l_det: LEMMA
    FORALL (l:real):
      k*v+vi = l*vo
      IMPLIES
        k*det(vo,v) = det(vi,vo) AND
        l*det(vo,v) = det(vi,v)

  k_l_det_nz: LEMMA
    FORALL (l:real):
      det(vo,v) /= 0 AND
      k = det(vi,vo)/det(vo,v) AND
      l = det(vi,v)/det(vo,v)
      IMPLIES
        k*v = l*vo-vi

  % Vector v is a directional vector
  % l=0 indicates no solution
  gs_only_line_k_l(v,vo,vi): {k:real,l:nnreal | l > 0 =>
                             k*v = l*vo-vi} =
    IF det(vo,v) /= 0 THEN
      LET k = det(vi,vo)/det(vo,v),
          l = det(vi,v)/det(vo,v) IN
        (k,max(l,0))
    ELSE
      (0,0)
    ENDIF

  gs_only_line_k_l_complete : LEMMA
    FORALL (l:posreal):
      (det(vi,vo) /= 0 OR det(vo,v) /= 0) AND
      k*v = l*vo-vi IMPLIES
      (k,l) = gs_only_line_k_l(v,vo,vi)

  % Vector v is a directional vector
  % Zero indicates no ground speed solution.
  gs_only_line(v,vo,vi): {k:real,nvo:Vect2 | nz_vect2?(nvo) =>
                                             gs_only?(vo)(nvo) AND
                                             k*v = nvo-vi} =
    LET (k,l) = gs_only_line_k_l(v,vo,vi) IN
      (k,l*vo)

  % gs_only_line computes the same resolutions for same dir vectors
  same_gs_only_line: LEMMA
    FORALL (l:posreal) :
      v2 = l*v1
      IMPLIES
        gs_only_line(v,v1,vi)`2 = gs_only_line(v,v2,vi)`2

  gs_only_line?(v,vo,vi)(k,nvo) : bool =
    nz_vect2?(nvo) AND
    (k,nvo) = gs_only_line(v,vo,vi)

  gs_only_line_complete : LEMMA
    gs_only?(vo)(nvo) AND
    (det(vi,vo) /= 0 OR det(vo,v) /= 0) AND
    k*v = nvo-vi IMPLIES
      gs_only_line?(v,vo,vi)(k,nvo)

  % Solves equation on nvo: u*(nvo-vi) = j, such that exists l > 0: nvo = l*vo
  gs_only_dot(u,vo,vi,j) : {nvo | nz_vect2?(nvo) =>
                                  gs_only?(vo)(nvo) AND
                                  u*(nvo-vi) = j} =
    gs_only_line(Vdir(u,vo-vi),vo,vi+W0(u,j))`2

  gs_only_dot_complete : LEMMA
    gs_only?(vnzo)(nvo) AND
    (det(vi+W0(u,j),vnzo) /= 0 OR det(vnzo,Vdir(u,vnzo-vi)) /= 0) AND
    u*(nvo-vi) = j IMPLIES
    nvo = gs_only_dot(u,vnzo,vi,j)

  %-----------------%
  % Circle Solution %
  %-----------------%

  %% zero indicates no solution

  gs_only_circle(s,vnzo,vi,t,dir,irt): {nvo | nz_vect2?(nvo) =>
                                              gs_only?(vnzo)(nvo) } =
    LET w = s-t*vi,
        a = sq(t)*sqv(vnzo),
        b = t*(w*vnzo),
        c = sqv(w)-sq(D) IN
    IF discr2b(a,b,c) >= 0 THEN
      LET l   = root2b(a,b,c,irt),
          nvo = max(l,0)*vnzo IN
      IF horizontal_dir_at?(s,nvo-vi,t,dir) THEN
        nvo
      ELSE
        zero
      ENDIF
    ELSE
      zero
    ENDIF

  gs_only_circle?(s,vnzo,vi,t,dir)(nvo): bool =
     nz_vect2?(nvo) AND
     EXISTS (irt:Sign): nvo = gs_only_circle(s,vnzo,vi,t,dir,irt)

  gs_only_circle_solution : LEMMA
    gs_only_circle?(s,vnzo,vi,t,dir)(nvo) IMPLIES
    circle_solution_2D?(s,nvo-vi,t,dir)

  gs_only_circle_complete : LEMMA
    gs_only?(vnzo)(nvo) AND
    circle_solution_2D?(s,nvo-vi,t,dir) IMPLIES
      gs_only_circle?(s,vnzo,vi,t,dir)(nvo)

  gs_only_vertical(s,vo,vi,t,dir): {nvo | nz_vect2?(nvo) =>
                                          gs_only?(vo)(nvo)} =
    LET v = vo-vi IN
    IF Delta(s,v) > 0 AND Theta_D(s,v,dir) > 0 THEN
      LET p = s + Theta_D(s,v,dir)*v IN
        gs_only_dot(t*p,vo,vi,sq(D)-s*p)
    ELSE
      zero
    ENDIF

  gs_only_vertical?(s,vo,vi,t,dir)(nvo) : bool =
    nz_vect2?(nvo) AND
    nvo =  gs_only_vertical(s,vo,vi,t,dir)

  gs_only_vertical_on_D: LEMMA
    LET v = vo-vi IN
    gs_only_vertical?(s,vo,vi,t,dir)(nvo) IMPLIES
      Delta(s,v) > 0 AND
      Theta_D(s,v,dir) > 0 AND
      (s+t*(nvo-vi))*(s+Theta_D(s,v,dir)*v) = sq(D)

END gs_only

¤ Dauer der Verarbeitung: 0.35 Sekunden (vorverarbeitet) ¤

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