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Datei: trk_bands_2D.pvs Sprache: PVS

Original von: PVS^©

%-----------------------------------------------------------------------------
% trk_bands_2D.pvs
% ACCoRD v.1.0
%-----------------------------------------------------------------------------

trk_bands_2D[D:posreal,B:nnreal,T: {AB: posreal|AB>B}] : THEORY
BEGIN

  IMPORTING analysis@interm_value_thm,
       vect_analysis@vect2_cont_dot[real],
            trig_fnd@sincos,
            omega_v2[D,B,T],
            trk_line[D],
     line_solutions[D],
            bands_util

  nvo,
  vo,vi : VAR Nz_vect2
  sp    : VAR Sp_vect2
  ss    : VAR Ss_vect2
  t     : VAR posreal
  band  : VAR ConnectedGeLt(0,2*pi)
  trk   : VAR real % Track with respect to true north.
  k     : VAR real

  pm,eps,
  irt,
  irt1,
  irt2   : VAR Sign
  s,
  nvo1,
  nvo2   : VAR Vect2

  trk_band : LEMMA
     nvo = trk_line_eps_irt(ss, vo, vi, eps, irt) IMPLIES
     eps*det(nvo,ss) >= eps*det(vi,ss)

  trk2v2_continuous : JUDGEMENT
    trk2v2(vo) HAS_TYPE continuous_rv_fun

  Vtrk_continuous : JUDGEMENT
    Vtrk(vo,vi) HAS_TYPE continuous_rv_fun

  %-----------------------------------------------%
  % 2-D Track Critical Points                     %
  %-----------------------------------------------%

  trk_critical?(s,vo,vi)(trk) : bool =
    LET trko = trk2v2(vo)(trk) IN
    (sqv(s)>=sq(D) AND trk_line?(s,vo,vi)(trko)) OR
    trk_only_circle?(s,vo,vi,T,Entry)(trko) OR
    (B>0 AND trk_only_circle?(s,vo,vi,B,Exit)(trko))

  Omega_trk_spc(vo,vi,t)(trk) : real =
    vi * trk2v2(vo)(trk) - sq(D/t)

  Omega_trk_spc : JUDGEMENT
    Omega_trk_spc(vo,vi,t) HAS_TYPE (continuous_rr?)

  trk_special_case_invariant: LEMMA
    trk_special_case?(s,vo,vi,t) IFF
    trk_special_case?(s,trk2v2(vo)(trk),vi,t)

  Omega_trk_spc_separated : LEMMA
    Omega_trk_spc(vo,vi,t)(trk) = 0 AND
    trk_special_case?(s,vo,vi,t) IMPLIES
    sqv(s)>=sq(D)

  Omega_trk_spc_critical : LEMMA
    Omega_trk_spc(vo,vi,t)(trk) = 0 AND
    trk_special_case?(s,vo,vi,t) IMPLIES
    line_solution?(s,Vtrk(vo,vi)(trk))

  trk_special_case_conflict_T : LEMMA
    trk_special_case?(s,vo,vi,T) IMPLIES
    (horizontal_entry_at?(s,Vtrk(vo,vi)(trk),T)
     IFF NOT conflict_2D?(s,Vtrk(vo,vi)(trk)))

  trk_special_case_conflict_B : LEMMA
    B>0 AND trk_special_case?(s,vo,vi,B) IMPLIES
    (horizontal_exit_at?(s,Vtrk(vo,vi)(trk),B)
     IFF NOT conflict_2D?(s,Vtrk(vo,vi)(trk)))

  Omega_trk_spc_conflict_T: LEMMA
    trk_special_case?(s,vo,vi,T) IMPLIES
    (Omega_trk_spc(vo,vi,T)(trk) < 0 IFF
    conflict_2D?(s,Vtrk(vo,vi)(trk)))

  Omega_trk_spc_conflict_B: LEMMA
    B>0 AND trk_special_case?(s,vo,vi,B) IMPLIES
    (-Omega_trk_spc(vo,vi,B)(trk) < 0 IFF
    conflict_2D?(s,Vtrk(vo,vi)(trk)))

  Omega_trk(s,vo,vi) : (continuous_rr?) =
    omega_v2(s) o Vtrk(vo,vi)

  Omega_trk_critical : LEMMA
    Omega_trk(s,vo,vi)(trk) = 0 AND
    NOT trk_special_case?(s,vo,vi,T) AND
    NOT (B>0 AND trk_special_case?(s,vo,vi,B)) IMPLIES
      trk_critical?(s,vo,vi)(trk)

  % Bands do not contain critical points
  trk_band?(s,vo,vi)(band) : bool =
    FORALL (trko:(band)) :
      NOT trk_critical?(s,vo,vi)(trko)

  trk_green?(s,vo,vi)(band) : bool =
    FORALL (trko:(band)) :
      NOT conflict_2D?(s,Vtrk(vo,vi)(trko))

  trk_red?(s,vo,vi)(band) : bool =
    FORALL (trko:(band)) :
      conflict_2D?(s,Vtrk(vo,vi)(trko))

  trk_line_sort_eps : LEMMA
    nvo /= vi AND
    trk_line_eps_irt?(ss,vo,vi,eps,irt)(nvo) IMPLIES
    eps*det(nvo,ss) > eps*det(vi,ss)

  trk_line_sort_eps_angle : LEMMA
    abs(det(vi,ss)/(norm(vo)*norm(ss))) <= 1 IMPLIES
    LET phi   = track(ss),
        psi   = asin(det(vi,ss)/(norm(vo)*norm(ss))),
        alpha = phi+psi,
        beta  = pi-2*psi IN
    nvo1 /= vi AND
    nvo2 /= vi AND
    trk_line_eps_irt?(ss,vo,vi,1,irt1)(nvo1) AND
    trk_line_eps_irt?(ss,vo,vi,-1,irt2)(nvo2)
    IMPLIES
      to2pi(track(nvo1)-alpha) < to2pi(track(nvo2)-alpha)

  trk_line_sort_eps_1_angle_case1 : LEMMA
    abs(det(vi,ss)/(norm(vo)*norm(ss))) <= 1 IMPLIES
    LET phi   = track(ss),
        psi   = asin(det(vi,ss)/(norm(vo)*norm(ss))),
        alpha = phi+psi,
        beta  = pi-2*psi IN
    (nvo1*ss) * (nvo2*ss) < 0 AND
    nvo1 /= vi AND
    nvo2 /= vi AND
    trk_line_eps_irt?(ss,vo,vi,1,-1)(nvo1) AND
    trk_line_eps_irt?(ss,vo,vi,1,1)(nvo2)
    IMPLIES
      to2pi(track(nvo1)-alpha) < to2pi(track(nvo2)-alpha)

  trk_line_sort_eps_1_angle_case2 : LEMMA
    abs(det(vi,ss)/(norm(vo)*norm(ss))) <= 1 IMPLIES
    LET phi   = track(ss),
        psi   = asin(det(vi,ss)/(norm(vo)*norm(ss))),
        alpha = phi+psi,
        beta  = pi-2*psi IN
    (nvo1*ss) * (nvo2*ss) >= 0 AND
    nvo1 /= vi AND
    nvo2 /= vi AND
    trk_line_eps_irt?(ss,vo,vi,1,-1)(nvo1) AND
    trk_line_eps_irt?(ss,vo,vi,1,1)(nvo2)
    IMPLIES
      (det(nvo1,nvo2) < 0 IFF to2pi(track(nvo1)-alpha) < to2pi(track(nvo2)-alpha))

  trk_line_color : LEMMA
    trk_line_eps_irt?(ss,vo,vi,eps,irt)(nvo) AND
    sq(Qs(ss,eps)*vi) > (sqv(ss)-sq(D))*(sqv(vi)-sqv(vo))
    IMPLIES
    (pm*(Q(ss,eps) * perpR(nvo)) > 0 IFF pm*(eps*irt) < 0)

  %-------------------------------------------%
  % THEOREM: Track Green Bands Correctness    %
  %-------------------------------------------%

  trk_green_band : THEOREM
    FORALL (trko:(band)) :
       trk_band?(s,vo,vi)(band) IMPLIES
        (NOT cd2d?(s,Vtrk(vo,vi)(trko)) IFF
         trk_green?(s,vo,vi)(band))

  %-------------------------------------------%
  % THEOREM: Track Red Bands Correctness      %
  %-------------------------------------------%

  trk_red_band : THEOREM
    FORALL (trko:(band)) :
      trk_band?(s,vo,vi)(band) IMPLIES
      (cd2d?(s,Vtrk(vo,vi)(trko)) IFF
        trk_red?(s,vo,vi)(band))

END trk_bands_2D

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