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Datei: bands_3D.pvs Sprache: PVS

Original von: PVS^©

bands_3D[D,H:posreal,B:nnreal,T:{x:posreal | x > B},
         gsmin:posreal,gsmax:{x:posreal | x > gsmin},
         vsmin:real,vsmax:{x:real | x > vsmin}] : THEORY
BEGIN

  IMPORTING bands_2D,
       bands_1D,
     cd3d[D,H,B,T],
            bands_util

  s   : VAR Vect3
  vo,vi   : VAR Nzv2_vect3
  trkb    : VAR (trk_fseq?[gsmin,gsmax])
  gsb     : VAR (gs_fseq?[gsmin,gsmax])
  vsb   : VAR (vs_fseq?[vsmin,vsmax])
  x,k,y   : VAR real
  trk   : VAR nnreal_lt_2pi
  trk2   : VAR {r: real | 0<=r AND r<=2*pi}
  gsp   : VAR {r: real | gsmin<=r AND r<=gsmax}
  vsp   : VAR {r: real | vsmin<=r AND r<=vsmax}

  %-------------%
  % Definitions %
  %-------------%

  % Track %

%  trk2v3(vo)(trkz:real) : (trk_only_3D?(vo)) =
%    trkgs2vect(trkz,norm(vect2(vo))) WITH [z |-> vo`z]

%  vect2_trk2v3 : LEMMA
%    vect2(trk2v3(vo)(trk)) = trk2v2[D,B,T](vo)(trk)

%  trk2v3_Nzv : JUDGEMENT
%    trk2v3(vo)(trk) HAS_TYPE Nzv2_vect3

%  Vtrk_3D(vo,vi)(trkz:real):(horizontal_only?(vo-vi)) =
%    trk2v3(vo)(trkz) - vi

%  trk2v3_trk2v3 : LEMMA
%    trk2v3(trk2v3(vo)(trk))(trk)=trk2v3(vo)(trk)

%  trk2v3_0 : LEMMA
%    vect2(vo) /= zero AND x >= 0 AND x < 2 * pi IMPLIES vect2(trk2v3(vo)(x)) /= zero

  % Ground Speed %

%  gs2v3(vo)(k) : Vect3 = gs2v2[D,B,T](vo)(k) WITH [z |-> vo`z]

%  gs2v3_gs_only : JUDGEMENT
%    gs2v3(vo)(gsp) HAS_TYPE (gs_only_3D?(vo))

%  vect2_gs2v3 : LEMMA
%    vect2(gs2v3(vo)(gsp)) = gs2v2[D,B,T](vo)(gsp)

%  Vgs_3D(vo,vi)(k) : (horizontal_only?(vo-vi)) =
%    gs2v3(vo)(k)-vi

  % Vertical Speed %

%  vs2v3(vo)(vsp) : (vs_only?(vo)) = vo WITH [z := vsp]

%  vect2_vs2v3 : LEMMA
%    vect2(vs2v3(vo)(vsp)) = vect2(vo)

%  Vs(vo,vi)(vsp) : (vs_only?(vo-vi)) =
%    vs2v3(vo)(vsp)-vi

  %------------%
  % ALGORITHMS %
  %------------%

  trk_bands_3D(s,vo,vi) : {fs : (trk_fseq?[gsmin,gsmax]) | increasing?(fs)} =
    IF vo`z = vi`z AND abs(s`z) < H
       THEN trk_bands[D,B,T,gsmin,gsmax](s,vo,vi)
    ELSIF vo`z /= vi`z THEN
      LET BB = max(Theta_H(s`z,vo`z-vi`z,Entry),B),
          TT = min(Theta_H(s`z,vo`z-vi`z,Exit),T) IN
    IF BB < TT THEN trk_bands[D,BB,TT,gsmin,gsmax](s,vo,vi)
          ELSE add(0,add(2*pi,emptyseq))
          ENDIF
    ELSE
     add(0,add(2*pi,emptyseq))
    ENDIF

  gs_bands_3D(s,vo,vi) : {fs : (gs_fseq?[gsmin,gsmax]) | increasing?(fs)} =
    IF vo`z = vi`z AND abs(s`z) < H
       THEN gs_bands[D,B,T,gsmin,gsmax](s,vo,vi)
    ELSIF vo`z /= vi`z THEN
      LET BB = max(Theta_H(s`z,vo`z-vi`z,Entry),B),
          TT = min(Theta_H(s`z,vo`z-vi`z,Exit),T) IN
    IF BB < TT THEN gs_bands[D,BB,TT,gsmin,gsmax](s,vo,vi)
          ELSE add(gsmin,add(gsmax,emptyseq))
          ENDIF
    ELSE
     add(gsmin,add(gsmax,emptyseq))
    ENDIF

  vs_bands_3D(s,vo,vi) : {fs : (vs_fseq?[vsmin,vsmax]) | increasing?(fs)} =
    IF vect2(vo-vi) = zero AND sqv(vect2(s)) < sq(D)
       THEN vs_bands[H,B,T,vsmin,vsmax](s`z,vi`z)
    ELSIF Delta(s,vo-vi) > 0 THEN
      LET BB = max(Theta_D(s,vo-vi,Entry),B),
          TT = min(Theta_D(s,vo-vi,Exit),T) IN
   IF BB < TT THEN vs_bands[H,BB,TT,vsmin,vsmax](s`z,vi`z)
   ELSE add(vsmin,add(vsmax,emptyseq))
   ENDIF
   ELSE
     add(vsmin,add(vsmax,emptyseq))
   ENDIF

  trk_bands_3D_not_empty : LEMMA
    trk_bands_3D(s,vo,vi)`length > 1

  trk_bands_last: LEMMA LET trkb = trk_bands_3D(s,vo,vi) IN
    trkb`seq(trkb`length-1) = 2*pi

  trk_bands_first: LEMMA LET trkb = trk_bands_3D(s,vo,vi) IN
    trkb`seq(0) = 0

  gs_bands_3D_not_empty : LEMMA
    gs_bands_3D(s,vo,vi)`length > 1

  gs_bands_last: LEMMA LET gsb = gs_bands_3D(s,vo,vi) IN
    gsb`seq(gsb`length-1) = gsmax

  gs_bands_first: LEMMA LET gsb = gs_bands_3D(s,vo,vi) IN
    gsb`seq(0) = gsmin

  vs_bands_3D_not_empty : LEMMA
    vs_bands_3D(s,vo,vi)`length > 1

  vs_bands_last: LEMMA LET vsb = vs_bands_3D(s,vo,vi) IN
    vsb`seq(vsb`length-1) = vsmax

  vs_bands_first: LEMMA LET vsb = vs_bands_3D(s,vo,vi) IN
    vsb`seq(0) = vsmin

  %-------------%
  % Definitions %
  %-------------%

  red_trk_band_3D?(s,vo,vi,trkb)(i:subrange(0,trkb`length-2)) : bool =
    LET mp = (trkb`seq(i)+trkb`seq(i+1))/2 in
    cd3d?(s,Vtrk_3D(vo,vi)(mp))

  red_gs_band_3D?(s,vo,vi,gsb)(i:subrange(0,gsb`length-2)) : bool =
    LET mp = (gsb`seq(i)+gsb`seq(i+1))/2 in
    cd3d?(s,Vgs_3D(vo,vi)(mp))

  red_vs_band_3D?(s,vo,vi,vsb)(i:subrange(0,vsb`length-2)) : bool =
    LET mp = (vsb`seq(i)+vsb`seq(i+1))/2 in
    cd3d?(s,Vs(vo,vi)(mp))

  %-------------%
  % THEOREMS    %
  %-------------%

  % Track %

  trk_red_bands_3D : THEOREM
    LET trkb = trk_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,trkb`length-2) | trkb`seq(i) /= trkb`seq(i+1)) :
       red_trk_band_3D?(s,vo,vi,trkb)(i) IFF
       (FORALL (x | trkb`seq(i) < x AND x < trkb`seq(i+1)):
         conflict_3D?(s,Vtrk_3D(vo,vi)(x)))

  trk_green_bands_3D : THEOREM
    LET trkb = trk_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,trkb`length-2) | trkb`seq(i) /= trkb`seq(i+1)) :
       NOT red_trk_band_3D?(s,vo,vi,trkb)(i) IFF
       (FORALL (x | trkb`seq(i) < x AND x < trkb`seq(i+1)):
         NOT conflict_3D?(s,Vtrk_3D(vo,vi)(x)))

  trk_bands_equivalence: LEMMA
     LET trkb = trk_bands_3D(s,vo,vi) IN
       FORALL (x,y): (0<=x AND x<=y AND y<=2*pi AND
         (FORALL (i:subrange(0,trkb`length-1)): trkb`seq(i)<x OR y<trkb`seq(i)))
  IMPLIES
         (conflict_3D?(s,Vtrk_3D(vo,vi)(x)) IFF conflict_3D?(s,Vtrk_3D(vo,vi)(y)))

  % Ground Speed %

  gs_red_bands_3D : THEOREM
    LET gsb = gs_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,gsb`length-2) | gsb`seq(i) /= gsb`seq(i+1)) :
       red_gs_band_3D?(s,vo,vi,gsb)(i) IFF
       (FORALL (x | gsb`seq(i) < x AND x < gsb`seq(i+1)):
         conflict_3D?(s,Vgs_3D(vo,vi)(x)))

  gs_green_bands_3D : THEOREM
    LET gsb = gs_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,gsb`length-2) | gsb`seq(i) /= gsb`seq(i+1)) :
       NOT red_gs_band_3D?(s,vo,vi,gsb)(i) IFF
       (FORALL (x | gsb`seq(i) < x AND x < gsb`seq(i+1)):
         NOT conflict_3D?(s,Vgs_3D(vo,vi)(x)))

  gs_bands_equivalence: LEMMA
     LET gsb = gs_bands_3D(s,vo,vi) IN
       FORALL (x,y): (gsmin<=x AND x<=y AND y<=gsmax AND
         (FORALL (i:subrange(0,gsb`length-1)): gsb`seq(i)<x OR y<gsb`seq(i)))
  IMPLIES
         (conflict_3D?(s,Vgs_3D(vo,vi)(x)) IFF conflict_3D?(s,Vgs_3D(vo,vi)(y)))

  % Vertical Seed %

  vs_red_bands_3D : THEOREM
    LET vsb = vs_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,vsb`length-2) | vsb`seq(i) /= vsb`seq(i+1)) :
       red_vs_band_3D?(s,vo,vi,vsb)(i) IFF
       (FORALL (x | vsb`seq(i) < x AND x < vsb`seq(i+1)):
         conflict_3D?(s,Vs(vo,vi)(x)))

  vs_green_bands_3D : THEOREM
    LET vsb = vs_bands_3D(s,vo,vi) IN
     FORALL (i:subrange(0,vsb`length-2) | vsb`seq(i) /= vsb`seq(i+1)) :
       NOT red_vs_band_3D?(s,vo,vi,vsb)(i) IFF
       (FORALL (x | vsb`seq(i) < x AND x < vsb`seq(i+1)):
         NOT conflict_3D?(s,Vs(vo,vi)(x)))

  vs_bands_equivalence: LEMMA
     LET vsb = vs_bands_3D(s,vo,vi) IN
       FORALL (x,y): (vsmin<=x AND x<=y AND y<=vsmax AND
         (FORALL (i:subrange(0,vsb`length-1)): vsb`seq(i)<x OR y<vsb`seq(i)))
  IMPLIES
         (conflict_3D?(s,Vs(vo,vi)(x)) IFF conflict_3D?(s,Vs(vo,vi)(y)))

END bands_3D

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