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Datei: horizontal_criterion_line.pvs Sprache: PVS

Original von: PVS^©

%------------------------------------------------------------------------------
% horizontal_criterion_line.pvs
% ACCoRD v1.0
%
% This theory simplifies the predicate horizontal_criterion and proves
% a stronger coordination theorem.
%------------------------------------------------------------------------------

horizontal_criterion_line[D: posreal]: THEORY
BEGIN

  IMPORTING horizontal_criteria[D],
            tangent_line[D],
            vectors@linear_transformations_2D

  v,nv,nv1,nv2         : VAR Vect2
  sp        : VAR Sp_vect2
  vo,vi,
  nvo,nvi   : VAR Vect2
  eps       : VAR Sign

  horizontal_criterion_rewrite: THEOREM
    horizontal_criterion?(sp,eps)(v) IFF
    (eps*det(v,tangent_line(sp,eps)) <= 0 AND
     eps*det(sp,v)<=0)

  % horizontal_criterion_rewrite2: THEOREM
  %   norm(sp)>D IMPLIES
  %   (horizontal_criterion?(sp,eps)(v) IFF
  %   eps*det(sp,v) >= norm(v)*D)

  horizontal_criterion_special_coordination: LEMMA
    horizontal_criterion?(-sp,eps)(nvi-vo) AND
    horizontal_criterion?(sp,eps)(nvo-vo)
    IMPLIES
      NOT horizontal_conflict?(sp,nvo-nvi)

  horizontal_criterion_dot_equality: LEMMA
    line_solution?(sp,nvo-vi,eps) AND
    line_solution?(sp,v,eps) IMPLIES
      det(vi-nvi,v) = det(nvo-nvi,v)

  horizontal_criterion_line_invariance: LEMMA
    line_solution?(sp,nvo-vi,eps) IMPLIES
     (horizontal_criterion?(sp,eps)(vi-nvi) IMPLIES
      horizontal_criterion?(sp,eps)(nvo-nvi))

  horizontal_criterion_line_coordination: LEMMA
    line_solution?(-sp,nvi-vo,eps) AND
    horizontal_criterion?(sp,eps)(nvo-vo)
    IMPLIES
      NOT horizontal_conflict?(sp,nvo-nvi)

  % The horizontal criterion is maximal!

  SCRIT: VAR SignedCrit2D

  % horizontal_criterion_maximal: LEMMA
  %   LET
  %       conflictpred = (LAMBDA (ss,v:Vect2): (LAMBDA (nv:Vect2): horizontal_conflict?(ss,nv)))
  %   IN
  %       (FORALL (sp:Sp_vect2,v:Vect2,eps:Sign):
  %       subset?(horizontal_criterion?(sp,eps),SCRIT(eps)(sp,v)) AND
  %       crit_symmetric_2D?(SCRIT(eps)) AND
  %       crit_independent_of_length_2D?(SCRIT(eps)) AND
  % coordinated_2D?(SCRIT(eps),SCRIT(eps),conflictpred))
  % IMPLIES
  % (FORALL (sp:Sp_vect2,v:Vect2,eps:Sign):
  %       horizontal_conflict?(sp,v) IMPLIES horizontal_criterion?(sp,eps)=SCRIT(eps)(sp,v))

  % The horizontal criterion is not coordinated if the
  % aircraft are not orginally in conflict. The following criteria,
  % which depends on s (original relative position),
  % v (original relative velocity), nv (new relative velocity), and eps (+-1),
  % is coordinated even if the aircraft are not originally in conflict.
  % In fact, if the original aircraft are in conflict, then this criteria is
  % equal to the horizontal_criterion. An even stronger result is true which
  % states that as long as the original trajectory does not
  % satisfy horizontal_criterion, this new criterion is equal to
  % the horizontal criterion.

  horizontal_criterion_ext?(sp,v,eps)(nv): bool =
    IF horizontal_criterion?(sp,eps)(v)
      THEN horizontal_criterion?(sp,eps)(nv-(1/2)*v)
    ELSIF
         horizontal_criterion?(sp,eps)(-v) OR
         (NOT horizontal_criterion?(sp,-eps)(v))
       THEN horizontal_criterion?(sp,eps)(nv)
    ELSE
      horizontal_criterion?(sp,eps)(nv) AND
      horizontal_criterion?(sp,eps)(nv-(1/2)*v)
    ENDIF

  horizontal_criterion_ext_subset: LEMMA
    horizontal_criterion_ext?(sp,v,eps)(nv) IMPLIES
    horizontal_criterion?(sp,eps)(nv)

  horizontal_criterion_ext_restrict: LEMMA
    horizontal_conflict?(sp,v) IMPLIES
    (horizontal_criterion_ext?(sp,v,eps)(nv) IFF
     horizontal_criterion?(sp,eps)(nv))

  horizontal_criterion_ext_restrict_conflict: LEMMA
    horizontal_conflict?(sp,v) IMPLIES
    (horizontal_criterion_ext?(sp,v,eps)(nv) IFF
     horizontal_criterion?(sp,eps)(nv))

  horizontal_criterion_ext_independence: LEMMA
    horizontal_criterion_ext?(sp,v,eps)(nv) IMPLIES
    NOT horizontal_conflict?(sp,nv)

  % horizontal_criterion_ext_coordinated_criterion: LEMMA
  %   horizontal_criterion_ext?(sp,vo-vi,eps)(nvo-vi) AND
  %   horizontal_criterion_ext?(-sp,vi-vo,eps)(nvi-vo)
  %   IMPLIES
  %   horizontal_criterion?(sp,eps)(nvo-nvi)

  horizontal_criterion_ext_coordinated: LEMMA
    horizontal_criterion_ext?(sp,vo-vi,eps)(nvo-vi) AND
    horizontal_criterion_ext?(-sp,vi-vo,eps)(nvi-vo)
    IMPLIES
    NOT horizontal_conflict?(sp,nvo-nvi)

  %%%%%%%%%%%%%%%%%%%%%%%%%%%

  horizontal_criterion_ext_sum_closed: LEMMA
    horizontal_criterion_ext?(sp,v,eps)(nv1) AND
    horizontal_criterion_ext?(sp,v,eps)(nv2)
    IMPLIES
    horizontal_criterion_ext?(sp,v,eps)(nv1+nv2)

  horizontal_criterion_ext_scal: LEMMA
    FORALL (c:real): c>=1 AND
    horizontal_criterion_ext?(sp,v,eps)(nv)
    IMPLIES
    horizontal_criterion_ext?(sp,v,eps)(c*nv)

  horizontal_criterion_ext_seg_convex: LEMMA
    FORALL (eps: Sign, sp: Sp_vect2[D], v, nv1, nv2: Vect2):
      horizontal_criterion_ext?(sp, v, eps)(nv1) AND
      horizontal_criterion_ext?(sp, v, eps)(nv2)
      IMPLIES
      (FORALL (t: real):
        0 <= t AND t <= 1 IMPLIES
horizontal_criterion_ext?(sp,v,eps)((1-t)*nv1 + t *nv2))

  % In next lemma, sp1 = rel position at time t1,
              % sp2 = rel position at time t2

  horizontal_criterion_ext_seg_independence: LEMMA
    FORALL (sp1,sp2:Vect2,t1,t2:real): t1<t2 AND
    horizontal_criterion_ext?(sp,v,eps)(sp1-sp) AND
    horizontal_criterion_ext?(sp,v,eps)(sp2-sp) IMPLIES
    FORALL (t:real): t1 <= t AND t <= t2
      IMPLIES NOT horizontal_los?(sp1 + ((t-t1)/(t2-t1))*(sp2-sp1))

  % IMPORANT NOTE: nso - so = (nso-si) - (so-si)
  %          nsi - si = (nsi-so) - (si-so)

  horizontal_criterion_ext_seg_coordination: LEMMA
    FORALL (so,si,vo,vi,nso1,nso2,nsi1,nsi2:Vect2,t1,t2:real):
    t1<t2 AND sqv(so-si) >= sq(D) AND
    horizontal_criterion_ext?(so-si,vo-vi,eps)(nso1-so) AND
    horizontal_criterion_ext?(so-si,vo-vi,eps)(nso2-so) AND
    horizontal_criterion_ext?(si-so,vi-vo,eps)(nsi1-si) AND
    horizontal_criterion_ext?(si-so,vi-vo,eps)(nsi2-si) IMPLIES
    FORALL (t:real):
      LET time01 = (t-t1)/(t2-t1),
         ownshippos = nso1 + time01*(nso2-nso1),
   intruderpos = nsi1 + time01*(nsi2-nsi1)
      IN
          t1 <= t AND t <= t2 IMPLIES NOT horizontal_los?(ownshippos-intruderpos)


END horizontal_criterion_line

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