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Datei: wedge_optimum_2D.pvs   Sprache: Unknown

wedge_optimum_2D: THEORY
%------------------------------------------------------------------------------
%  This file defines an algorithm that finds the point on a line
%  with the largest norm such that it lies between two half planes 
%  that are inputs
%
%             /-half plane 1
%           /
%         / |  
%       /   | 
%     0     |   
%       \   | 
%   \ X-point    
%     \   
%       \
%         \
%
%------------------------------------------------------------------------------

BEGIN

   IMPORTING vectors@vectors_2D,
         horizontal,
      vectors@law_cos_pos_2D,
      vectors@basis_2D,
      analysis@interm_value_thm,
      analysis@continuous_lambda,
      reals@quad_minmax

   v,u,w,c,p,s : VAR Vect2
   e1,e2  : VAR Nz_vect2
   R,k    : VAR posreal
   t    : VAR nnreal
   a,b  : VAR real

  intersects_segment?(p,w,v,a,b,e1,e2): bool =
    EXISTS (t:real): a<=t AND t<=b AND
              (w+t*v-p)*e1>=0 AND
       (w+t*v-p)*e2>=0

  intersects_segment_fun?(p,w,v,a,b,e1,e2): bool =
    LET
      tlow1  = IF v*e1>0 THEN ((p-w)*e1)/(v*e1) ELSIF v*e1 = 0 AND (w-p)*e1<0 THEN b+1 ELSE a ENDIF,
      thigh1 = IF v*e1<0 THEN ((p-w)*e1)/(v*e1) ELSIF v*e1 = 0 AND (w-p)*e1<0 THEN a-1 ELSE b ENDIF,
      tlow2  = IF v*e2>0 THEN ((p-w)*e2)/(v*e2) ELSIF v*e2 = 0 AND (w-p)*e2<0 THEN b+2 ELSE a ENDIF,
      thigh2 = IF v*e2<0 THEN ((p-w)*e2)/(v*e2) ELSIF v*e2 = 0 AND (w-p)*e2<0 THEN a-2 ELSE b ENDIF,
      tlow   = max(max(tlow1,tlow2),a),
      thigh  = min(min(thigh1,thigh2),b)
    IN (tlow <= thigh)

  intersects_segment_fun_def: LEMMA
    intersects_segment_fun? = intersects_segment?

  on_segment?(w,v,a,b)(s): bool = 
    EXISTS (t:real): a<=t AND t<=b AND
              s = w+t*v

  on_segment_in_wedge?(p,w,v,a,b,e1,e2)(s): bool = 
    EXISTS (t:real): a<=t AND t<=b AND
              s = w+t*v AND
       (s-p)*e1>=0 AND
       (s-p)*e2>=0

  intersects_segment_sym: LEMMA
    intersects_segment?(p,w,v,a,b,e2,e1) IFF
    intersects_segment?(p,w,v,a,b,e1,e2)

  on_segment_in_wedge_sym: LEMMA
    on_segment_in_wedge?(p,w,v,a,b,e2,e1) =
    on_segment_in_wedge?(p,w,v,a,b,e1,e2)


  %%% The line segment is w+tv, where t is in [a,b]

  max_segment_point_in_slice(p,w,v,a,b,e1,e2): Vect2 =
    LET
      tlow1  = IF v*e1>0 THEN ((p-w)*e1)/(v*e1) ELSIF v*e1 = 0 AND (w-p)*e1<0 THEN b+1 ELSE a ENDIF,
      thigh1 = IF v*e1<0 THEN ((p-w)*e1)/(v*e1) ELSIF v*e1 = 0 AND (w-p)*e1<0 THEN a-1 ELSE b ENDIF,
      tlow2  = IF v*e2>0 THEN ((p-w)*e2)/(v*e2) ELSIF v*e2 = 0 AND (w-p)*e2<0 THEN b+2 ELSE a ENDIF,
      thigh2 = IF v*e2<0 THEN ((p-w)*e2)/(v*e2) ELSIF v*e2 = 0 AND (w-p)*e2<0 THEN a-2 ELSE b ENDIF,
      tlow   = max(max(tlow1,tlow2),a),
      thigh  = min(min(thigh1,thigh2),b),
      aa     = sqv(v),
      bb     = 2*(v*(w-p)),
      cc     = sqv(w-p),
      qmint  = IF v = zero THEN 0 ELSE -bb/(2*aa) ENDIF,
      maxt   = IF aa*tlow^2 + bb*tlow <= aa*thigh^2+bb*thigh THEN thigh
              ELSE tlow ENDIF
    IN w + maxt*v

  % max_segment_point_in_slice_symm: LEMMA
  %   max_segment_point_in_slice(p,w,v,a,b,e1,e2) =
  %   max_segment_point_in_slice(p,w,v,a,b,e2,e1)

  max_segment_point_in_slice_def: LEMMA
    Let s = max_segment_point_in_slice(p,w,v,a,b,e1,e2) IN
    (intersects_segment?(p,w,v,a,b,e1,e2)
    IMPLIES on_segment_in_wedge?(p,w,v,a,b,e1,e2)(s)) AND
    (FORALL (u): on_segment_in_wedge?(p,w,v,a,b,e1,e2)(u)
           IMPLIES norm(u-p) <= norm(s-p))

  max_segment_point_in_slice_complete: LEMMA 
    LET s = max_segment_point_in_slice(p,w,v,a,b,e1,e2) IN
    FORALL (u): on_segment_in_wedge?(p,w,v,a,b,e1,e2)(u)
         IMPLIES
  on_segment_in_wedge?(p,w,v,a,b,e1,e2)(s)



END wedge_optimum_2D

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