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Datei: vect2_Heine.pvs Sprache: PVS

Original von: PVS^©

%------------------------------------------------------------------------------------------------
%
%       A Special Result On Continuous Functions R x R^2 -> R
%
%       Author: Anthony Narkawicz, NASA Langley
%
%
%       Version 1.0                     February 24, 2010
%
%------------------------------------------------------------------------------------------------

vect2_Heine: THEORY
BEGIN

   IMPORTING vect2_metric_space,
             analysis@real_metric_space,
             analysis@cross_metric_real_fun[real,real_dist,Vect2,dist],
             analysis@continuity_ms[Vect2,dist,real,real_dist]

   f: VAR [[real,Vect2] -> real]

   curried_min_exists_2D: LEMMA FORALL (A: real, B: {x: real | A < x}):
    continuous?(f,fullset[[(closed_intv(A,B)),Vect2]])
     IMPLIES (FORALL (y: Vect2):
        nonempty?[real]({r: real | EXISTS (t: (LAMBDA (x: real): A <= x AND x <= B)):r = f(t, y)})
    AND below_bounded[real]({r: real | EXISTS (t: (LAMBDA (x: real): A <= x AND x <= B)):r = f(t, y)})
    AND ext[real]({r: real | EXISTS (t: (LAMBDA (x: real): A <= x AND x <= B)): r = f(t, y)})
              (inf[real]({r: real | EXISTS (t: (LAMBDA (x: real): A <= x AND x <= B)): r = f(t, y)}))
    AND (EXISTS (tmin: (closed_intv(A,B))): FORALL (s: (closed_intv(A,B))): f(tmin,y) <= f(s,y)))

   curried_min_is_cont_2D: THEOREM FORALL (A: real, B: {x: real | A < x}):
      continuous?(f,fullset[[(closed_intv(A,B)),Vect2]])
      IMPLIES
          LET unif_min_first(y: Vect2): real
              = min({r: real | EXISTS (t: (closed_intv(A,B))): r = f(t,y)})
          IN continuous?(unif_min_first)

   curried_min_is_cont_2D_ed: THEOREM FORALL (A: real, B: {x: real | A < x}):
       (FORALL (x: (closed_intv(A,B)), y: Vect2, epsilon: posreal):
          EXISTS (delta: posreal):
             FORALL (z: (closed_intv(A,B)), w: Vect2):
               (real_dist(x,z) <= delta AND dist(y,w) <= delta) IMPLIES real_dist(f(x,y),f(z,w))<epsilon)
       IMPLIES
           LET unif_min_first(y: Vect2): real
               = min({r: real | EXISTS (t: (closed_intv(A,B))): r = f(t,y)})
           IN
              (FORALL (y: Vect2): FORALL (epsilon: posreal):
                  EXISTS (delta: posreal): FORALL (q:Vect2):
                       dist(y,q) < delta IMPLIES real_dist(unif_min_first(y),unif_min_first(q)) < epsilon)

   % Uniform Continuity In The First Variable

   multiary_Heine_2D:   THEOREM FORALL (A: real, B: {x: real | A < x}):
      continuous?(f,fullset[[(closed_intv(A,B)),Vect2]])
      IMPLIES uniformly_continuous_in_first?(f,closed_intv(A,B),fullset[Vect2])

   multiary_Heine_2D_ed:   THEOREM FORALL (A: real, B: {x: real | A < x}):
     (FORALL (x: (closed_intv(A,B)), y: Vect2, epsilon: posreal):
        EXISTS (delta: posreal):
           FORALL (z: (closed_intv(A,B)), w: Vect2): (real_dist(x,z) <= delta AND dist(y,w) <= delta)
                                                          IMPLIES real_dist(f(x,y),f(z,w))<epsilon)
     IMPLIES
        FORALL (y1: Vect2, epsilon: posreal): EXISTS (delta: posreal):
           FORALL (x1,x2: (closed_intv(A,B)), y2: Vect2): (real_dist(x1,x2) <= delta AND dist(y1,y2) <= delta)
              IMPLIES real_dist(f(x1,y1),f(x2,y2))<epsilon

END vect2_Heine

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