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Datei: composition_borel.pvs Sprache: PVS

Original von: PVS^©

real_finite_sequences: THEORY

  BEGIN

  IMPORTING ordered_finite_sequences[real, <=], reals@sigma_below

  s, s1, s2: VAR ne_seqs
  cf, cf1, cf2: VAR consensus_function

  eps, delta: VAR nnreal
  k: VAR posnat
  i: VAR nat

  diameter?(s): nnreal = max(s) - min(s)

% diameter?(s1 o s2)

  agreement_propagation: LEMMA
    cf1(s1) - cf2(s2) <= max(s1) - min(s2)

  similar?(s1, s2, eps): bool =
    (s1`length = s2`length) AND
    FORALL (i: below(s1`length)):
      s1`seq(i) - s2`seq(i) <= eps

  similar_sort: LEMMA
    similar?(s1, s2, eps) IMPLIES similar?(sort(s1), sort(s2), eps)

%  similar_reduce: LEMMA
%    similar?(s1, s2, eps) IMPLIES similar?(reduce(tau)(s1), reduce(tau)(s2), eps)

  inexact_consensus?(cf): bool =
    FORALL s1, s2, eps: similar?(s1, s2, eps) IMPLIES cf(s1) - cf(s2) <= eps

  sum(s): real = sigma[below(s`length)](0, s`length - 1, s`seq)

  sum_lower: LEMMA max(s) + (s`length - 1) * min(s) <= sum(s)
  sum_upper: LEMMA sum(s) <= (s`length - 1) * max(s) + min(s)

  mean(s): real = sum(s) / s`length

  min_le_mean: LEMMA min(s) <= mean(s)
  mean_le_max: LEMMA mean(s) <= max(s)

  midpoint(s) : real = mean(minmax(s))

  midpoint_def: LEMMA midpoint(s) = (min(s) + max(s)) / 2

  mean_lower: LEMMA s`length <= k IMPLIES (max(s) + (k - 1) * min(s)) / k <= mean(s)
  mean_upper: LEMMA s`length <= k IMPLIES mean(s) <= ((k - 1) * max(s) + min(s)) / k

  mean_consensus:     JUDGEMENT mean     HAS_TYPE consensus_function
  midpoint_consensus: JUDGEMENT midpoint HAS_TYPE consensus_function

  inexact_min:      LEMMA inexact_consensus?(min)
  inexact_max:      LEMMA inexact_consensus?(max)
  inexact_choose:   LEMMA inexact_consensus?(choose(i))
  inexact_mean:     LEMMA inexact_consensus?(mean)
  inexact_midpoint: LEMMA inexact_consensus?(midpoint)

  mean_convergence: THEOREM
      s1`length <= k AND
      s2`length <= k AND
      max(s1) - min(s2) <= delta + eps AND
      min(s1) - max(s2) <= eps
    IMPLIES
      mean(s1) - mean(s2) <= delta * (k - 1) / k + eps

  midpoint_convergence: COROLLARY
      max(s1) - min(s2) <= delta + eps AND
      min(s1) - max(s2) <= eps
    IMPLIES
      midpoint(s1) - midpoint(s2) <= delta / 2 + eps

  convergent?(cf, i, k): bool =
    FORALL s1, s2, delta, eps:
      s1`length <= i AND
      s2`length <= i AND
      max(s1) - min(s2) <= delta + eps AND
      min(s1) - max(s2) <= eps
    IMPLIES
      cf(s1) - cf(s2) <= delta * (k - 1) / k + eps

  convergent_mean    : LEMMA convergent?(mean, k, k + i)
  convergent_midpoint: LEMMA convergent?(midpoint, i, 2)


  END real_finite_sequences

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