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Datei: majority.pvs Sprache: PVS

Original von: PVS^©

%
%
% Purpose : validity and agreement properties for a multi-stage majority
%           exchange using the results-based fault model.
%
%

majority
[
    N   : [nat -> posnat],
    T   : TYPE+
] : THEORY

  BEGIN

  IMPORTING
    majority_stage,
    exact_comm[N, T],
    protocol[N, T]

  i, j, k: VAR nat
  v: VAR T

  common_check: VAR [j : nat -> non_empty_finite_set[below(N(j))]]

  check: VAR check_function
  sent:  VAR sent_vec
  rcvd:  VAR rcvd_matrix

  mf: VAR [nat -> majority_function[T]]

  majority_vote(mf): choice_function =
    LAMBDA i: majority_vote[N(i), N(i + 1), T](mf(i))

  % Main results
  consensus_validity: THEOREM
      protocol(majority_vote(mf), sent, rcvd, check, j, j + k) AND
      majority_correct?(sent, rcvd, check, j, j + k) AND
      enabled_within?(rcvd, check, j, j + k)(common_check) AND
      uniform?(sent(j), v)(common_check(j))
    IMPLIES
      uniform?(sent(j + k), v)(common_check(j + k))

  agreement_generation: THEOREM
      protocol(majority_vote(mf), sent, rcvd, check, j, j + k) AND
      exists_all_symmetric?(sent, rcvd, check, j, j + k) AND
      enabled_within?(rcvd, check, j, j + k)(common_check)
    IMPLIES
      EXISTS v: uniform?(sent(j + k), v)(common_check(j + k))

  % diagnostic results

  faulty_source: THEOREM % cross-channel compare at any stage
    FORALL (s: below(N(j)), d: below(N(j + 1))):
      protocol(majority_vote(mf), sent, rcvd, check, j, j + 1) AND
      enabled_within?(rcvd, check, j, j + 1)(common_check) AND
      uniform?(sent(j), v)(common_check(j)) AND
      enabled(rcvd(j), check(j))(d)(s) AND
      rcvd(j)(d)(s) /= sent(j + 1)(d)
    IMPLIES
      NOT majority_correct_element?(sent(j), rcvd(j), check(j))(s)

  faulty_stage: COROLLARY  % to consensus_validity
      protocol(majority_vote(mf), sent, rcvd, check, j, j + k) AND
      majority_correct?(sent, rcvd, check, j, j + k) AND
      enabled_within?(rcvd, check, j, j+ k)(common_check) AND
      (EXISTS (d: (common_check(j + k))): sent(j + k)(d) /= v)
    IMPLIES
      NOT uniform?(sent(j), v)(common_check(j))

  asymmetric_source: THEOREM %
      protocol(majority_vote(mf), sent, rcvd, check, j, j + 2) AND
      majority_correct?(sent, rcvd, check, j + 1, j + 2) AND
      enabled_within?(rcvd, check, j, j + 2)(common_check) AND
      (EXISTS (d: below(N(j + 2))):
        NOT majority_exists?(rcvd(j + 1)(d),
                             enabled(rcvd(j + 1), check(j + 1))(d)))
    IMPLIES
      EXISTS (s: below(N(j))): NOT symmetric_source?(rcvd(j), check(j))(s)

END majority

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