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Datei: csequence_flatten.pvs Sprache: PVS

Original von: PVS^©

%-----------------------------------------------------------------------------
% Flatten, an operator that takes a sequence of sequences, and concatenates
% the elements to form a single sequence of countable length.
%
% Author: Jerry James <[email protected]>
%
% This file and its accompanying proof file are distributed under the CC0 1.0
% Universal license: http://creativecommons.org/publicdomain/zero/1.0/.
%
% Version history:
%   2007 Feb 14: PVS 4.0 version
%   2011 May  6: PVS 5.0 version
%   2013 Jan 14: PVS 6.0 version
%-----------------------------------------------------------------------------
csequence_flatten[T: TYPE]: THEORY
BEGIN

  IMPORTING csequence_prefix_suffix[T], csequence_filter[csequence[T]]

  nonempty_seq_seq: TYPE = {x: csequence[csequence[T]] | every(nonempty?)(x)}

  p: VAR pred[T]
  n: VAR nat
  tseq: VAR csequence[T]
  cseq: VAR csequence[csequence[T]]
  eseq: VAR empty_csequence[csequence[T]]
  nseq: VAR nonempty_seq_seq

  % A result we need several times in the ensuing proofs
  some_every_empty: LEMMA
    FORALL cseq: every(empty?)(cseq) IFF NOT some(nonempty?)(cseq)

  flatten_struct(nseq): csequence_struct[T, nonempty_seq_seq] =
      IF empty?(nseq) THEN inj_empty_cseq[T, nonempty_seq_seq]
      ELSE inj_add[T, nonempty_seq_seq]
               (first(first(nseq)),
                IF empty?(rest(first(nseq))) THEN rest(nseq)
                ELSE add(rest(first(nseq)), rest(nseq))
                ENDIF)
      ENDIF

  flatten(cseq): csequence[T] =
      coreduce(flatten_struct)(filter(cseq, nonempty?))

  flatten_empty: THEOREM
    FORALL cseq: empty?(flatten(cseq)) IFF every(empty?)(cseq)

  flatten_empty_cseq: COROLLARY FORALL eseq: empty?(flatten(eseq))

  flatten_nonempty: THEOREM
    FORALL cseq: nonempty?(flatten(cseq)) IFF some(nonempty?)(cseq)

  flatten_filter: THEOREM
    FORALL cseq: flatten(filter(cseq, nonempty?)) = flatten(cseq)

  flatten_reduce: THEOREM
    FORALL cseq:
      empty?(cseq) OR
       nonempty?(first(cseq)) OR flatten(cseq) = flatten(rest(cseq))

  flatten_rest: THEOREM
    FORALL cseq:
      nonempty?(flatten(cseq)) IMPLIES
       rest(flatten(cseq)) =
        flatten(add(rest(first(filter(cseq, nonempty?))),
                    rest(filter(cseq, nonempty?))))

  flatten_rest2: THEOREM
    FORALL cseq:
      empty?(cseq) OR
       empty?(first(cseq)) OR
        rest(flatten(cseq)) = flatten(add(rest(first(cseq)), rest(cseq)))

  flatten_first: THEOREM
    FORALL cseq:
      nonempty?(flatten(cseq)) IMPLIES
       first(flatten(cseq)) = first(first(filter(cseq, nonempty?)))

  flatten_suffix: THEOREM
    FORALL cseq:
      some(nonempty?)(cseq) IMPLIES
       flatten(cseq) = flatten(suffix(cseq, first_p(nonempty?, cseq)))

  % I would really like to define flatten like this:
  %   flatten(cseq) = reduce(o)(cseq)
  % but this theorem is as close as I can come
  flatten_concatenate: THEOREM
    FORALL cseq:
      nonempty?(cseq) IMPLIES
       flatten(cseq) = first(cseq) o flatten(rest(cseq))

  flatten_rest_suffix: THEOREM
    FORALL cseq:
      nonempty?(flatten(cseq)) IMPLIES
       LET first = first_p(nonempty?, cseq) IN
         rest(flatten(cseq)) =
          flatten(add(rest(nth(cseq, first)), suffix(cseq, first + 1)))

  flatten_finite: THEOREM
    FORALL cseq:
      is_finite(flatten(cseq)) IFF
       is_finite(filter(cseq, nonempty?)) AND every(is_finite)(cseq)

  flatten_infinite: THEOREM
    FORALL cseq:
      is_infinite(flatten(cseq)) IFF
       is_infinite(filter(cseq, nonempty?)) OR some(is_infinite)(cseq)

  finite_flatten_csequence: TYPE = {cseq | is_finite(flatten(cseq))}
  fseq: VAR finite_flatten_csequence

  length_of_flatten(fseq): RECURSIVE nat =
    IF empty?(flatten(fseq)) THEN 0
    ELSE length(nth(fseq, first_p(nonempty?, fseq))) +
          length_of_flatten(suffix(fseq, first_p(nonempty?, fseq) + 1))
    ENDIF
     MEASURE length(filter(fseq, nonempty?))

  length_of_flatten_add: THEOREM
    FORALL fseq, tseq:
      is_finite(tseq) IMPLIES
       length_of_flatten(add(tseq, fseq)) =
        length(tseq) + length_of_flatten(fseq)

  flatten_length: THEOREM
    FORALL cseq:
      is_finite(flatten(cseq)) IMPLIES
       length(flatten(cseq)) = length_of_flatten(cseq)

  flatten_some: THEOREM
    FORALL cseq, p: some(p)(flatten(cseq)) IMPLIES some(some(p))(cseq)

  flatten_some_finite: THEOREM
    FORALL cseq, p:
      every(is_finite)(cseq) IMPLIES
       (some(p)(flatten(cseq)) IFF some(some(p))(cseq))

  flatten_every: THEOREM
    FORALL cseq, p: every(every(p))(cseq) IMPLIES every(p)(flatten(cseq))

  flatten_every_finite: THEOREM
    FORALL cseq, p:
      every(is_finite)(cseq) IMPLIES
       (every(p)(flatten(cseq)) IFF every(every(p))(cseq))

  % I believe the following two theorems are true.  Proving them, however,
  % is a different matter.  If you have a proof strategy to suggest, or a
  % nicer way of stating these theorems, please contact me.
  %
  %   flatten_some_converse: THEOREM
  %     FORALL cseq, p:
  %       (EXISTS (i: indexes(cseq)):
  %          some(p)(nth(cseq, i)) AND
  %           (FORALL (j: indexes(cseq)):
  %              j < i IMPLIES is_finite(nth(cseq, j))))
  %        IMPLIES some(p)(flatten(cseq))
  %
  %   flatten_every_converse: THEOREM
  %     FORALL cseq, p:
  %       every(p)(flatten(cseq)) IMPLIES
  %        (FORALL (i: indexes(cseq)):
  %           (FORALL (j: indexes(cseq)):
  %              j < i IMPLIES is_finite(nth(cseq, j)))
  %            IMPLIES every(p)(nth(cseq, i)))

END csequence_flatten

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