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Datei: csequence_limit.pvs Sprache: PVS

Original von: PVS^©

%-----------------------------------------------------------------------------
% The limit of a function that produces prefixes of a 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_limit[T: TYPE]: THEORY
BEGIN

  IMPORTING csequence_prefix_suffix[T]
  IMPORTING ascending_chains[csequence, prefix?]
  IMPORTING csequence_codt_coreduce[T, ascending_chain]

  t: VAR T
  n, m, k: VAR nat
  chain: VAR ascending_chain
  cseq: VAR csequence
  fseq: VAR finite_csequence
  p: VAR pred[csequence]

  % FIXME: prefix_chain and suffix_chain sound like converses, but they are not

  prefix_chain(cseq): ascending_chain = LAMBDA n: prefix(cseq, n)

  suffix_chain(chain, n): ascending_chain = LAMBDA m: suffix(chain(m), n)

  rest_chain(chain): ascending_chain =
      LAMBDA n:
        IF empty?(chain(n)) THEN empty_cseq ELSE rest(chain(n)) ENDIF

  ascending_chain?_nth: THEOREM
    FORALL chain, n, m, k:
      n <= m AND index?(chain(n))(k) IMPLIES
       index?(chain(m))(k) AND nth(chain(n), k) = nth(chain(m), k)

  limit_struct(chain): csequence_struct[T, ascending_chain] =
      IF (FORALL n: empty?(chain(n))) THEN inj_empty_cseq
      ELSE inj_add(first(chain(min({n | nonempty?(chain(n))}))),
                   rest_chain(chain))
      ENDIF

  limit(chain): csequence = coreduce(limit_struct)(chain)

  limit_empty: THEOREM
    FORALL chain: empty?(limit(chain)) IFF (FORALL n: empty?(chain(n)))

  limit_nonempty: THEOREM
    FORALL chain:
      nonempty?(limit(chain)) IFF (EXISTS n: nonempty?(chain(n)))

  limit_first: THEOREM
    FORALL chain, n:
      nonempty?(chain(n)) IMPLIES first(chain(n)) = first(limit(chain))

  limit_rest_chain: THEOREM
    FORALL chain:
      nonempty?(limit(chain)) IMPLIES
       rest(limit(chain)) = limit(rest_chain(chain))

  limit_suffix_chain: THEOREM
    FORALL chain, n: suffix(limit(chain), n) = limit(suffix_chain(chain, n))

  limit_lub: THEOREM FORALL chain: least_upperbound?(chain)(limit(chain))

  limit_nth: THEOREM
    FORALL chain, cseq, n, m:
      index?(chain(n))(m) IMPLIES
       index?(limit(chain))(m) AND nth(chain(n), m) = nth(limit(chain), m)

  limit_def: THEOREM
    FORALL chain, cseq:
      limit(chain) = cseq IFF
       (FORALL t, n:
          index?(cseq)(n) AND nth(cseq, n) = t IFF
           (EXISTS m: index?(chain(m))(n) AND nth(chain(m), n) = t))

  limit_prefix_chain: THEOREM FORALL cseq: limit(prefix_chain(cseq)) = cseq

  limit_prefix_compact: THEOREM
    FORALL cseq, n, chain:
      (n = 0 OR index?(cseq)(n - 1)) IMPLIES
       (prefix?(prefix(cseq, n), limit(chain)) IMPLIES
         (EXISTS m: prefix?(prefix(cseq, n), chain(m))))

  limit_finite_compact: THEOREM
    FORALL cseq:
      is_finite(cseq) IFF
       (FORALL chain:
          prefix?(cseq, limit(chain)) IMPLIES
           (EXISTS m: prefix?(cseq, chain(m))))

  continuous?(p): bool =
      FORALL chain: (FORALL n: p(chain(n))) IMPLIES p(limit(chain))

  continuous?_infinite: THEOREM
    FORALL p:
      continuous?(p) AND (FORALL fseq: p(fseq)) IMPLIES
       (FORALL cseq: p(cseq))

END csequence_limit

¤ Dauer der Verarbeitung: 0.16 Sekunden (vorverarbeitet) ¤

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