Quelle csequence_rest.pvs Sprache: PVS

%-----------------------------------------------------------------------------
% Properties of the rest function as applied to a sequence of countable
% length.
%
% Author: Jerry James <loganjerry@gmail.com>
%
% 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_rest[T: TYPE]: THEORY
BEGIN

  IMPORTING csequence_nth[T]

  n: VAR nat
  t: VAR T
  p: VAR pred[T]
  iseq: VAR infinite_csequence
  nseq: VAR nonempty_csequence
  nfseq: VAR nonempty_finite_csequence

  rest_finite: JUDGEMENT rest(nfseq) HAS_TYPE finite_csequence

  rest_infinite: JUDGEMENT rest(iseq) HAS_TYPE infinite_csequence

  rest_empty_add: THEOREM
    FORALL nseq:
      empty?(rest(nseq)) IFF (EXISTS t: nseq = add(t, empty_cseq))

  rest_empty: THEOREM
    FORALL nseq: empty?(rest(nseq)) IFF is_finite(nseq) AND length(nseq) = 1

  rest_nonempty: THEOREM
    FORALL nseq:
      nonempty?(rest(nseq)) IFF is_infinite(nseq) OR length(nseq) > 1

  rest_first: THEOREM
    FORALL nseq:
      nonempty?(rest(nseq)) IMPLIES first(rest(nseq)) = nth(nseq, 1)

  rest_length: THEOREM FORALL nfseq: length(rest(nfseq)) = length(nfseq) - 1

  rest_index: THEOREM
    FORALL nseq, n: index?(rest(nseq))(n) IFF index?(nseq)(n + 1)

  rest_nth: THEOREM
    FORALL nseq, (n: indexes(rest(nseq))):
      nth(rest(nseq), n) = nth(nseq, n + 1)

  rest_some: THEOREM
    FORALL nseq, p: some(p)(nseq) IFF p(first(nseq)) OR some(p)(rest(nseq))

  rest_every: THEOREM
    FORALL nseq, p:
      every(p)(nseq) IFF p(first(nseq)) AND every(p)(rest(nseq))

END csequence_rest

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