Quelle csequence_finseq.pvs Sprache: PVS

%-----------------------------------------------------------------------------
% Conversions between sequences of countable length defined as coalgebraic
% datatypes and the prelude finseq type.
%
% 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_finseq[T: TYPE]: THEORY
BEGIN

  IMPORTING finite_sequences[T], csequence_nth[T]

  n: VAR nat
  fs: VAR finseq
  fseq: VAR finite_csequence

  % Convert a finseq to a finite_csequence
  from_finseq(fs): RECURSIVE finite_csequence =
    IF fs`length = 0 THEN empty_cseq
    ELSE add(fs`seq(0),
             from_finseq((# length := fs`length - 1,
                            seq
                              := LAMBDA (x: below[fs`length - 1]):
                                   fs`seq(x + 1) #)))
    ENDIF
     MEASURE fs`length

  from_finseq_length: THEOREM FORALL fs: length(from_finseq(fs)) = fs`length

  from_finseq_index: THEOREM
    FORALL fs, n: index?(from_finseq(fs))(n) IFF n < fs`length

  from_finseq_nth: THEOREM
    FORALL fs, (n: below[fs`length]): nth(from_finseq(fs), n) = fs`seq(n)

  % Convert a finite_csequence to a finseq
  to_finseq(fseq): finseq =
      (# length := length(fseq),
         seq := LAMBDA (x: below[length(fseq)]): nth(fseq, x) #)

  to_finseq_length: THEOREM
    FORALL fseq: to_finseq(fseq)`length = length(fseq)

  to_finseq_index: THEOREM
    FORALL fseq, n: n < to_finseq(fseq)`length IFF index?(fseq)(n)

  to_finseq_nth: THEOREM
    FORALL fseq, (n: indexes(fseq)): to_finseq(fseq)`seq(n) = nth(fseq, n)

  % Applying both conversions gives you back what you started with

  to_from_finseq: THEOREM FORALL fs: to_finseq(from_finseq(fs)) = fs

  from_to_finseq: THEOREM FORALL fseq: from_finseq(to_finseq(fseq)) = fseq

END csequence_finseq

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