Quellcode-Bibliothek csequence_map_props.pvs Sprache: PVS

%-----------------------------------------------------------------------------
% Properties of mapping functions on sequences 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_map_props[T1, T2: TYPE]: THEORY
BEGIN

  IMPORTING csequence_nth[T1], csequence_nth[T2]
  IMPORTING csequence_codt_map[T1, T2]

  t: VAR T1
  p: VAR pred[T2]
  f: VAR [T1 -> T2]
  cseq, cseq1, cseq2: VAR csequence[T1]
  fseq: VAR finite_csequence[T1]
  iseq: VAR infinite_csequence[T1]
  nseq: VAR nonempty_csequence[T1]
  nfseq: VAR nonempty_finite_csequence[T1]

  map_empty: THEOREM FORALL f, cseq: empty?(map(f, cseq)) IFF empty?(cseq)

  map_nonempty: THEOREM
    FORALL f, cseq: nonempty?(map(f, cseq)) IFF nonempty?(cseq)

  map_finite: JUDGEMENT map(f, fseq) HAS_TYPE finite_csequence[T2]

  map_infinite: JUDGEMENT map(f, iseq) HAS_TYPE infinite_csequence[T2]

  map_first: THEOREM FORALL f, nseq: first(map(f, nseq)) = f(first(nseq))

  map_rest: THEOREM FORALL f, nseq: rest(map(f, nseq)) = map(f, rest(nseq))

  map_length: THEOREM FORALL f, fseq: length(map(f, fseq)) = length(fseq)

  map_index: THEOREM FORALL f, cseq: index?(map(f, cseq)) = index?(cseq)

  map_nth: THEOREM
    FORALL f, cseq, (n: indexes(cseq)):
      nth(map(f, cseq), n) = f(nth(cseq, n))

  map_add: THEOREM
    FORALL f, cseq, t: add(f(t), map(f, cseq)) = map(f, add(t, cseq))

  map_last: THEOREM FORALL f, nfseq: last(map(f, nfseq)) = f(last(nfseq))

  % The generated map theory includes both curried and uncurried definitions
  % of map, but doesn't prove that they are equal.
  map_map: THEOREM FORALL f, cseq: map(f)(cseq) = map(f, cseq)

  map_injective: THEOREM FORALL f: injective?(f) IMPLIES injective?(map(f))

  map_extensionality: THEOREM
    FORALL f, cseq1, cseq2:
      injective?(f) AND map(f, cseq1) = map(f, cseq2) IMPLIES cseq1 = cseq2

  map_some: THEOREM
    FORALL f, cseq, p: some(p)(map(f, cseq)) IFF some(p o f)(cseq)

  map_every: THEOREM
    FORALL f, cseq, p: every(p)(map(f, cseq)) IFF every(p o f)(cseq)

END csequence_map_props

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