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Datei: countability.pvs Sprache: PVS

Original von: PVS^©

%-------------------------------------------------------------------------
%
%  Countable sets, and a recharacterization of infinite sets based on
%  the definition of countability.
%
%  For PVS version 3.2.  March 1, 2005
%  ---------------------------------------------------------------------
%      Author: Jerry James ([email protected]), University of Kansas
%
%  EXPORTS
%  -------
%  prelude: infinite_sets_def[T]
%  sets_aux: card_comp_set[nat,T], card_comp_set[T,nat],
%    card_comp_set_props[T,nat], countability[T]
%
%-------------------------------------------------------------------------
countability[T: TYPE]: THEORY
BEGIN

  IMPORTING infinite_sets_def[T]
  IMPORTING orders@integer_enumerations[nat] % Proofs only

  S: VAR set[T]

  % ==========================================================================
  % Basic definitions
  % ==========================================================================

  is_countable(S): bool = EXISTS (f: (injective?[(S), nat])): TRUE
  is_countably_infinite(S): bool = EXISTS (f: (bijective?[(S), nat])): TRUE
  is_uncountable(S): MACRO bool = NOT is_countable(S)

  countable_set: TYPE+ = (is_countable) CONTAINING emptyset[T]
  countably_infinite_set: TYPE = (is_countably_infinite)
  uncountable_set: TYPE = (is_uncountable)

  countably_infinite_countable: JUDGEMENT
    countably_infinite_set SUBTYPE_OF countable_set

  % ==========================================================================
  % Infinite subsets of a countably infinite set are countably infinite
  % ==========================================================================

  countably_infinite_subset: THEOREM
    FORALL (CountInf: countably_infinite_set), (Inf: infinite_set[T]):
      subset?(Inf, CountInf) IMPLIES is_countably_infinite(Inf)

  countable_subset: THEOREM
    FORALL S, (Count: countable_set):
      subset?(S, Count) IMPLIES is_countable(S)

  % ==========================================================================
  % Countability of the base type
  % ==========================================================================

  is_countable_type: bool = EXISTS (f: (injective?[T, nat])): TRUE
  is_countably_infinite_type: bool = EXISTS (f: (bijective?[T, nat])): TRUE
  is_uncountable_type: MACRO bool = NOT is_countable_type

  countable_type_is_countably_infinite: LEMMA
    is_countably_infinite_type IMPLIES is_countable_type

  countable_full: LEMMA is_countable_type IFF is_countable(fullset[T])
  countably_infinite_full: LEMMA
    is_countably_infinite_type IFF is_countably_infinite(fullset[T])
  uncountable_full: COROLLARY
    is_uncountable_type IFF is_uncountable(fullset[T])

  countable_type_set: LEMMA
    is_countable_type IMPLIES (FORALL S: is_countable(S))
  countably_infinite_type_set: LEMMA
    is_countably_infinite_type IMPLIES (FORALL S: is_countable(S))
  uncountably_infinite_type_set: LEMMA
    (EXISTS S: is_uncountable(S)) IMPLIES is_uncountable_type

  countable_complement: LEMMA
    is_countable_type IMPLIES (FORALL S: is_countable(complement(S)))
  countably_infinite_complement: LEMMA
    is_countably_infinite_type IMPLIES (FORALL S: is_countable(complement(S)))

END countability

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