Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Tarski.summary Sprache: Unknown

Original von: PVS^©

sort_fseq[T: TYPE+, <= : (total_order?[T]) ]: THEORY
%------------------------------------------------------------------------
%
%  sort_fseq (basic definitions and properties)
%  -------------------------------------------
%
%      Author: Ricky W. Butler
%
%  This theory defines the sort function over an seq of
%  values.  Most of the text of this theory is scaffolding to
%  demonstrate that the definition of "sort" is sound.  In particular,
%  the function "asort" is a witness function and not intended
%  to be used externally.
%
%  The following definitions should be used exclusively:
%
%    is_permutation(A1,A2): bool = (EXISTS (f: [below(N) -> below(N)]):
%                      bijective?(f) AND (FORALL ii: A1(ii) = A2(f(ii))))
%
%    increasing(A): bool = (FORALL i,j: 0 <= i AND i < j AND j < N
%                                 IMPLIES A(i) <= A(j))
%
%    sort(A): {a: seqs | is_permutation(A,a) AND increasing(a)}
%
%  Note:
%     The properties of sort are readily obtained via
%     TYPEPRED "sort" or through use of sort_lem.
%
%------------------------------------------------------------------------

  EXPORTING ALL BUT seq_to_array % i.e. do not export sort_array
        WITH permutations_fseq[T,<=], fseqs[T], fsq[T]

BEGIN

  IMPORTING fseqs[T]

  s,s1,s2,ss: VAR fseq
  x, t: VAR T
  i,j: VAR nat

%%  <(x,y: T): bool = x <= y AND x /= y                         %% CAUSED SOME PROBLEMS

  IMPORTING sort_array, below_arrays, permutations_fseq

  seq_to_array(s): below_array[length(s),T] =
             (LAMBDA (i: below(length(s))): seq(s)(i))

  increasing?(s): bool = (FORALL i,j: 0 <= i AND i < j AND j < length(s)
                            IMPLIES seq(s)(i) <= seq(s)(j))

  strictly_increasing?(s): bool = (FORALL i,j: 0 <= i AND i < j AND j < length(s)
                            IMPLIES (seq(s)(i) <= seq(s)(j) AND seq(s)(i) /= seq(s)(j)))

  sort(s): {ss: fseq | permutation?[T,<=](s,ss) AND increasing?(ss)}

  sort_length  : LEMMA length(sort(s)) = length(s)

  AUTO_REWRITE+ sort_length

  sort_fseq_lem  : LEMMA permutation?[T,<=](s,sort(s)) AND increasing?(sort(s))

  sort_fseq_in?  : LEMMA in?(x,s) IFF in?(x,sort(s))

  sort_fseq_in   : LEMMA FORALL (ii: below(length(s))):
                                     in?(seq(sort(s))(ii),s)

  sort_fseq_in2  : LEMMA FORALL (ii: below(length(s))): in?(seq(s)(ii),sort(s))

  sort_fseq_lt   : LEMMA FORALL (i,j: below(length(s))):
                         sort(s)(i) <= sort(s)(j) AND sort(s)(i) /= sort(s)(j) IMPLIES i < j

  sort_singleton : LEMMA sort(singleton(x)) = singleton(x)

  member_sort  : LEMMA member(x,s) IFF member(x,sort(s))

  %----------------- Strictly Sorting ----------------%

  strictly_sort(s): {ss: fseq | strictly_increasing?(ss) AND (FORALL (x:T): member(x,s) IFF member(x,ss))}

  strictly_sort_length: LEMMA strictly_sort(s)`length <= s`length

  strictly_sort_subint_eq: LEMMA LET sss = strictly_sort(s) IN increasing?(s) IMPLIES FORALL (ii:below(sss`length-1)):
        EXISTS (jj:below(s`length-1)): sss`seq(ii+1)=s`seq(jj+1) AND sss`seq(ii) = s`seq(jj)

  strictly_sort_map(s): {sq: [below(strictly_sort(s)`length)->below(s`length)] |
          LET sss = strictly_sort(s) IN
        (increasing?(s) AND sss`length >= 1 IMPLIES sq(sss`length-1) = s`length-1) AND
        (FORALL (ii:below(sss`length)): (sss`seq(ii) = s`seq(sq(ii)) AND
        (increasing?(s) AND ii<sss`length-1 IMPLIES sss`seq(ii+1) = s`seq(sq(ii)+1))))}

  strictly_sort_map_between: LEMMA LET sss = strictly_sort(s) IN
              FORALL (ii:below(sss`length-1)): FORALL (jj:below(s`length)):
       increasing?(s) AND strictly_sort_map(s)(ii)<jj AND jj<=strictly_sort_map(s)(ii+1) IMPLIES
       s`seq(strictly_sort_map(s)(ii+1)) = s`seq(jj)

  strictly_sort_map_increasing: LEMMA increasing?(s) IMPLIES FORALL (ii,jj:below(strictly_sort(s)`length)):
      LET sq = strictly_sort_map(s) IN ii<jj IMPLIES sq(ii)<sq(jj)

% ------------------ NEW ------------------%

  sort_map(s): { map: [below(length(s)) -> below(length(s))] |
            bijective?(map) AND
              (FORALL (i: below(length(s))): seq(s)(i) = seq(sort(s))(map(i)))}

  sort_map_def: LEMMA FORALL (i: below(length(s))):
                            seq(s)(i) = seq(sort(s))(sort_map(s)(i))

  sort_map_bij: LEMMA bijective?(sort_map(s))

  isort_map(s: {ss: fseq | length(ss) > 0}):
       { map: [below(length(s)) -> below(length(s))] |
             bijective?(map) AND
               (FORALL (i:below(length(s))): seq(s)(map(i)) = seq(sort(s))(i))}

  isort_map_def: LEMMA FORALL (i: below(length(s))):
                             seq(s)(isort_map(s)(i)) = seq(sort(s))(i)

  isort_map_bij: LEMMA FORALL (s: {ss: fseq  | length(ss) > 0}):
                            bijective?(isort_map(s))

END sort_fseq

¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤

vermutete Sprache: Sekunden
vermutete Sprache: sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.