Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: sort_fseq_lems.pvs Sprache: PVS

Original von: PVS^©

%-------------------------------------------------------------------------
%
%  Monotonically increasing, decreasing, nonincreasing, and nondecreasing
%  functions.
%
%  For PVS version 3.2.  January 13, 2005
%  ---------------------------------------------------------------------
%      Author: Jerry James ([email protected]), University of Kansas
%
%  EXPORTS
%  -------
%  prelude: orders[D], orders[R], relation_props2[D, D, D, D],
%    relation_props2[R, R, R, R], relations[D], relations[R]
%  orders: closure_ops[D], closure_ops[R], indexed_sets_extra,
%    monotone_functions[D,R], relation_iterate[D], relation_iterate[R],
%    relations_extra[D], relations_extra[R]
%
%-------------------------------------------------------------------------
monotone_functions[D: TYPE, R: TYPE]: THEORY
BEGIN

  IMPORTING closure_ops[D], closure_ops[R]

  d1, d2: VAR D
  r1, r2: VAR R
  f: VAR [D -> R]
  Drel: VAR pred[[D, D]]
  Rrel: VAR pred[[R, R]]
  Dtot: VAR (total_order?[D])
  Dstr: VAR (strict_total_order?[D])

  increasing?(Drel, Rrel)(f): bool =
      preserves(f, irreflexive_kernel(Drel), irreflexive_kernel(Rrel))

  nonincreasing?(Drel, Rrel)(f): bool =
      inverts(f, reflexive_closure(Drel), reflexive_closure(Rrel))

  decreasing?(Drel, Rrel)(f): bool =
      inverts(f, irreflexive_kernel(Drel), irreflexive_kernel(Rrel))

  nondecreasing?(Drel, Rrel)(f): bool =
      preserves(f, reflexive_closure(Drel), reflexive_closure(Rrel))

  increasing_is_nondecreasing: THEOREM
    FORALL Drel, Rrel, f:
      increasing?(Drel, Rrel)(f) IMPLIES nondecreasing?(Drel, Rrel)(f)

  decreasing_is_nonincreasing: THEOREM
    FORALL Drel, Rrel, f:
      decreasing?(Drel, Rrel)(f) IMPLIES nonincreasing?(Drel, Rrel)(f)

  not_increasing_and_nonincreasing: THEOREM
    (EXISTS d1, d2: d1 /= d2) IMPLIES
     (FORALL (Drel: (trichotomous?[D])), (Rrel: (antisymmetric?[R])), f:
        NOT (increasing?(Drel, Rrel)(f) AND nonincreasing?(Drel, Rrel)(f)))

  not_decreasing_and_nondecreasing: THEOREM
    (EXISTS d1, d2: d1 /= d2) IMPLIES
     (FORALL (Drel: (trichotomous?[D])), (Rrel: (antisymmetric?[R])), f:
        NOT (decreasing?(Drel, Rrel)(f) AND nondecreasing?(Drel, Rrel)(f)))

  increasing_converse_decreasing1: THEOREM
    FORALL Drel, Rrel:
      increasing?(Drel, Rrel) = decreasing?(Drel, converse(Rrel))

  increasing_converse_decreasing2: THEOREM
    FORALL Drel, Rrel:
      increasing?(Drel, Rrel) = decreasing?(converse(Drel), Rrel)

  nonincreasing_converse_nondecreasing1: THEOREM
    FORALL Drel, Rrel:
      nonincreasing?(Drel, Rrel) = nondecreasing?(Drel, converse(Rrel))

  nonincreasing_converse_nondecreasing2: THEOREM
    FORALL Drel, Rrel:
      nonincreasing?(Drel, Rrel) = nondecreasing?(converse(Drel), Rrel)

  increasing_total_order_injective: THEOREM
    FORALL Dtot, Rrel, f: increasing?(Dtot, Rrel)(f) IMPLIES injective?(f)

  increasing_strict_total_order_injective: THEOREM
    FORALL Dstr, Rrel, f: increasing?(Dstr, Rrel)(f) IMPLIES injective?(f)

  decreasing_total_order_injective: THEOREM
    FORALL Dtot, Rrel, f: decreasing?(Dtot, Rrel)(f) IMPLIES injective?(f)

  decreasing_strict_total_order_injective: THEOREM
    FORALL Dstr, Rrel, f: decreasing?(Dstr, Rrel)(f) IMPLIES injective?(f)

END monotone_functions

¤ Dauer der Verarbeitung: 0.12 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des 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.