helm/software/matita/dama/ordered_set.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/ordered_set/".
  16
  17 include "excedence.ma".
  18
  19 record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ {
  20   por_reflexive: reflexive ? le;
  21   por_transitive: transitive ? le;
  22   por_antisimmetric: antisymmetric ? le eq
  23 }.
  24
  25 record pordered_set: Type ≝ {
  26   pos_carr:> excedence;
  27   pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
  28 }.
  29
  30 lemma pordered_set_of_excedence: excedence → pordered_set.
  31 intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation);
  32 [apply le_reflexive|apply le_transitive|apply le_antisymmetric]
  33 qed.
  34
  35 alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con".
  36 alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con".
  37 alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
  38
  39 definition total_order_property : ∀E:excedence. Type ≝
  40   λE:excedence. ∀a,b:E. a ≰ b → b < a.
  41
  42 record tordered_set : Type ≝ {
  43  tos_poset:> pordered_set;
  44  tos_totality: total_order_property tos_poset
  45 }.