helm/software/matita/library/formal_topology/saturations_reductions.ma.dontcompile

   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 include "datatypes/subsets.ma".
  16 include "formal_topology/relations.ma".
  17
  18 definition is_saturation ≝
  19  λC:REL.λA:unary_morphism (Ω \sup C) (Ω \sup C).
  20   ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
  21
  22 definition is_reduction ≝
  23  λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C).
  24   ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
  25
  26 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
  27  intros; apply (fi ?? (H ??)); apply subseteq_refl.
  28 qed.
  29
  30 theorem saturation_monotone:
  31  ∀C,A. is_saturation C A →
  32   ∀U,V. U ⊆ V → A U ⊆ A V.
  33  intros; apply (if ?? (H ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
  34  assumption.
  35 qed.
  36
  37 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
  38  intros; split;
  39   [ apply (if ?? (H ??)); apply subseteq_refl
  40   | apply saturation_expansive; assumption]
  41 qed.