helm/software/matita/library/formal_topology/saturations.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "formal_topology/relations.ma".
  16
  17 definition is_saturation: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp1 ≝
  18  λC:REL.λA:Ω^C ⇒_1 Ω^C. ∀U,V. (U ⊆ A V) =_1 (A U ⊆ A V).
  19
  20 definition is_reduction: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp1 ≝
  21  λC:REL.λJ: Ω^C ⇒_1 Ω^C. ∀U,V. (J U ⊆ V) =_1 (J U ⊆ J V).
  22
  23 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
  24  intros; apply (fi ?? (i ??)); apply subseteq_refl.
  25 qed.
  26
  27 theorem saturation_monotone:
  28  ∀C,A. is_saturation C A →
  29   ∀U,V. U ⊆ V → A U ⊆ A V.
  30  intros; apply (if ?? (i ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
  31  assumption.
  32 qed.
  33
  34 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
  35  intros; split;
  36   [ apply (if ?? (i ??)); apply subseteq_refl
  37   | apply saturation_expansive; assumption]
  38 qed.