helm/software/matita/contribs/formal_topology/overlap/saturations.ma

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  14
  15 include "relations.ma".
  16
  17 definition is_saturation: ∀C:REL. unary_morphism1 (Ω \sup C) (Ω \sup C) → CProp1 ≝
  18  λC:REL.λA:unary_morphism1 (Ω \sup C) (Ω \sup C).
  19   ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
  20
  21 definition is_reduction: ∀C:REL. unary_morphism1 (Ω \sup C) (Ω \sup C) → CProp1 ≝
  22  λC:REL.λJ:unary_morphism1 (Ω \sup C) (Ω \sup C).
  23   ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
  24
  25 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
  26  intros; apply (fi ?? (i ??)); apply subseteq_refl.
  27 qed.
  28
  29 theorem saturation_monotone:
  30  ∀C,A. is_saturation C A →
  31   ∀U,V. U ⊆ V → A U ⊆ A V.
  32  intros; apply (if ?? (i ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
  33  assumption.
  34 qed.
  35
  36 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
  37  intros; split;
  38   [ apply (if ?? (i ??)); apply subseteq_refl
  39   | apply saturation_expansive; assumption]
  40 qed.