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

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  14
  15 include "o-algebra.ma".
  16
  17 definition hint1: OA → Type ≝ λc:OA.carr (oa_P c).
  18 coercion hint1.
  19
  20 definition hint2: ∀C.hint1 C → carr1 ((λx.x) (setoid1_of_setoid (oa_P C))).
  21 intros; assumption;
  22 qed.
  23 coercion hint2.
  24
  25 alias symbol "eq" = "setoid1 eq".
  26 definition is_saturation ≝
  27  λC:OA.λA:C → C.
  28   ∀U,V. (U ≤ A V) = (A U ≤ A V).
  29
  30 definition is_reduction ≝
  31  λC:OA.λJ:C → C.
  32     ∀U,V. (J U ≤ V) = (J U ≤ J V).
  33
  34 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ≤ A U.
  35  intros; apply (fi ?? (H ??)); apply (oa_leq_refl C).
  36 qed.
  37
  38 theorem saturation_monotone:
  39  ∀C,A. is_saturation C A →
  40   ∀U,V:C. U ≤ V → A U ≤ A V.
  41  intros; apply (if ?? (H ??)); apply (oa_leq_trans C);
  42   [apply V|3: apply saturation_expansive ]
  43  assumption.
  44 qed.
  45
  46 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U.
  47  eq (oa_P C) (A (A U)) (A U).
  48  intros; apply (oa_leq_antisym C);
  49   [ apply (if ?? (H (A U) U)); apply (oa_leq_refl C).
  50   | apply saturation_expansive; assumption]
  51 qed.