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

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  14
  15 include "formal_topology/o-algebra.ma".
  16
  17 definition is_o_saturation: ∀C:OA. C ⇒_1 C → CProp1 ≝
  18  λC:OA.λA:C ⇒_1 C.∀U,V. (U ≤ A V) =_1 (A U ≤ A V).
  19
  20 definition is_o_reduction: ∀C:OA. C ⇒_1 C → CProp1 ≝
  21  λC:OA.λJ:C ⇒_1 C.∀U,V. (J U ≤ V) =_1 (J U ≤ J V).
  22
  23 theorem o_saturation_expansive: ∀C,A. is_o_saturation C A → ∀U. U ≤ A U.
  24  intros; apply (fi ?? (i ??)); apply (oa_leq_refl C).
  25 qed.
  26
  27 theorem o_saturation_monotone: ∀C:OA.∀A:C ⇒_1 C. is_o_saturation C A → ∀U,V. U ≤ V → A U ≤ A V.
  28  intros; apply (if ?? (i ??)); apply (oa_leq_trans C);
  29   [apply V|3: apply o_saturation_expansive ]
  30  assumption.
  31 qed.
  32
  33 theorem o_saturation_idempotent: ∀C:OA.∀A:C ⇒_1 C. is_o_saturation C A → ∀U. A (A U) =_1 A U.
  34  intros; apply (oa_leq_antisym C);
  35   [ apply (if ?? (i (A U) U)); apply (oa_leq_refl C).
  36   | apply o_saturation_expansive; assumption]
  37 qed.