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

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  14
  15 include "o-algebra.ma".
  16
  17 alias symbol "eq" = "setoid1 eq".
  18 definition is_o_saturation: ∀C:OA. unary_morphism1 C C → CProp1 ≝
  19  λC:OA.λA:unary_morphism1 C C.
  20   ∀U,V. (U ≤ A V) = (A U ≤ A V).
  21
  22 definition is_o_reduction: ∀C:OA. unary_morphism1 C C → CProp1 ≝
  23  λC:OA.λJ:unary_morphism1 C C.
  24     ∀U,V. (J U ≤ V) = (J U ≤ J V).
  25
  26 theorem o_saturation_expansive: ∀C,A. is_o_saturation C A → ∀U. U ≤ A U.
  27  intros; apply (fi ?? (i ??)); apply (oa_leq_refl C).
  28 qed.
  29
  30 theorem o_saturation_monotone:
  31  ∀C,A. is_o_saturation C A →
  32   ∀U,V. U ≤ V → A U ≤ A V.
  33  intros; apply (if ?? (i ??)); apply (oa_leq_trans C);
  34   [apply V|3: apply o_saturation_expansive ]
  35  assumption.
  36 qed.
  37
  38 theorem o_saturation_idempotent: ∀C,A. is_o_saturation C A → ∀U.
  39  eq1 C (A (A U)) (A U).
  40  intros; apply (oa_leq_antisym C);
  41   [ apply (if ?? (i (A U) U)); apply (oa_leq_refl C).
  42   | apply o_saturation_expansive; assumption]
  43 qed.