foo overlap

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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma b/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "o-algebra.ma".
+
+definition hint1: OA → Type ≝ λc:OA.carr (oa_P c).
+coercion hint1.
+
+definition hint2: ∀C.hint1 C → carr1 ((λx.x) (setoid1_of_setoid (oa_P C))).
+intros; assumption;
+qed.
+coercion hint2. 
+
+alias symbol "eq" = "setoid1 eq".
+definition is_saturation ≝
+ λC:OA.λA:C → C.
+  ∀U,V. (U ≤ A V) = (A U ≤ A V).
+
+definition is_reduction ≝
+ λC:OA.λJ:C → C.
+    ∀U,V. (J U ≤ V) = (J U ≤ J V).
+
+theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ≤ A U.
+ intros; apply (fi ?? (H ??)); apply (oa_leq_refl C).
+qed.
+
+theorem saturation_monotone:
+ ∀C,A. is_saturation C A →
+  ∀U,V:C. U ≤ V → A U ≤ A V.
+ intros; apply (if ?? (H ??)); apply (oa_leq_trans C);
+  [apply V|3: apply saturation_expansive ]
+ assumption.
+qed.
+
+theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. 
+ eq (oa_P C) (A (A U)) (A U).
+ intros; apply (oa_leq_antisym C);
+  [ apply (if ?? (H (A U) U)); apply (oa_leq_refl C).
+  | apply saturation_expansive; assumption]
+qed.