- lambda_delta: "conversion" and "equivalence" components started

[helm.git] / matita / matita / lib / formal_topology / saturations.ma

diff --git a/matita/matita/lib/formal_topology/saturations.ma b/matita/matita/lib/formal_topology/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 "formal_topology/relations.ma".
+
+definition is_saturation: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp[1] ≝
+ λC:REL.λA:Ω^C ⇒_1 Ω^C. ∀U,V. (U ⊆ A V) =_1 (A U ⊆ A V).
+
+definition is_reduction: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp[1] ≝
+ λC:REL.λJ: Ω^C ⇒_1 Ω^C. ∀U,V. (J U ⊆ V) =_1 (J U ⊆ J V).
+
+theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
+ intros; apply (fi ?? (i ??)); apply subseteq_refl.
+qed.
+
+theorem saturation_monotone:
+ ∀C,A. is_saturation C A →
+  ∀U,V. U ⊆ V → A U ⊆ A V.
+ intros; apply (if ?? (i ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
+ assumption.
+qed.
+
+theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
+ intros; split;
+  [ apply (if ?? (i ??)); apply subseteq_refl
+  | apply saturation_expansive; assumption]
+qed.