Fixing universe levels for saturations and (partially) basic_topologies.

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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/saturations.ma b/helm/software/matita/contribs/formal_topology/overlap/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 "relations.ma".
+
+definition is_saturation: ∀C:REL. unary_morphism1 (Ω \sup C) (Ω \sup C) → CProp1 ≝
+ λC:REL.λA:unary_morphism1 (Ω \sup C) (Ω \sup C).
+  ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
+
+definition is_reduction: ∀C:REL. unary_morphism1 (Ω \sup C) (Ω \sup C) → CProp1 ≝
+ λC:REL.λJ:unary_morphism1 (Ω \sup C) (Ω \sup 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 ?? (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.