First experiment in Padua about formal topologies.

diff --git a/helm/software/matita/library/demo/formal_topology.ma b/helm/software/matita/library/demo/formal_topology.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 "logic/connectives.ma".
+
+coinductive fish (A:Type) (i: A → Type) (C: ∀a:A.i a → A → Prop) (U: A → Prop)
+               : A → Prop
+≝
+ mk_foo: ∀a:A. (U a ∧ ∀j: i a. ∃y: A. C a j y ∧ fish A i C U y) → fish A i C U a.
+
+let corec fish_rec (A:Type) (i: A → Type) (C: ∀a:A.i a → A → Prop) (U: A → Prop)
+ (P: A → Prop) (H1: ∀a:A. P a → U a)
+  (H2: ∀a:A. P a → ∀j: i a. ∃y: A. C a j y ∧ P y) :
+   ∀a:A. ∀p: P a. fish A i C U a ≝
+ λa,p.
+  mk_foo A i C U a
+   (conj ? ? (H1 ? p)
+   (λj: i a.
+    match H2 a p j with
+     [ ex_intro (y: A) (Ha: C a j y ∧ P y) ⇒
+        match Ha with
+         [ conj (fHa: C a j y) (sHa: P y) ⇒
+            ex_intro A (λy.C a j y ∧ fish A i C U y) y
+             (conj ? ? fHa (fish_rec A i C U P H1 H2 y sHa))
+         ]
+     ])).
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