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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
new file mode 100644
index 000000000..3b9e3fbec
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+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/syntax/path_closed.ma".
+include "delayed_updating/syntax/path_height.ma".
+include "delayed_updating/syntax/path_depth.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+(* Destructions with height and depth ***************************************)
+
+lemma path_closed_des_depth (o) (e) (q) (n):
+      q ϵ 𝐂❨o,n,e❩ → ♯q + n = ♭q + e.
+#o #e #q #n #Hq elim Hq -q -n //
+#q #n #_ #IH
+<nplus_succ_dx <nplus_succ_sn //
+qed-.
+
+lemma path_closed_des_succ_zero_depth (o) (q) (n):
+      q ϵ 𝐂❨o,↑n,𝟎❩ → ♭q = ↑↓♭q.
+#o #q #n #Hq
+>(nplus_zero_dx (♭q))
+<(path_closed_des_depth … Hq) -Hq
+<nplus_succ_dx <npred_succ //
+qed-.