--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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-.