--- /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/substitution/lift_prototerm.ma".
+include "delayed_updating/substitution/lift_path_proper.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_term_proper (f) (t):
+ t ϵ 𝐏 → ↑[f]t ϵ 𝐏.
+#f #t #Ht #p * #q #Hq #H0 destruct
+@lift_path_proper @Ht -Ht // (**) (* auto fails *)
+qed.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_term_inv_proper (f) (t):
+ ↑[f]t ϵ 𝐏 → t ϵ 𝐏.
+#f #t #Ht #p #Hp
+@(lift_path_inv_proper f)
+@Ht -Ht @in_comp_lift_bi // (**) (* auto fails *)
+qed-.