--- /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_path.ma".
+include "delayed_updating/syntax/path_closed.ma".
+include "ground/relocation/xap.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with pcc ***************************************************)
+
+lemma lift_path_closed (f) (q) (n):
+ q ϵ 𝐂❨n❩ → ↑[f]q ϵ 𝐂❨↑[q]f@❨n❩❩.
+#f #q #n #Hq elim Hq -Hq //
+#q #n [ #k ] #_ #IH
+/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+lemma lift_path_rmap_closed (f) (p) (q) (n):
+ q ϵ 𝐂❨n❩ → ↑[↑[p]f]q ϵ 𝐂❨↑[p●q]f@❨n❩❩.
+/2 width=1 by lift_path_closed/
+qed.