--- /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_eq.ma".
+include "ground/relocation/tr_compose_pap.ma".
+include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_tls.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma lift_path_after (p) (f1) (f2):
+ ↑[f2]↑[f1]p = ↑[f2∘f1]p.
+#p elim p -p [| * ] // [ #n ] #p #IH #f1 #f2
+[ <lift_path_d_sn <lift_path_d_sn <lift_path_d_sn //
+| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+ >tr_compose_push_bi //
+]
+qed.