--- /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".
+
+axiom lift_path_after (p) (f1) (f2):
+ ↑[f2]↑[f1]p = ↑[f2∘f1]p.
+(*
+#p @(path_ind_lift … p) -p // [ #n | #n #l #p | #p ] #IH #f1 #f2
+[ <lift_path_d_empty_sn <lift_path_d_empty_sn
+| <lift_path_d_lcons_sn <lift_path_d_lcons_sn
+| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+*)
+include "delayed_updating/substitution/lift_prototerm.ma".
+
+axiom lift_term_after (t) (f1) (f2):
+ ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
+
+include "delayed_updating/syntax/prototerm_constructors.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+lemma lift_iref_after_sn (f) (t) (n:pnat):
+ ↑[f∘𝐮❨n❩]t ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q #Hq #H0 destruct
+@(ex2_intro … (𝗱n◗𝗺◗q))
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma lift_iref_after_dx (f) (t) (n:pnat):
+ ↑[f](𝛗n.t) ⊆ ↑[f∘𝐮❨n❩]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_iref_after (f) (t) (n:pnat):
+ ↑[f∘𝐮❨n❩]t ⇔ ↑[f](𝛗n.t).
+/3 width=1 by conj, lift_iref_after_sn, lift_iref_after_dx/
+qed.
+
+lemma lift_iref (f) (t) (n:pnat):
+ ↑[f]↑[𝐮❨n❩]t ⇔ ↑[f](𝛗n.t).
+/3 width=3 by lift_term_after, subset_eq_trans/
+qed.