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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
new file mode 100644
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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.