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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_tls_nexts_eq.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_tls_nexts_eq.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 "ground/relocation/pr_nexts.ma".
+include "ground/relocation/pr_tls_eq.ma".
+
+(* ITERATED TAIL FOR PARTIAL RELOCATION MAPS ********************************)
+
+(* Inversions with pr_nexts and pr_eq ***************************************)
+
+(*** eq_inv_nexts_sn *)
+lemma pr_eq_inv_nexts_sn (n):
+      ∀f1,g2. ↑*[n] f1 ≡ g2 →
+      ∧∧ f1 ≡ ⫰*[n]g2 & ↑*[n]⫰*[n]g2 = g2.
+#n @(nat_ind_succ … n) -n /2 width=1 by conj/
+#n #IH #f1 #g2 #H
+elim (pr_eq_inv_next_sn … H) -H [|*: // ] #Hf10 *
+elim (IH … Hf10) -IH -Hf10 #Hf12 #H2
+<pr_tls_succ /2 width=1 by conj/
+qed-.
+
+(*** eq_inv_nexts_dx *)
+lemma pr_eq_inv_nexts_dx (n):
+      ∀f2,g1. g1 ≡ ↑*[n] f2 →
+      ∧∧ ⫰*[n]g1 ≡ f2 & ↑*[n]⫰*[n]g1 = g1.
+#n @(nat_ind_succ … n) -n /2 width=1 by conj/
+#n #IH #f2 #g1 #H
+elim (pr_eq_inv_next_dx … H) -H [|*: // ] #Hf02 *
+elim (IH … Hf02) -IH -Hf02 #Hf12 #H2
+<pr_tls_succ /2 width=1 by conj/
+qed-.