--- /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 "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-.