--- /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/notation/functions/uparrowstar_2.ma".
+include "ground/arith/nat_succ_iter.ma".
+include "ground/relocation/pr_map.ma".
+
+(* ITERATED NEXT FOR PARTIAL RELOCATION MAPS ********************************)
+
+(*** nexts *)
+definition pr_nexts (f:pr_map) (n:nat) ≝
+ (pr_next^n) f.
+
+interpretation
+ "iterated next (partial relocation maps)"
+ 'UpArrowStar n f = (pr_nexts f n).
+
+(* Basic constructions ******************************************************)
+
+(*** nexts_O *)
+lemma pr_nexts_zero:
+ ∀f. f = ↑*[𝟎] f.
+// qed.
+
+(*** nexts_swap *)
+lemma pr_nexts_next (n):
+ ∀f. ↑↑*[n] f = ↑*[n] ↑f.
+#f #n @(niter_appl … pr_next)
+qed.
+
+(*** nexts_S *)
+lemma pr_nexts_succ (n):
+ ∀f. ↑↑*[n] f = ↑*[↑n] f.
+#f #n @(niter_succ … pr_next)
+qed.
+
+(*** nexts_xn *)
+lemma pr_nexts_swap (n):
+ ∀f. ↑*[n] ↑f = ↑*[↑n] f.
+// qed.