--- /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/upspoonstar_2.ma".
+include "ground/arith/nat_succ_iter.ma".
+include "ground/relocation/pr_map.ma".
+
+(* ITERATED PUSH FOR PARTIAL RELOCATION MAPS ********************************)
+
+(*** pushs *)
+definition pr_pushs (f:pr_map) (n:nat) ≝
+ (pr_push^n) f.
+
+interpretation
+ "iterated push (partial relocation maps)"
+ 'UpSpoonStar n f = (pr_pushs f n).
+
+(* Basic constructions ******************************************************)
+
+(*** pushs_O *)
+lemma pr_pushs_zero:
+ ∀f. f = ⫯*[𝟎] f.
+// qed.
+
+(*** push_swap *)
+lemma pr_pushs_push (n):
+ ∀f. ⫯⫯*[n] f = ⫯*[n] ⫯f.
+#n #f @(niter_appl … pr_push)
+qed.
+
+(*** pushs_S *)
+lemma pr_pushs_succ (n):
+ ∀f. ⫯⫯*[n] f = ⫯*[↑n] f.
+#f #n @(niter_succ … pr_push)
+qed.
+
+(*** pushs_xn *)
+lemma pr_pushs_swap (n):
+ ∀f. ⫯*[n] ⫯f = ⫯*[↑n] f.
+// qed.