--- /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_2/relocation/rtmap_after.ma".
+include "ground_2/relocation/rtmap_basic.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+(* Properties with composition **********************************************)
+
+lemma after_basic_rc (m2,m1,n2,n1):
+ m1 ≤ m2 → m2 ≤ m1+n1 → 𝐁❴m2,n2❵ ⊚ 𝐁❴m1,n1❵ ≘ 𝐁❴m1,n2+n1❵.
+#m2 elim m2 -m2
+[ #m1 #n2 #n1 #Hm21 #_
+ <(le_n_O_to_eq … Hm21) -m1 //
+| #m2 #IH *
+ [ #n2 #n1 #_ < plus_O_n #H
+ elim (le_inv_S1 … H) -H #x #Hx #H destruct
+ <plus_n_Sm
+ @after_push [4:|*: // ]
+ @(IH 0 … Hx) -IH -n2 -x //
+ | #m1 #n2 #n1 #H1 #H2
+ lapply (le_S_S_to_le … H1) -H1 #H1
+ lapply (le_S_S_to_le … H2) -H2 #H2
+ /3 width=7 by after_refl/
+ ]
+]
+qed.