--- /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/arith/nat_plus_rplus.ma".
+include "ground/relocation/rtmap_basic_nat.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+(* Prioerties with application **********************************************)
+
+lemma at_basic_lt (m) (n) (i):
+ ninj i ≤ m → @❪i, 𝐁❨m,n❩❫ ≘ i.
+#m #n #i >(npsucc_pred i) #Hmi
+/2 width=1 by rm_nat_basic_lt/
+qed.
+
+lemma at_basic_ge (m) (n) (i):
+ m < ninj i → @❪i, 𝐁❨m,n❩❫ ≘ i+n.
+#m #n #i >(npsucc_pred i) #Hmi <nrplus_npsucc_sn
+/3 width=1 by rm_nat_basic_ge, nlt_inv_succ_dx/
+qed.
+
+(* Inversion lemmas with application ****************************************)
+
+lemma at_basic_inv_lt (m) (n) (i) (j):
+ ninj i ≤ m → @❪i, 𝐁❨m,n❩❫ ≘ j → i = j.
+/3 width=4 by at_basic_lt, at_mono/ qed-.
+
+lemma at_basic_inv_ge (m) (n) (i) (j):
+ m < ninj i → @❪i, 𝐁❨m,n❩❫ ≘ j → i+n = j.
+/3 width=4 by at_basic_ge, at_mono/ qed-.