+(**************************************************************************)
+(* ___ *)
+(* ||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 "basic_2/syntax/lenv_length.ma".
+include "basic_2/syntax/voids.ma".
+
+(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
+
+(* Properties with length for local environments ****************************)
+
+lemma length_void: ∀L,n. n+|L| = |ⓧ*[n]L|.
+#L #n elim n -n //
+#n #IH <voids_succ >length_bind <IH -IH //
+qed.
+
+lemma length_void_le: ∀L,n. |L| ≤ |ⓧ*[n]L|.
+// qed.
+
+(* Main forward properties with length for local environments ***************)
+
+theorem voids_inj_length: ∀n1,n2,L1,L2. ⓧ*[n1]L1 = ⓧ*[n2]L2 →
+ |L1| = |L2| → n1 = n2 ∧ L1 = L2.
+#n1 elim n1 -n1
+[ * /2 width=1 by conj/ #n2 #L1 #L2 | #n1 #IH * [ | #n2 ] #L1 #L2 ]
+[ <voids_zero #H destruct
+ <length_void <commutative_plus #H
+ elim (plus_xSy_x_false … H)
+| <voids_zero #H destruct
+ <length_void <commutative_plus #H
+ elim (plus_xSy_x_false … (sym_eq … H))
+| <voids_succ <voids_succ #H #HL12
+ elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
+ #H #_ elim (IH … H HL12) -IH -H -HL12 /2 width=1 by conj/
+]
+qed-.