--- /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 "basic_2/computation/lfprs_aaa.ma".
+include "basic_2/equivalence/lfpcs_lfpcs.ma".
+
+(* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
+
+(* Main properties about atomic arity assignment on terms *******************)
+
+theorem aaa_lfpcs_mono: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
+ ∀T,A1. L1 ⊢ T ⁝ A1 → ∀A2. L2 ⊢ T ⁝ A2 →
+ A1 = A2.
+#L1 #L2 #HL12 #T #A1 #HT1 #A2 #HT2
+elim (lfpcs_inv_lfprs … HL12) -HL12 #L #HL1 #HL2
+lapply (aaa_lfprs_conf … HT1 … HL1) -L1 #HT1
+lapply (aaa_lfprs_conf … HT2 … HL2) -L2 #HT2
+lapply (aaa_mono … HT1 … HT2) -L -T //
+qed-.