--- /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/fprs_aaa.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Main properties about atomic arity assignment on terms *******************)
+
+theorem aaa_fpcs_mono: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
+ ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 →
+ A1 = A2.
+#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1
+lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2
+lapply (aaa_mono … HT1 … HT2) -L -T //
+qed-.