--- /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/scpds_aaa.ma".
+include "basic_2/equivalence/scpes.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL EQUIVALENCE FOR TERMS ***********************)
+
+(* Main inversion lemmas about atomic arity assignment on terms *************)
+
+theorem scpes_aaa_mono: ∀h,o,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
+ ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
+ A1 = A2.
+#h #o #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
+lapply (scpds_aaa_conf … HA1 … HT1) -T1 #HA1
+lapply (scpds_aaa_conf … HA2 … HT2) -T2 #HA2
+lapply (aaa_mono … HA1 … HA2) -L -T //
+qed-.