--- /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/static/lsuba_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Main properties **********************************************************)
+
+theorem lsuba_trans: ∀L1,L. L1 ⁝⊑ L → ∀L2. L ⁝⊑ L2 → L1 ⁝⊑ L2.
+#L1 #L #H elim H -L1 -L
+[ #X #H >(lsuba_inv_atom1 … H) -H //
+| #I #L1 #L #V #HL1 #IHL1 #X #H
+ elim (lsuba_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=1/
+ | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/
+ ]
+| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H
+ elim (lsuba_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=5/
+ | #V #A2 #_ #_ #_ #_ #H destruct
+ ]
+]
+qed.