--- /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_2A/static/lsubd_da.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+(* Main properties **********************************************************)
+
+theorem lsubd_trans: ∀h,g,G. Transitive … (lsubd h g G).
+#h #g #G #L1 #L #H elim H -L1 -L
+[ #X #H >(lsubd_inv_atom1 … H) -H //
+| #I #L1 #L #Y #HL1 #IHL1 #X #H
+ elim (lsubd_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=1 by lsubd_pair/
+ | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct
+ /3 width=3 by lsubd_beta, lsubd_da_trans/
+ ]
+| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H
+ elim (lsubd_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/
+ | #W0 #V0 #d0 #_ #_ #_ #H destruct
+ ]
+]
+qed-.