--- /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/grammar/lcoeq.ma".
+
+(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+theorem lcoeq_trans: ∀d,e. Transitive … (lcoeq d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e //
+[ #I #L1 #L #V #HL1 #_ #X #H >(lcoeq_inv_O2 … HL1) -HL1 //
+| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 … H) -H //
+ #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lcoeq_pair/
+| #I #L1 #L #V #d #e #_ #IHL1 #X #H elim (lcoeq_inv_succ1 … H) -H //
+ #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/
+]
+qed-.