--- /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/lsubr.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+(* Auxiliary inversion lemmas ***********************************************)
+
+fact lsubr_inv_pair1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = Abbr & X = ⓝW.V.
+#L1 #L2 * -L1 -L2
+[ #L #J #K1 #X #H destruct /2 width=1 by or3_intro0/
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by or3_intro1, ex2_intro/
+| #L1 #L2 #V #W #HL12 #J #K1 #X #H destruct /3 width=6 by or3_intro2, ex4_3_intro/
+]
+qed-.
+
+lemma lsubr_inv_pair1: ∀I,K1,L2,X. K1.ⓑ{I}X ⫃ L2 →
+ ∨∨ L2 = ⋆
+ | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = Abbr & X = ⓝW.V.
+/2 width=3 by lsubr_inv_pair1_aux/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H
+ lapply (lsubr_inv_atom1 … H) -H //
+| #I #L1 #L #V #_ #IHL1 #X #H
+ elim (lsubr_inv_pair1 … H) -H // *
+ #L2 [2: #V2 #W2 ] #HL2 #H1 [ #H2 #H3 ] destruct /3 width=1 by lsubr_pair, lsubr_beta/
+| #L1 #L #V1 #W #_ #IHL1 #X #H
+ elim (lsubr_inv_abst1 … H) -H // *
+ #L2 #HL2 #H destruct /3 width=1 by lsubr_beta/
+]
+qed-.