--- /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/relocation/lexs_lexs.ma".
+include "basic_2/relocation/lreq.ma".
+
+(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
+
+(* Main properties **********************************************************)
+
+theorem lreq_trans: ∀f. Transitive … (lreq f).
+/2 width=3 by lexs_trans/ qed-.
+
+theorem lreq_canc_sn: ∀f. left_cancellable … (lreq f).
+/3 width=3 by lexs_canc_sn, lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_canc_dx: ∀f. right_cancellable … (lreq f).
+/3 width=3 by lexs_canc_dx, lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_join: ∀L1,L2,f1. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
+ ∀f. f1 ⋓ f2 ≡ f → L1 ≡[f] L2.
+/2 width=5 by lexs_join/ qed-.
+
+theorem lreq_meet: ∀L1,L2,f1. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
+ ∀f. f1 ⋒ f2 ≡ f → L1 ≡[f] L2.
+/2 width=5 by lexs_meet/ qed-.