--- /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/multiple/llpx_sn_lreq.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L →
+ ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L.
+/2 width=3 by lreq_llpx_sn_trans/ qed-.
+
+lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 →
+ ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2.
+/2 width=3 by llpx_sn_lreq_trans/ qed-.
+
+lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 →
+ ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2.
+/2 width=5 by llpx_sn_lreq_repl/ qed-.
+
+lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 →
+ ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2.
+/2 width=5 by llpx_sn_bind_repl_SO/ qed-.