--- /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 "static_2/static/rex_fqup.ma".
+include "static_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Advanced properties ******************************************************)
+
+lemma rdeq_refl: ∀h,o,T. reflexive … (rdeq h o T).
+/2 width=1 by rex_refl/ qed.
+
+lemma rdeq_pair_refl: ∀h,o,V1,V2. V1 ≛[h, o] V2 →
+ ∀I,L. ∀T:term. L.ⓑ{I}V1 ≛[h, o, T] L.ⓑ{I}V2.
+/2 width=1 by rex_pair_refl/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rdeq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
+ L1 ≛[h, o, V] L2 ∧ L1.ⓧ ≛[h, o, T] L2.ⓧ.
+/2 width=3 by rex_inv_bind_void/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma rdeq_fwd_bind_dx_void: ∀h,o,p,I,L1,L2,V,T.
+ L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ.
+/2 width=4 by rex_fwd_bind_dx_void/ qed-.