--- /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/lifts_tdeq.ma".
+include "basic_2/static/rex_length.ma".
+include "basic_2/static/rex_fsle.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Advanved properties with free variables inclusion ************************)
+
+lemma rdeq_fsge_comp (h) (o): rex_fsge_compatible (cdeq h o).
+#h #o #L1 #L2 #T * #f1 #Hf1 #HL12
+lapply (frees_rdeq_conf h o … Hf1 … HL12)
+lapply (sex_fwd_length … HL12)
+/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
+qed-.
+
+(* Properties with length for local environments ****************************)
+
+(* Basic_2A1: uses: lleq_sort *)
+lemma rdeq_sort_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀s. L1 ≛[h, o, ⋆s] L2.
+/2 width=1 by rex_sort_length/ qed.
+
+(* Basic_2A1: uses: lleq_gref *)
+lemma rdeq_gref_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀l. L1 ≛[h, o, §l] L2.
+/2 width=1 by rex_gref_length/ qed.
+
+lemma rdeq_unit_length (h) (o): ∀L1,L2. |L1| = |L2| →
+ ∀I. L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+/2 width=1 by rex_unit_length/ qed.
+
+(* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *)
+lemma rdeq_lifts_bi (h) (o): ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[h, o, T] K2 →
+ ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ≛[h, o, U] L2.
+/3 width=9 by rex_lifts_bi, tdeq_lifts_sn/ qed-.
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: lleq_fwd_length *)
+lemma rdeq_fwd_length (h) (o): ∀L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
+/2 width=3 by rex_fwd_length/ qed-.