--- /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/lfxs_drops.ma".
+include "basic_2/static/lfdeq_fqup.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: includes: lleq_lift_le lleq_lift_ge *)
+lemma lfdeq_lifts: ∀h,o. dedropable_sn (cdeq h o).
+/3 width=5 by lfxs_liftable_dedropable, tdeq_lifts/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: restricts: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
+lemma lfdeq_inv_lifts: ∀h,o. dropable_sn (cdeq h o).
+/2 width=5 by lfxs_dropable_sn/ qed-.
+
+(* Note: missing in basic_2A1 *)
+lemma lfdeq_inv_lifts_dx: ∀h,o. dropable_dx (cdeq h o).
+/2 width=5 by lfxs_dropable_dx/ qed-.
+
+(* Note: missing in basic_2A1 *)
+lemma lfdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≡[h, o, U] L2 →
+ ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
+ ∀T. ⬆*[i] T ≡ U → K1 ≡[h, o, T] K2.
+/2 width=8 by lfxs_inv_lifts_bi/ qed-.
+
+lemma lfdeq_inv_lref_sn: ∀h,o,L1,L2,i. L1 ≡[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ≡[h, o, V1] K2 & V1 ≡[h, o] V2.
+/2 width=3 by lfxs_inv_lref_sn/ qed-.
+
+lemma lfdeq_inv_lref_dx: ∀h,o,L1,L2,i. L1 ≡[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ≡[h, o, V1] K2 & V1 ≡[h, o] V2.
+/2 width=3 by lfxs_inv_lref_dx/ qed-.