--- /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/notation/constructors/star_0.ma".
+include "basic_2A/notation/constructors/dxbind2_3.ma".
+include "basic_2A/notation/constructors/dxabbr_2.ma".
+include "basic_2A/notation/constructors/dxabst_2.ma".
+include "basic_2A/grammar/term.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+(* local environments *)
+inductive lenv: Type[0] ≝
+| LAtom: lenv (* empty *)
+| LPair: lenv → bind2 → term → lenv (* binary binding construction *)
+.
+
+interpretation "sort (local environment)"
+ 'Star = LAtom.
+
+interpretation "local environment binding construction (binary)"
+ 'DxBind2 L I T = (LPair L I T).
+
+interpretation "abbreviation (local environment)"
+ 'DxAbbr L T = (LPair L Abbr T).
+
+interpretation "abstraction (local environment)"
+ 'DxAbst L T = (LPair L Abst T).
+
+(* Basic properties *********************************************************)
+
+lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ] * [2,4: #L2 #I2 #V2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_bind2_dec I1 I2) #HI
+ [ elim (eq_term_dec V1 V2) #HV
+ [ elim (IHL1 L2) -IHL1 /2 width=1 by or_introl/ #HL ]
+ ]
+ @or_intror #H destruct /2 width=1 by/
+| /2 width=1 by or_introl/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_lpair_lpair_aux: ∀I1,I2,L1,L2,V1,V2. L1.ⓑ{I1}V1 = L2.ⓑ{I2}V2 →
+ ∧∧L1 = L2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma discr_lpair_x_xy: ∀I,V,L. L = L.ⓑ{I}V → ⊥.
+#I #V #L elim L -L
+[ #H destruct
+| #L #J #W #IHL #H
+ elim (destruct_lpair_lpair_aux … H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.