--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/length.ma".
+
+(* LOCAL ENVIRONMENT EQUALITY ***********************************************)
+
+inductive leq: lenv → nat → nat → lenv → Prop ≝
+| leq_sort: ∀d,e. leq (⋆) d e (⋆)
+| leq_comp: ∀L1,L2,I1,I2,V1,V2.
+ leq L1 0 0 L2 → leq (L1. 𝕓{I1} V1) 0 0 (L2. 𝕓{I2} V2)
+| leq_eq: ∀L1,L2,I,V,e. leq L1 0 e L2 → leq (L1. 𝕓{I} V) 0 (e + 1) (L2.𝕓{I} V)
+| leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
+ leq L1 d e L2 → leq (L1. 𝕓{I1} V1) (d + 1) e (L2. 𝕓{I2} V2)
+.
+
+interpretation "local environment equality" 'Eq L1 d e L2 = (leq L1 d e L2).
+
+(* Basic properties *********************************************************)
+
+lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
+#d elim d -d
+[ #e elim e -e [ #L elim L -L /2/ | #e #IHe #L elim L -L /2/ ]
+| #d #IHd #e #L elim L -L /2/
+]
+qed.
+
+lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
+qed.
+
+lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
+ ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
+
+#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
+qed.
+
+lemma leq_fwd_length: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e; normalize //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma leq_inv_sort1_aux: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e
+[ //
+| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #H destruct
+| #L1 #L2 #I #V #e #_ #_ #H destruct
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #_ #H destruct
+qed.
+
+lemma leq_inv_sort1: ∀L2,d,e. ⋆ [d, e] ≈ L2 → L2 = ⋆.
+/2 width=5/ qed.
+
+lemma leq_inv_sort2: ∀L1,d,e. L1 [d, e] ≈ ⋆ → L1 = ⋆.
+/3/ qed.