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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/static/aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+inductive lsuba: relation lenv ≝
+| lsuba_atom: lsuba (⋆) (⋆)
+| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ⁝ A → L2 ⊢ W ⁝ A →
+              lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+  "local environment refinement (atomic arity assigment)"
+  'CrSubEqA L1 L2 = (lsuba L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ⁝⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsuba_inv_atom1: ∀L2. ⋆ ⁝⊑ L2 → L2 = ⋆.
+/2 width=3/ qed-.
+
+fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+                          (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+                          ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                    L2 = K2. ⓛW & I = Abbr.
+#L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+]
+qed.
+
+lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ⁝⊑ L2 →
+                       (∃∃K2. K1 ⁝⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+                       ∃∃K2,W,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                 L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ⁝⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubc_inv_atom2: ∀L1. L1 ⁝⊑ ⋆ → L1 = ⋆.
+/2 width=3/ qed-.
+
+fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ⁝⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+                          (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+                          ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                    L1 = K1. ⓓV & I = Abst.
+#L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+]
+qed.
+
+lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ⁝⊑ K2. ⓑ{I} W →
+                       (∃∃K1. K1 ⁝⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+                       ∃∃K1,V,A. K1 ⊢ V ⁝ A & K2 ⊢ W ⁝ A & K1 ⁝⊑ K2 &
+                                 L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuba_refl: ∀L. L ⁝⊑ L.
+#L elim L -L // /2 width=1/
+qed.