--- /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 "static_2/notation/relations/lrsubeqa_3.ma".
+include "static_2/static/aaa.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+inductive lsuba (G:genv): relation lenv ≝
+| lsuba_atom: lsuba G (⋆) (⋆)
+| lsuba_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (atomic arity assignment)"
+ 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
+/2 width=4 by lsuba_inv_atom1_aux/ qed-.
+
+fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃⁝ K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+#G #L1 #L2 * -L1 -L2
+[ #J #K1 #H destruct
+| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #H destruct /3 width=9 by ex5_4_intro, or_intror/
+]
+qed-.
+
+lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
+ (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+/2 width=3 by lsuba_inv_bind1_aux/ qed-.
+
+fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
+/2 width=4 by lsuba_inv_atom2_aux/ qed-.
+
+fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K2 #H destruct
+| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #H destruct /3 width=9 by ex5_4_intro, or_intror/
+]
+qed-.
+
+lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_bind2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
+#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_bind/
+qed.