+(**************************************************************************)
+(* ___ *)
+(* ||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/static/aaa_aaa.ma".
+include "basic_2A/static/lsuba.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Properties concerning atomic arity assignment ****************************)
+
+lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
+ ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
+#G #L1 #V #A #H elim H -G -L1 -V -A
+[ //
+| #I #G #L1 #K1 #V #A #i #HLK1 #HV #IHV #L2 #HL12
+ elim (lsuba_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+ elim (lsuba_inv_pair1 … H) -H * #K2
+ [ #HK12 #H destruct /3 width=5 by aaa_lref/
+ | #W0 #V0 #A0 #HV0 #HW0 #_ #H1 #H2 #H3 destruct
+ lapply (aaa_mono … HV0 … HV) #H destruct -V0 /2 width=5 by aaa_lref/
+ ]
+| /4 width=2 by lsuba_pair, aaa_abbr/
+| /4 width=1 by lsuba_pair, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
+ ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
+#G #L2 #V #A #H elim H -G -L2 -V -A
+[ //
+| #I #G #L2 #K2 #V #A #i #HLK2 #H1V #IHV #L1 #HL12
+ elim (lsuba_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsuba_inv_pair2 … H) -H * #K1
+ [ #HK12 #H destruct /3 width=5 by aaa_lref/
+ | #V0 #A0 #HV0 #H2V #_ #H1 #H2 destruct
+ lapply (aaa_mono … H2V … H1V) #H destruct -K2 /2 width=5 by aaa_lref/
+ ]
+| /4 width=2 by lsuba_pair, aaa_abbr/
+| /4 width=1 by lsuba_pair, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /3 width=1 by aaa_cast/
+]
+qed-.