(* Properties concerning atomic arity assignment ****************************)
-lemma lsuba_aaa_conf: ∀L1,V,A. L1 ⊢ V ⁝ A → ∀L2. L1 ⁝⊑ L2 → L2 ⊢ V ⁝ A.
-#L1 #V #A #H elim H -L1 -V -A
+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 #L1 #K1 #V #A #i #HLK1 #HV #IHV #L2 #HL12
+| #I #G #L1 #K1 #V #A #i #HLK1 #HV #IHV #L2 #HL12
elim (lsuba_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
elim (lsuba_inv_pair1 … H) -H * #K2
[ #HK12 #H destruct /3 width=5/
]
qed-.
-lemma lsuba_aaa_trans: ∀L2,V,A. L2 ⊢ V ⁝ A → ∀L1. L1 ⁝⊑ L2 → L1 ⊢ V ⁝ A.
-#L2 #V #A #H elim H -L2 -V -A
+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 #L2 #K2 #V #A #i #HLK2 #H1V #IHV #L1 #HL12
+| #I #G #L2 #K2 #V #A #i #HLK2 #H1V #IHV #L1 #HL12
elim (lsuba_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
elim (lsuba_inv_pair2 … H) -H * #K1
[ #HK12 #H destruct /3 width=5/