(* Properties about atomic arity assignment on terms ************************)
-lemma aaa_lpxs_conf: ∀h,g,L1,T,A.
- L1 ⊢ T ⁝ A → ∀L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → L2 ⊢ T ⁝ A.
-#h #g #L1 #T #A #HT #L2 #HL12
-@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_lpx_conf/
+lemma aaa_lpxs_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+#h #g #G #L1 #T #A #HT #L2 #HL12
+@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_lpx_conf/
qed-.
-lemma aaa_lprs_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ➡* L2 → L2 ⊢ T ⁝ A.
-#L1 #T #A #HT #L2 #HL12
-@(aaa_lpxs_conf … HT) -A -T /2 width=3/
-qed-.
+lemma aaa_lprs_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by lprs_lpxs, aaa_lpxs_conf/ qed-.