(* Properties on context-sensitive free variables ***************************)
fact llpx_sn_frees_trans_aux: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
- ∀L2,U,d,i. L2 ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- ∀L1. llpx_sn R d U L1 L2 → L1 ⊢ i ϵ 𝐅*[d]⦃U⦄.
-#R #H1R #H2R #L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
-#I2 #L2 #K2 #U #W2 #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+ ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+ ∀L1. llpx_sn R l U L1 L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#R #H1R #H2R #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
elim (llpx_sn_inv_alt_r … HL12) -HL12 #HL12 #IH
lapply (drop_fwd_length_lt2 … HLK2) #Hj
elim (drop_O1_lt (Ⓕ) L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
qed-.
lemma llpx_sn_frees_trans: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
- ∀L1,L2,U,d. llpx_sn R d U L1 L2 →
- ∀i. L2 ⊢ i ϵ 𝐅*[d]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[d]⦃U⦄.
+ ∀L1,L2,U,l. llpx_sn R l U L1 L2 →
+ ∀i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
/2 width=6 by llpx_sn_frees_trans_aux/ qed-.