--- /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 "basic_2/multiple/frees.ma".
+include "basic_2/multiple/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* 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
+elim (llpx_sn_inv_alt_r … HL12) -HL12 #HL12 #IH
+lapply (ldrop_fwd_length_lt2 … HLK2) #Hj
+elim (ldrop_O1_lt (Ⓕ) L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
+elim (IH … HnU HLK1 HLK2) // -IH -HLK2 /5 width=11 by frees_be/
+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⦄.
+/2 width=6 by llpx_sn_frees_trans_aux/ qed-.