--- /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_2A/substitution/lpx_sn_alt.ma".
+include "basic_2A/multiple/llor.ma".
+include "basic_2A/multiple/lleq_alt.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Inversion lemmas on pointwise union for local environments ****************)
+
+lemma llpx_sn_llor_fwd_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T,l. llpx_sn R l T L1 L2 →
+ ∀L. L1 ⋓[T, l] L2 ≡ L → lpx_sn R L1 L.
+#R #HR #L1 #L2 #T #l #H1 #L #H2
+elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
+elim H2 -H2 #_ #HL1 #IH2
+@lpx_sn_intro_alt // #I1 #I #K1 #K #V1 #V #i #HLK1 #HLK
+lapply (drop_fwd_length_lt2 … HLK) #HiL
+elim (drop_O1_lt (Ⓕ) L2 i) // -HiL -HL1 -HL12 #I2 #K2 #V2 #HLK2
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK * /2 width=1 by conj/
+#HnT #H1 #H2 elim (IH1 … HnT … HLK1 HLK2) -IH1 -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed-.