+(**************************************************************************)
+(* ___ *)
+(* ||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/grammar/lenv_px.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on pointwise extension ****************************************)
+
+lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R).
+#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5/
+]
+qed.
+
+lemma lpx_liftable_dedropable: ∀R. reflexive ? R →
+ t_liftable R → dedropable_sn (lpx R).
+#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+]
+qed.
+
+fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2.
+#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/
+| #K2 #I #V2 #X #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
+| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
+ elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
+| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ >commutative_plus normalize #H destruct
+]
+qed-.
+
+lemma lpx_dropable: ∀R. dropable_dx (lpx R).
+/2 width=5 by lpx_dropable_aux/ qed.