-fact lpx_sn_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
- d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e
-[ #d #X #H >(lpx_sn_inv_atom2 … H) -H /2 width=3/
-| #K2 #I #V2 #X #H
- elim (lpx_sn_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
-| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
- elim (lpx_sn_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
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+ #K1 #V1 #HK12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+ #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ <plus_n_Sm #H destruct