fact lpx_sn_dropable_aux: ∀R,L2,K2,s,l,m. ⬇[s, l, m] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
l = 0 → ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & lpx_sn R K1 K2.
#R #L2 #K2 #s #l #m #H elim H -L2 -K2 -l -m
-[ #l #m #Hm #X #H >(lpx_sn_inv_atom2 … H) -H
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom2 … H) -H
/4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
#K1 #V1 #HK12 #HV12 #H destruct
| #I #L2 #K2 #V2 #m #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
#L1 #V1 #HL12 #HV12 #H destruct
elim (IHLK2 … HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_
- <plus_n_Sm #H destruct
+| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_ #H elim (ysucc_inv_O_dx … H)
]
qed-.