-(* Basic_1: was: drop_conf_lt *)
-theorem ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
- ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 &
- ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
-#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
-[ #d #e #e2 #K2 #I #V2 #H
- lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #L #I #V #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
+(* Note: apparently this was missing in basic_1 *)
+theorem ldrop_conf_be: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. ⇩[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #L0 #K0 #I #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HL20
+ [ -IHLK0 -He21 destruct <minus_n_O /3 width=3/
+ | -HLK0 <minus_le_minus_minus_comm //
+ elim (IHLK0 … HL20 ? ?) -L0 // /2 width=1/ /2 width=3/
+ ]
+| #L0 #K0 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HL02
+ elim (IHLK0 … HL02 ? ?) -L0 /2 width=1/ /3 width=3/
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem ldrop_conf_le: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ⇩[0, e2] L1 ≡ L & ⇩[d1 - e2, e1] L2 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct /2 width=3/
+| #K0 #I #V0 #L2 #e2 #H1 #H2
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct
+ lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /2 width=3/
+| #K0 #K1 #I #V0 #e1 #HK01 #_ #L2 #e2 #H1 #H2
+ lapply (le_n_O_to_eq … H2) -H2 #H destruct
+ lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #e2 #H #He2d1