+lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L1| = |L2| + e.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1 by/
+qed-.
+
+lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by plus_minus, le_n/
+qed-.
+
+lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e = |L1| - |L2|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → e ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
+qed-.
+
+lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e.
+ ⇩[Ⓕ, d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
+#L1 #I2 #K2 #V2 #d #e #H
+lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
+qed-.
+
+lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|.
+#L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1 by lt_minus_to_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq1: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |L1| = |L2| → |K1| = |K2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2
+/2 width=2 by injective_plus_r/
+qed-.
+
+lemma ldrop_fwd_length_eq2: ∀L1,L2,K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ |K1| = |K2| → |L1| = |L2|.
+#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+lapply (ldrop_fwd_length … HLK1) -HLK1
+lapply (ldrop_fwd_length … HLK2) -HLK2 //
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e // normalize
+[ /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #H >H -H //
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #e #HL12 #_ #_
+ lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
+ @(le_to_lt_to_lt … HL12) -HL12 //
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+ >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
+]
+qed-.
+
+lemma ldrop_fwd_rfw: ∀I,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#I #L #K #V #i #HLK lapply (ldrop_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact ldrop_inv_O2_aux: ∀L1,L2,s,d,e. ⇩[s, d, e] L1 ≡ L2 → e = 0 → L1 = L2.
+#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+[ //
+| //
+| #I #L1 #L2 #V #e #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H
+ >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e //