+
+(* Advanced inversion lemmas ************************************************)
+
+lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| →
+ ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K.
+#d >commutative_plus @(nat_ind_plus … d) -d
+[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+ >(length_inv_zero_sn … H1) -K
+ @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *)
+| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+ >H1 in IHd; -H1 #IHd
+ elim (IHd K ?) -IHd // #J #L #W #H1 #H2 destruct
+ @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *)
+ >append_length /2 width=1/
+]
+qed-.