Definition remove_In (X: Set) n (v: Vector.t X (S n)) x i (p: List.In x (remove v i)): List.In x v
:= incl_In x p (SkipList_incl (SkipList_remove i v)).
(* nats *)

Lemma In_nats_S v u w: List.In w (nats u v) -> List.In (Snb w) (nats (S u) v).
Proof with auto. (* todo: rename *)
induction v in u, w |- *...