(* Even when it is not known which of [k] and [s] is greater, [insert] commutes with itself. The formula is slightly horrid, but can be very useful. *)

Lemma insert_insert_always:
forall A k s (a b : option A) e,
raw_insert k a (raw_insert s b e) =
raw_insert (shift k s) b (raw_insert (if le_gt_dec k s then k else k - 1) a e).