induction w...
simpl.
intros.
rewrite IHw...
Qed.
Require vec.

(* Specifications: *)
Lemma filtered_sort (T: Set) (R: relation T) (P: preorder T R) (p: T -> T -> bool) (pc: forall x y, p y x = true -> ~ R y x) (l: list T): vec.sorted R l ->
elemsR le (map (fun x => length (filter (p (fst x)) (snd x))) (splits l)) (nats 0 (length l)).