#[global]
Instance split_ok1 s x `(Ok s) : Ok (split x s)#l.#[global]
Instance split_ok2 s x `(Ok s) : Ok (split x s)#r.(** ** Intersection *)

Ltac destruct_split := match goal with
 | H : split ?x ?s = << ?u, ?v, ?w >> |- _ =>
   assert ((split x s)#l = u) by (rewrite H; auto);
   assert ((split x s)#b = v) by (rewrite H; auto);
   assert ((split x s)#r = w) by (rewrite H; auto);
   clear H; subst u w
 end.

Lemma inter_spec_ok : forall s1 s2 `{Ok s1, Ok s2}, Ok (inter s1 s2) /\ (forall y, InT y (inter s1 s2) <-> InT y s1 /\ InT y s2).