(** ** Insertion *)
(** The next lemmas combine simultaneous results about rbt and arbt. A first solution here: statement with [if ... then ... else] *)
Definition ifred s (A B:Prop) := rcase (fun _ _ _ => A) (fun _ => B) s.

Lemma ins_rr_rb x s n : rbt n s ->
ifred s (rrt n (ins x s)) (rbt n (ins x s)).