#[global]
Instance add_ok s x `(Ok s) : Ok (add x s).(** ** Join *)

(** Function/Functional Scheme can't deal with internal fix.
    Let's do its job by hand: *)

Ltac join_tac :=
 let l := fresh "l" in
 intro l; induction l as [| lh ll _ lx lr Hlr];
   [ | intros x r; induction r as [| rh rl Hrl rx rr _]; unfold join;
     [ | destruct ((rh+2) <? lh) eqn:LT;
       [ match goal with |- context b [ bal ?a ?b ?c] =>
           replace (bal a b c)
           with (bal ll lx (join lr x (Node rh rl rx rr))); [ | auto]
         end
       | destruct ((lh+2) <? rh) eqn:LT';
         [ match goal with |- context b [ bal ?a ?b ?c] =>
             replace (bal a b c)
             with (bal (join (Node lh ll lx lr) x rl) rx rr); [ | auto]
           end
         | ] ] ] ]; intros.

Lemma join_spec : forall l x r y, InT y (join l x r) <-> X.eq y x \/ InT y l \/ InT y r.