Fixpoint xfoldi (m : t A) (v : B) (i : key) :=
      match m with
        | Leaf _ => v
        | Node l (Some x) r =>
          xfoldi r (f i x (xfoldi l v (append i 2))) (append i 3)
        | Node l None r =>
          xfoldi r (xfoldi l v (append i 2)) (append i 3)
      end.

Lemma xfoldi_1 : forall m v i, xfoldi m v i = fold_left (fun a p => f (fst p) (snd p) a) (xelements m i) v.