Fixpoint map1 (A B: Type) (f: A -> B) (m: t A) {struct m} : t B :=
    match m with
    | Leaf => Leaf
    | Node l o r => Node (map1 f l) (option_map f o) (map1 f r)
    end.

(* Specifications: *)
Theorem gmap1: forall (A B: Type) (f: A -> B) (i: elt) (m: t A), get i (map1 f m) = option_map f (get i m).
Theorem gmap1: forall (A B: Type) (f: A -> B) (i: elt) (m: t A), get i (map1 f m) = option_map f (get i m).