Fixpoint xmapi (m : t A) (i : key) : t B :=
       match m with
        | Leaf => @Leaf B
        | Node l o r => Node (xmapi l (append i (xO xH)))
                             (option_map (f i) o)
                             (xmapi r (append i (xI xH)))
       end.Definition mapi m := xmapi m xH.Definition map (f : A -> B) m := mapi (fun _ => f) m.

(* Specifications: *)
Lemma xgmapi: forall (A B: Type) (f: key -> A -> B) (i j : key) (m: t A), find i (xmapi f m j) = option_map (f (append j i)) (find i m).
Theorem gmapi: forall (A B: Type) (f: key -> A -> B) (i: key) (m: t A), find i (mapi f m) = option_map (f i) (find i m).
Lemma mapi_1 : forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:key->elt->elt'), MapsTo x e m -> exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m).
Lemma mapi_2 : forall (elt elt':Type)(m: t elt)(x:key)(f:key->elt->elt'), In x (mapi f m) -> In x m.
Lemma map_1 : forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:elt->elt'), MapsTo x e m -> MapsTo x (f e) (map f m).
Lemma map_2 : forall (elt elt':Type)(m: t elt)(x:key)(f:elt->elt'), In x (map f m) -> In x m.