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.

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).