Section Elt.
Variable elt elt' elt'': Type.
Implicit Types m : t elt.
Implicit Types x y : key.
Implicit Types e : elt.
Definition empty : t elt := Bbst (empty_bst elt) (empty_avl elt).
Definition is_empty m : bool := is_empty (this m).
Definition add x e m : t elt :=
Bbst (add_bst x e (is_bst m)) (add_avl x e (is_avl m)).
Definition remove x m : t elt :=
Bbst (remove_bst x (is_bst m)) (remove_avl x (is_avl m)).
Definition mem x m : bool := mem x (this m).
Definition find x m : option elt := find x (this m).
Definition map f m : t elt' :=
Bbst (map_bst f (is_bst m)) (map_avl f (is_avl m)).
Definition mapi (f:key->elt->elt') m : t elt' :=
Bbst (mapi_bst f (is_bst m)) (mapi_avl f (is_avl m)).
Definition map2 f m (m':t elt') : t elt'' :=
Bbst (map2_bst f (is_bst m) (is_bst m')) (map2_avl f (is_avl m) (is_avl m')).
Definition elements m : list (key*elt) := elements (this m).
Definition cardinal m := cardinal (this m).
Definition fold (A:Type) (f:key->elt->A->A) m i := fold (A:=A) f (this m) i.
Definition equal cmp m m' : bool := equal cmp (this m) (this m').
Definition MapsTo x e m : Prop := MapsTo x e (this m).
Definition In x m : Prop := In0 x (this m).
Definition Empty m : Prop := Empty (this m).
Definition eq_key : (key*elt) -> (key*elt) -> Prop := @PX.eqk elt.
Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop := @PX.eqke elt.
Definition lt_key : (key*elt) -> (key*elt) -> Prop := @PX.ltk elt.
End Elt.

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).