End Map2.
End Proofs.
End Raw.
(** * Encapsulation  Now, in order to really provide a functor implementing [S], we need to encapsulate everything into a type of balanced binary search trees. *)
Module IntMake (I:Int)(X: OrderedType) <: S with Module E := X.
Module E := X.
Module Raw := Raw I X.
#[universes(template)]
Record bst (elt:Type) :=
Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.
Definition t := bst.
Definition key := E.t.
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 := Bst (empty_bst elt).
Definition is_empty m : bool := Raw.is_empty (this m).
Definition add x e m : t elt := Bst (add_bst x e (is_bst m)).
Definition remove x m : t elt := Bst (remove_bst x (is_bst m)).
Definition mem x m : bool := Raw.mem x (this m).
Definition find x m : option elt := Raw.find x (this m).
Definition map f m : t elt' := Bst (map_bst f (is_bst m)).
Definition mapi (f:key->elt->elt') m : t elt' :=
Bst (mapi_bst f (is_bst m)).
Definition map2 f m (m':t elt') : t elt'' :=
Bst (map2_bst f (is_bst m) (is_bst m')).
Definition elements m : list (key*elt) := Raw.elements (this m).
Definition cardinal m := Raw.cardinal (this m).
Definition fold (A:Type) (f:key->elt->A->A) m i := Raw.fold (A:=A) f (this m) i.
Definition equal cmp m m' : bool := Raw.equal cmp (this m) (this m').
Definition MapsTo x e m : Prop := Raw.MapsTo x e (this m).
Definition In x m : Prop := Raw.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.

Lemma remove_2 : forall m x y e, ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).