Module E := X.Module Raw := Raw I X.#[universes(template)]
 Record bst (elt:Type) :=
  Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.#[universes(template)]
 Record bst (elt:Type) :=
  Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.#[universes(template)]
 Record bst (elt:Type) :=
  Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.#[universes(template)]
 Record bst (elt:Type) :=
  Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.Definition t := bst.Definition key := E.t.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.

(* Specifications: *)
Lemma MapsTo_1 : forall m x y e, E.eq x y -> MapsTo x e m -> MapsTo y e m.
Lemma mem_1 : forall m x, In x m -> mem x m = true.
Lemma mem_2 : forall m x, mem x m = true -> In x m.
Lemma empty_1 : Empty empty.
Lemma is_empty_1 : forall m, Empty m -> is_empty m = true.
Lemma is_empty_2 : forall m, is_empty m = true -> Empty m.
Lemma add_1 : forall m x y e, E.eq x y -> MapsTo y e (add x e m).
Lemma add_2 : forall m x y e e', ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
Lemma add_3 : forall m x y e e', ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
Lemma remove_1 : forall m x y, E.eq x y -> ~ In y (remove x m).
Lemma remove_2 : forall m x y e, ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
Lemma remove_3 : forall m x y e, MapsTo y e (remove x m) -> MapsTo y e m.
Lemma find_1 : forall m x e, MapsTo x e m -> find x m = Some e.
Lemma find_2 : forall m x e, find x m = Some e -> MapsTo x e m.
Lemma fold_1 : forall m (A : Type) (i : A) (f : key -> elt -> A -> A), fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.
Lemma elements_1 : forall m x e, MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
Lemma elements_2 : forall m x e, InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
Lemma elements_3 : forall m, sort lt_key (elements m).
Lemma elements_3w : forall m, NoDupA eq_key (elements m).
Lemma cardinal_1 : forall m, cardinal m = length (elements m).