Module E := X.Module Import AvlProofs := AvlProofs I X.#[universes(template)]
 Record bbst (elt:Type) :=
  Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.#[universes(template)]
 Record bbst (elt:Type) :=
  Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.#[universes(template)]
 Record bbst (elt:Type) :=
  Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.#[universes(template)]
 Record bbst (elt:Type) :=
  Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.#[universes(template)]
 Record bbst (elt:Type) :=
  Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.Definition t := bbst.Definition key := E.t.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.

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