Module Raw := Raw X.Module E := X.Definition key := E.t.Record slist (elt:Type) :=
  {this :> Raw.t elt; sorted : sort (@Raw.PX.ltk elt) this}.Record slist (elt:Type) :=
  {this :> Raw.t elt; sorted : sort (@Raw.PX.ltk elt) this}.Record slist (elt:Type) :=
  {this :> Raw.t elt; sorted : sort (@Raw.PX.ltk elt) this}.Definition t (elt:Type) : Type := slist elt.Implicit Types m : t elt.Implicit Types x y : key.Implicit Types e : elt.Definition empty : t elt := Build_slist (Raw.empty_sorted elt).Definition is_empty m : bool := Raw.is_empty (this m).Definition add x e m : t elt := Build_slist (Raw.add_sorted (sorted m) x e).Definition find x m : option elt := Raw.find x (this m).Definition remove x m : t elt := Build_slist (Raw.remove_sorted (sorted m) x).Definition mem x m : bool := Raw.mem x (this m).Definition map f m : t elt' := Build_slist (Raw.map_sorted (sorted m) f).Definition mapi (f:key->elt->elt') m : t elt' := Build_slist (Raw.mapi_sorted (sorted m) f).Definition map2 f m (m':t elt') : t elt'' :=
   Build_slist (Raw.map2_sorted f (sorted m) (sorted m')).Definition elements m : list (key*elt) := @Raw.elements elt (this m).Definition cardinal m := length (this m).Definition fold (A:Type)(f:key->elt->A->A) m (i:A) : A := @Raw.fold elt A f (this m) i.Definition equal cmp m m' : bool := @Raw.equal elt cmp (this m) (this m').Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e (this m).Definition In x m : Prop := Raw.PX.In x (this m).Definition Empty m : Prop := Raw.Empty (this m).Definition Equal m m' := forall y, find y m = find y m'.Definition Equiv (eq_elt:elt->elt->Prop) m m' :=
         (forall k, In k m <-> In k m') /\
         (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e').Definition Equivb cmp m m' : Prop := @Raw.Equivb elt cmp (this m) (this m').Definition eq_key : (key*elt) -> (key*elt) -> Prop := @Raw.PX.eqk elt.Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop:= @Raw.PX.eqke elt.Definition lt_key : (key*elt) -> (key*elt) -> Prop := @Raw.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).