Global Instance Foldable_option {T} : Foldable (option T) T :=
  fun _ f d v =>
    match v with
      | None => d
      | Some x => f x d
    end.Global Instance Traversable_option : Traversable option :=
{| mapT := fun F _ _ _ f o =>
  match o with
    | None => pure None
    | Some o => ap (pure (@Some _)) (f o)
  end
|}.Global Instance Applicative_option : Applicative option :=
{| pure := @Some
 ; ap := fun _ _ f x =>
           match f , x with
             | Some f , Some x => Some (f x)
             | _ , _ => None
           end
|}.Global Instance Functor_option : Functor option :=
{| fmap := fun _ _ f x => match x with
                            | None => None
                            | Some x => Some (f x)
                          end |}.Inductive Roption : Relation_Definitions.relation (option T) :=
  | Roption_None : Roption None None
  | Roption_Some : forall x y, R x y -> Roption (Some x) (Some y).Inductive Roption : Relation_Definitions.relation (option T) :=
  | Roption_None : Roption None None
  | Roption_Some : forall x y, R x y -> Roption (Some x) (Some y).Inductive Roption : Relation_Definitions.relation (option T) :=
  | Roption_None : Roption None None
  | Roption_Some : forall x y, R x y -> Roption (Some x) (Some y).

Lemma Transitive_Roption : Transitive R -> Transitive Roption.