Definition complete_option {A : Type} x0 (x : option A) : A :=
match x with
| None => x0
| Some x => x
end.Definition option_le x y : Prop := 
 exists vx vy, x = Some vx /\ y = Some vy /\ le vx vy.Notation "x ≤p y" := (option_le x y) (at level 60).Definition option_choice {A B: Type} (f g: A -> option B) x :=
match f x with 
| Some v => Some v
| None => g x
end.Notation " f ;; g " := (option_choice f g) (at level 60).

Lemma option_map_Some {A B : Type} (f : A -> B) x v: option_map f x = Some v -> exists v', x = Some v'.