Definition Annotation_sub (A1 A2: option Annotation):Prop := 
  match A1, A2 with
    _, None => True
  | Some (StrongAnnotation _), Some (WeakAnnotation _) => True
  | Some (StrongAnnotation X), Some (StrongAnnotation Y) => X=Y (*maybe have entailment here?*)
  | X, Y => X=Y 
  end.

Lemma Annotation_sub_trans a1 a2 a3: Annotation_sub a1 a2 -> Annotation_sub a2 a3 -> Annotation_sub a1 a3.