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 tycontext_sub_trans: forall Delta1 Delta2 Delta3, tycontext_sub Delta1 Delta2 -> tycontext_sub Delta2 Delta3 -> tycontext_sub Delta1 Delta3.