Definition union_default_filter m :=
  match m with
  | nil => fun _ => false
  | hd :: _ => fun m => if member_dec hd m then true else false
  end.Definition is_default_filter {A} f (l: list A) :=
  match l with
  | nil => True
  | hd :: _ => f hd = true
  end.

Lemma union_default_filter_is_default_filter: forall m, is_default_filter (union_default_filter m) m.