Definition Mux := If BV.Definition lowbit (l : list bool) :=
  match l with
  | nil => false
  | b :: _ => b
  end.Definition highs (l : list bool) :=
  match l with
  | nil => nil (A:=bool)
  | _ :: v => v
  end.

Lemma app_lowbit_highs : forall v : BV, v <> nilbv -> appbv (consbv (lowbit v) nilbv) (highs v) = v.