Fixpoint partial_stream_clen {A: Type} (h: nat -> stream A) (n: nat): nat * nat :=
  match n with
  | 0 => (0, 0)
  | S n => let (k, m) := partial_stream_clen h n in
           match h k (S m) with
           | Some _ => (k, S m)
           | None => (S k, 0)
           end
  end.Fixpoint partial_stream_clen' {A: Type} (h: nat -> stream A) (n: nat): option (nat * nat) :=
  let (k, m) := partial_stream_clen h n in
  match n with
  | 0 => match h k m with
         | Some _ => Some (k, m)
         | None => None
         end
  | S n => match h k m, partial_stream_clen' h n with
           | Some _, Some _ => Some (k, m)
           | _, _ => None
           end
  end.

Lemma partial_stream_clen_inf0 {A: Type}: forall (h: nat -> stream A) (n: nat), is_inf_stream (h 0) -> partial_stream_clen h n = (0, n).