Lemma cut2_exists {A: Type}: forall (h: stream A) (P: A -> Prop), (exists h1 h2, is_fin_stream h1 /\ h = stream_app h1 h2 /\ (forall n a, h1 n = Some a -> ~ P a) /\ (exists a, h2 0 = Some a /\ P a)) \/ (forall n a, h n = Some a -> ~ P a).