Definition is_n_stream {A: Type} (n: nat) (h: stream A): Prop :=
  h n = None /\ forall n', n' < n -> h n' <> None.Definition is_fin_stream {A: Type} (h: stream A): Prop :=
  exists n, h n = None.Definition is_inf_stream {A: Type} (h: stream A): Prop :=
  forall n, h n <> None.Definition is_at_least_n_stream {A: Type} (n: nat) (h: stream A): Prop :=
  forall n', n' < n -> h n' <> None.Definition is_at_most_n_stream {A: Type} (n: nat) (h: stream A): Prop :=
  h n = None.Definition is_empty_stream {A: Type}: stream A -> Prop :=
  is_n_stream 0.

Lemma at_least_n_stream_Sn {A: Type}: forall (h: stream A) (n: nat), is_at_least_n_stream (S n) h <-> is_at_least_n_stream n h /\ ~ is_n_stream n h.