Definition fin_stream {A: Type} (h: list A) : stream A.
exists (fun n => nth_error h n).
intros.
rewrite nth_error_None_iff in H0 |- *.
omega.
Defined.
Definition inf_stream {A: Type} (h: nat -> A) : stream A.
exists (fun n => Some (h n)).
intros.
congruence.
Defined.
Definition empty_stream {A: Type}: stream A := fin_stream nil.
Definition empty_stream_spec {A: Type}: forall n, @empty_stream A n = None.
Program Definition stream_map {A B: Type} (f: A -> B) (h: stream A): stream B :=
fun n => match h n return option B with
| Some a => Some (f a)
| None => None
end.
Next Obligation.
pose proof proj2_sig h x y H.
change (proj1_sig h) with (stream_get h) in H1.
destruct (h x), (h y); try congruence.
specialize (H1 eq_refl); congruence.
Qed.
Program Definition fstn_stream {A: Type} (n: nat) (h: stream A) : stream A :=
fun m => if le_lt_dec n m then None else h m.
Next Obligation.
destruct (le_lt_dec n x), (le_lt_dec n y); try congruence.
+ omega.
+ apply (stream_sound1 h x y); auto; omega.
Qed.
Definition skipn_stream {A: Type} (n: nat) (h: stream A) : stream A.
exists (fun m => h (m + n)).
intros.
revert H0; apply (stream_sound1 h).
omega.
Defined.
Definition partial_stream_len {A: Type} (h: stream A) (n: nat): option nat :=
match h n with
| Some _ => None
| None => Some
((fix f (m: nat): nat :=
match h m with
| Some _ => S m
| None => match m with
| 0 => 0
| S m0 => f m0
end
end) n)
end.
Program Definition stream_app {A: Type} (h1 h2: stream A): stream A :=
fun n: nat =>
match partial_stream_len h1 n return option A with
| None => h1 n
| Some m => h2 (n - m)
end.
Next Obligation.
destruct (partial_stream_len h1 x) eqn:?H.
+ pose proof H1.
apply (partial_stream_len_mono2 h1 x y n) in H1; [| omega].
rewrite H1.
apply (stream_sound1 h2 (x - n) (y - n)); auto.
omega.
+ unfold partial_stream_len in H1.
rewrite H0 in H1.
congruence.
Qed.
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.
Program Definition stream_capp {A: Type} (h: nat -> stream A): stream A :=
fun n: nat =>
match partial_stream_clen' h n return option A with
| Some (k, m) => h k m
| None => None
end.
Next Obligation.
assert (x <= y) by omega; clear H.
rewrite partial_stream_clen'_None_iff in H0 |- *.
induction H1; auto.
clear H0 H1 x.
Opaque partial_stream_clen.
simpl.
Transparent partial_stream_clen.
destruct (partial_stream_clen h (S m)) as [k n].
rewrite IHle.
destruct (h k n); auto.
Qed.

Lemma stream_capp_cons {A: Type}: forall (h: nat -> stream A),
~ is_empty_stream (h 0) ->
stream_capp h = stream_app (h 0) (stream_capp (fun n => h (S n))).