(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Set Implicit Arguments.
(** Streams *)
CoInductive Stream (A : Type) :=
Cons : A -> Stream A -> Stream A.
Section Streams.
Variable A : Type.
Notation Stream := (Stream A).
Definition hd (x:Stream) := match x with
| Cons a _ => a
end.
Definition tl (x:Stream) := match x with
| Cons _ s => s
end.
Fixpoint Str_nth_tl (n:nat) (s:Stream) : Stream :=
match n with
| O => s
| S m => Str_nth_tl m (tl s)
end.
Definition Str_nth (n:nat) (s:Stream) : A := hd (Str_nth_tl n s).

(* Specifications: *)
Lemma unfold_Stream :
forall x:Stream, x = match x with
| Cons a s => Cons a s
end.
Lemma tl_nth_tl :
forall (n:nat) (s:Stream), tl (Str_nth_tl n s) = Str_nth_tl n (tl s).