--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground/notation/functions/downspoon_2.ma".
+include "ground/lib/stream_eq.ma".
+include "ground/lib/arith.ma".
+
+(* STREAMS ******************************************************************)
+
+definition hd (A:Type[0]): stream A → A ≝
+ λt. match t with [ seq a _ ⇒ a ].
+
+definition tl (A:Type[0]): stream A → stream A ≝
+ λt. match t with [ seq _ t ⇒ t ].
+
+interpretation "tail (stream)" 'DownSpoon A t = (tl A t).
+
+(* basic properties *********************************************************)
+
+lemma hd_rew (A) (a) (t): a = hd A (a⨮t).
+// qed.
+
+lemma tl_rew (A) (a) (t): t = tl A (a⨮t).
+// qed.
+
+lemma eq_stream_split (A) (t): (hd … t) ⨮ ⫰t ≗{A} t.
+#A * //
+qed.