(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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)         *)
(************************************************************************)
Local Open Scope R_scope.
(** TT Ak; 0<=k<=N *)
Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R :=
match N with
| O => f O
| S p => prod_f_R0 f p * f (S p)
end.
Notation prod_f_SO := (fun An N => prod_f_R0 (fun n => An (S n)) N).

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Lemma prod_SO_split :
forall (An:nat -> R) (n k:nat),
(k < n)%nat ->
prod_f_R0 An n =
prod_f_R0 An k * prod_f_R0 (fun l:nat => An (k +1+l)%nat) (n - k -1).