(************************************************************************)
(*         *   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)         *)
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Local Open Scope R_scope.

(* The main result *)
Theorem cauchy_finite :
forall (An Bn:nat -> R) (N:nat),
(0 < N)%nat ->
sum_f_R0 An N * sum_f_R0 Bn N =
sum_f_R0 (fun k:nat => sum_f_R0 (fun p:nat => An p * Bn (k - p)%nat) k) N +
sum_f_R0
(fun k:nat =>
sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (N - l)%nat)
(pred (N - k))) (pred N).