# Library Coq.Reals.Cauchy_prod

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import PartSum.
Local Open Scope R_scope.
Lemma sum_N_predN :
forall (An:nat -> R) (N:nat),
(0 < N)%nat -> sum_f_R0 An N = sum_f_R0 An (pred N) + An N.
Lemma sum_plus :
forall (An Bn:nat -> R) (N:nat),
sum_f_R0 (fun l:nat => An l + Bn l) N = sum_f_R0 An N + sum_f_R0 Bn N.
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).