Library Stdlib.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).