Library Coq.Reals.Rsigma

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import PartSum.
Require Import Lia.
Local Open Scope R_scope.

Set Implicit Arguments.

Section Sigma.

  Variable f : nat -> R.

  Definition sigma (low high:nat) : R :=
    sum_f_R0 (fun k:nat => f (low + k)) (high - low).

  Theorem sigma_split :
    forall low high k:nat,
      (low <= k)%nat ->
      (k < high)%nat -> sigma low high = sigma low k + sigma (S k) high.

  Theorem sigma_diff :
    forall low high k:nat,
      (low <= k)%nat ->
      (k < high)%nat -> sigma low high - sigma low k = sigma (S k) high.

  Theorem sigma_diff_neg :
    forall low high k:nat,
      (low <= k)%nat ->
      (k < high)%nat -> sigma low k - sigma low high = - sigma (S k) high.

  Theorem sigma_first :
    forall low high:nat,
      (low < high)%nat -> sigma low high = f low + sigma (S low) high.

  Theorem sigma_last :
    forall low high:nat,
      (low < high)%nat -> sigma low high = f high + sigma low (pred high).

  Theorem sigma_eq_arg : forall low:nat, sigma low low = f low.

End Sigma.