# Library Coq.Reals.Rsigma

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import PartSum.
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.