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

(* Specifications: *)
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.