(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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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).

(* 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.
End Sigma.