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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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.
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(*            Newton's Integral            *)
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Definition Newton_integrable (f:R -> R) (a b:R) : Type :=
{ g:R -> R | antiderivative f g a b \/ antiderivative f g b a }.
Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R :=
let (g,_) := pr in g b - g a.

(* $\int_a^b \lambda f + g = \lambda \int_a^b f + \int_a^b f *)
Lemma NewtonInt_P6 :
forall (f g:R -> R) (l a b:R) (pr1:Newton_integrable f a b)
(pr2:Newton_integrable g a b),
NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) =
l * NewtonInt f a b pr1 + NewtonInt g a b pr2.