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(*         *      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.
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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.

Lemma antiderivative_P2 :
forall (f F0 F1:R -> R) (a b c:R),
antiderivative f F0 a b ->
antiderivative f F1 b c ->
antiderivative f
(fun x:R =>
match Rle_dec x b with
| left _ => F0 x
| right _ => F1 x + (F0 b - F1 b)
end) a c.