(************************************************************************)
(*         *   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)         *)
(************************************************************************)
Local Open Scope R_scope.

(* If f has a null derivative in ]a,b[ and is continue in [a,b], *)
(* then f is constant on [a,b] *)
Lemma null_derivative_loc :
forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x),
(forall x:R, a <= x <= b -> continuity_pt f x) ->
(forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) ->
constant_D_eq f (fun x:R => a <= x <= b) (f a).