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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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Lemma derive_increasing_interv_ax :
forall (a b:R) (f:R -> R) (pr:derivable f),
a < b ->
((forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) ->
forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
((forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y).