(************************************************************************)
(*         *   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.
(* begin hide *)
Notation Rmin_pos := Rmin_pos (only parsing). (* compat *)
(* end hide *)

Lemma maj_term1 :
forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal)
(f1 f2:R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h:R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f1d ->
Rabs h < Rmin eps_f2 alp_f2 ->
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.