(************************************************************************)
(*         *   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)         *)
(************************************************************************)

Require Import Rbase.
Require Import Ranalysis_reg.
Require Import Rfunctions.
Require Import Rseries.
Require Import Lra.
Require Import RiemannInt.
Require Import SeqProp.
Require Import Max.
Require Import Omega.
Require Import Lra.
Local Open Scope R_scope.

Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub,
       lb < ub ->
       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
       (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
       (forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
       (forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y).
forall (f g : R -> R) (lb ub : R),
lb < ub ->
(forall x y : R,
 lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x : R,
 f lb <= x -> x <= f ub -> comp f g x = id x) ->
(forall x : R,
 f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
forall x y : R,
f lb <= x -> x < y -> y <= f ub -> g x < g y
Proof.
forall (f g : R -> R) (lb ub : R),
lb < ub ->
(forall x y : R,
 lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x : R,
 f lb <= x -> x <= f ub -> comp f g x = id x) ->
(forall x : R,
 f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
forall x y : R,
f lb <= x -> x < y -> y <= f ub -> g x < g y
  intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
g x < g y
  assert (x_encad : f lb <= x <= f ub) by lra.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
g x < g y
  assert (y_encad : f lb <= y <= f ub) by lra.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
g x < g y
  assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
g x < g y
  assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
g x < g y
  case (Rlt_dec (g x) (g y)); [ easy |].
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
~ g x < g y -> g x < g y
  intros Hfalse.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
g x < g y
  assert (Temp := Rnot_lt_le _ _ Hfalse).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
g x < g y
  enough (y <= x) by lra.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
y <= x
  replace y with (id y) by easy.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
id y <= x
  replace x with (id x) by easy.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
id y <= id x
  rewrite <- f_eq_g by easy.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
comp f g y <= id x
  rewrite <- f_eq_g by easy.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
comp f g y <= comp f g x
  assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
forall x0 y0 : R,
lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
comp f g y <= comp f g x {
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
forall x0 y0 : R,
lb <= x0 -> x0 <= y0 -> y0 < ub -> f x0 <= f y0
    intros m n lb_le_m m_le_n n_lt_ub.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
m, n:R
lb_le_m:lb <= m
m_le_n:m <= n
n_lt_ub:n < ub
f m <= f n
    case (m_le_n).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
m, n:R
lb_le_m:lb <= m
m_le_n:m <= n
n_lt_ub:n < ub
m < n -> f m <= f n
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
m, n:R
lb_le_m:lb <= m
m_le_n:m <= n
n_lt_ub:n < ub
m = n -> f m <= f n
    -
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
m, n:R
lb_le_m:lb <= m
m_le_n:m <= n
n_lt_ub:n < ub
m < n -> f m <= f n intros; apply Rlt_le, f_incr, Rlt_le; assumption.
    -
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
m, n:R
lb_le_m:lb <= m
m_le_n:m <= n
n_lt_ub:n < ub
m = n -> f m <= f n intros Hyp; rewrite Hyp; apply Req_le; reflexivity.
  }
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
x_encad:f lb <= x <= f ub
y_encad:f lb <= y <= f ub
gx_encad:lb <= g x <= ub
gy_encad:lb <= g y <= ub
Hfalse:~ g x < g y
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
comp f g y <= comp f g x
  apply f_incr2; intuition.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
g x < ub
  enough (g x <> ub) by lra.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
g x <> ub
  intro Hf.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
False
  assert (Htemp : (comp f g) x = f ub).
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
comp f g x = f ub
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
Htemp:comp f g x = f ub
False {
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
comp f g x = f ub
    unfold comp; rewrite Hf; reflexivity.
  }
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
Htemp:comp f g x = f ub
False
  rewrite f_eq_g in Htemp by easy.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
Htemp:id x = f ub
False
  unfold id in Htemp.
f, g:R -> R
lb, ub:R
lb_lt_ub:lb < ub
f_incr:forall x0 y0 : R,
lb <= x0 ->
x0 < y0 -> y0 <= ub -> f x0 < f y0
f_eq_g:forall x0 : R,
f lb <= x0 ->
x0 <= f ub -> comp f g x0 = id x0
g_ok:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x, y:R
lb_le_x:f lb <= x
x_lt_y:x < y
y_le_ub:y <= f ub
Hfalse:g x < g y -> False
Temp:g y <= g x
f_incr2:forall x0 y0 : R,
lb <= x0 ->
x0 <= y0 -> y0 < ub -> f x0 <= f y0
H:f lb <= x
H0:x <= f ub
H1:f lb <= y
H2:y <= f ub
H3:lb <= g x
H4:g x <= ub
H5:lb <= g y
H6:g y <= ub
Hf:g x = ub
Htemp:x = f ub
False
  lra.
Qed.

Lemma derivable_pt_id_interv : forall (lb ub x:R),
       lb <= x <= ub ->
       derivable_pt id x.
forall lb ub x : R, lb <= x <= ub -> derivable_pt id x
Proof.
forall lb ub x : R, lb <= x <= ub -> derivable_pt id x
intros.
lb, ub, x:R
H:lb <= x <= ub
derivable_pt id x
 reg.
Qed.

Lemma pr_nu_var2_interv : forall (f g : R -> R) (lb ub x : R) (pr1 : derivable_pt f x)
       (pr2 : derivable_pt g x),
       lb < ub ->
       lb < x < ub ->
       (forall h : R, lb < h < ub -> f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2.
forall (f g : R -> R) (lb ub x : R)
  (pr1 : derivable_pt f x) (pr2 : derivable_pt g x),
lb < ub ->
lb < x < ub ->
(forall h : R, lb < h < ub -> f h = g h) ->
derive_pt f x pr1 = derive_pt g x pr2
Proof.
forall (f g : R -> R) (lb ub x : R)
  (pr1 : derivable_pt f x) (pr2 : derivable_pt g x),
lb < ub ->
lb < x < ub ->
(forall h : R, lb < h < ub -> f h = g h) ->
derive_pt f x pr1 = derive_pt g x pr2
intros f g lb ub x Prf Prg lb_lt_ub x_encad local_eq.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
derive_pt f x Prf = derive_pt g x Prg
assert (forall x l, lb < x < ub -> (derivable_pt_abs f x l <-> derivable_pt_abs g x l)).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <-> derivable_pt_abs g x0 l
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 intros a l a_encad.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
derivable_pt_abs f a l <-> derivable_pt_abs g a l
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 unfold derivable_pt_abs, derivable_pt_lim.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((f (a + h) - f a) / h - l) < eps) <->
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 split.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((f (a + h) - f a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 intros Hyp eps eps_pos.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 elim (Hyp eps eps_pos) ; intros delta Hyp2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Pos_cond : Rmin delta (Rmin (ub - a) (a - lb)) > 0).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
Rmin delta (Rmin (ub - a) (a - lb)) > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  clear-a lb ub a_encad delta.
lb, ub, a:R
a_encad:lb < a < ub
delta:posreal
Rmin delta (Rmin (ub - a) (a - lb)) > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  apply Rmin_pos ; [exact ((cond_pos delta)) | apply Rmin_pos ] ; apply Rlt_Rminus ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 exists (mkposreal (Rmin delta (Rmin (ub - a) (a - lb))) Pos_cond).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (a + h) - f a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |} ->
Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 intros h h_neq h_encad.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 replace (g (a + h) - g a) with (f (a + h) - f a).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Hyp2 ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rabs h < delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rabs h < Rmin delta (Rmin (ub - a) (a - lb))
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rmin delta (Rmin (ub - a) (a - lb)) <= delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rmin delta (Rmin (ub - a) (a - lb)) <= delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg apply Rmin_l.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (local_eq2 : forall h : R, lb < h < ub -> - f h = - g h).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
forall h0 : R, lb < h0 < ub -> - f h0 = - g h0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; apply Ropp_eq_compat ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
f (a + h) - f a = g (a + h) - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 rewrite local_eq ; unfold Rminus.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) + - f a = g (a + h) + - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg rewrite local_eq2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) + - g a = g (a + h) + - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg reflexivity.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma2 : forall x y, Rabs x < Rabs y -> y > 0 -> x < y).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < y
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros m n Hyp_abs y_pos.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
m < n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg apply Rlt_le_trans with (r2:=Rabs n).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
m < Rabs n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
Rabs n <= n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
   apply Rle_lt_trans with (r2:=Rabs m) ; [ | assumption] ; apply RRle_abs.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
Rabs n <= n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
   apply Req_le ; apply Rabs_right ; apply Rgt_ge ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 split.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma : forall x y z, -z < y - x -> x < y + z).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
forall x0 y z : R, - z < y - x0 -> x0 < y + z
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; lra.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
- h < a - lb
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
Rabs (- h) < Rabs (a - lb)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg rewrite Rabs_Ropp.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
Rabs h < Rabs (a - lb)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=a-lb) ; [| apply RRle_abs] ;
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma : forall x y z, y < z - x -> x + y < z).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
forall x0 y z : R, y < z - x0 -> x0 + y < z
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; lra.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
h < ub - a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
Rabs h < Rabs (ub - a)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
ub - a > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=ub-a) ; [| apply RRle_abs] ;
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((f (a + h0) - f a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((f (a + h0) - f a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
ub - a > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
(forall eps : R,
 0 < eps ->
 exists delta : posreal,
   forall h : R,
   h <> 0 ->
   Rabs h < delta ->
   Rabs ((g (a + h) - g a) / h - l) < eps) ->
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 intros Hyp eps eps_pos.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 elim (Hyp eps eps_pos) ; intros delta Hyp2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Pos_cond : Rmin delta (Rmin (ub - a) (a - lb)) > 0).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
Rmin delta (Rmin (ub - a) (a - lb)) > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  clear-a lb ub a_encad delta.
lb, ub, a:R
a_encad:lb < a < ub
delta:posreal
Rmin delta (Rmin (ub - a) (a - lb)) > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  apply Rmin_pos ; [exact ((cond_pos delta)) | apply Rmin_pos ] ; apply Rlt_Rminus ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 exists (mkposreal (Rmin delta (Rmin (ub - a) (a - lb))) Pos_cond).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta0 ->
  Rabs ((g (a + h) - g a) / h - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((g (a + h) - g a) / h - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |} ->
Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 intros h h_neq h_encad.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
Rabs ((f (a + h) - f a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 replace (f (a + h) - f a) with (g (a + h) - g a).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
Rabs ((g (a + h) - g a) / h - l) < eps
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Hyp2 ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rabs h < delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rabs h < Rmin delta (Rmin (ub - a) (a - lb))
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rmin delta (Rmin (ub - a) (a - lb)) <= delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  (h0 = 0 -> False) ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h = 0 -> False
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
H:lb < x
H0:x < ub
H1:lb < a
H2:a < ub
Rmin delta (Rmin (ub - a) (a - lb)) <= delta
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg apply Rmin_l.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (local_eq2 : forall h : R, lb < h < ub -> - f h = - g h).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
forall h0 : R, lb < h0 < ub -> - f h0 = - g h0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; apply Ropp_eq_compat ; intuition.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) - g a = f (a + h) - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 rewrite local_eq ; unfold Rminus.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) + - g a = g (a + h) + - f a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg rewrite local_eq2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
g (a + h) + - g a = g (a + h) + - g a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg reflexivity.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma2 : forall x y, Rabs x < Rabs y -> y > 0 -> x < y).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
forall x0 y : R, Rabs x0 < Rabs y -> y > 0 -> x0 < y
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros m n Hyp_abs y_pos.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
m < n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg apply Rlt_le_trans with (r2:=Rabs n).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
m < Rabs n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
Rabs n <= n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
   apply Rle_lt_trans with (r2:=Rabs m) ; [ | assumption] ; apply RRle_abs.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
m, n:R
Hyp_abs:Rabs m < Rabs n
y_pos:n > 0
Rabs n <= n
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
   apply Req_le ; apply Rabs_right ; apply Rgt_ge ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 split.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma : forall x y z, -z < y - x -> x < y + z).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
forall x0 y z : R, - z < y - x0 -> x0 < y + z
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; lra.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
lb < a + h
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
- h < a - lb
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
Rabs (- h) < Rabs (a - lb)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg rewrite Rabs_Ropp.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
Rabs h < Rabs (a - lb)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=a-lb) ; [| apply RRle_abs] ;
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R,
- z < y - x0 -> x0 < y + z
a - lb > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 assert (Sublemma : forall x y z, y < z - x -> x + y < z).
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
forall x0 y z : R, y < z - x0 -> x0 + y < z
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  intros ; lra.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
a + h < ub
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
h < ub - a
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Sublemma2.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
Rabs h < Rabs (ub - a)
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
ub - a > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=ub-a) ; [| apply RRle_abs] ;
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h0 : R, lb < h0 < ub -> f h0 = g h0
a, l:R
a_encad:lb < a < ub
Hyp:forall eps0 : R,
0 < eps0 ->
exists delta0 : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta0 ->
  Rabs ((g (a + h0) - g a) / h0 - l) < eps0
eps:R
eps_pos:0 < eps
delta:posreal
Hyp2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((g (a + h0) - g a) / h0 - l) < eps
Pos_cond:Rmin delta (Rmin (ub - a) (a - lb)) > 0
h:R
h_neq:h <> 0
h_encad:Rabs h <
{|
pos := Rmin delta (Rmin (ub - a) (a - lb));
cond_pos := Pos_cond |}
local_eq2:forall h0 : R,
lb < h0 < ub -> - f h0 = - g h0
Sublemma2:forall x0 y : R,
Rabs x0 < Rabs y -> y > 0 -> x0 < y
Sublemma:forall x0 y z : R, y < z - x0 -> x0 + y < z
ub - a > 0
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
 apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
f, g:R -> R
lb, ub, x:R
Prf:derivable_pt f x
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
 unfold derivable_pt in Prf.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:derivable_pt g x
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  unfold derivable_pt in Prg.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x0 l : R,
lb < x0 < ub ->
derivable_pt_abs f x0 l <->
derivable_pt_abs g x0 l
derive_pt f x Prf = derive_pt g x Prg
  elim Prf; intros x0 p.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x1 l : R,
lb < x1 < ub ->
derivable_pt_abs f x1 l <->
derivable_pt_abs g x1 l
x0:R
p:derivable_pt_abs f x x0
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 p) =
derive_pt g x Prg
  elim Prg; intros x1 p0.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 p) =
derive_pt g x
  (exist (fun l : R => derivable_pt_abs g x l) x1 p0)
  assert (Temp := p); rewrite H in Temp.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs g x x0
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 p) =
derive_pt g x
  (exist (fun l : R => derivable_pt_abs g x l) x1 p0)
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs f x x0
lb < x < ub
  unfold derivable_pt_abs in p.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_lim f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs g x x0
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 p) =
derive_pt g x
  (exist (fun l : R => derivable_pt_abs g x l) x1 p0)
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs f x x0
lb < x < ub
  unfold derivable_pt_abs in p0.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_lim f x x0
x1:R
p0:derivable_pt_lim g x x1
Temp:derivable_pt_abs g x x0
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 p) =
derive_pt g x
  (exist (fun l : R => derivable_pt_abs g x l) x1 p0)
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs f x x0
lb < x < ub
  simpl in |- *.
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_lim f x x0
x1:R
p0:derivable_pt_lim g x x1
Temp:derivable_pt_abs g x x0
x0 = x1
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs f x x0
lb < x < ub
  apply (uniqueness_limite g x x0 x1 Temp p0).
f, g:R -> R
lb, ub, x:R
Prf:{l : R | derivable_pt_abs f x l}
Prg:{l : R | derivable_pt_abs g x l}
lb_lt_ub:lb < ub
x_encad:lb < x < ub
local_eq:forall h : R, lb < h < ub -> f h = g h
H:forall x2 l : R,
lb < x2 < ub ->
derivable_pt_abs f x2 l <->
derivable_pt_abs g x2 l
x0:R
p:derivable_pt_abs f x x0
x1:R
p0:derivable_pt_abs g x x1
Temp:derivable_pt_abs f x x0
lb < x < ub
  assumption.
Qed.


(* begin hide *)
Lemma leftinv_is_rightinv : forall (f g:R->R),
       (forall x y, x < y -> f x < f y) ->
       (forall x, (comp f g) x = id x) ->
       (forall x, (comp g f) x = id x).
forall f g : R -> R,
(forall x y : R, x < y -> f x < f y) ->
(forall x : R, comp f g x = id x) ->
forall x : R, comp g f x = id x
Proof.
forall f g : R -> R,
(forall x y : R, x < y -> f x < f y) ->
(forall x : R, comp f g x = id x) ->
forall x : R, comp g f x = id x
intros f g f_incr Hyp x.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
comp g f x = id x
 assert (forall x, f (g (f x)) = f x).
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
forall x0 : R, f (g (f x0)) = f x0
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
comp g f x = id x
  intros ; apply Hyp.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
comp g f x = id x
 assert(f_inj : forall x y, f x = f y -> x = y).
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
forall x0 y : R, f x0 = f y -> x0 = y
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  intros a b fa_eq_fb.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  case(total_order_T a b).
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
{a < b} + {a = b} -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  intro s ; case s ; clear s.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a < b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  intro Hf.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a < b
a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  assert (Hfalse := f_incr a b Hf).
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a < b
Hfalse:f a < f b
a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  apply False_ind.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a < b
Hfalse:f a < f b
False
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x apply (Rlt_not_eq (f a) (f b)) ; assumption.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  intuition.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  intro Hf.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a > b
a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x assert (Hfalse := f_incr b a Hf).
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a > b
Hfalse:f b < f a
a = b
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
  apply False_ind.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
a, b:R
fa_eq_fb:f a = f b
Hf:a > b
Hfalse:f b < f a
False
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x apply (Rlt_not_eq (f b) (f a)) ; [|symmetry] ; assumption.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
comp g f x = id x
 apply f_inj.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
f (comp g f x) = f (id x) unfold comp.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, comp f g x0 = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
f (g (f x)) = f (id x)
 unfold comp in Hyp.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, f (g x0) = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
f (g (f x)) = f (id x)
 rewrite Hyp.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, f (g x0) = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
id (f x) = f (id x)
 unfold id.
f, g:R -> R
f_incr:forall x0 y : R, x0 < y -> f x0 < f y
Hyp:forall x0 : R, f (g x0) = id x0
x:R
H:forall x0 : R, f (g (f x0)) = f x0
f_inj:forall x0 y : R, f x0 = f y -> x0 = y
f x = f x
 reflexivity.
Qed.
(* end hide *)

Lemma leftinv_is_rightinv_interv : forall (f g:R->R) (lb ub:R),
       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
       (forall y, f lb <= y -> y <= f ub -> (comp f g) y = id y) ->
       (forall x, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
       forall x,
       lb <= x <= ub ->
       (comp g f) x = id x.
forall (f g : R -> R) (lb ub : R),
(forall x y : R,
 lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall y : R,
 f lb <= y -> y <= f ub -> comp f g y = id y) ->
(forall x : R,
 f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
forall x : R, lb <= x <= ub -> comp g f x = id x
Proof.
forall (f g : R -> R) (lb ub : R),
(forall x y : R,
 lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall y : R,
 f lb <= y -> y <= f ub -> comp f g y = id y) ->
(forall x : R,
 f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
forall x : R, lb <= x <= ub -> comp g f x = id x
intros f g lb ub f_incr_interv Hyp g_wf x x_encad.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
comp g f x = id x
 assert(f_inj : forall x y, lb <= x <= ub -> lb <= y <= ub -> f x = f y -> x = y).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  intros a b a_encad b_encad fa_eq_fb.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  case(total_order_T a b).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
{a < b} + {a = b} -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  intro s ; case s ; clear s.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a < b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  intro Hf.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a < b
a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  assert (Hfalse := f_incr_interv a b (proj1 a_encad) Hf (proj2 b_encad)).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a < b
Hfalse:f a < f b
a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  apply False_ind.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a < b
Hfalse:f a < f b
False
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x apply (Rlt_not_eq (f a) (f b)) ; assumption.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a = b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  intuition.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
a > b -> a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  intro Hf.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a > b
a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x assert (Hfalse := f_incr_interv b a (proj1 b_encad) Hf (proj2 a_encad)).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a > b
Hfalse:f b < f a
a = b
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
  apply False_ind.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
a, b:R
a_encad:lb <= a <= ub
b_encad:lb <= b <= ub
fa_eq_fb:f a = f b
Hf:a > b
Hfalse:f b < f a
False
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x apply (Rlt_not_eq (f b) (f a)) ; [|symmetry] ; assumption.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
comp g f x = id x
 assert (f_incr_interv2 : forall x y, lb <= x -> x <= y -> y <= ub -> f x <= f y).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
forall x0 y : R,
lb <= x0 -> x0 <= y -> y <= ub -> f x0 <= f y
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x
  intros m n cond1 cond2 cond3.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x
   case cond2.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
m < n -> f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
m = n -> f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x
   intro cond.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
cond:m < n
f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
m = n -> f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x apply Rlt_le ; apply f_incr_interv ; assumption.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
m, n:R
cond1:lb <= m
cond2:m <= n
cond3:n <= ub
m = n -> f m <= f n
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x
   intro cond ; right ; rewrite cond ; reflexivity.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
comp g f x = id x
 assert (Hyp2:forall x, lb <= x <= ub -> f (g (f x)) = f x).
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
forall x0 : R, lb <= x0 <= ub -> f (g (f x0)) = f x0
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
comp g f x = id x
  intros ; apply Hyp.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x1 y : R,
lb <= x1 ->
x1 < y -> y <= ub -> f x1 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x1 : R,
f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x1 y : R,
lb <= x1 <= ub ->
lb <= y <= ub -> f x1 = f y -> x1 = y
f_incr_interv2:forall x1 y : R,
lb <= x1 ->
x1 <= y -> y <= ub -> f x1 <= f y
x0:R
H:lb <= x0 <= ub
f lb <= f x0
f, g:R -> R
lb, ub:R
f_incr_interv:forall x1 y : R,
lb <= x1 ->
x1 < y -> y <= ub -> f x1 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x1 : R,
f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x1 y : R,
lb <= x1 <= ub ->
lb <= y <= ub -> f x1 = f y -> x1 = y
f_incr_interv2:forall x1 y : R,
lb <= x1 ->
x1 <= y -> y <= ub -> f x1 <= f y
x0:R
H:lb <= x0 <= ub
f x0 <= f ub
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
comp g f x = id x  apply f_incr_interv2 ; intuition.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x1 y : R,
lb <= x1 ->
x1 < y -> y <= ub -> f x1 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x1 : R,
f lb <= x1 -> x1 <= f ub -> lb <= g x1 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x1 y : R,
lb <= x1 <= ub ->
lb <= y <= ub -> f x1 = f y -> x1 = y
f_incr_interv2:forall x1 y : R,
lb <= x1 ->
x1 <= y -> y <= ub -> f x1 <= f y
x0:R
H:lb <= x0 <= ub
f x0 <= f ub
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
comp g f x = id x 
 apply f_incr_interv2 ; intuition.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> comp f g y = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
comp g f x = id x
 unfold comp ; unfold comp in Hyp.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
g (f x) = id x
 apply f_inj.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
lb <= g (f x) <= ub
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
lb <= id x <= ub
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
f (g (f x)) = f (id x)
 apply g_wf ; apply f_incr_interv2 ; intuition.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
lb <= id x <= ub
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
f (g (f x)) = f (id x)
 unfold id ; assumption.
f, g:R -> R
lb, ub:R
f_incr_interv:forall x0 y : R,
lb <= x0 ->
x0 < y -> y <= ub -> f x0 < f y
Hyp:forall y : R,
f lb <= y -> y <= f ub -> f (g y) = id y
g_wf:forall x0 : R,
f lb <= x0 -> x0 <= f ub -> lb <= g x0 <= ub
x:R
x_encad:lb <= x <= ub
f_inj:forall x0 y : R,
lb <= x0 <= ub ->
lb <= y <= ub -> f x0 = f y -> x0 = y
f_incr_interv2:forall x0 y : R,
lb <= x0 ->
x0 <= y -> y <= ub -> f x0 <= f y
Hyp2:forall x0 : R,
lb <= x0 <= ub -> f (g (f x0)) = f x0
f (g (f x)) = f (id x)
 apply Hyp2 ; unfold id ; assumption.
Qed.