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Datei: Max.v Sprache: Coq

Original von: Coq^©

(************************************************************************)
(* * 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) *)
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(* // * 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 Rfunctions.
Require Import Ranalysis1.
Require Import Rtopology.
Local Open Scope R_scope.

(* The Mean Value Theorem *)
Theorem MVT :
 forall (f g:R -> R) (a b:R) (pr1:forall c:R, a < c derivable_pt f c)
 (pr2:forall c:R, a < c derivable_pt g c),
 a 
 (forall c:R, a <= c <= b -> continuity_pt f c) ->
 (forall c:R, a <= c <= b -> continuity_pt g c) ->
 exists c : R,
 (exists P : a < c < b,
 (g b - g a) * derive_pt f c (pr1 c P) =
 (f b - f a) * derive_pt g c (pr2 c P)).
Proof.
 intros; assert (H2 := Rlt_le _ _ H).
 set (h := fun y:R => (g b - g a) * f y - (f b - f a) * g y).
 cut (forall c:R, a < c derivable_pt h c).
 intro X; cut (forall c:R, a <= c <= b -> continuity_pt h c).
 intro; assert (H4 := continuity_ab_maj h a b H2 H3).
 assert (H5 := continuity_ab_min h a b H2 H3).
 elim H4; intros Mx H6.
 elim H5; intros mx H7.
 cut (h a = h b).
 intro; set (M := h Mx); set (m := h mx).
 cut
 (forall (c:R) (P:a < c < b),
 derive_pt h c (X c P) =
 (g b - g a) * derive_pt f c (pr1 c P) -
 (f b - f a) * derive_pt g c (pr2 c P)).
 intro; case (Req_dec (h a) M); intro.
 case (Req_dec (h a) m); intro.
 cut (forall c:R, a <= c <= b -> h c = M).
 intro; cut (a < (a + b) / 2 < b).
(*** h constant ***)
 intro; exists ((a + b) / 2).
 exists H13.
 apply Rminus_diag_uniq; rewrite <- H9; apply deriv_constant2 with a b.
 elim H13; intros; assumption.
 elim H13; intros; assumption.
 intros; rewrite (H12 ((a + b) / 2)).
 apply H12; split; left; assumption.
 elim H13; intros; split; left; assumption.
 split.
 apply Rmult_lt_reg_l with 2.
 prove_sup0.
 unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
 rewrite <- Rinv_r_sym.
 rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H.
 discrR.
 apply Rmult_lt_reg_l with 2.
 prove_sup0.
 unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
 rewrite <- Rinv_r_sym.
 rewrite Rmult_1_l; rewrite Rplus_comm; rewrite double;
 apply Rplus_lt_compat_l; apply H.
 discrR.
 intros; elim H6; intros H13 _.
 elim H7; intros H14 _.
 apply Rle_antisym.
 apply H13; apply H12.
 rewrite H10 in H11; rewrite H11; apply H14; apply H12.
 cut (a < mx < b).
(*** h admet un minimum global sur [a,b] ***)
 intro; exists mx.
 exists H12.
 apply Rminus_diag_uniq; rewrite <- H9; apply deriv_minimum with a b.
 elim H12; intros; assumption.
 elim H12; intros; assumption.
 intros; elim H7; intros.
 apply H15; split; left; assumption.
 elim H7; intros _ H12; elim H12; intros; split.
 inversion H13.
 apply H15.
 rewrite H15 in H11; elim H11; reflexivity.
 inversion H14.
 apply H15.
 rewrite H8 in H11; rewrite <- H15 in H11; elim H11; reflexivity.
 cut (a < Mx < b).
(*** h admet un maximum global sur [a,b] ***)
 intro; exists Mx.
 exists H11.
 apply Rminus_diag_uniq; rewrite <- H9; apply deriv_maximum with a b.
 elim H11; intros; assumption.
 elim H11; intros; assumption.
 intros; elim H6; intros; apply H14.
 split; left; assumption.
 elim H6; intros _ H11; elim H11; intros; split.
 inversion H12.
 apply H14.
 rewrite H14 in H10; elim H10; reflexivity.
 inversion H13.
 apply H14.
 rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity.
 intros; unfold h;
 replace
 (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P))
 with
 (derive_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) c
 (derivable_pt_minus _ _ _
 (derivable_pt_mult _ _ _ (derivable_pt_const (g b - g a) c) (pr1 c P))
 (derivable_pt_mult _ _ _ (derivable_pt_const (f b - f a) c) (pr2 c P))));
 [ idtac | apply pr_nu ].
 rewrite derive_pt_minus; do 2 rewrite derive_pt_mult;
 do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l;
 do 2 rewrite Rplus_0_l; reflexivity.
 unfold h; ring.
 intros; unfold h;
 change
 (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
 c).
 apply continuity_pt_minus; apply continuity_pt_mult.
 apply derivable_continuous_pt; apply derivable_const.
 apply H0; apply H3.
 apply derivable_continuous_pt; apply derivable_const.
 apply H1; apply H3.
 intros;
 change
 (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
 c).
 apply derivable_pt_minus; apply derivable_pt_mult.
 apply derivable_pt_const.
 apply (pr1 _ H3).
 apply derivable_pt_const.
 apply (pr2 _ H3).
Qed.

(* Corollaries ... *)
Lemma MVT_cor1 :
 forall (f:R -> R) (a b:R) (pr:derivable f),
 a 
 exists c : R, f b - f a = derive_pt f c (pr c) * (b - a) /\ a < c < b.
Proof.
 intros f a b pr H; cut (forall c:R, a < c derivable_pt f c);
 [ intro X | intros; apply pr ].
 cut (forall c:R, a < c derivable_pt id c);
 [ intro X0 | intros; apply derivable_pt_id ].
 cut (forall c:R, a <= c <= b -> continuity_pt f c);
 [ intro | intros; apply derivable_continuous_pt; apply pr ].
 cut (forall c:R, a <= c <= b -> continuity_pt id c);
 [ intro | intros; apply derivable_continuous_pt; apply derivable_id ].
 assert (H2 := MVT f id a b X X0 H H0 H1).
 destruct H2 as (c & P & H4).
 exists c; split.
 cut (derive_pt id c (X0 c P) = derive_pt id c (derivable_pt_id c));
 [ intro H5 | apply pr_nu ].
 rewrite H5 in H4; rewrite (derive_pt_id c) in H4; rewrite Rmult_1_r in H4;
 rewrite <- H4; replace (derive_pt f c (X c P)) with (derive_pt f c (pr c));
 [ idtac | apply pr_nu ]; apply Rmult_comm.
 apply P.
Qed.

Theorem MVT_cor2 :
 forall (f f':R -> R) (a b:R),
 a 
 (forall c:R, a <= c <= b -> derivable_pt_lim f c (f' c)) ->
 exists c : R, f b - f a = f' c * (b - a) /\ a < c < b.
Proof.
 intros f f' a b H H0; cut (forall c:R, a <= c <= b -> derivable_pt f c).
 intro X; cut (forall c:R, a < c derivable_pt f c).
 intro X0; cut (forall c:R, a <= c <= b -> continuity_pt f c).
 intro; cut (forall c:R, a <= c <= b -> derivable_pt id c).
 intro X1; cut (forall c:R, a < c derivable_pt id c).
 intro X2; cut (forall c:R, a <= c <= b -> continuity_pt id c).
 intro; elim (MVT f id a b X0 X2 H H1 H2); intros x (P,H3).
 exists x; split.
 cut (derive_pt id x (X2 x P) = 1).
 cut (derive_pt f x (X0 x P) = f' x).
 intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3;
 rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry ;
 assumption.
 apply derive_pt_eq_0; apply H0; elim P; intros; split; left; assumption.
 apply derive_pt_eq_0; apply derivable_pt_lim_id.
 assumption.
 intros; apply derivable_continuous_pt; apply X1; assumption.
 intros; apply derivable_pt_id.
 intros; apply derivable_pt_id.
 intros; apply derivable_continuous_pt; apply X; assumption.
 intros; elim H1; intros; apply X; split; left; assumption.
 intros; unfold derivable_pt; exists (f' c); apply H0;
 apply H1.
Qed.

Lemma MVT_cor3 :
 forall (f f':R -> R) (a b:R),
 a 
 (forall x:R, a <= x -> x <= b -> derivable_pt_lim f x (f' x)) ->
 exists c : R, a <= c /\ c <= b /\ f b = f a + f' c * (b - a).
Proof.
 intros f f' a b H H0;
 assert (H1 : exists c : R, f b - f a = f' c * (b - a) /\ a < c < b);
 [ apply MVT_cor2; [ apply H | intros; elim H1; intros; apply (H0 _ H2 H3) ]
 | elim H1; intros; exists x; elim H2; intros; elim H4; intros; split;
 [ left; assumption | split; [ left; assumption | rewrite <- H3; ring ] ] ].
Qed.

Lemma Rolle :
 forall (f:R -> R) (a b:R) (pr:forall x:R, a < x derivable_pt f x),
 (forall x:R, a <= x <= b -> continuity_pt f x) ->
 a 
 f a = f b ->
 exists c : R, (exists P : a < c < b, derive_pt f c (pr c P) = 0).
Proof.
 intros; assert (H2 : forall x:R, a < x derivable_pt id x).
 intros; apply derivable_pt_id.
 assert (H3 := MVT f id a b pr H2 H0 H);
 assert (H4 : forall x:R, a <= x <= b -> continuity_pt id x).
 intros; apply derivable_continuous; apply derivable_id.
 destruct (H3 H4) as (c & P & H6). exists c; exists P; rewrite H1 in H6.
 unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6.
 rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a);
 [ rewrite Rmult_0_r; apply H6
 | apply Rminus_eq_contra; red; intro H7; rewrite H7 in H0;
 elim (Rlt_irrefl _ H0) ].
Qed.

(**********)
Lemma nonneg_derivative_1 :
 forall (f:R -> R) (pr:derivable f),
 (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f.
Proof.
 intros.
 unfold increasing.
 intros.
 destruct (total_order_T x y) as [[H1| ->]|H1].
 apply Rplus_le_reg_l with (- f x).
 rewrite Rplus_opp_l; rewrite Rplus_comm.
 pose proof (MVT_cor1 f _ _ pr H1) as (c & H3 & H4).
 unfold Rminus in H3.
 rewrite H3.
 apply Rmult_le_pos.
 apply H.
 apply Rplus_le_reg_l with x.
 rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
 right; reflexivity.
 elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 H1)).
Qed.

(**********)
Lemma nonpos_derivative_0 :
 forall (f:R -> R) (pr:derivable f),
 decreasing f -> forall x:R, derive_pt f x (pr x) <= 0.
Proof.
 intros f pr H x; assert (H0 := H); unfold decreasing in H0;
 generalize (derivable_derive f x (pr x)); intro; elim H1;
 intros l H2.
 rewrite H2; case (Rtotal_order l 0); intro.
 left; assumption.
 elim H3; intro.
 right; assumption.
 generalize (derive_pt_eq_1 f x l (pr x) H2); intros; cut (0 < l / 2).
 intro; elim (H5 (l / 2) H6); intros delta H7;
 cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).
 intro; decompose [and] H8; intros; generalize (H7 (delta / 2) H9 H12);
 cut ((f (x + delta / 2) - f x) / (delta / 2) <= 0).
 intro; cut (0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)).
 intro; unfold Rabs;
 case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)) as [Hlt|Hge].
 intros;
 generalize
 (Rplus_lt_compat_r (- l) (- ((f (x + delta / 2) - f x) / (delta / 2) - l))
 (l / 2) H14); unfold Rminus.
 replace (l / 2 + - l) with (- (l / 2)).
 replace (- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) + - l) with
 (- ((f (x + delta / 2) + - f x) / (delta / 2))).
 intro.
 generalize
 (Ropp_lt_gt_contravar (- ((f (x + delta / 2) + - f x) / (delta / 2)))
 (- (l / 2)) H15).
 repeat rewrite Ropp_involutive.
 intro.
 generalize
 (Rlt_trans 0 (l / 2) ((f (x + delta / 2) - f x) / (delta / 2)) H6 H16);
 intro.
 elim
 (Rlt_irrefl 0
 (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)).
 ring.
 pattern l at 3; rewrite double_var.
 ring.
 intros.
 generalize
 (Ropp_ge_le_contravar ((f (x + delta / 2) - f x) / (delta / 2) - l) _ Hge).
 rewrite Ropp_0.
 intro.
 elim
 (Rlt_irrefl 0
 (Rlt_le_trans 0 (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) 0 H13
 H15)).
 replace (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) with
 ((f x - f (x + delta / 2)) / (delta / 2) + l).
 unfold Rminus.
 apply Rplus_le_lt_0_compat.
 unfold Rdiv; apply Rmult_le_pos.
 cut (x <= x + delta * / 2).
 intro; generalize (H0 x (x + delta * / 2) H13); intro;
 generalize
 (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14);
 rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
 pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
 left; assumption.
 left; apply Rinv_0_lt_compat; assumption.
 assumption.
 rewrite Ropp_minus_distr.
 unfold Rminus.
 rewrite (Rplus_comm l).
 unfold Rdiv.
 rewrite <- Ropp_mult_distr_l_reverse.
 rewrite Ropp_plus_distr.
 rewrite Ropp_involutive.
 rewrite (Rplus_comm (f x)).
 reflexivity.
 replace ((f (x + delta / 2) - f x) / (delta / 2)) with
 (- ((f x - f (x + delta / 2)) / (delta / 2))).
 rewrite <- Ropp_0.
 apply Ropp_ge_le_contravar.
 apply Rle_ge.
 unfold Rdiv; apply Rmult_le_pos.
 cut (x <= x + delta * / 2).
 intro; generalize (H0 x (x + delta * / 2) H10); intro.
 generalize
 (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13);
 rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
 pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
 left; assumption.
 left; apply Rinv_0_lt_compat; assumption.
 unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.
 rewrite Ropp_minus_distr.
 reflexivity.
 split.
 unfold Rdiv; apply prod_neq_R0.
 generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H8;
 elim (Rlt_irrefl 0 H8).
 apply Rinv_neq_0_compat; discrR.
 split.
 unfold Rdiv; apply Rmult_lt_0_compat;
 [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
 rewrite Rabs_right.
 unfold Rdiv; apply Rmult_lt_reg_l with 2.
 prove_sup0.
 rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
 rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1;
 rewrite <- Rplus_0_r.
 apply Rplus_lt_compat_l; apply (cond_pos delta).
 discrR.
 apply Rle_ge; unfold Rdiv; left; apply Rmult_lt_0_compat.
 apply (cond_pos delta).
 apply Rinv_0_lt_compat; prove_sup0.
 unfold Rdiv; apply Rmult_lt_0_compat;
 [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ].
Qed.

(**********)
Lemma increasing_decreasing_opp :
 forall f:R -> R, increasing f -> decreasing (- f)%F.
Proof.
 unfold increasing, decreasing, opp_fct; intros; generalize (H x y H0);
 intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption.
Qed.

(**********)
Lemma nonpos_derivative_1 :
 forall (f:R -> R) (pr:derivable f),
 (forall x:R, derive_pt f x (pr x) <= 0) -> decreasing f.
Proof.
 intros.
 cut (forall h:R, - - f h = f h).
 intro.
 generalize (increasing_decreasing_opp (- f)%F).
 unfold decreasing.
 unfold opp_fct.
 intros.
 rewrite <- (H0 x); rewrite <- (H0 y).
 apply H1.
 cut (forall x:R, 0 <= derive_pt (- f) x (derivable_opp f pr x)).
 intros.
 replace (fun x:R => - f x) with (- f)%F; [ idtac | reflexivity ].
 apply (nonneg_derivative_1 (- f)%F (derivable_opp f pr) H3).
 intro.
 assert (H3 := derive_pt_opp f x0 (pr x0)).
 cut
 (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
 derive_pt (- f) x0 (derivable_opp f pr x0)).
 intro.
 rewrite <- H4.
 rewrite H3.
 rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; apply (H x0).
 apply pr_nu.
 assumption.
 intro; ring.
Qed.

(**********)
Lemma positive_derivative :
 forall (f:R -> R) (pr:derivable f),
 (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f.
Proof.
 intros.
 unfold strict_increasing.
 intros.
 apply Rplus_lt_reg_l with (- f x).
 rewrite Rplus_opp_l; rewrite Rplus_comm.
 assert (H1 := MVT_cor1 f _ _ pr H0).
 elim H1; intros.
 elim H2; intros.
 unfold Rminus in H3.
 rewrite H3.
 apply Rmult_lt_0_compat.
 apply H.
 apply Rplus_lt_reg_l with x.
 rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
Qed.

(**********)
Lemma strictincreasing_strictdecreasing_opp :
 forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F.
Proof.
 unfold strict_increasing, strict_decreasing, opp_fct; intros;
 generalize (H x y H0); intro; apply Ropp_lt_gt_contravar;
 assumption.
Qed.

(**********)
Lemma negative_derivative :
 forall (f:R -> R) (pr:derivable f),
 (forall x:R, derive_pt f x (pr x) < 0) -> strict_decreasing f.
Proof.
 intros.
 cut (forall h:R, - - f h = f h).
 intros.
 generalize (strictincreasing_strictdecreasing_opp (- f)%F).
 unfold strict_decreasing, opp_fct.
 intros.
 rewrite <- (H0 x).
 rewrite <- (H0 y).
 apply H1; [ idtac | assumption ].
 cut (forall x:R, 0 < derive_pt (- f) x (derivable_opp f pr x)).
 intros; eapply positive_derivative; apply H3.
 intro.
 assert (H3 := derive_pt_opp f x0 (pr x0)).
 cut
 (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
 derive_pt (- f) x0 (derivable_opp f pr x0)).
 intro.
 rewrite <- H4; rewrite H3.
 rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply (H x0).
 apply pr_nu.
 intro; ring.
Qed.

(**********)
Lemma null_derivative_0 :
 forall (f:R -> R) (pr:derivable f),
 constant f -> forall x:R, derive_pt f x (pr x) = 0.
Proof.
 intros.
 unfold constant in H.
 apply derive_pt_eq_0.
 intros; exists (mkposreal 1 Rlt_0_1); simpl; intros.
 rewrite (H x (x + h)); unfold Rminus; unfold Rdiv;
 rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r;
 rewrite Rabs_R0; assumption.
Qed.

(**********)
Lemma increasing_decreasing :
 forall f:R -> R, increasing f -> decreasing f -> constant f.
Proof.
 unfold increasing, decreasing, constant; intros;
 case (Rtotal_order x y); intro.
 generalize (Rlt_le x y H1); intro;
 apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)).
 elim H1; intro.
 rewrite H2; reflexivity.
 generalize (Rlt_le y x H2); intro; symmetry ;
 apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)).
Qed.

(**********)
Lemma null_derivative_1 :
 forall (f:R -> R) (pr:derivable f),
 (forall x:R, derive_pt f x (pr x) = 0) -> constant f.
Proof.
 intros.
 cut (forall x:R, derive_pt f x (pr x) <= 0).
 cut (forall x:R, 0 <= derive_pt f x (pr x)).
 intros.
 assert (H2 := nonneg_derivative_1 f pr H0).
 assert (H3 := nonpos_derivative_1 f pr H1).
 apply increasing_decreasing; assumption.
 intro; right; symmetry ; apply (H x).
 intro; right; apply (H x).
Qed.

(**********)
Lemma derive_increasing_interv_ax :
 forall (a b:R) (f:R -> R) (pr:derivable f),
 a 
 ((forall t:R, a < t 0 < derive_pt f t (pr t)) ->
 forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
 ((forall t:R, a < t 0 <= derive_pt f t (pr t)) ->
 forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y).
Proof.
 intros.
 split; intros.
 apply Rplus_lt_reg_l with (- f x).
 rewrite Rplus_opp_l; rewrite Rplus_comm.
 assert (H4 := MVT_cor1 f _ _ pr H3).
 elim H4; intros.
 elim H5; intros.
 unfold Rminus in H6.
 rewrite H6.
 apply Rmult_lt_0_compat.
 apply H0.
 elim H7; intros.
 split.
 elim H1; intros.
 apply Rle_lt_trans with x; assumption.
 elim H2; intros.
 apply Rlt_le_trans with y; assumption.
 apply Rplus_lt_reg_l with x.
 rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
 apply Rplus_le_reg_l with (- f x).
 rewrite Rplus_opp_l; rewrite Rplus_comm.
 assert (H4 := MVT_cor1 f _ _ pr H3).
 elim H4; intros.
 elim H5; intros.
 unfold Rminus in H6.
 rewrite H6.
 apply Rmult_le_pos.
 apply H0.
 elim H7; intros.
 split.
 elim H1; intros.
 apply Rle_lt_trans with x; assumption.
 elim H2; intros.
 apply Rlt_le_trans with y; assumption.
 apply Rplus_le_reg_l with x.
 rewrite Rplus_0_r; replace (x + (y + - x)) with y;
 [ left; assumption | ring ].
Qed.

(**********)
Lemma derive_increasing_interv :
 forall (a b:R) (f:R -> R) (pr:derivable f),
 a 
 (forall t:R, a < t 0 < derive_pt f t (pr t)) ->
 forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y.
Proof.
 intros.
 generalize (derive_increasing_interv_ax a b f pr H); intro.
 elim H4; intros H5 _; apply (H5 H0 x y H1 H2 H3).
Qed.

(**********)
Lemma derive_increasing_interv_var :
 forall (a b:R) (f:R -> R) (pr:derivable f),
 a 
 (forall t:R, a < t 0 <= derive_pt f t (pr t)) ->
 forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y.
Proof.
 intros a b f pr H H0 x y H1 H2 H3;
 generalize (derive_increasing_interv_ax a b f pr H);
 intro; elim H4; intros _ H5; apply (H5 H0 x y H1 H2 H3).
Qed.

(**********)
(**********)
Theorem IAF :
 forall (f:R -> R) (a b k:R) (pr:derivable f),
 a <= b ->
 (forall c:R, a <= c <= b -> derive_pt f c (pr c) <= k) ->
 f b - f a <= k * (b - a).
Proof.
 intros.
 destruct (total_order_T a b) as [[H1| -> ]|H1].
 pose proof (MVT_cor1 f _ _ pr H1) as (c & -> & H4).
 do 2 rewrite <- (Rmult_comm (b - a)).
 apply Rmult_le_compat_l.
 apply Rplus_le_reg_l with a; rewrite Rplus_0_r.
 replace (a + (b - a)) with b; [ assumption | ring ].
 apply H0.
 elim H4; intros.
 split; left; assumption.
 unfold Rminus; do 2 rewrite Rplus_opp_r.
 rewrite Rmult_0_r; right; reflexivity.
 elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H H1)).
Qed.

Lemma IAF_var :
 forall (f g:R -> R) (a b:R) (pr1:derivable f) (pr2:derivable g),
 a <= b ->
 (forall c:R, a <= c <= b -> derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) ->
 g b - g a <= f b - f a.
Proof.
 intros.
 cut (derivable (g - f)).
 intro X.
 cut (forall c:R, a <= c <= b -> derive_pt (g - f) c (X c) <= 0).
 intro.
 assert (H2 := IAF (g - f)%F a b 0 X H H1).
 rewrite Rmult_0_l in H2; unfold minus_fct in H2.
 apply Rplus_le_reg_l with (- f b + f a).
 replace (- f b + f a + (f b - f a)) with 0; [ idtac | ring ].
 replace (- f b + f a + (g b - g a)) with (g b - f b - (g a - f a));
 [ apply H2 | ring ].
 intros.
 cut
 (derive_pt (g - f) c (X c) =
 derive_pt (g - f) c (derivable_pt_minus _ _ _ (pr2 c) (pr1 c))).
 intro.
 rewrite H2.
 rewrite derive_pt_minus.
 apply Rplus_le_reg_l with (derive_pt f c (pr1 c)).
 rewrite Rplus_0_r.
 replace
 (derive_pt f c (pr1 c) + (derive_pt g c (pr2 c) - derive_pt f c (pr1 c)))
 with (derive_pt g c (pr2 c)); [ idtac | ring ].
 apply H0; assumption.
 apply pr_nu.
 apply derivable_minus; assumption.
Qed.

(* If f has a null derivative in ]a,b[ and is continue in [a,b], *)
(* then f is constant on [a,b] *)
Lemma null_derivative_loc :
 forall (f:R -> R) (a b:R) (pr:forall x:R, a < x derivable_pt f x),
 (forall x:R, a <= x <= b -> continuity_pt f x) ->
 (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) ->
 constant_D_eq f (fun x:R => a <= x <= b) (f a).
Proof.
 intros; unfold constant_D_eq; intros; destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].
 assert (H2 : forall y:R, a < y < x -> derivable_pt id y).
 intros; apply derivable_pt_id.
 assert (H3 : forall y:R, a <= y <= x -> continuity_pt id y).
 intros; apply derivable_continuous; apply derivable_id.
 assert (H4 : forall y:R, a < y < x -> derivable_pt f y).
 intros; apply pr; elim H4; intros; split.
 assumption.
 elim H1; intros; apply Rlt_le_trans with x; assumption.
 assert (H5 : forall y:R, a <= y <= x -> continuity_pt f y).
 intros; apply H; elim H5; intros; split.
 assumption.
 elim H1; intros; apply Rle_trans with x; assumption.
 elim H1; clear H1; intros; elim H1; clear H1; intro.
 assert (H7 := MVT f id a x H4 H2 H1 H5 H3).
 destruct H7 as (c & P & H9).
 assert (H10 : a < c < b).
 split.
 apply P.
 apply Rlt_le_trans with x; [apply P|assumption].
 assert (H11 : derive_pt f c (H4 c P) = 0).
 replace (derive_pt f c (H4 c P)) with (derive_pt f c (pr c H10));
 [ apply H0 | apply pr_nu ].
 assert (H12 : derive_pt id c (H2 c P) = 1).
 apply derive_pt_eq_0; apply derivable_pt_lim_id.
 rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9;
 rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry ;
 assumption.
 rewrite H1; reflexivity.
 assert (H2 : x = a).
 rewrite <- Heq in H1; elim H1; intros; apply Rle_antisym; assumption.
 rewrite H2; reflexivity.
 elim H1; intros;
 elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H2 H3) Hgt)).
Qed.

(* Unicity of the antiderivative *)
Lemma antiderivative_Ucte :
 forall (f g1 g2:R -> R) (a b:R),
 antiderivative f g1 a b ->
 antiderivative f g2 a b ->
 exists c : R, (forall x:R, a <= x <= b -> g1 x = g2 x + c).
Proof.
 unfold antiderivative; intros; elim H; clear H; intros; elim H0;
 clear H0; intros H0 _; exists (g1 a - g2 a); intros;
 assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x).
 intros; unfold derivable_pt; exists (f x0); elim (H x0 H3);
 intros; eapply derive_pt_eq_1; symmetry ;
 apply H4.
 assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x).
 intros; unfold derivable_pt; exists (f x0);
 elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry ;
 apply H5.
 assert (H5 : forall x:R, a < x derivable_pt (g1 - g2) x).
 intros; elim H5; intros; apply derivable_pt_minus;
 [ apply H3; split; left; assumption | apply H4; split; left; assumption ].
 assert (H6 : forall x:R, a <= x <= b -> continuity_pt (g1 - g2) x).
 intros; apply derivable_continuous_pt; apply derivable_pt_minus;
 [ apply H3 | apply H4 ]; assumption.
 assert (H7 : forall (x:R) (P:a < x < b), derive_pt (g1 - g2) x (H5 x P) = 0).
 intros; elim P; intros; apply derive_pt_eq_0; replace 0 with (f x0 - f x0);
 [ idtac | ring ].
 assert (H9 : a <= x0 <= b).
 split; left; assumption.
 apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros;
 eapply derive_pt_eq_1; symmetry ; apply H10.
 assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7);
 unfold constant_D_eq in H8; assert (H9 := H8 _ H2);
 unfold minus_fct in H9; rewrite <- H9; ring.
Qed.

(* A variant of MVT using absolute values. *)
Lemma MVT_abs :
 forall (f f' : R -> R) (a b : R),
 (forall c : R, Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
 exists c : R, Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
 Rmin a b <= c <= Rmax a b.
Proof.
intros f f' a b.
destruct (Rle_dec a b) as [aleb | blta].
destruct (Req_dec a b) as [ab | anb].
 unfold Rminus; intros _; exists a; split.
 now rewrite <- ab, !Rplus_opp_r, Rabs_R0, Rmult_0_r.
 split;[apply Rmin_l | apply Rmax_l].
rewrite Rmax_right, Rmin_left; auto; intros derv.
destruct (MVT_cor2 f f' a b) as [c [hc intc]];
 [destruct aleb;[assumption | contradiction] | apply derv | ].
exists c; rewrite hc, Rabs_mult;split;
 [reflexivity | unfold Rle; tauto].
assert (b < a) by (apply Rnot_le_gt; assumption).
assert (b <= a) by (apply Rlt_le; assumption).
rewrite Rmax_left, Rmin_right; try assumption; intros derv.
destruct (MVT_cor2 f f' b a) as [c [hc intc]];
[assumption | apply derv | ].
exists c; rewrite <- Rabs_Ropp, Ropp_minus_distr, hc, Rabs_mult.
split;[now rewrite <- (Rabs_Ropp (b - a)), Ropp_minus_distr| unfold Rle; tauto].
Qed.

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