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(************************************************************************)
(* * 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 Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo1.
Require Import Ranalysis1.
Require Import Ranalysis3.
Require Import Exp_prop.
Require Import MVT.
Local Open Scope R_scope.
(**********)
Lemma derivable_pt_inv :
forall (f:R -> R) (x:R),
f x <> 0 -> derivable_pt f x -> derivable_pt (/ f) x.
Proof.
intros f x H X; cut (derivable_pt (fct_cte 1 / f) x -> derivable_pt (/ f) x).
intro X0; apply X0.
apply derivable_pt_div.
apply derivable_pt_const.
assumption.
assumption.
unfold div_fct, inv_fct, fct_cte; intros (x0,p);
unfold derivable_pt; exists x0;
unfold derivable_pt_abs; unfold derivable_pt_lim;
unfold derivable_pt_abs in p; unfold derivable_pt_lim in p;
intros; elim (p eps H0); intros; exists x1; intros;
unfold Rdiv in H1; unfold Rdiv; rewrite <- (Rmult_1_l (/ f x));
rewrite <- (Rmult_1_l (/ f (x + h))).
apply H1; assumption.
Qed.
(**********)
Lemma pr_nu_var :
forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x),
f = g -> derive_pt f x pr1 = derive_pt g x pr2.
Proof.
unfold derivable_pt, derive_pt; intros f g x (x0,p0) (x1,p1) ->.
apply uniqueness_limite with g x; assumption.
Qed.
(**********)
Lemma pr_nu_var2 :
forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x),
(forall h:R, f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2.
Proof.
unfold derivable_pt, derive_pt; intros f g x (x0,p0) (x1,p1) H.
assert (H0 := uniqueness_step2 _ _ _ p0).
assert (H1 := uniqueness_step2 _ _ _ p1).
cut (limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) x1 0).
intro H2; assert (H3 := uniqueness_step1 _ _ _ _ H0 H2).
assumption.
unfold limit1_in; unfold limit_in; unfold dist;
simpl; unfold R_dist; unfold limit1_in in H1;
unfold limit_in in H1; unfold dist in H1; simpl in H1;
unfold R_dist in H1.
intros; elim (H1 eps H2); intros.
elim H3; intros.
exists x2.
split.
assumption.
intros; do 2 rewrite H; apply H5; assumption.
Qed.
(**********)
Lemma derivable_inv :
forall f:R -> R, (forall x:R, f x <> 0) -> derivable f -> derivable (/ f).
Proof.
intros f H X.
unfold derivable; intro x.
apply derivable_pt_inv.
apply (H x).
apply (X x).
Qed.
Lemma derive_pt_inv :
forall (f:R -> R) (x:R) (pr:derivable_pt f x) (na:f x <> 0),
derive_pt (/ f) x (derivable_pt_inv f x na pr) =
- derive_pt f x pr / Rsqr (f x).
Proof.
intros;
replace (derive_pt (/ f) x (derivable_pt_inv f x na pr)) with
(derive_pt (fct_cte 1 / f) x
(derivable_pt_div (fct_cte 1) f x (derivable_pt_const 1 x) pr na)).
rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte;
rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus;
rewrite Rplus_0_l; reflexivity.
apply pr_nu_var2.
intro; unfold div_fct, fct_cte, inv_fct.
unfold Rdiv; ring.
Qed.
(** Rabsolu *)
Lemma Rabs_derive_1 : forall x:R, 0 < x -> derivable_pt_lim Rabs x 1.
Proof.
intros.
unfold derivable_pt_lim; intros.
exists (mkposreal x H); intros.
rewrite (Rabs_right x).
rewrite (Rabs_right (x + h)).
rewrite Rplus_comm.
unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r.
rewrite Rplus_0_r; unfold Rdiv; rewrite <- Rinv_r_sym.
rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0.
apply H1.
apply Rle_ge.
destruct (Rcase_abs h) as [Hlt|Hgt].
rewrite (Rabs_left h Hlt) in H2.
left; rewrite Rplus_comm; apply Rplus_lt_reg_l with (- h); rewrite Rplus_0_r;
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
apply H2.
apply Rplus_le_le_0_compat.
left; apply H.
apply Rge_le; apply Hgt.
left; apply H.
Qed.
Lemma Rabs_derive_2 : forall x:R, x < 0 -> derivable_pt_lim Rabs x (-1).
Proof.
intros.
unfold derivable_pt_lim; intros.
cut (0 < - x).
intro; exists (mkposreal (- x) H1); intros.
rewrite (Rabs_left x).
rewrite (Rabs_left (x + h)).
replace ((-(x + h) - - x) / h - -1) with 0 by now field.
rewrite Rabs_R0; apply H0.
destruct (Rcase_abs h) as [Hlt|Hgt].
apply Ropp_lt_cancel.
rewrite Ropp_0; rewrite Ropp_plus_distr; apply Rplus_lt_0_compat.
apply H1.
apply Ropp_0_gt_lt_contravar; apply Hlt.
rewrite (Rabs_right h Hgt) in H3.
apply Rplus_lt_reg_l with (- x); rewrite Rplus_0_r; rewrite <- Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_l; apply H3.
apply H.
apply Ropp_0_gt_lt_contravar; apply H.
Qed.
(** Rabsolu is derivable for all x <> 0 *)
Lemma Rderivable_pt_abs : forall x:R, x <> 0 -> derivable_pt Rabs x.
Proof.
intros.
destruct (total_order_T x 0) as [[Hlt|Heq]|Hgt].
unfold derivable_pt; exists (-1).
apply (Rabs_derive_2 x Hlt).
elim H; exact Heq.
unfold derivable_pt; exists 1.
apply (Rabs_derive_1 x Hgt).
Qed.
(** Rabsolu is continuous for all x *)
Lemma Rcontinuity_abs : continuity Rabs.
Proof.
unfold continuity; intro.
case (Req_dec x 0); intro.
unfold continuity_pt; unfold continue_in;
unfold limit1_in; unfold limit_in;
simpl; unfold R_dist; intros; exists eps;
split.
apply H0.
intros; rewrite H; rewrite Rabs_R0; unfold Rminus; rewrite Ropp_0;
rewrite Rplus_0_r; rewrite Rabs_Rabsolu; elim H1;
intros; rewrite H in H3; unfold Rminus in H3; rewrite Ropp_0 in H3;
rewrite Rplus_0_r in H3; apply H3.
apply derivable_continuous_pt; apply (Rderivable_pt_abs x H).
Qed.
(** Finite sums : Sum a_k x^k *)
Lemma continuity_finite_sum :
forall (An:nat -> R) (N:nat),
continuity (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N).
Proof.
intros; unfold continuity; intro.
induction N as [| N HrecN].
simpl.
apply continuity_pt_const.
unfold constant; intros; reflexivity.
replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with
((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) +
(fun y:R => (An (S N) * y ^ S N)%R))%F.
apply continuity_pt_plus.
apply HrecN.
replace (fun y:R => An (S N) * y ^ S N) with
(mult_real_fct (An (S N)) (fun y:R => y ^ S N)).
apply continuity_pt_scal.
apply derivable_continuous_pt.
apply derivable_pt_pow.
reflexivity.
reflexivity.
Qed.
Lemma derivable_pt_lim_fs :
forall (An:nat -> R) (x:R) (N:nat),
(0 < N)%nat ->
derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x
(sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N)).
Proof.
intros; induction N as [| N HrecN].
elim (lt_irrefl _ H).
cut (N = 0%nat \/ (0 < N)%nat).
intro; elim H0; intro.
rewrite H1.
simpl.
replace (fun y:R => An 0%nat * 1 + An 1%nat * (y * 1)) with
(fct_cte (An 0%nat * 1) + mult_real_fct (An 1%nat) (id * fct_cte 1))%F.
replace (1 * An 1%nat * 1) with (0 + An 1%nat * (1 * fct_cte 1 x + id x * 0)).
apply derivable_pt_lim_plus.
apply derivable_pt_lim_const.
apply derivable_pt_lim_scal.
apply derivable_pt_lim_mult.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte, id; ring.
reflexivity.
replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with
((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) +
(fun y:R => (An (S N) * y ^ S N)%R))%F.
replace (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N)))
with
(sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N) +
An (S N) * (INR (S (pred (S N))) * x ^ pred (S N))).
apply derivable_pt_lim_plus.
apply HrecN.
assumption.
replace (fun y:R => An (S N) * y ^ S N) with
(mult_real_fct (An (S N)) (fun y:R => y ^ S N)).
apply derivable_pt_lim_scal.
replace (pred (S N)) with N; [ idtac | reflexivity ].
pattern N at 3; replace N with (pred (S N)).
apply derivable_pt_lim_pow.
reflexivity.
reflexivity.
cut (pred (S N) = S (pred N)).
intro; rewrite H2.
rewrite tech5.
apply Rplus_eq_compat_l.
rewrite <- H2.
replace (pred (S N)) with N; [ idtac | reflexivity ].
ring.
simpl.
apply S_pred with 0%nat; assumption.
unfold plus_fct.
simpl; reflexivity.
inversion H.
left; reflexivity.
right; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ].
Qed.
Lemma derivable_pt_lim_finite_sum :
forall (An:nat -> R) (x:R) (N:nat),
derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x
match N with
| O => 0
| _ => sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N)
end.
Proof.
intros.
induction N as [| N HrecN].
simpl.
rewrite Rmult_1_r.
replace (fun _:R => An 0%nat) with (fct_cte (An 0%nat));
[ apply derivable_pt_lim_const | reflexivity ].
apply derivable_pt_lim_fs; apply lt_O_Sn.
Qed.
Lemma derivable_pt_finite_sum :
forall (An:nat -> R) (N:nat) (x:R),
derivable_pt (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x.
Proof.
intros.
unfold derivable_pt.
assert (H := derivable_pt_lim_finite_sum An x N).
induction N as [| N HrecN].
exists 0; apply H.
exists
(sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N)));
apply H.
Qed.
Lemma derivable_finite_sum :
forall (An:nat -> R) (N:nat),
derivable (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N).
Proof.
intros; unfold derivable; intro; apply derivable_pt_finite_sum.
Qed.
(** Regularity of hyperbolic functions *)
Lemma derivable_pt_lim_cosh : forall x:R, derivable_pt_lim cosh x (sinh x).
Proof.
intro.
unfold cosh, sinh; unfold Rdiv.
replace (fun x0:R => (exp x0 + exp (- x0)) * / 2) with
((exp + comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ].
replace ((exp x - exp (- x)) * / 2) with
((exp x + exp (- x) * -1) * fct_cte (/ 2) x +
(exp + comp exp (- id))%F x * 0).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_plus.
apply derivable_pt_lim_exp.
apply derivable_pt_lim_comp.
apply derivable_pt_lim_opp.
apply derivable_pt_lim_id.
apply derivable_pt_lim_exp.
apply derivable_pt_lim_const.
unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring.
Qed.
Lemma derivable_pt_lim_sinh : forall x:R, derivable_pt_lim sinh x (cosh x).
Proof.
intro.
unfold cosh, sinh; unfold Rdiv.
replace (fun x0:R => (exp x0 - exp (- x0)) * / 2) with
((exp - comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ].
replace ((exp x + exp (- x)) * / 2) with
((exp x - exp (- x) * -1) * fct_cte (/ 2) x +
(exp - comp exp (- id))%F x * 0).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_minus.
apply derivable_pt_lim_exp.
apply derivable_pt_lim_comp.
apply derivable_pt_lim_opp.
apply derivable_pt_lim_id.
apply derivable_pt_lim_exp.
apply derivable_pt_lim_const.
unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring.
Qed.
Lemma derivable_pt_exp : forall x:R, derivable_pt exp x.
Proof.
intro.
unfold derivable_pt.
exists (exp x).
apply derivable_pt_lim_exp.
Qed.
Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x.
Proof.
intro.
unfold derivable_pt.
exists (sinh x).
apply derivable_pt_lim_cosh.
Qed.
Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x.
Proof.
intro.
unfold derivable_pt.
exists (cosh x).
apply derivable_pt_lim_sinh.
Qed.
Lemma derivable_exp : derivable exp.
Proof.
unfold derivable; apply derivable_pt_exp.
Qed.
Lemma derivable_cosh : derivable cosh.
Proof.
unfold derivable; apply derivable_pt_cosh.
Qed.
Lemma derivable_sinh : derivable sinh.
Proof.
unfold derivable; apply derivable_pt_sinh.
Qed.
Lemma derive_pt_exp :
forall x:R, derive_pt exp x (derivable_pt_exp x) = exp x.
Proof.
intro; apply derive_pt_eq_0.
apply derivable_pt_lim_exp.
Qed.
Lemma derive_pt_cosh :
forall x:R, derive_pt cosh x (derivable_pt_cosh x) = sinh x.
Proof.
intro; apply derive_pt_eq_0.
apply derivable_pt_lim_cosh.
Qed.
Lemma derive_pt_sinh :
forall x:R, derive_pt sinh x (derivable_pt_sinh x) = cosh x.
Proof.
intro; apply derive_pt_eq_0.
apply derivable_pt_lim_sinh.
Qed.
Lemma sinh_lt : forall x y, x < y -> sinh x < sinh y.
intros x y xy; destruct (MVT_cor2 sinh cosh x y xy) as [c [Pc _]].
intros; apply derivable_pt_lim_sinh.
apply Rplus_lt_reg_l with (Ropp (sinh x)); rewrite Rplus_opp_l, Rplus_comm.
unfold Rminus at 1 in Pc; rewrite Pc; apply Rmult_lt_0_compat;[ | ].
unfold cosh; apply Rmult_lt_0_compat;[|apply Rinv_0_lt_compat, Rlt_0_2].
now apply Rplus_lt_0_compat; apply exp_pos.
now apply Rlt_Rminus; assumption.
Qed.
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