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Diffstat (limited to 'theories/Reals/Ranalysis4.v')
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diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v new file mode 100644 index 00000000..86f49cd4 --- /dev/null +++ b/theories/Reals/Ranalysis4.v @@ -0,0 +1,384 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis4.v,v 1.19.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import Ranalysis3. +Require Import Exp_prop. Open Local Scope R_scope. + +(**********) +Lemma derivable_pt_inv : + forall (f:R -> R) (x:R), + f x <> 0 -> derivable_pt f x -> derivable_pt (/ f) x. +intros; cut (derivable_pt (fct_cte 1 / f) x -> derivable_pt (/ f) x). +intro; apply X0. +apply derivable_pt_div. +apply derivable_pt_const. +assumption. +assumption. +unfold div_fct, inv_fct, fct_cte in |- *; intro; elim X0; intros; + unfold derivable_pt in |- *; apply existT with x0; + unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *; + 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 in |- *; 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. +unfold derivable_pt, derive_pt in |- *; intros. +elim pr1; intros. +elim pr2; intros. +simpl in |- *. +rewrite H in p. +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. +unfold derivable_pt, derive_pt in |- *; intros. +elim pr1; intros. +elim pr2; intros. +simpl in |- *. +assert (H0 := uniqueness_step2 _ _ _ p). +assert (H1 := uniqueness_step2 _ _ _ p0). +cut (limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) x1 0). +intro; assert (H3 := uniqueness_step1 _ _ _ _ H0 H2). +assumption. +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; 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). +intros. +unfold derivable in |- *; intro. +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). +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 in |- *; + rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Rplus_0_l; reflexivity. +apply pr_nu_var2. +intro; unfold div_fct, fct_cte, inv_fct in |- *. +unfold Rdiv in |- *; ring. +Qed. + +(* Rabsolu *) +Lemma Rabs_derive_1 : forall x:R, 0 < x -> derivable_pt_lim Rabs x 1. +intros. +unfold derivable_pt_lim in |- *; intros. +exists (mkposreal x H); intros. +rewrite (Rabs_right x). +rewrite (Rabs_right (x + h)). +rewrite Rplus_comm. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r. +rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. +rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0. +apply H1. +apply Rle_ge. +case (Rcase_abs h); intro. +rewrite (Rabs_left h r) in H2. +left; rewrite Rplus_comm; apply Rplus_lt_reg_r 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 r. +left; apply H. +Qed. + +Lemma Rabs_derive_2 : forall x:R, x < 0 -> derivable_pt_lim Rabs x (-1). +intros. +unfold derivable_pt_lim in |- *; intros. +cut (0 < - x). +intro; exists (mkposreal (- x) H1); intros. +rewrite (Rabs_left x). +rewrite (Rabs_left (x + h)). +rewrite Rplus_comm. +rewrite Ropp_plus_distr. +unfold Rminus in |- *; rewrite Ropp_involutive; rewrite Rplus_assoc; + rewrite Rplus_opp_l. +rewrite Rplus_0_r; unfold Rdiv in |- *. +rewrite Ropp_mult_distr_l_reverse. +rewrite <- Rinv_r_sym. +rewrite Ropp_involutive; rewrite Rplus_opp_l; rewrite Rabs_R0; apply H0. +apply H2. +case (Rcase_abs h); intro. +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 r. +rewrite (Rabs_right h r) in H3. +apply Rplus_lt_reg_r 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. +intros. +case (total_order_T x 0); intro. +elim s; intro. +unfold derivable_pt in |- *; apply existT with (-1). +apply (Rabs_derive_2 x a). +elim H; exact b. +unfold derivable_pt in |- *; apply existT with 1. +apply (Rabs_derive_1 x r). +Qed. + +(* Rabsolu is continuous for all x *) +Lemma Rcontinuity_abs : continuity Rabs. +unfold continuity in |- *; intro. +case (Req_dec x 0); intro. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; exists eps; + split. +apply H0. +intros; rewrite H; rewrite Rabs_R0; unfold Rminus in |- *; 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). +intros; unfold continuity in |- *; intro. +induction N as [| N HrecN]. +simpl in |- *. +apply continuity_pt_const. +unfold constant in |- *; 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)). +intros; induction N as [| N HrecN]. +elim (lt_irrefl _ H). +cut (N = 0%nat \/ (0 < N)%nat). +intro; elim H0; intro. +rewrite H1. +simpl in |- *. +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 in |- *; 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 in |- *; 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 in |- *. +apply S_pred with 0%nat; assumption. +unfold plus_fct in |- *. +simpl in |- *; 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. +intros. +induction N as [| N HrecN]. +simpl in |- *. +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. +intros. +unfold derivable_pt in |- *. +assert (H := derivable_pt_lim_finite_sum An x N). +induction N as [| N HrecN]. +apply existT with 0; apply H. +apply existT with + (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). +intros; unfold derivable in |- *; 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). +intro. +unfold cosh, sinh in |- *; unfold Rdiv in |- *. +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 in |- *; ring. +Qed. + +Lemma derivable_pt_lim_sinh : forall x:R, derivable_pt_lim sinh x (cosh x). +intro. +unfold cosh, sinh in |- *; unfold Rdiv in |- *. +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 in |- *; ring. +Qed. + +Lemma derivable_pt_exp : forall x:R, derivable_pt exp x. +intro. +unfold derivable_pt in |- *. +apply existT with (exp x). +apply derivable_pt_lim_exp. +Qed. + +Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x. +intro. +unfold derivable_pt in |- *. +apply existT with (sinh x). +apply derivable_pt_lim_cosh. +Qed. + +Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x. +intro. +unfold derivable_pt in |- *. +apply existT with (cosh x). +apply derivable_pt_lim_sinh. +Qed. + +Lemma derivable_exp : derivable exp. +unfold derivable in |- *; apply derivable_pt_exp. +Qed. + +Lemma derivable_cosh : derivable cosh. +unfold derivable in |- *; apply derivable_pt_cosh. +Qed. + +Lemma derivable_sinh : derivable sinh. +unfold derivable in |- *; apply derivable_pt_sinh. +Qed. + +Lemma derive_pt_exp : + forall x:R, derive_pt exp x (derivable_pt_exp x) = exp x. +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. +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. +intro; apply derive_pt_eq_0. +apply derivable_pt_lim_sinh. +Qed.
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