diff options
Diffstat (limited to 'theories/Reals/Rpower.v')
| -rw-r--r-- | theories/Reals/Rpower.v | 165 |
1 files changed, 145 insertions, 20 deletions
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index 014d7025..e30ea334 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -20,8 +20,10 @@ Require Import Ranalysis1. Require Import Exp_prop. Require Import Rsqrt_def. Require Import R_sqrt. +Require Import Sqrt_reg. Require Import MVT. Require Import Ranalysis4. +Require Import Fourier. Local Open Scope R_scope. Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y). @@ -43,7 +45,7 @@ Proof. rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)). - unfold exp; case (exist_exp (-1)); intros; simpl; + unfold exp; case (exist_exp (-1)) as (?,e); simpl in |- *; unfold exp_in in e; assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1). cut @@ -137,7 +139,7 @@ Qed. Lemma exp_ineq1 : forall x:R, 0 < x -> 1 + x < exp x. Proof. - intros; apply Rplus_lt_reg_r with (- exp 0); rewrite <- (Rplus_comm (exp x)); + intros; apply Rplus_lt_reg_l with (- exp 0); rewrite <- (Rplus_comm (exp x)); assert (H0 := MVT_cor1 exp 0 x derivable_exp H); elim H0; intros; elim H1; intros; unfold Rminus in H2; rewrite H2; rewrite Ropp_0; rewrite Rplus_0_r; @@ -178,13 +180,13 @@ Qed. (**********) Lemma ln_exists : forall y:R, 0 < y -> { z:R | y = exp z }. Proof. - intros; case (Rle_dec 1 y); intro. - apply (ln_exists1 _ r). + intros; destruct (Rle_dec 1 y) as [Hle|Hnle]. + apply (ln_exists1 _ Hle). assert (H0 : 1 <= / y). apply Rmult_le_reg_l with y. apply H. rewrite <- Rinv_r_sym. - rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n). + rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ Hnle). red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). destruct (ln_exists1 _ H0) as (x,p); exists (- x); apply Rmult_eq_reg_l with (exp x / y). @@ -213,12 +215,10 @@ Definition ln (x:R) : R := Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x. Proof. - intros; unfold ln; case (Rlt_dec 0 x); intro. + intros; unfold ln; decide (Rlt_dec 0 x) with H. unfold Rln; - case (ln_exists (mkposreal x r) (cond_pos (mkposreal x r))); - intros. - simpl in e; symmetry ; apply e. - elim n; apply H. + case (ln_exists (mkposreal x H) (cond_pos (mkposreal x H))) as (?,Hex). + symmetry; apply Hex. Qed. Theorem exp_inv : forall x y:R, exp x = exp y -> x = y. @@ -313,12 +313,12 @@ Proof. red; apply P_Rmin. apply Rmult_lt_0_compat. assumption. - apply Rplus_lt_reg_r with 1. + apply Rplus_lt_reg_l with 1. rewrite Rplus_0_r; replace (1 + (exp eps - 1)) with (exp eps); [ apply H1 | ring ]. apply Rmult_lt_0_compat. assumption. - apply Rplus_lt_reg_r with (exp (- eps)). + apply Rplus_lt_reg_l with (exp (- eps)). rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1; [ apply H2 | ring ]. unfold dist, R_met, R_dist; simpl. @@ -335,7 +335,7 @@ Proof. apply H. rewrite Hxyy. apply Ropp_lt_cancel. - apply Rplus_lt_reg_r with (r := y). + apply Rplus_lt_reg_l with (r := y). replace (y + - (y * exp (- eps))) with (y * (1 - exp (- eps))); [ idtac | ring ]. replace (y + - x) with (Rabs (x - y)). @@ -358,7 +358,7 @@ Proof. apply Rmult_lt_reg_l with (r := y). apply H. rewrite Hxyy. - apply Rplus_lt_reg_r with (r := - y). + apply Rplus_lt_reg_l with (r := - y). replace (- y + y * exp eps) with (y * (exp eps - 1)); [ idtac | ring ]. replace (- y + x) with (Rabs (x - y)). apply Rlt_le_trans with (1 := H5); apply Rmin_l. @@ -610,7 +610,7 @@ Proof. replace h with (x + h - x); [ idtac | ring ]. apply H3; split. unfold D_x; split. - case (Rcase_abs h); intro. + destruct (Rcase_abs h) as [Hlt|Hgt]. assert (H7 : Rabs h < x / 2). apply Rlt_le_trans with alp. apply H6. @@ -619,13 +619,13 @@ Proof. unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. rewrite Rabs_left in H7. - apply Rplus_lt_reg_r with (- h - x / 2). + apply Rplus_lt_reg_l with (- h - x / 2). replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ]. pattern x at 2; rewrite double_var. replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ]. - apply r. - apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply r ]. - apply (not_eq_sym (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h; + apply Hlt. + apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply Hgt ]. + apply (sym_not_eq (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h; [ apply H5 | ring ]. replace (x + h - x) with h; [ apply Rlt_le_trans with alp; @@ -703,3 +703,128 @@ Proof. ring. apply derivable_pt_lim_exp. Qed. + +(* added later. *) + +Lemma Rpower_mult_distr : + forall x y z, 0 < x -> 0 < y -> + Rpower x z * Rpower y z = Rpower (x * y) z. +intros x y z x0 y0; unfold Rpower. +rewrite <- exp_plus, ln_mult, Rmult_plus_distr_l; auto. +Qed. + +Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> Rpower a c <= Rpower b c. +Proof. +intros [c0 | c0]; + [ | intros; rewrite <- c0, !Rpower_O; [apply Rle_refl | |] ]. + intros [a0 [ab|ab]]. + left; apply exp_increasing. + now apply Rmult_lt_compat_l; auto; apply ln_increasing; fourier. + rewrite ab; apply Rle_refl. + apply Rlt_le_trans with a; tauto. +tauto. +Qed. + +(* arcsinh function *) + +Definition arcsinh x := ln (x + sqrt (x ^ 2 + 1)). + +Lemma arcsinh_sinh : forall x, arcsinh (sinh x) = x. +intros x; unfold sinh, arcsinh. +assert (Rminus_eq_0 : forall r, r - r = 0) by (intros; ring). +pattern 1 at 5; rewrite <- exp_0, <- (Rminus_eq_0 x); unfold Rminus. +rewrite exp_plus. +match goal with |- context[sqrt ?a] => + replace a with (((exp x + exp(-x))/2)^2) by field +end. +rewrite sqrt_pow2; + [|apply Rlt_le, Rmult_lt_0_compat;[apply Rplus_lt_0_compat; apply exp_pos | + apply Rinv_0_lt_compat, Rlt_0_2]]. +match goal with |- context[ln ?a] => replace a with (exp x) by field end. +rewrite ln_exp; reflexivity. +Qed. + +Lemma sinh_arcsinh x : sinh (arcsinh x) = x. +unfold sinh, arcsinh. +assert (cmp : 0 < x + sqrt (x ^ 2 + 1)). + destruct (Rle_dec x 0). + replace (x ^ 2) with ((-x) ^ 2) by ring. + assert (sqrt ((- x) ^ 2) < sqrt ((-x)^2+1)). + apply sqrt_lt_1_alt. + split;[apply pow_le | ]; fourier. + pattern x at 1; replace x with (- (sqrt ((- x) ^ 2))). + assert (t:= sqrt_pos ((-x)^2)); fourier. + simpl; rewrite Rmult_1_r, sqrt_square, Ropp_involutive;[reflexivity | fourier]. + apply Rplus_lt_le_0_compat;[apply Rnot_le_gt; assumption | apply sqrt_pos]. +rewrite exp_ln;[ | assumption]. +rewrite exp_Ropp, exp_ln;[ | assumption]. +assert (Rmult_minus_distr_r : + forall x y z, (x - y) * z = x * z - y * z) by (intros; ring). +apply Rminus_diag_uniq; unfold Rdiv; rewrite Rmult_minus_distr_r. +assert (t: forall x y z, x - z = y -> x - y - z = 0);[ | apply t; clear t]. + intros a b c H; rewrite <- H; ring. +apply Rmult_eq_reg_l with (2 * (x + sqrt (x ^ 2 + 1)));[ | + apply Rgt_not_eq, Rmult_lt_0_compat;[apply Rlt_0_2 | assumption]]. +assert (pow2_sqrt : forall x, 0 <= x -> sqrt x ^ 2 = x) by + (intros; simpl; rewrite Rmult_1_r, sqrt_sqrt; auto). +field_simplify;[rewrite pow2_sqrt;[field | ] | apply Rgt_not_eq; fourier]. +apply Rplus_le_le_0_compat;[simpl; rewrite Rmult_1_r; apply (Rle_0_sqr x)|apply Rlt_le, Rlt_0_1]. +Qed. + +Lemma derivable_pt_lim_arcsinh : + forall x, derivable_pt_lim arcsinh x (/sqrt (x ^ 2 + 1)). +intros x; unfold arcsinh. +assert (0 < x + sqrt (x ^ 2 + 1)). + destruct (Rle_dec x 0); + [ | assert (0 < x) by (apply Rnot_le_gt; assumption); + apply Rplus_lt_le_0_compat; auto; apply sqrt_pos]. + replace (x ^ 2) with ((-x) ^ 2) by ring. + assert (sqrt ((- x) ^ 2) < sqrt ((-x)^2+1)). + apply sqrt_lt_1_alt. + split;[apply pow_le|]; fourier. + pattern x at 1; replace x with (- (sqrt ((- x) ^ 2))). + assert (t:= sqrt_pos ((-x)^2)); fourier. + simpl; rewrite Rmult_1_r, sqrt_square, Ropp_involutive; auto; fourier. +assert (0 < x ^ 2 + 1). + apply Rplus_le_lt_0_compat;[simpl; rewrite Rmult_1_r; apply Rle_0_sqr|fourier]. +replace (/sqrt (x ^ 2 + 1)) with + (/(x + sqrt (x ^ 2 + 1)) * + (1 + (/(2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)))). +apply (derivable_pt_lim_comp (fun x => x + sqrt (x ^ 2 + 1)) ln). + apply (derivable_pt_lim_plus). + apply derivable_pt_lim_id. + apply (derivable_pt_lim_comp (fun x => x ^ 2 + 1) sqrt x). + apply derivable_pt_lim_plus. + apply derivable_pt_lim_pow. + apply derivable_pt_lim_const. + apply derivable_pt_lim_sqrt; assumption. + apply derivable_pt_lim_ln; assumption. + replace (INR 2 * x ^ 1 + 0) with (2 * x) by (simpl; ring). +replace (1 + / (2 * sqrt (x ^ 2 + 1)) * (2 * x)) with + (((sqrt (x ^ 2 + 1) + x))/sqrt (x ^ 2 + 1)); + [ | field; apply Rgt_not_eq, sqrt_lt_R0; assumption]. +apply Rmult_eq_reg_l with (x + sqrt (x ^ 2 + 1)); + [ | apply Rgt_not_eq; assumption]. +rewrite <- Rmult_assoc, Rinv_r;[field | ]; apply Rgt_not_eq; auto; + apply sqrt_lt_R0; assumption. +Qed. + +Lemma arcsinh_lt : forall x y, x < y -> arcsinh x < arcsinh y. +intros x y xy. +case (Rle_dec (arcsinh y) (arcsinh x));[ | apply Rnot_le_lt ]. +intros abs; case (Rlt_not_le _ _ xy). +rewrite <- (sinh_arcsinh y), <- (sinh_arcsinh x). +destruct abs as [lt | q];[| rewrite q; fourier]. +apply Rlt_le, sinh_lt; assumption. +Qed. + +Lemma arcsinh_le : forall x y, x <= y -> arcsinh x <= arcsinh y. +intros x y [xy | xqy]. + apply Rlt_le, arcsinh_lt; assumption. +rewrite xqy; apply Rle_refl. +Qed. + +Lemma arcsinh_0 : arcsinh 0 = 0. + unfold arcsinh; rewrite pow_ne_zero, !Rplus_0_l, sqrt_1, ln_1; + [reflexivity | discriminate]. +Qed. |