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Diffstat (limited to 'theories/Reals/Rpower.v')
| -rw-r--r-- | theories/Reals/Rpower.v | 127 |
1 files changed, 127 insertions, 0 deletions
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index 2eb485188..9099ec057 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -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). @@ -701,3 +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. |