(* Title: HOL/NthRoot.thy Author: Jacques D. Fleuriot, 1998 Author: Lawrence C Paulson, 2004*)section \<open>Nth Roots of Real Numbers\<close>theory NthRoot imports Derivbeginsubsection \<open>Existence of Nth Root\<close>text \<open>Existence follows from the Intermediate Value Theorem\<close>lemma realpow_pos_nth: fixes a :: real assumes n: "0 < n" and a: "0 < a" shows "\<exists>r>0. r ^ n = a"proof - have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a" proof (rule IVT) show "0 ^ n \<le> a" using n a by (simp add: power_0_left) show "0 \<le> max 1 a" by simp from n have n1: "1 \<le> n" by simp have "a \<le> max 1 a ^ 1" by simp also have "max 1 a ^ 1 \<le> max 1 a ^ n" using n1 by (rule power_increasing) simp finally show "a \<le> max 1 a ^ n" . show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r" by simp qed then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast with n a have "r \<noteq> 0" by (auto simp add: power_0_left) with r have "0 < r \<and> r ^ n = a" by simp then show ?thesis ..qed(* Used by Integration/RealRandVar.thy in AFP *)lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a" by (blast intro: realpow_pos_nth)text \<open>Uniqueness of nth positive root.\<close>lemma realpow_pos_nth_unique: "0 < n \<Longrightarrow> 0 < a \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = a" for a :: real by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base)subsection \<open>Nth Root\<close>text \<open> We define roots of negative reals such that \<open>root n (- x) = - root n x\<close>. This allows us to omit side conditions from many theorems.\<close>lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f")proof (rule injI) have x: "(0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" for a b :: real by auto fix x y assume "?f x = ?f y" with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] \<open>0 < n\<close> show "x = y" by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]]) (simp_all add: x)qedlemma sgn_power_injE: "sgn a * \<bar>a\<bar> ^ n = x \<Longrightarrow> x = sgn b * \<bar>b\<bar> ^ n \<Longrightarrow> 0 < n \<Longrightarrow> a = b" for a b :: real using inj_sgn_power[THEN injD, of n a b] by simpdefinition root :: "nat \<Rightarrow> real \<Rightarrow> real" where "root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)"lemma root_0 [simp]: "root 0 x = 0" by (simp add: root_def)lemma root_sgn_power: "0 < n \<Longrightarrow> root n (sgn y * \<bar>y\<bar>^n) = y" using the_inv_f_f[OF inj_sgn_power] by (simp add: root_def)lemma sgn_power_root: assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x")proof (cases "x = 0") case True with assms root_sgn_power[of n 0] show ?thesis by simpnext case False with realpow_pos_nth[OF \<open>0 < n\<close>, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto with \<open>x \<noteq> 0\<close> have S: "x \<in> range ?f" by (intro image_eqI[of _ _ "sgn x * r"]) (auto simp: abs_mult sgn_mult power_mult_distrib abs_sgn_eq mult_sgn_abs) from \<open>0 < n\<close> f_the_inv_into_f[OF inj_sgn_power[OF \<open>0 < n\<close>] this] show ?thesis by (simp add: root_def)qedlemma split_root: "P (root n x) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar>^n = x \<longrightarrow> P y))"proof (cases "n = 0") case True then show ?thesis by simpnext case False then show ?thesis by simp (metis root_sgn_power sgn_power_root)qedlemma real_root_zero [simp]: "root n 0 = 0" by (simp split: split_root add: sgn_zero_iff)lemma real_root_minus: "root n (- x) = - root n x" by (clarsimp split: split_root elim!: sgn_power_injE simp: sgn_minus)lemma real_root_less_mono: "0 < n \<Longrightarrow> x < y \<Longrightarrow> root n x < root n y"proof (clarsimp split: split_root) have *: "0 < b \<Longrightarrow> a < 0 \<Longrightarrow> \<not> a > b" for a b :: real by auto fix a b :: real assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n" then show "a < b" using power_less_imp_less_base[of a n b] power_less_imp_less_base[of "- b" n "- a"] by (simp add: sgn_real_def * [of "a ^ n" "- ((- b) ^ n)"] split: if_split_asm)qedlemma real_root_gt_zero: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> 0 < root n x" using real_root_less_mono[of n 0 x] by simplemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x" using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less)lemma real_root_pow_pos: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x ^ n = x" (* TODO: rename *) using sgn_power_root[of n x] real_root_gt_zero[of n x] by simplemma real_root_pow_pos2 [simp]: "0 < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> root n x ^ n = x" (* TODO: rename *) by (auto simp add: order_le_less real_root_pow_pos)lemma sgn_root: "0 < n \<Longrightarrow> sgn (root n x) = sgn x" by (auto split: split_root simp: sgn_real_def)lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x" using sgn_power_root[of n x] by (simp add: odd_pos sgn_real_def split: if_split_asm)lemma real_root_power_cancel: "0 < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> root n (x ^ n) = x" using root_sgn_power[of n x] by (auto simp add: le_less power_0_left)lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x" using root_sgn_power[of n x] by (simp add: odd_pos sgn_real_def power_0_left split: if_split_asm)lemma real_root_pos_unique: "0 < n \<Longrightarrow> 0 \<le> y \<Longrightarrow> y ^ n = x \<Longrightarrow> root n x = y" using root_sgn_power[of n y] by (auto simp add: le_less power_0_left)lemma odd_real_root_unique: "odd n \<Longrightarrow> y ^ n = x \<Longrightarrow> root n x = y" by (erule subst, rule odd_real_root_power_cancel)lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" by (simp add: real_root_pos_unique)text \<open>Root function is strictly monotonic, hence injective.\<close>lemma real_root_le_mono: "0 < n \<Longrightarrow> x \<le> y \<Longrightarrow> root n x \<le> root n y" by (auto simp add: order_le_less real_root_less_mono)lemma real_root_less_iff [simp]: "0 < n \<Longrightarrow> root n x < root n y \<longleftrightarrow> x < y" by (cases "x < y") (simp_all add: real_root_less_mono linorder_not_less real_root_le_mono)lemma real_root_le_iff [simp]: "0 < n \<Longrightarrow> root n x \<le> root n y \<longleftrightarrow> x \<le> y" by (cases "x \<le> y") (simp_all add: real_root_le_mono linorder_not_le real_root_less_mono)lemma real_root_eq_iff [simp]: "0 < n \<Longrightarrow> root n x = root n y \<longleftrightarrow> x = y" by (simp add: order_eq_iff)lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified]lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> 1 < root n y \<longleftrightarrow> 1 < y" using real_root_less_iff [where x=1] by simplemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> root n x < 1 \<longleftrightarrow> x < 1" using real_root_less_iff [where y=1] by simplemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> 1 \<le> root n y \<longleftrightarrow> 1 \<le> y" using real_root_le_iff [where x=1] by simplemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> root n x \<le> 1 \<longleftrightarrow> x \<le> 1" using real_root_le_iff [where y=1] by simplemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> root n x = 1 \<longleftrightarrow> x = 1" using real_root_eq_iff [where y=1] by simptext \<open>Roots of multiplication and division.\<close>lemma real_root_mult: "root n (x * y) = root n x * root n y" by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib)lemma real_root_inverse: "root n (inverse x) = inverse (root n x)" by (auto split: split_root elim!: sgn_power_injE simp: inverse_sgn power_inverse)lemma real_root_divide: "root n (x / y) = root n x / root n y" by (simp add: divide_inverse real_root_mult real_root_inverse)lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" by (simp add: abs_if real_root_minus)lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" by (induct k) (simp_all add: real_root_mult)text \<open>Roots of roots.\<close>lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x" by (simp add: odd_real_root_unique)lemma real_root_mult_exp: "root (m * n) x = root m (root n x)" by (auto split: split_root elim!: sgn_power_injE simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq)lemma real_root_commute: "root m (root n x) = root n (root m x)" by (simp add: real_root_mult_exp [symmetric] mult.commute)text \<open>Monotonicity in first argument.\<close>lemma real_root_strict_decreasing: assumes "0 < n" "n < N" "1 < x" shows "root N x < root n x"proof - from assms have "root n (root N x) ^ n < root N (root n x) ^ N" by (simp add: real_root_commute power_strict_increasing del: real_root_pow_pos2) with assms show ?thesis by simpqedlemma real_root_strict_increasing: assumes "0 < n" "n < N" "0 < x" "x < 1" shows "root n x < root N x"proof - from assms have "root N (root n x) ^ N < root n (root N x) ^ n" by (simp add: real_root_commute power_strict_decreasing del: real_root_pow_pos2) with assms show ?thesis by simpqedlemma real_root_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 \<le> x \<Longrightarrow> root N x \<le> root n x" by (auto simp add: order_le_less real_root_strict_decreasing)lemma real_root_increasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> root n x \<le> root N x" by (auto simp add: order_le_less real_root_strict_increasing)text \<open>Continuity and derivatives.\<close>lemma isCont_real_root: "isCont (root n) x"proof (cases "n > 0") case True let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n" have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else - ((-x) ^ n) :: real)" using True by (intro continuous_on_If continuous_intros) auto then have "continuous_on UNIV ?f" by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less le_less True) then have [simp]: "isCont ?f x" for x by (simp add: continuous_on_eq_continuous_at) have "isCont (root n) (?f (root n x))" by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power True) then show ?thesis by (simp add: sgn_power_root True)next case False then show ?thesis by (simp add: root_def[abs_def])qedlemma tendsto_real_root [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) \<longlongrightarrow> root n x) F" using isCont_tendsto_compose[OF isCont_real_root, of f x F] .lemma continuous_real_root [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))" unfolding continuous_def by (rule tendsto_real_root)lemma continuous_on_real_root [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))" unfolding continuous_on_def by (auto intro: tendsto_real_root)lemma DERIV_real_root: assumes n: "0 < n" and x: "0 < x" shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"proof (rule DERIV_inverse_function) show "0 < x" using x . show "x < x + 1" by simp show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" using n by simp show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" by (rule DERIV_pow) show "real n * root n x ^ (n - Suc 0) \<noteq> 0" using n x by simp show "isCont (root n) x" by (rule isCont_real_root)qedlemma DERIV_odd_real_root: assumes n: "odd n" and x: "x \<noteq> 0" shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"proof (rule DERIV_inverse_function) show "x - 1 < x" by simp show "x < x + 1" by simp show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" using n by (simp add: odd_real_root_pow) show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)" by (rule DERIV_pow) show "real n * root n x ^ (n - Suc 0) \<noteq> 0" using odd_pos [OF n] x by simp show "isCont (root n) x" by (rule isCont_real_root)qedlemma DERIV_even_real_root: assumes n: "0 < n" and "even n" and x: "x < 0" shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"proof (rule DERIV_inverse_function) show "x - 1 < x" by simp show "x < 0" using x . show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y" proof (rule allI, rule impI, erule conjE) fix y assume "x - 1 < y" and "y < 0" then have "root n (-y) ^ n = -y" using \<open>0 < n\<close> by simp with real_root_minus and \<open>even n\<close> show "- (root n y ^ n) = y" by simp qed show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)" by (auto intro!: derivative_eq_intros) show "- real n * root n x ^ (n - Suc 0) \<noteq> 0" using n x by simp show "isCont (root n) x" by (rule isCont_real_root)qedlemma DERIV_real_root_generic: assumes "0 < n" and "x \<noteq> 0" and "even n \<Longrightarrow> 0 < x \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))" and "even n \<Longrightarrow> x < 0 \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))" and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))" shows "DERIV (root n) x :> D" using assms by (cases "even n", cases "0 < x") (auto intro: DERIV_real_root[THEN DERIV_cong] DERIV_odd_real_root[THEN DERIV_cong] DERIV_even_real_root[THEN DERIV_cong])subsection \<open>Square Root\<close>definition sqrt :: "real \<Rightarrow> real" where "sqrt = root 2"lemma pos2: "0 < (2::nat)" by simplemma real_sqrt_unique: "y\<^sup>2 = x \<Longrightarrow> 0 \<le> y \<Longrightarrow> sqrt x = y" unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>" apply (rule real_sqrt_unique) apply (rule power2_abs) apply (rule abs_ge_zero) donelemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x" unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])lemma real_sqrt_pow2_iff [simp]: "(sqrt x)\<^sup>2 = x \<longleftrightarrow> 0 \<le> x" apply (rule iffI) apply (erule subst) apply (rule zero_le_power2) apply (erule real_sqrt_pow2) donelemma real_sqrt_zero [simp]: "sqrt 0 = 0" unfolding sqrt_def by (rule real_root_zero)lemma real_sqrt_one [simp]: "sqrt 1 = 1" unfolding sqrt_def by (rule real_root_one [OF pos2])lemma real_sqrt_four [simp]: "sqrt 4 = 2" using real_sqrt_abs[of 2] by simplemma real_sqrt_minus: "sqrt (- x) = - sqrt x" unfolding sqrt_def by (rule real_root_minus)lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" unfolding sqrt_def by (rule real_root_mult)lemma real_sqrt_mult_self[simp]: "sqrt a * sqrt a = \<bar>a\<bar>" using real_sqrt_abs[of a] unfolding power2_eq_square real_sqrt_mult .lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" unfolding sqrt_def by (rule real_root_inverse)lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" unfolding sqrt_def by (rule real_root_divide)lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" unfolding sqrt_def by (rule real_root_power [OF pos2])lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" unfolding sqrt_def by (rule real_root_ge_zero)lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" unfolding sqrt_def by (rule real_root_less_mono [OF pos2])lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" unfolding sqrt_def by (rule real_root_le_mono [OF pos2])lemma real_sqrt_less_iff [simp]: "sqrt x < sqrt y \<longleftrightarrow> x < y" unfolding sqrt_def by (rule real_root_less_iff [OF pos2])lemma real_sqrt_le_iff [simp]: "sqrt x \<le> sqrt y \<longleftrightarrow> x \<le> y" unfolding sqrt_def by (rule real_root_le_iff [OF pos2])lemma real_sqrt_eq_iff [simp]: "sqrt x = sqrt y \<longleftrightarrow> x = y" unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])lemma real_less_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y\<^sup>2 \<Longrightarrow> sqrt x < y" using real_sqrt_less_iff[of x "y\<^sup>2"] by simplemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y" using real_sqrt_le_iff[of x "y\<^sup>2"] by simplemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y" using real_sqrt_le_mono[of "x\<^sup>2" y] by simplemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y" using real_sqrt_less_mono[of "x\<^sup>2" y] by simplemma real_sqrt_power_even: assumes "even n" "x \<ge> 0" shows "sqrt x ^ n = x ^ (n div 2)"proof - from assms obtain k where "n = 2*k" by (auto elim!: evenE) with assms show ?thesis by (simp add: power_mult)qedlemma sqrt_le_D: "sqrt x \<le> y \<Longrightarrow> x \<le> y\<^sup>2" by (meson not_le real_less_rsqrt)lemma sqrt_even_pow2: assumes n: "even n" shows "sqrt (2 ^ n) = 2 ^ (n div 2)"proof - from n obtain m where m: "n = 2 * m" .. from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)" by (simp only: power_mult[symmetric] mult.commute) then show ?thesis using m by simpqedlemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, unfolded real_sqrt_zero]lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, unfolded real_sqrt_zero]lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, unfolded real_sqrt_zero]lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, unfolded real_sqrt_zero]lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, unfolded real_sqrt_zero]lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, unfolded real_sqrt_one]lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, unfolded real_sqrt_one]lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, unfolded real_sqrt_one]lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, unfolded real_sqrt_one]lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, unfolded real_sqrt_one]lemma sqrt_add_le_add_sqrt: assumes "0 \<le> x" "0 \<le> y" shows "sqrt (x + y) \<le> sqrt x + sqrt y" by (rule power2_le_imp_le) (simp_all add: power2_sum assms)lemma isCont_real_sqrt: "isCont sqrt x" unfolding sqrt_def by (rule isCont_real_root)lemma tendsto_real_sqrt [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) \<longlongrightarrow> sqrt x) F" unfolding sqrt_def by (rule tendsto_real_root)lemma continuous_real_sqrt [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))" unfolding sqrt_def by (rule continuous_real_root)lemma continuous_on_real_sqrt [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))" unfolding sqrt_def by (rule continuous_on_real_root)lemma DERIV_real_sqrt_generic: assumes "x \<noteq> 0" and "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2" and "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2" shows "DERIV sqrt x :> D" using assms unfolding sqrt_def by (auto intro!: DERIV_real_root_generic)lemma DERIV_real_sqrt: "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" using DERIV_real_sqrt_generic by simpdeclare DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros] DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]lemma not_real_square_gt_zero [simp]: "\<not> 0 < x * x \<longleftrightarrow> x = 0" for x :: real apply auto using linorder_less_linear [where x = x and y = 0] apply (simp add: zero_less_mult_iff) donelemma real_sqrt_abs2 [simp]: "sqrt (x * x) = \<bar>x\<bar>" apply (subst power2_eq_square [symmetric]) apply (rule real_sqrt_abs) donelemma real_inv_sqrt_pow2: "0 < x \<Longrightarrow> (inverse (sqrt x))\<^sup>2 = inverse x" by (simp add: power_inverse)lemma real_sqrt_eq_zero_cancel: "0 \<le> x \<Longrightarrow> sqrt x = 0 \<Longrightarrow> x = 0" by simplemma real_sqrt_ge_one: "1 \<le> x \<Longrightarrow> 1 \<le> sqrt x" by simplemma sqrt_divide_self_eq: assumes nneg: "0 \<le> x" shows "sqrt x / x = inverse (sqrt x)"proof (cases "x = 0") case True then show ?thesis by simpnext case False then have pos: "0 < x" using nneg by arith show ?thesis proof (rule right_inverse_eq [THEN iffD1, symmetric]) show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg False) show "inverse (sqrt x) / (sqrt x / x) = 1" by (simp add: divide_inverse mult.assoc [symmetric] power2_eq_square [symmetric] real_inv_sqrt_pow2 pos False) qedqedlemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x" by (cases "x = 0") (simp_all add: sqrt_divide_self_eq [of x] field_simps)lemma real_divide_square_eq [simp]: "(r * a) / (r * r) = a / r" for a r :: real by (cases "r = 0") (simp_all add: divide_inverse ac_simps)lemma lemma_real_divide_sqrt_less: "0 < u \<Longrightarrow> u / sqrt 2 < u" by (simp add: divide_less_eq)lemma four_x_squared: "4 * x\<^sup>2 = (2 * x)\<^sup>2" for x :: real by (simp add: power2_eq_square)lemma sqrt_at_top: "LIM x at_top. sqrt x :: real :> at_top" by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="power2"]) (auto intro: eventually_gt_at_top)subsection \<open>Square Root of Sum of Squares\<close>lemma sum_squares_bound: "2 * x * y \<le> x\<^sup>2 + y\<^sup>2" for x y :: "'a::linordered_field"proof - have "(x - y)\<^sup>2 = x * x - 2 * x * y + y * y" by algebra then have "0 \<le> x\<^sup>2 - 2 * x * y + y\<^sup>2" by (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square) then show ?thesis by arithqedlemma arith_geo_mean: fixes u :: "'a::linordered_field" assumes "u\<^sup>2 = x * y" "x \<ge> 0" "y \<ge> 0" shows "u \<le> (x + y)/2" apply (rule power2_le_imp_le) using sum_squares_bound assms apply (auto simp: zero_le_mult_iff) apply (auto simp: algebra_simps power2_eq_square) donelemma arith_geo_mean_sqrt: fixes x :: real assumes "x \<ge> 0" "y \<ge> 0" shows "sqrt (x * y) \<le> (x + y)/2" apply (rule arith_geo_mean) using assms apply (auto simp: zero_le_mult_iff) donelemma real_sqrt_sum_squares_mult_ge_zero [simp]: "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2))" by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)lemma real_sqrt_sum_squares_mult_squared_eq [simp]: "(sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)))\<^sup>2 = (x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)" by (simp add: zero_le_mult_iff)lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0" by (drule arg_cong [where f = "\<lambda>x. x\<^sup>2"]) simplemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0" by (drule arg_cong [where f = "\<lambda>x. x\<^sup>2"]) simplemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)" by (rule power2_le_imp_le) simp_alllemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)" by (rule power2_le_imp_le) simp_alllemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)" by (rule power2_le_imp_le) simp_alllemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)" by (rule power2_le_imp_le) simp_alllemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)" by (simp add: power2_eq_square [symmetric])lemma real_sqrt_sum_squares_triangle_ineq: "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)" apply (rule power2_le_imp_le) apply simp apply (simp add: power2_sum) apply (simp only: mult.assoc distrib_left [symmetric]) apply (rule mult_left_mono) apply (rule power2_le_imp_le) apply (simp add: power2_sum power_mult_distrib) apply (simp add: ring_distribs) apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)") apply simp apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans) apply (rule zero_le_power2) apply (simp add: power2_diff power_mult_distrib) apply simp apply simp apply (simp add: add_increasing) donelemma real_sqrt_sum_squares_less: "\<bar>x\<bar> < u / sqrt 2 \<Longrightarrow> \<bar>y\<bar> < u / sqrt 2 \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u" apply (rule power2_less_imp_less) apply simp apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) apply (simp add: power_divide) apply (drule order_le_less_trans [OF abs_ge_zero]) apply (simp add: zero_less_divide_iff) donelemma sqrt2_less_2: "sqrt 2 < (2::real)" by (metis not_less not_less_iff_gr_or_eq numeral_less_iff real_sqrt_four real_sqrt_le_iff semiring_norm(75) semiring_norm(78) semiring_norm(85))lemma sqrt_sum_squares_half_less: "x < u/2 \<Longrightarrow> y < u/2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u" apply (rule real_sqrt_sum_squares_less) apply (auto simp add: abs_if field_simps) apply (rule le_less_trans [where y = "x*2"]) using less_eq_real_def sqrt2_less_2 apply force apply assumption apply (rule le_less_trans [where y = "y*2"]) using less_eq_real_def sqrt2_less_2 mult_le_cancel_left apply auto donelemma LIMSEQ_root: "(\<lambda>n. root n n) \<longlonglongrightarrow> 1"proof - define x where "x n = root n n - 1" for n have "x \<longlonglongrightarrow> sqrt 0" proof (rule tendsto_sandwich[OF _ _ tendsto_const]) show "(\<lambda>x. sqrt (2 / x)) \<longlonglongrightarrow> sqrt 0" by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially]) (simp_all add: at_infinity_eq_at_top_bot) have "x n \<le> sqrt (2 / real n)" if "2 < n" for n :: nat proof - have "1 + (real (n - 1) * n) / 2 * (x n)\<^sup>2 = 1 + of_nat (n choose 2) * (x n)\<^sup>2" by (auto simp add: choose_two of_nat_div mod_eq_0_iff_dvd) also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)" by (simp add: x_def) also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)" using \<open>2 < n\<close> by (intro sum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq) also have "\<dots> = (x n + 1) ^ n" by (simp add: binomial_ring) also have "\<dots> = n" using \<open>2 < n\<close> by (simp add: x_def) finally have "real (n - 1) * (real n / 2 * (x n)\<^sup>2) \<le> real (n - 1) * 1" by simp then have "(x n)\<^sup>2 \<le> 2 / real n" using \<open>2 < n\<close> unfolding mult_le_cancel_left by (simp add: field_simps) from real_sqrt_le_mono[OF this] show ?thesis by simp qed then show "eventually (\<lambda>n. x n \<le> sqrt (2 / real n)) sequentially" by (auto intro!: exI[of _ 3] simp: eventually_sequentially) show "eventually (\<lambda>n. sqrt 0 \<le> x n) sequentially" by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def) qed from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by (simp add: x_def)qedlemma LIMSEQ_root_const: assumes "0 < c" shows "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"proof - have ge_1: "(\<lambda>n. root n c) \<longlonglongrightarrow> 1" if "1 \<le> c" for c :: real proof - define x where "x n = root n c - 1" for n have "x \<longlonglongrightarrow> 0" proof (rule tendsto_sandwich[OF _ _ tendsto_const]) show "(\<lambda>n. c / n) \<longlonglongrightarrow> 0" by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially]) (simp_all add: at_infinity_eq_at_top_bot) have "x n \<le> c / n" if "1 < n" for n :: nat proof - have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1" by (simp add: choose_one) also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)" by (simp add: x_def) also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)" using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (intro sum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq) also have "\<dots> = (x n + 1) ^ n" by (simp add: binomial_ring) also have "\<dots> = c" using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (simp add: x_def) finally show ?thesis using \<open>1 \<le> c\<close> \<open>1 < n\<close> by (simp add: field_simps) qed then show "eventually (\<lambda>n. x n \<le> c / n) sequentially" by (auto intro!: exI[of _ 3] simp: eventually_sequentially) show "eventually (\<lambda>n. 0 \<le> x n) sequentially" using \<open>1 \<le> c\<close> by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def) qed from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by (simp add: x_def) qed show ?thesis proof (cases "1 \<le> c") case True with ge_1 show ?thesis by blast next case False with \<open>0 < c\<close> have "1 \<le> 1 / c" by simp then have "(\<lambda>n. 1 / root n (1 / c)) \<longlonglongrightarrow> 1 / 1" by (intro tendsto_divide tendsto_const ge_1 \<open>1 \<le> 1 / c\<close> one_neq_zero) then show ?thesis by (rule filterlim_cong[THEN iffD1, rotated 3]) (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide) qedqedtext "Legacy theorem names:"lemmas real_root_pos2 = real_root_power_cancellemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]lemmas real_root_pos_pos_le = real_root_ge_zerolemmas real_sqrt_mult_distrib = real_sqrt_multlemmas real_sqrt_mult_distrib2 = real_sqrt_multlemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iffend