(*  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 Deriv
begin


subsection \<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)
qed

lemma 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 simp

definition 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 simp
next
  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)
qed

lemma 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 simp
next
  case False
  then show ?thesis
    by simp (metis root_sgn_power sgn_power_root)
qed

lemma 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)
qed

lemma real_root_gt_zero: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> 0 < root n x"
  using real_root_less_mono[of n 0 x] by simp

lemma 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 simp

lemma 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 simp

lemma 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 simp

lemma 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 simp

lemma 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 simp

lemma 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 simp


text \<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: 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 root_abs_power: "n > 0 \<Longrightarrow> abs (root n (y ^n)) = abs y"
  using root_sgn_power [of n]
  by (metis abs_ge_zero power_abs real_root_abs real_root_power_cancel)

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 simp
qed

lemma 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 simp
qed

lemma real_root_decreasing: "0 < n \<Longrightarrow> n \<le> 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 \<le> 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])
qed

lemma 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 "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)
qed (use n in auto)

lemma 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" "x < x + 1"
    by auto
  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)
qed (use n odd_real_root_pow in auto)

lemma 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 "- (root n y ^ n) = y" if "x - 1 < y" and "y < 0" for y
  proof -
    have "root n (-y) ^ n = -y" 
      using that \<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)
qed

lemma 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])

lemma power_tendsto_0_iff [simp]:
  fixes f :: "'a \<Rightarrow> real"
  assumes "n > 0"
  shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
proof -
  have "((\<lambda>x. \<bar>root n (f x ^ n)\<bar>) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
    by (auto simp: assms root_abs_power tendsto_rabs_zero_iff)
  then have "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
    by (metis tendsto_real_root abs_0 real_root_zero tendsto_rabs)
  with assms show ?thesis
    by (auto simp: tendsto_null_power)
qed

subsection \<open>Square Root\<close>

definition sqrt :: "real \<Rightarrow> real"
  where "sqrt = root 2"

lemma pos2: "0 < (2::nat)"
  by simp

lemma 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>"
  by (metis power2_abs abs_ge_zero real_sqrt_unique)

lemma 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"
  by (metis real_sqrt_pow2 zero_le_power2)

lemma 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 simp

lemma 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 simp

lemma 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 simp

lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
  using real_sqrt_le_mono[of "x\<^sup>2" y] by simp

lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y"
  using real_sqrt_less_mono[of "x\<^sup>2" y] by simp

lemma 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)
qed

lemma sqrt_le_D: "sqrt x \<le> y \<Longrightarrow> x \<le> y\<^sup>2"
  by (meson not_le real_less_rsqrt)

lemma sqrt_ge_absD: "\<bar>x\<bar> \<le> sqrt y \<Longrightarrow> x\<^sup>2 \<le> y"
  using real_sqrt_le_iff[of "x\<^sup>2"] by simp

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 simp
qed

lemmas 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 simp

declare
  DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
  DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]

lemmas has_derivative_real_sqrt[derivative_intros] = DERIV_real_sqrt[THEN DERIV_compose_FDERIV]

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)
  done

lemma real_sqrt_abs2 [simp]: "sqrt (x * x) = \<bar>x\<bar>"
  apply (subst power2_eq_square [symmetric])
  apply (rule real_sqrt_abs)
  done

lemma 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 simp

lemma real_sqrt_ge_one: "1 \<le> x \<Longrightarrow> 1 \<le> sqrt x"
  by simp

lemma 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 simp
next
  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)
  qed
qed

lemma 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 arith
qed

lemma 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)
  done

lemma 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)
  done

lemma 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"]) simp

lemma 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"]) simp

lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
  by (rule power2_le_imp_le) simp_all

lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
  by (rule power2_le_imp_le) simp_all

lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
  by (rule power2_le_imp_le) simp_all

lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
  by (rule power2_le_imp_le) simp_all

lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
  by (simp add: power2_eq_square [symmetric])

lemma sqrt_sum_squares_le_sum:
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) \<le> x + y"
  by (rule power2_le_imp_le) (simp_all add: power2_sum)

lemma L2_set_mult_ineq_lemma:
  fixes a b c d :: real
  shows "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
proof -
  have "0 \<le> (a * d - b * c)\<^sup>2" by simp
  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * d) * (b * c)"
    by (simp only: power2_diff power_mult_distrib)
  also have "\<dots> = a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2 - 2 * (a * c) * (b * d)"
    by simp
  finally show "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
    by simp
qed

lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<^sup>2 + y\<^sup>2) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
  by (rule power2_le_imp_le) (simp_all add: power2_sum)

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)"
proof -
  have "(a * c + b * d) \<le> (sqrt (a\<^sup>2 + b\<^sup>2) * sqrt (c\<^sup>2 + d\<^sup>2))"
    by (rule power2_le_imp_le) (simp_all add: power2_sum power_mult_distrib ring_distribs L2_set_mult_ineq_lemma add.commute)
  then have "(a + c)\<^sup>2 + (b + d)\<^sup>2 \<le> (sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2))\<^sup>2"
    by (simp add: power2_sum)
  then show ?thesis
    by (auto intro: power2_le_imp_le)
qed

lemma 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)
  done

lemma 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
  done

lemma 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 field_char_0_class.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\<le>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)
qed

lemma 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\<le>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)
  qed
qed


text "Legacy theorem names:"
lemmas real_root_pos2 = real_root_power_cancel
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
lemmas real_root_pos_pos_le = real_root_ge_zero
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff

end
