src/HOL/NthRoot.thy
 author nipkow Thu, 22 Oct 2020 08:39:08 +0200 changeset 72501 70b420065a07 parent 70722 ae2528273eeb permissions -rw-r--r--
tuned names: t_ -> T_
```
(*  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"
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]])
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"

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
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"

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"

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>"

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"

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
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
next
case False
then show ?thesis
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]

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]
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"

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"

lemma four_x_squared: "4 * x\<^sup>2 = (2 * x)\<^sup>2"
for x :: real

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"
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)"

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)"

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))"
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"
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 (drule order_le_less_trans [OF abs_ge_zero])
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])
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)"
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"
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
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])
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"
also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
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"
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
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
```