# Theory NthRoot

theory NthRoot
imports Deriv
```(*  Title:      HOL/NthRoot.thy
Author:     Jacques D. Fleuriot, 1998
Author:     Lawrence C Paulson, 2004
*)

section ‹Nth Roots of Real Numbers›

theory NthRoot
imports Deriv
begin

subsection ‹Existence of Nth Root›

text ‹Existence follows from the Intermediate Value Theorem›

lemma realpow_pos_nth:
fixes a :: real
assumes n: "0 < n"
and a: "0 < a"
shows "∃r>0. r ^ n = a"
proof -
have "∃r≥0. r ≤ (max 1 a) ∧ r ^ n = a"
proof (rule IVT)
show "0 ^ n ≤ a"
using n a by (simp add: power_0_left)
show "0 ≤ max 1 a"
by simp
from n have n1: "1 ≤ n"
by simp
have "a ≤ max 1 a ^ 1"
by simp
also have "max 1 a ^ 1 ≤ max 1 a ^ n"
using n1 by (rule power_increasing) simp
finally show "a ≤ max 1 a ^ n" .
show "∀r. 0 ≤ r ∧ r ≤ max 1 a ⟶ isCont (λx. x ^ n) r"
by simp
qed
then obtain r where r: "0 ≤ r ∧ r ^ n = a"
by fast
with n a have "r ≠ 0"
with r have "0 < r ∧ r ^ n = a"
by simp
then show ?thesis ..
qed

(* Used by Integration/RealRandVar.thy in AFP *)
lemma realpow_pos_nth2: "(0::real) < a ⟹ ∃r>0. r ^ Suc n = a"
by (blast intro: realpow_pos_nth)

text ‹Uniqueness of nth positive root.›
lemma realpow_pos_nth_unique: "0 < n ⟹ 0 < a ⟹ ∃!r. 0 < r ∧ r ^ n = a" for a :: real
by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base)

subsection ‹Nth Root›

text ‹
We define roots of negative reals such that ‹root n (- x) = - root n x›.
This allows us to omit side conditions from many theorems.
›

lemma inj_sgn_power:
assumes "0 < n"
shows "inj (λy. sgn y * ¦y¦^n :: real)"
(is "inj ?f")
proof (rule injI)
have x: "(0 < a ∧ b < 0) ∨ (a < 0 ∧ 0 < b) ⟹ a ≠ b" for a b :: real
by auto
fix x y
assume "?f x = ?f y"
with power_eq_iff_eq_base[of n "¦x¦" "¦y¦"] ‹0 < n› 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 * ¦a¦ ^ n = x ⟹ x = sgn b * ¦b¦ ^ n ⟹ 0 < n ⟹ a = b"
for a b :: real
using inj_sgn_power[THEN injD, of n a b] by simp

definition root :: "nat ⇒ real ⇒ real"
where "root n x = (if n = 0 then 0 else the_inv (λy. sgn y * ¦y¦^n) x)"

lemma root_0 [simp]: "root 0 x = 0"

lemma root_sgn_power: "0 < n ⟹ root n (sgn y * ¦y¦^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) * ¦(root n x)¦^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 ‹0 < n›, of "¦x¦"]
obtain r where "0 < r" "r ^ n = ¦x¦"
by auto
with ‹x ≠ 0› have S: "x ∈ 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 ‹0 < n› f_the_inv_into_f[OF inj_sgn_power[OF ‹0 < n›] this]  show ?thesis
qed

lemma split_root: "P (root n x) ⟷ (n = 0 ⟶ P 0) ∧ (0 < n ⟶ (∀y. sgn y * ¦y¦^n = x ⟶ 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 ⟹ x < y ⟹ root n x < root n y"
proof (clarsimp split: split_root)
have *: "0 < b ⟹ a < 0 ⟹ ¬ a > b" for a b :: real
by auto
fix a b :: real
assume "0 < n" "sgn a * ¦a¦ ^ n < sgn b * ¦b¦ ^ 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 ⟹ 0 < x ⟹ 0 < root n x"
using real_root_less_mono[of n 0 x] by simp

lemma real_root_ge_zero: "0 ≤ x ⟹ 0 ≤ 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 ⟹ 0 < x ⟹ 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 ⟹ 0 ≤ x ⟹ root n x ^ n = x"  (* TODO: rename *)
by (auto simp add: order_le_less real_root_pow_pos)

lemma sgn_root: "0 < n ⟹ sgn (root n x) = sgn x"
by (auto split: split_root simp: sgn_real_def)

lemma odd_real_root_pow: "odd n ⟹ 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 ⟹ 0 ≤ x ⟹ 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 ⟹ 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 ⟹ 0 ≤ y ⟹ y ^ n = x ⟹ 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 ⟹ y ^ n = x ⟹ root n x = y"
by (erule subst, rule odd_real_root_power_cancel)

lemma real_root_one [simp]: "0 < n ⟹ root n 1 = 1"

text ‹Root function is strictly monotonic, hence injective.›

lemma real_root_le_mono: "0 < n ⟹ x ≤ y ⟹ root n x ≤ root n y"
by (auto simp add: order_le_less real_root_less_mono)

lemma real_root_less_iff [simp]: "0 < n ⟹ root n x < root n y ⟷ 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 ⟹ root n x ≤ root n y ⟷ x ≤ y"
by (cases "x ≤ y") (simp_all add: real_root_le_mono linorder_not_le real_root_less_mono)

lemma real_root_eq_iff [simp]: "0 < n ⟹ root n x = root n y ⟷ 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 ⟹ 1 < root n y ⟷ 1 < y"
using real_root_less_iff [where x=1] by simp

lemma real_root_lt_1_iff [simp]: "0 < n ⟹ root n x < 1 ⟷ x < 1"
using real_root_less_iff [where y=1] by simp

lemma real_root_ge_1_iff [simp]: "0 < n ⟹ 1 ≤ root n y ⟷ 1 ≤ y"
using real_root_le_iff [where x=1] by simp

lemma real_root_le_1_iff [simp]: "0 < n ⟹ root n x ≤ 1 ⟷ x ≤ 1"
using real_root_le_iff [where y=1] by simp

lemma real_root_eq_1_iff [simp]: "0 < n ⟹ root n x = 1 ⟷ x = 1"
using real_root_eq_iff [where y=1] by simp

text ‹Roots of multiplication and division.›

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 ⟹ root n ¦x¦ = ¦root n x¦"

lemma real_root_power: "0 < n ⟹ root n (x ^ k) = root n x ^ k"
by (induct k) (simp_all add: real_root_mult)

text ‹Roots of roots.›

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 ‹Monotonicity in first argument.›

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 ⟹ n < N ⟹ 1 ≤ x ⟹ root N x ≤ root n x"
by (auto simp add: order_le_less real_root_strict_decreasing)

lemma real_root_increasing: "0 < n ⟹ n < N ⟹ 0 ≤ x ⟹ x ≤ 1 ⟹ root n x ≤ root N x"
by (auto simp add: order_le_less real_root_strict_increasing)

text ‹Continuity and derivatives.›

lemma isCont_real_root: "isCont (root n) x"
proof (cases "n > 0")
case True
let ?f = "λy::real. sgn y * ¦y¦^n"
have "continuous_on ({0..} ∪ {.. 0}) (λx. 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 ⤏ x) F ⟹ ((λx. root n (f x)) ⤏ 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 ⟹ continuous F (λx. root n (f x))"
unfolding continuous_def by (rule tendsto_real_root)

lemma continuous_on_real_root [continuous_intros]:
"continuous_on s f ⟹ continuous_on s (λx. 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 "∀y. 0 < y ∧ y < x + 1 ⟶ root n y ^ n = y"
using n by simp
show "DERIV (λx. 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) ≠ 0"
using n x by simp
show "isCont (root n) x"
by (rule isCont_real_root)
qed

lemma DERIV_odd_real_root:
assumes n: "odd 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 < x + 1"
by simp
show "∀y. x - 1 < y ∧ y < x + 1 ⟶ root n y ^ n = y"
using n by (simp add: odd_real_root_pow)
show "DERIV (λx. 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) ≠ 0"
using odd_pos [OF n] x by simp
show "isCont (root n) x"
by (rule isCont_real_root)
qed

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 "∀y. x - 1 < y ∧ y < 0 ⟶ - (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 ‹0 < n› by simp
with real_root_minus and ‹even n›
show "- (root n y ^ n) = y" by simp
qed
show "DERIV (λx. - (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) ≠ 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 ≠ 0"
and "even n ⟹ 0 < x ⟹ D = inverse (real n * root n x ^ (n - Suc 0))"
and "even n ⟹ x < 0 ⟹ D = - inverse (real n * root n x ^ (n - Suc 0))"
and "odd n ⟹ 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 ‹Square Root›

definition sqrt :: "real ⇒ real"
where "sqrt = root 2"

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

lemma real_sqrt_unique: "y⇧2 = x ⟹ 0 ≤ y ⟹ sqrt x = y"
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])

lemma real_sqrt_abs [simp]: "sqrt (x⇧2) = ¦x¦"
apply (rule real_sqrt_unique)
apply (rule power2_abs)
apply (rule abs_ge_zero)
done

lemma real_sqrt_pow2 [simp]: "0 ≤ x ⟹ (sqrt x)⇧2 = x"
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])

lemma real_sqrt_pow2_iff [simp]: "(sqrt x)⇧2 = x ⟷ 0 ≤ x"
apply (rule iffI)
apply (erule subst)
apply (rule zero_le_power2)
apply (erule real_sqrt_pow2)
done

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 = ¦a¦"
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 ⟹ 0 < sqrt x"
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])

lemma real_sqrt_ge_zero: "0 ≤ x ⟹ 0 ≤ sqrt x"
unfolding sqrt_def by (rule real_root_ge_zero)

lemma real_sqrt_less_mono: "x < y ⟹ sqrt x < sqrt y"
unfolding sqrt_def by (rule real_root_less_mono [OF pos2])

lemma real_sqrt_le_mono: "x ≤ y ⟹ sqrt x ≤ sqrt y"
unfolding sqrt_def by (rule real_root_le_mono [OF pos2])

lemma real_sqrt_less_iff [simp]: "sqrt x < sqrt y ⟷ x < y"
unfolding sqrt_def by (rule real_root_less_iff [OF pos2])

lemma real_sqrt_le_iff [simp]: "sqrt x ≤ sqrt y ⟷ x ≤ y"
unfolding sqrt_def by (rule real_root_le_iff [OF pos2])

lemma real_sqrt_eq_iff [simp]: "sqrt x = sqrt y ⟷ x = y"
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])

lemma real_less_lsqrt: "0 ≤ x ⟹ 0 ≤ y ⟹ x < y⇧2 ⟹ sqrt x < y"
using real_sqrt_less_iff[of x "y⇧2"] by simp

lemma real_le_lsqrt: "0 ≤ x ⟹ 0 ≤ y ⟹ x ≤ y⇧2 ⟹ sqrt x ≤ y"
using real_sqrt_le_iff[of x "y⇧2"] by simp

lemma real_le_rsqrt: "x⇧2 ≤ y ⟹ x ≤ sqrt y"
using real_sqrt_le_mono[of "x⇧2" y] by simp

lemma real_less_rsqrt: "x⇧2 < y ⟹ x < sqrt y"
using real_sqrt_less_mono[of "x⇧2" y] by simp

lemma real_sqrt_power_even:
assumes "even n" "x ≥ 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 ≤ y ⟹ x ≤ y⇧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)⇧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 ≤ x" "0 ≤ y"
shows "sqrt (x + y) ≤ 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 ⤏ x) F ⟹ ((λx. sqrt (f x)) ⤏ sqrt x) F"
unfolding sqrt_def by (rule tendsto_real_root)

lemma continuous_real_sqrt [continuous_intros]:
"continuous F f ⟹ continuous F (λx. sqrt (f x))"
unfolding sqrt_def by (rule continuous_real_root)

lemma continuous_on_real_sqrt [continuous_intros]:
"continuous_on s f ⟹ continuous_on s (λx. sqrt (f x))"
unfolding sqrt_def by (rule continuous_on_real_root)

lemma DERIV_real_sqrt_generic:
assumes "x ≠ 0"
and "x > 0 ⟹ D = inverse (sqrt x) / 2"
and "x < 0 ⟹ 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 ⟹ 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]

lemma not_real_square_gt_zero [simp]: "¬ 0 < x * x ⟷ 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) = ¦x¦"
apply (subst power2_eq_square [symmetric])
apply (rule real_sqrt_abs)
done

lemma real_inv_sqrt_pow2: "0 < x ⟹ (inverse (sqrt x))⇧2 = inverse x"

lemma real_sqrt_eq_zero_cancel: "0 ≤ x ⟹ sqrt x = 0 ⟹ x = 0"
by simp

lemma real_sqrt_ge_one: "1 ≤ x ⟹ 1 ≤ sqrt x"
by simp

lemma sqrt_divide_self_eq:
assumes nneg: "0 ≤ 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 ≠ 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 ≤ x ⟹ 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 ⟹ u / sqrt 2 < u"

lemma four_x_squared: "4 * x⇧2 = (2 * x)⇧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="λx. True" and P="λx. 0 < x" and g="power2"])
(auto intro: eventually_gt_at_top)

subsection ‹Square Root of Sum of Squares›

lemma sum_squares_bound: "2 * x * y ≤ x⇧2 + y⇧2"
for x y :: "'a::linordered_field"
proof -
have "(x - y)⇧2 = x * x - 2 * x * y + y * y"
by algebra
then have "0 ≤ x⇧2 - 2 * x * y + y⇧2"
then show ?thesis
by arith
qed

lemma arith_geo_mean:
fixes u :: "'a::linordered_field"
assumes "u⇧2 = x * y" "x ≥ 0" "y ≥ 0"
shows "u ≤ (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 ≥ 0" "y ≥ 0"
shows "sqrt (x * y) ≤ (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 ≤ sqrt ((x⇧2 + y⇧2) * (xa⇧2 + ya⇧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⇧2 + y⇧2) * (xa⇧2 + ya⇧2)))⇧2 = (x⇧2 + y⇧2) * (xa⇧2 + ya⇧2)"

lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x⇧2 + y⇧2) = x ⟹ y = 0"
by (drule arg_cong [where f = "λx. x⇧2"]) simp

lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x⇧2 + y⇧2) = y ⟹ x = 0"
by (drule arg_cong [where f = "λx. x⇧2"]) simp

lemma real_sqrt_sum_squares_ge1 [simp]: "x ≤ sqrt (x⇧2 + y⇧2)"
by (rule power2_le_imp_le) simp_all

lemma real_sqrt_sum_squares_ge2 [simp]: "y ≤ sqrt (x⇧2 + y⇧2)"
by (rule power2_le_imp_le) simp_all

lemma real_sqrt_ge_abs1 [simp]: "¦x¦ ≤ sqrt (x⇧2 + y⇧2)"
by (rule power2_le_imp_le) simp_all

lemma real_sqrt_ge_abs2 [simp]: "¦y¦ ≤ sqrt (x⇧2 + y⇧2)"
by (rule power2_le_imp_le) simp_all

lemma le_real_sqrt_sumsq [simp]: "x ≤ sqrt (x * x + y * y)"

lemma real_sqrt_sum_squares_triangle_ineq:
"sqrt ((a + c)⇧2 + (b + d)⇧2) ≤ sqrt (a⇧2 + b⇧2) + sqrt (c⇧2 + d⇧2)"
apply (rule power2_le_imp_le)
apply simp
apply (simp only: mult.assoc distrib_left [symmetric])
apply (rule mult_left_mono)
apply (rule power2_le_imp_le)
apply (subgoal_tac "0 ≤ b⇧2 * c⇧2 + a⇧2 * d⇧2 - 2 * (a * c) * (b * d)")
apply simp
apply (rule_tac b="(a * d - b * c)⇧2" in ord_le_eq_trans)
apply (rule zero_le_power2)
apply simp
apply simp
done

lemma real_sqrt_sum_squares_less: "¦x¦ < u / sqrt 2 ⟹ ¦y¦ < u / sqrt 2 ⟹ sqrt (x⇧2 + y⇧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 ⟹ y < u/2 ⟹ 0 ≤ x ⟹ 0 ≤ y ⟹ sqrt (x⇧2 + y⇧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: "(λn. root n n) ⇢ 1"
proof -
define x where "x n = root n n - 1" for n
have "x ⇢ sqrt 0"
proof (rule tendsto_sandwich[OF _ _ tendsto_const])
show "(λx. sqrt (2 / x)) ⇢ sqrt 0"
by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
have "x n ≤ sqrt (2 / real n)" if "2 < n" for n :: nat
proof -
have "1 + (real (n - 1) * n) / 2 * (x n)⇧2 = 1 + of_nat (n choose 2) * (x n)⇧2"
by (auto simp add: choose_two of_nat_div mod_eq_0_iff_dvd)
also have "… ≤ (∑k∈{0, 2}. of_nat (n choose k) * x n^k)"
also have "… ≤ (∑k=0..n. of_nat (n choose k) * x n^k)"
using ‹2 < n›
by (intro sum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
also have "… = (x n + 1) ^ n"
also have "… = n"
using ‹2 < n› by (simp add: x_def)
finally have "real (n - 1) * (real n / 2 * (x n)⇧2) ≤ real (n - 1) * 1"
by simp
then have "(x n)⇧2 ≤ 2 / real n"
using ‹2 < n› 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 (λn. x n ≤ sqrt (2 / real n)) sequentially"
by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
show "eventually (λn. sqrt 0 ≤ 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 "(λn. root n c) ⇢ 1"
proof -
have ge_1: "(λn. root n c) ⇢ 1" if "1 ≤ c" for c :: real
proof -
define x where "x n = root n c - 1" for n
have "x ⇢ 0"
proof (rule tendsto_sandwich[OF _ _ tendsto_const])
show "(λn. c / n) ⇢ 0"
by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
have "x n ≤ 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 "… ≤ (∑k∈{0, 1}. of_nat (n choose k) * x n^k)"
also have "… ≤ (∑k=0..n. of_nat (n choose k) * x n^k)"
using ‹1 < n› ‹1 ≤ c›
by (intro sum_mono2)
(auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
also have "… = (x n + 1) ^ n"
also have "… = c"
using ‹1 < n› ‹1 ≤ c› by (simp add: x_def)
finally show ?thesis
using ‹1 ≤ c› ‹1 < n› by (simp add: field_simps)
qed
then show "eventually (λn. x n ≤ c / n) sequentially"
by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
show "eventually (λn. 0 ≤ x n) sequentially"
using ‹1 ≤ c›
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 ≤ c")
case True
with ge_1 show ?thesis by blast
next
case False
with ‹0 < c› have "1 ≤ 1 / c"
by simp
then have "(λn. 1 / root n (1 / c)) ⇢ 1 / 1"
by (intro tendsto_divide tendsto_const ge_1 ‹1 ≤ 1 / c› 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_mult_distrib = real_sqrt_mult
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff

end
```