# Theory NthRoot

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theory NthRoot
imports Deriv
`(*  Title       : NthRoot.thy    Author      : Jacques D. Fleuriot    Copyright   : 1998  University of Cambridge    Conversion to Isar and new proofs by Lawrence C Paulson, 2004*)header {* Nth Roots of Real Numbers *}theory NthRootimports Parity Derivbeginsubsection {* Existence of Nth Root *}text {* Existence follows from the Intermediate Value Theorem *}lemma realpow_pos_nth:  assumes n: "0 < n"  assumes a: "0 < a"  shows "∃r>0. r ^ n = (a::real)"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" by (auto simp add: power_0_left)  with r have "0 < r ∧ r ^ n = a" by simp  thus ?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::real)"apply (auto intro!: realpow_pos_nth)apply (rule_tac n=n in power_eq_imp_eq_base, simp_all)donesubsection {* Nth Root *}text {* We define roots of negative reals such that  @{term "root n (- x) = - root n x"}. This allows  us to omit side conditions from many theorems. *}definition  root :: "[nat, real] => real" where  "root n x = (if 0 < x then (THE u. 0 < u ∧ u ^ n = x) else               if x < 0 then - (THE u. 0 < u ∧ u ^ n = - x) else 0)"lemma real_root_zero [simp]: "root n 0 = 0"unfolding root_def by simplemma real_root_minus: "0 < n ==> root n (- x) = - root n x"unfolding root_def by simplemma real_root_gt_zero: "[|0 < n; 0 < x|] ==> 0 < root n x"apply (simp add: root_def)apply (drule (1) realpow_pos_nth_unique)apply (erule theI' [THEN conjunct1])donelemma real_root_pow_pos: (* TODO: rename *)  "[|0 < n; 0 < x|] ==> root n x ^ n = x"apply (simp add: root_def)apply (drule (1) realpow_pos_nth_unique)apply (erule theI' [THEN conjunct2])donelemma real_root_pow_pos2 [simp]: (* TODO: rename *)  "[|0 < n; 0 ≤ x|] ==> root n x ^ n = x"by (auto simp add: order_le_less real_root_pow_pos)lemma odd_real_root_pow: "odd n ==> root n x ^ n = x"apply (rule_tac x=0 and y=x in linorder_le_cases)apply (erule (1) real_root_pow_pos2 [OF odd_pos])apply (subgoal_tac "root n (- x) ^ n = - x")apply (simp add: real_root_minus odd_pos)apply (simp add: odd_pos)donelemma real_root_ge_zero: "[|0 < n; 0 ≤ x|] ==> 0 ≤ root n x"by (auto simp add: order_le_less real_root_gt_zero)lemma real_root_power_cancel: "[|0 < n; 0 ≤ x|] ==> root n (x ^ n) = x"apply (subgoal_tac "0 ≤ x ^ n")apply (subgoal_tac "0 ≤ root n (x ^ n)")apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n")apply (erule (3) power_eq_imp_eq_base)apply (erule (1) real_root_pow_pos2)apply (erule (1) real_root_ge_zero)apply (erule zero_le_power)donelemma odd_real_root_power_cancel: "odd n ==> root n (x ^ n) = x"apply (rule_tac x=0 and y=x in linorder_le_cases)apply (erule (1) real_root_power_cancel [OF odd_pos])apply (subgoal_tac "root n ((- x) ^ n) = - x")apply (simp add: real_root_minus odd_pos)apply (erule real_root_power_cancel [OF odd_pos], simp)donelemma real_root_pos_unique:  "[|0 < n; 0 ≤ y; y ^ n = x|] ==> root n x = y"by (erule subst, rule real_root_power_cancel)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"by (simp add: real_root_pos_unique)text {* Root function is strictly monotonic, hence injective *}lemma real_root_less_mono_lemma:  "[|0 < n; 0 ≤ x; x < y|] ==> root n x < root n y"apply (subgoal_tac "0 ≤ y")apply (subgoal_tac "root n x ^ n < root n y ^ n")apply (erule power_less_imp_less_base)apply (erule (1) real_root_ge_zero)apply simpapply simpdonelemma real_root_less_mono: "[|0 < n; x < y|] ==> root n x < root n y"apply (cases "0 ≤ x")apply (erule (2) real_root_less_mono_lemma)apply (cases "0 ≤ y")apply (rule_tac y=0 in order_less_le_trans)apply (subgoal_tac "0 < root n (- x)")apply (simp add: real_root_minus)apply (simp add: real_root_gt_zero)apply (simp add: real_root_ge_zero)apply (subgoal_tac "root n (- y) < root n (- x)")apply (simp add: real_root_minus)apply (simp add: real_root_less_mono_lemma)donelemma 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)"apply (cases "x < y")apply (simp add: real_root_less_mono)apply (simp add: linorder_not_less real_root_le_mono)donelemma real_root_le_iff [simp]:  "0 < n ==> (root n x ≤ root n y) = (x ≤ y)"apply (cases "x ≤ y")apply (simp add: real_root_le_mono)apply (simp add: linorder_not_le real_root_less_mono)donelemma real_root_eq_iff [simp]:  "0 < n ==> (root n x = root n y) = (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 ==> (1 < root n y) = (1 < y)"by (insert real_root_less_iff [where x=1], simp)lemma real_root_lt_1_iff [simp]: "0 < n ==> (root n x < 1) = (x < 1)"by (insert real_root_less_iff [where y=1], simp)lemma real_root_ge_1_iff [simp]: "0 < n ==> (1 ≤ root n y) = (1 ≤ y)"by (insert real_root_le_iff [where x=1], simp)lemma real_root_le_1_iff [simp]: "0 < n ==> (root n x ≤ 1) = (x ≤ 1)"by (insert real_root_le_iff [where y=1], simp)lemma real_root_eq_1_iff [simp]: "0 < n ==> (root n x = 1) = (x = 1)"by (insert real_root_eq_iff [where y=1], simp)text {* Roots of roots *}lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"by (simp add: odd_real_root_unique)lemma real_root_pos_mult_exp:  "[|0 < m; 0 < n; 0 < x|] ==> root (m * n) x = root m (root n x)"by (rule real_root_pos_unique, simp_all add: power_mult)lemma real_root_mult_exp:  "[|0 < m; 0 < n|] ==> root (m * n) x = root m (root n x)"apply (rule linorder_cases [where x=x and y=0])apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))")apply (simp add: real_root_minus)apply (simp_all add: real_root_pos_mult_exp)donelemma real_root_commute:  "[|0 < m; 0 < n|] ==> 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:  "[|0 < n; n < N; 1 < x|] ==> root N x < root n x"apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)apply (simp add: real_root_commute power_strict_increasing            del: real_root_pow_pos2)donelemma real_root_strict_increasing:  "[|0 < n; n < N; 0 < x; x < 1|] ==> root n x < root N x"apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)apply (simp add: real_root_commute power_strict_decreasing            del: real_root_pow_pos2)donelemma 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 {* Roots of multiplication and division *}lemma real_root_mult_lemma:  "[|0 < n; 0 ≤ x; 0 ≤ y|] ==> root n (x * y) = root n x * root n y"by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib)lemma real_root_inverse_lemma:  "[|0 < n; 0 ≤ x|] ==> root n (inverse x) = inverse (root n x)"by (simp add: real_root_pos_unique power_inverse [symmetric])lemma real_root_mult:  assumes n: "0 < n"  shows "root n (x * y) = root n x * root n y"proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases)  assume "0 ≤ x" and "0 ≤ y"  thus ?thesis by (rule real_root_mult_lemma [OF n])next  assume "0 ≤ x" and "y ≤ 0"  hence "0 ≤ x" and "0 ≤ - y" by simp_all  hence "root n (x * - y) = root n x * root n (- y)"    by (rule real_root_mult_lemma [OF n])  thus ?thesis by (simp add: real_root_minus [OF n])next  assume "x ≤ 0" and "0 ≤ y"  hence "0 ≤ - x" and "0 ≤ y" by simp_all  hence "root n (- x * y) = root n (- x) * root n y"    by (rule real_root_mult_lemma [OF n])  thus ?thesis by (simp add: real_root_minus [OF n])next  assume "x ≤ 0" and "y ≤ 0"  hence "0 ≤ - x" and "0 ≤ - y" by simp_all  hence "root n (- x * - y) = root n (- x) * root n (- y)"    by (rule real_root_mult_lemma [OF n])  thus ?thesis by (simp add: real_root_minus [OF n])qedlemma real_root_inverse:  assumes n: "0 < n"  shows "root n (inverse x) = inverse (root n x)"proof (rule linorder_le_cases)  assume "0 ≤ x"  thus ?thesis by (rule real_root_inverse_lemma [OF n])next  assume "x ≤ 0"  hence "0 ≤ - x" by simp  hence "root n (inverse (- x)) = inverse (root n (- x))"    by (rule real_root_inverse_lemma [OF n])  thus ?thesis by (simp add: real_root_minus [OF n])qedlemma real_root_divide:  "0 < n ==> root n (x / y) = root n x / root n y"by (simp add: divide_inverse real_root_mult real_root_inverse)lemma real_root_power:  "0 < n ==> root n (x ^ k) = root n x ^ k"by (induct k, simp_all add: real_root_mult)lemma real_root_abs: "0 < n ==> root n ¦x¦ = ¦root n x¦"by (simp add: abs_if real_root_minus)text {* Continuity and derivatives *}lemma isCont_root_pos:  assumes n: "0 < n"  assumes x: "0 < x"  shows "isCont (root n) x"proof -  have "isCont (root n) (root n x ^ n)"  proof (rule isCont_inverse_function [where f="λa. a ^ n"])    show "0 < root n x" using n x by simp    show "∀z. ¦z - root n x¦ ≤ root n x --> root n (z ^ n) = z"      by (simp add: abs_le_iff real_root_power_cancel n)    show "∀z. ¦z - root n x¦ ≤ root n x --> isCont (λa. a ^ n) z"      by simp  qed  thus ?thesis using n x by simpqedlemma isCont_root_neg:  "[|0 < n; x < 0|] ==> isCont (root n) x"apply (subgoal_tac "isCont (λx. - root n (- x)) x")apply (simp add: real_root_minus)apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]])apply (simp add: isCont_root_pos)donelemma isCont_root_zero:  "0 < n ==> isCont (root n) 0"unfolding isCont_defapply (rule LIM_I)apply (rule_tac x="r ^ n" in exI, safe)apply (simp)apply (simp add: real_root_abs [symmetric])apply (rule_tac n="n" in power_less_imp_less_base, simp_all)donelemma isCont_real_root: "0 < n ==> isCont (root n) x"apply (rule_tac x=x and y=0 in linorder_cases)apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero)donelemma DERIV_real_root:  assumes n: "0 < n"  assumes 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"    using n by (rule isCont_real_root)qedlemma DERIV_odd_real_root:  assumes n: "odd n"  assumes 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"    using odd_pos [OF n] by (rule isCont_real_root)qedlemma DERIV_even_real_root:  assumes n: "0 < n" and "even n"  assumes 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 .next  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"    hence "root n (-y) ^ n = -y" using `0 < n` by simp    with real_root_minus[OF `0 < n`] and `even n`    show "- (root n y ^ n) = y" by simp  qednext  show "DERIV (λx. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"    by  (auto intro!: DERIV_intros)  show "- real n * root n x ^ (n - Suc 0) ≠ 0"    using n x by simp  show "isCont (root n) x"    using n by (rule isCont_real_root)qedlemma 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 simplemma real_sqrt_unique: "[|y² = x; 0 ≤ y|] ==> sqrt x = y"unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])lemma real_sqrt_abs [simp]: "sqrt (x²) = ¦x¦"apply (rule real_sqrt_unique)apply (rule power2_abs)apply (rule abs_ge_zero)donelemma 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)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_minus: "sqrt (- x) = - sqrt x"unfolding sqrt_def by (rule real_root_minus [OF pos2])lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"unfolding sqrt_def by (rule real_root_mult [OF pos2])lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"unfolding sqrt_def by (rule real_root_inverse [OF pos2])lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"unfolding sqrt_def by (rule real_root_divide [OF pos2])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 [OF pos2])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])lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]lemma isCont_real_sqrt: "isCont sqrt x"unfolding sqrt_def by (rule isCont_real_root [OF pos2])lemma DERIV_real_sqrt_generic:  assumes "x ≠ 0"  assumes "x > 0 ==> D = inverse (sqrt x) / 2"  assumes "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 simpdeclare  DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_real_root_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"apply autoapply (cut_tac x = x and y = 0 in linorder_less_linear)apply (simp add: zero_less_mult_iff)donelemma real_sqrt_abs2 [simp]: "sqrt(x*x) = ¦x¦"apply (subst power2_eq_square [symmetric])apply (rule real_sqrt_abs)donelemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"by (simp add: power_inverse [symmetric])lemma real_sqrt_eq_zero_cancel: "[| 0 ≤ x; sqrt(x) = 0|] ==> x = 0"by simplemma real_sqrt_ge_one: "1 ≤ x ==> 1 ≤ sqrt x"by simplemma sqrt_divide_self_eq:  assumes nneg: "0 ≤ x"  shows "sqrt x / x = inverse (sqrt x)"proof cases  assume "x=0" thus ?thesis by simpnext  assume nz: "x≠0"   hence pos: "0<x" using nneg by arith  show ?thesis  proof (rule right_inverse_eq [THEN iffD1, THEN sym])     show "sqrt x / x ≠ 0" by (simp add: divide_inverse nneg nz)     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 nz)   qedqedlemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"apply (simp add: divide_inverse)apply (case_tac "r=0")apply (auto simp add: mult_ac)donelemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"by (simp add: divide_less_eq)lemma four_x_squared:   fixes x::real  shows "4 * x² = (2 * x)²"by (simp add: power2_eq_square)subsection {* Square Root of Sum of Squares *}lemma real_sqrt_sum_squares_ge_zero: "0 ≤ sqrt (x² + y²)"  by simp (* TODO: delete *)declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]lemma real_sqrt_sum_squares_mult_ge_zero [simp]:     "0 ≤ sqrt ((x² + y²)*(xa² + ya²))"  by (simp add: zero_le_mult_iff)lemma real_sqrt_sum_squares_mult_squared_eq [simp]:     "sqrt ((x² + y²) * (xa² + ya²)) ^ 2 = (x² + y²) * (xa² + ya²)"  by (simp add: zero_le_mult_iff)lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x² + y²) = x ==> y = 0"by (drule_tac f = "%x. x²" in arg_cong, simp)lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x² + y²) = y ==> x = 0"by (drule_tac f = "%x. x²" in arg_cong, simp)lemma real_sqrt_sum_squares_ge1 [simp]: "x ≤ sqrt (x² + y²)"by (rule power2_le_imp_le, simp_all)lemma real_sqrt_sum_squares_ge2 [simp]: "y ≤ sqrt (x² + y²)"by (rule power2_le_imp_le, simp_all)lemma real_sqrt_ge_abs1 [simp]: "¦x¦ ≤ sqrt (x² + y²)"by (rule power2_le_imp_le, simp_all)lemma real_sqrt_ge_abs2 [simp]: "¦y¦ ≤ sqrt (x² + y²)"by (rule power2_le_imp_le, simp_all)lemma le_real_sqrt_sumsq [simp]: "x ≤ sqrt (x * x + y * y)"by (simp add: power2_eq_square [symmetric])lemma real_sqrt_sum_squares_triangle_ineq:  "sqrt ((a + c)² + (b + d)²) ≤ sqrt (a² + b²) + sqrt (c² + d²)"apply (rule power2_le_imp_le, 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 ≤ b² * c² + a² * d² - 2 * (a * c) * (b * d)", simp)apply (rule_tac b="(a * d - b * c)²" in ord_le_eq_trans)apply (rule zero_le_power2)apply (simp add: power2_diff power_mult_distrib)apply (simp add: mult_nonneg_nonneg)apply simpapply (simp add: add_increasing)donelemma real_sqrt_sum_squares_less:  "[|¦x¦ < u / sqrt 2; ¦y¦ < u / sqrt 2|] ==> sqrt (x² + y²) < u"apply (rule power2_less_imp_less, 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)donetext{*Needed for the infinitely close relation over the nonstandard    complex numbers*}lemma lemma_sqrt_hcomplex_capprox:     "[| 0 < u; x < u/2; y < u/2; 0 ≤ x; 0 ≤ y |] ==> sqrt (x² + y²) < u"apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)apply (erule_tac [2] lemma_real_divide_sqrt_less)apply (rule power2_le_imp_le)apply (auto simp add: zero_le_divide_iff power_divide)apply (rule_tac t = "u²" in real_sum_of_halves [THEN subst])apply (rule add_mono)apply (auto simp add: four_x_squared intro: power_mono)donetext "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_iff(* needed for CauchysMeanTheorem.het_base from AFP *)lemma real_root_pos: "0 < x ==> root (Suc n) (x ^ (Suc n)) = x"by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])end`