author  paulson <lp15@cam.ac.uk> 
Fri, 07 Mar 2014 12:35:06 +0000  
changeset 55967  5dadc93ff3df 
parent 54413  88a036a95967 
child 56371  fb9ae0727548 
permissions  rwrr 
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(* Title : NthRoot.thy 
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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 
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*) 
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header {* Nth Roots of Real Numbers *} 
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theory NthRoot 
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imports Parity Deriv 
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begin 
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lemma abs_sgn_eq: "abs (sgn x :: real) = (if x = 0 then 0 else 1)" 
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by (simp add: sgn_real_def) 
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lemma inverse_sgn: "sgn (inverse a) = inverse (sgn a :: real)" 
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by (simp add: sgn_real_def) 
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lemma power_eq_iff_eq_base: 
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fixes a b :: "_ :: linordered_semidom" 
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shows "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b" 
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using power_eq_imp_eq_base[of a n b] by auto 
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subsection {* Existence of Nth Root *} 
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text {* Existence follows from the Intermediate Value Theorem *} 
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lemma realpow_pos_nth: 
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assumes n: "0 < n" 
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assumes a: "0 < a" 
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shows "\<exists>r>0. r ^ n = (a::real)" 
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proof  
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have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a" 
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proof (rule IVT) 
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show "0 ^ n \<le> a" using n a by (simp add: power_0_left) 
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show "0 \<le> max 1 a" by simp 
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from n have n1: "1 \<le> n" by simp 
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have "a \<le> max 1 a ^ 1" by simp 
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also have "max 1 a ^ 1 \<le> max 1 a ^ n" 
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using n1 by (rule power_increasing, simp) 
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finally show "a \<le> max 1 a ^ n" . 
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show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r" 
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by simp 
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qed 
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then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast 
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with n a have "r \<noteq> 0" by (auto simp add: power_0_left) 
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with r have "0 < r \<and> r ^ n = a" by simp 
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thus ?thesis .. 
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qed 
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23047  51 
(* Used by Integration/RealRandVar.thy in AFP *) 
52 
lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a" 

53 
by (blast intro: realpow_pos_nth) 

54 

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text {* Uniqueness of nth positive root *} 
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lemma realpow_pos_nth_unique: "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)" 
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by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base) 
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subsection {* Nth Root *} 
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text {* We define roots of negative reals such that 
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@{term "root n ( x) =  root n x"}. This allows 
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us to omit side conditions from many theorems. *} 
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lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f") 
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proof (rule injI) 
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have x: "\<And>a b :: real. (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" by auto 
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fix x y assume "?f x = ?f y" with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] `0<n` show "x = y" 
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by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]]) 
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(simp_all add: x) 
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qed 
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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::real)" 
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using inj_sgn_power[THEN injD, of n a b] by simp 
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definition root :: "nat \<Rightarrow> real \<Rightarrow> real" where 
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"root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)" 
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lemma root_0 [simp]: "root 0 x = 0" 
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by (simp add: root_def) 
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lemma root_sgn_power: "0 < n \<Longrightarrow> root n (sgn y * \<bar>y\<bar>^n) = y" 
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using the_inv_f_f[OF inj_sgn_power] by (simp add: root_def) 
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lemma sgn_power_root: 
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assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x") 
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proof cases 
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assume "x \<noteq> 0" 
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with realpow_pos_nth[OF `0 < n`, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto 
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with `x \<noteq> 0` have S: "x \<in> range ?f" 
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by (intro image_eqI[of _ _ "sgn x * r"]) 
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(auto simp: abs_mult sgn_mult power_mult_distrib abs_sgn_eq mult_sgn_abs) 
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from `0 < n` f_the_inv_into_f[OF inj_sgn_power[OF `0 < n`] this] show ?thesis 
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95 
by (simp add: root_def) 
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qed (insert `0 < n` root_sgn_power[of n 0], simp) 
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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))" 
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apply (cases "n = 0") 
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apply simp_all 
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101 
apply (metis root_sgn_power sgn_power_root) 
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done 
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lemma real_root_zero [simp]: "root n 0 = 0" 
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by (simp split: split_root add: sgn_zero_iff) 
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106 

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lemma real_root_minus: "root n ( x) =  root n x" 
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by (clarsimp split: split_root elim!: sgn_power_injE simp: sgn_minus) 
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lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" 
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proof (clarsimp split: split_root) 
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112 
have x: "\<And>a b :: real. (0 < b \<and> a < 0) \<Longrightarrow> \<not> a > b" by auto 
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fix a b :: real assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n" then show "a < b" 
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using power_less_imp_less_base[of a n b] power_less_imp_less_base[of "b" n "a"] 
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by (simp add: sgn_real_def power_less_zero_eq x[of "a ^ n" " (( b) ^ n)"] split: split_if_asm) 
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116 
qed 
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lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x" 
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119 
using real_root_less_mono[of n 0 x] by simp 
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120 

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lemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x" 
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122 
using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less) 
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123 

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lemma real_root_pow_pos: (* TODO: rename *) 
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"\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x" 
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using sgn_power_root[of n x] real_root_gt_zero[of n x] by simp 
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lemma real_root_pow_pos2 [simp]: (* TODO: rename *) 
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"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x" 
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by (auto simp add: order_le_less real_root_pow_pos) 
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lemma sgn_root: "0 < n \<Longrightarrow> sgn (root n x) = sgn x" 
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by (auto split: split_root simp: sgn_real_def power_less_zero_eq) 
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23046  135 
lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x" 
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using sgn_power_root[of n x] by (simp add: odd_pos sgn_real_def split: split_if_asm) 
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lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x" 
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using root_sgn_power[of n x] by (auto simp add: le_less power_0_left) 
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23046  141 
lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x" 
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using root_sgn_power[of n x] by (simp add: odd_pos sgn_real_def power_0_left split: split_if_asm) 
23046  143 

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lemma real_root_pos_unique: "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" 
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using root_sgn_power[of n y] by (auto simp add: le_less power_0_left) 
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23046  147 
lemma odd_real_root_unique: 
148 
"\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" 

149 
by (erule subst, rule odd_real_root_power_cancel) 

150 

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lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" 
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by (simp add: real_root_pos_unique) 
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text {* Root function is strictly monotonic, hence injective *} 
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155 

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lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" 
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by (auto simp add: order_le_less real_root_less_mono) 
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lemma real_root_less_iff [simp]: 
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"0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" 
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apply (cases "x < y") 
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apply (simp add: real_root_less_mono) 
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apply (simp add: linorder_not_less real_root_le_mono) 
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done 
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165 

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lemma real_root_le_iff [simp]: 
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"0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" 
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apply (cases "x \<le> y") 
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apply (simp add: real_root_le_mono) 
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apply (simp add: linorder_not_le real_root_less_mono) 
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done 
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172 

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lemma real_root_eq_iff [simp]: 
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"0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" 
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by (simp add: order_eq_iff) 
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176 

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lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] 
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lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] 
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lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] 
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lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] 
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lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] 
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23257  183 
lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)" 
184 
by (insert real_root_less_iff [where x=1], simp) 

185 

186 
lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)" 

187 
by (insert real_root_less_iff [where y=1], simp) 

188 

189 
lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)" 

190 
by (insert real_root_le_iff [where x=1], simp) 

191 

192 
lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)" 

193 
by (insert real_root_le_iff [where y=1], simp) 

194 

195 
lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)" 

196 
by (insert real_root_eq_iff [where y=1], simp) 

197 

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text {* Roots of multiplication and division *} 
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199 

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lemma real_root_mult: "root n (x * y) = root n x * root n y" 
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201 
by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib) 
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202 

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lemma real_root_inverse: "root n (inverse x) = inverse (root n x)" 
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by (auto split: split_root elim!: sgn_power_injE simp: inverse_sgn power_inverse) 
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205 

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lemma real_root_divide: "root n (x / y) = root n x / root n y" 
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207 
by (simp add: divide_inverse real_root_mult real_root_inverse) 
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208 

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lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>" 
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210 
by (simp add: abs_if real_root_minus) 
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211 

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lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" 
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213 
by (induct k) (simp_all add: real_root_mult) 
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214 

23257  215 
text {* Roots of roots *} 
216 

217 
lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x" 

218 
by (simp add: odd_real_root_unique) 

219 

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lemma real_root_mult_exp: "root (m * n) x = root m (root n x)" 
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221 
by (auto split: split_root elim!: sgn_power_injE 
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simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq) 
23257  223 

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lemma real_root_commute: "root m (root n x) = root n (root m x)" 
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225 
by (simp add: real_root_mult_exp [symmetric] mult_commute) 
23257  226 

227 
text {* Monotonicity in first argument *} 

228 

229 
lemma real_root_strict_decreasing: 

230 
"\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x" 

231 
apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp) 

232 
apply (simp add: real_root_commute power_strict_increasing 

233 
del: real_root_pow_pos2) 

234 
done 

235 

236 
lemma real_root_strict_increasing: 

237 
"\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x" 

238 
apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp) 

239 
apply (simp add: real_root_commute power_strict_decreasing 

240 
del: real_root_pow_pos2) 

241 
done 

242 

243 
lemma real_root_decreasing: 

244 
"\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x" 

245 
by (auto simp add: order_le_less real_root_strict_decreasing) 

246 

247 
lemma real_root_increasing: 

248 
"\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x" 

249 
by (auto simp add: order_le_less real_root_strict_increasing) 

250 

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text {* Continuity and derivatives *} 
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252 

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lemma isCont_real_root: "isCont (root n) x" 
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254 
proof cases 
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assume n: "0 < n" 
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let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n" 
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257 
have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else  ((x) ^ n) :: real)" 
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258 
using n by (intro continuous_on_If continuous_on_intros) auto 
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259 
then have "continuous_on UNIV ?f" 
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by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less real_sgn_neg le_less n) 
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261 
then have [simp]: "\<And>x. isCont ?f x" 
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262 
by (simp add: continuous_on_eq_continuous_at) 
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263 

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264 
have "isCont (root n) (?f (root n x))" 
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by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power n) 
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266 
then show ?thesis 
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267 
by (simp add: sgn_power_root n) 
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268 
qed (simp add: root_def[abs_def]) 
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269 

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lemma tendsto_real_root[tendsto_intros]: 
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"(f > x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) > root n x) F" 
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using isCont_tendsto_compose[OF isCont_real_root, of f x F] . 
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lemma continuous_real_root[continuous_intros]: 
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"continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))" 
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lemma continuous_on_real_root[continuous_on_intros]: 
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"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))" 
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unfolding continuous_on_def by (auto intro: tendsto_real_root) 
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lemma DERIV_real_root: 
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assumes n: "0 < n" 
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assumes x: "0 < x" 
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shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 
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proof (rule DERIV_inverse_function) 
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show "0 < x" using x . 
288 
show "x < x + 1" by simp 

289 
show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

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using n by simp 
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show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 
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by (rule DERIV_pow) 
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show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 
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using n x by simp 
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qed (rule isCont_real_root) 
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23046  297 
lemma DERIV_odd_real_root: 
298 
assumes n: "odd n" 

299 
assumes x: "x \<noteq> 0" 

300 
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n  Suc 0))" 

301 
proof (rule DERIV_inverse_function) 

302 
show "x  1 < x" by simp 

303 
show "x < x + 1" by simp 

304 
show "\<forall>y. x  1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y" 

305 
using n by (simp add: odd_real_root_pow) 

306 
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n  Suc 0)" 

307 
by (rule DERIV_pow) 

308 
show "real n * root n x ^ (n  Suc 0) \<noteq> 0" 

309 
using odd_pos [OF n] x by simp 

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qed (rule isCont_real_root) 
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31880  312 
lemma DERIV_even_real_root: 
313 
assumes n: "0 < n" and "even n" 

314 
assumes x: "x < 0" 

315 
shows "DERIV (root n) x :> inverse ( real n * root n x ^ (n  Suc 0))" 

316 
proof (rule DERIV_inverse_function) 

317 
show "x  1 < x" by simp 

318 
show "x < 0" using x . 

319 
next 

320 
show "\<forall>y. x  1 < y \<and> y < 0 \<longrightarrow>  (root n y ^ n) = y" 

321 
proof (rule allI, rule impI, erule conjE) 

322 
fix y assume "x  1 < y" and "y < 0" 

323 
hence "root n (y) ^ n = y" using `0 < n` by simp 

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324 
with real_root_minus and `even n` 
31880  325 
show " (root n y ^ n) = y" by simp 
326 
qed 

327 
next 

328 
show "DERIV (\<lambda>x.  (x ^ n)) (root n x) :>  real n * root n x ^ (n  Suc 0)" 

329 
by (auto intro!: DERIV_intros) 

330 
show " real n * root n x ^ (n  Suc 0) \<noteq> 0" 

331 
using n x by simp 

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qed (rule isCont_real_root) 
31880  333 

334 
lemma DERIV_real_root_generic: 

335 
assumes "0 < n" and "x \<noteq> 0" 

49753  336 
and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 
337 
and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D =  inverse (real n * root n x ^ (n  Suc 0))" 

338 
and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n  Suc 0))" 

31880  339 
shows "DERIV (root n) x :> D" 
340 
using assms by (cases "even n", cases "0 < x", 

341 
auto intro: DERIV_real_root[THEN DERIV_cong] 

342 
DERIV_odd_real_root[THEN DERIV_cong] 

343 
DERIV_even_real_root[THEN DERIV_cong]) 

344 

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subsection {* Square Root *} 
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346 

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definition sqrt :: "real \<Rightarrow> real" where 
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"sqrt = root 2" 
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349 

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lemma pos2: "0 < (2::nat)" by simp 
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351 

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lemma real_sqrt_unique: "\<lbrakk>y\<^sup>2 = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" 
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353 
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) 
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354 

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lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>" 
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356 
apply (rule real_sqrt_unique) 
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357 
apply (rule power2_abs) 
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358 
apply (rule abs_ge_zero) 
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359 
done 
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360 

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lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x" 
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362 
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) 
22856  363 

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364 
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)" 
22856  365 
apply (rule iffI) 
366 
apply (erule subst) 

367 
apply (rule zero_le_power2) 

368 
apply (erule real_sqrt_pow2) 

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369 
done 
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370 

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lemma real_sqrt_zero [simp]: "sqrt 0 = 0" 
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372 
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373 

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lemma real_sqrt_one [simp]: "sqrt 1 = 1" 
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375 
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376 

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377 
lemma real_sqrt_minus: "sqrt ( x) =  sqrt x" 
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unfolding sqrt_def by (rule real_root_minus) 
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379 

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380 
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" 
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381 
unfolding sqrt_def by (rule real_root_mult) 
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382 

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383 
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" 
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384 
unfolding sqrt_def by (rule real_root_inverse) 
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385 

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386 
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" 
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387 
unfolding sqrt_def by (rule real_root_divide) 
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388 

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389 
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" 
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390 
unfolding sqrt_def by (rule real_root_power [OF pos2]) 
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391 

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392 
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" 
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393 
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394 

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395 
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" 
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396 
unfolding sqrt_def by (rule real_root_ge_zero) 
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397 

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398 
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" 
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399 
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) 
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400 

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401 
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" 
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402 
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) 
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403 

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404 
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" 
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405 
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) 
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406 

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407 
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" 
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408 
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) 
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409 

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410 
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" 
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411 
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) 
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412 

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413 
lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y" 
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414 
using real_sqrt_le_iff[of x "y\<^sup>2"] by simp 
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415 

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416 
lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y" 
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417 
using real_sqrt_le_mono[of "x\<^sup>2" y] by simp 
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418 

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419 
lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y" 
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parents:
53594
diff
changeset

420 
using real_sqrt_less_mono[of "x\<^sup>2" y] by simp 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

421 

88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

422 
lemma sqrt_even_pow2: 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
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diff
changeset

423 
assumes n: "even n" 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
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diff
changeset

424 
shows "sqrt (2 ^ n) = 2 ^ (n div 2)" 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

425 
proof  
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

426 
from n obtain m where m: "n = 2 * m" 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

427 
unfolding even_mult_two_ex .. 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

428 
from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)" 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

429 
by (simp only: power_mult[symmetric] mult_commute) 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
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53594
diff
changeset

430 
then show ?thesis 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
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diff
changeset

431 
using m by simp 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

432 
qed 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

433 

53594  434 
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, unfolded real_sqrt_zero] 
435 
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, unfolded real_sqrt_zero] 

436 
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, unfolded real_sqrt_zero] 

437 
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, unfolded real_sqrt_zero] 

438 
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, unfolded real_sqrt_zero] 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

439 

53594  440 
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, unfolded real_sqrt_one] 
441 
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, unfolded real_sqrt_one] 

442 
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, unfolded real_sqrt_one] 

443 
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, unfolded real_sqrt_one] 

444 
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, unfolded real_sqrt_one] 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

445 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

446 
lemma isCont_real_sqrt: "isCont sqrt x" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

447 
unfolding sqrt_def by (rule isCont_real_root) 
23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

448 

51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

449 
lemma tendsto_real_sqrt[tendsto_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

450 
"(f > x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) > sqrt x) F" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

451 
unfolding sqrt_def by (rule tendsto_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

452 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

453 
lemma continuous_real_sqrt[continuous_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

454 
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

455 
unfolding sqrt_def by (rule continuous_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

456 

270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

457 
lemma continuous_on_real_sqrt[continuous_on_intros]: 
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

458 
"continuous_on s f \<Longrightarrow> 0 < n \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))" 
51483
dc39d69774bb
modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0th root is constant 0
hoelzl
parents:
51478
diff
changeset

459 
unfolding sqrt_def by (rule continuous_on_real_root) 
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
49962
diff
changeset

460 

31880  461 
lemma DERIV_real_sqrt_generic: 
462 
assumes "x \<noteq> 0" 

463 
assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2" 

464 
assumes "x < 0 \<Longrightarrow> D =  inverse (sqrt x) / 2" 

465 
shows "DERIV sqrt x :> D" 

466 
using assms unfolding sqrt_def 

467 
by (auto intro!: DERIV_real_root_generic) 

468 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

469 
lemma DERIV_real_sqrt: 
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

470 
"0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2" 
31880  471 
using DERIV_real_sqrt_generic by simp 
472 

473 
declare 

474 
DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

475 
DERIV_real_root_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] 

23042
492514b39956
add lemmas about continuity and derivatives of roots
huffman
parents:
23009
diff
changeset

476 

20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

477 
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

478 
apply auto 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

479 
apply (cut_tac x = x and y = 0 in linorder_less_linear) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

480 
apply (simp add: zero_less_mult_iff) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

481 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

482 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

483 
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" 
22856  484 
apply (subst power2_eq_square [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

485 
apply (rule real_sqrt_abs) 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

486 
done 
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

487 

53076  488 
lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x" 
22856  489 
by (simp add: power_inverse [symmetric]) 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

490 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

491 
lemma real_sqrt_eq_zero_cancel: "[ 0 \<le> x; sqrt(x) = 0] ==> x = 0" 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

492 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

493 

fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

494 
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" 
22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

495 
by simp 
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset

496 

22443  497 
lemma sqrt_divide_self_eq: 
498 
assumes nneg: "0 \<le> x" 

499 
shows "sqrt x / x = inverse (sqrt x)" 

500 
proof cases 

501 
assume "x=0" thus ?thesis by simp 

502 
next 

503 
assume nz: "x\<noteq>0" 

504 
hence pos: "0<x" using nneg by arith 

505 
show ?thesis 

506 
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

507 
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

508 
show "inverse (sqrt x) / (sqrt x / x) = 1" 

509 
by (simp add: divide_inverse mult_assoc [symmetric] 

510 
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

511 
qed 

512 
qed 

513 

54413
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

514 
lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x" 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

515 
apply (cases "x = 0") 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

516 
apply simp_all 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

517 
using sqrt_divide_self_eq[of x] 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

518 
apply (simp add: inverse_eq_divide field_simps) 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

519 
done 
88a036a95967
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents:
53594
diff
changeset

520 

22721
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

521 
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

522 
apply (simp add: divide_inverse) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

523 
apply (case_tac "r=0") 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

524 
apply (auto simp add: mult_ac) 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

525 
done 
d9be18bd7a28
moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
22630
diff
changeset

526 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

527 
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u" 
35216  528 
by (simp add: divide_less_eq) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

529 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

530 
lemma four_x_squared: 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

531 
fixes x::real 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

532 
shows "4 * x\<^sup>2 = (2 * x)\<^sup>2" 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

533 
by (simp add: power2_eq_square) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

534 

22856  535 
subsection {* Square Root of Sum of Squares *} 
536 

55967  537 
lemma sum_squares_bound: 
538 
fixes x:: "'a::linordered_field" 

539 
shows "2*x*y \<le> x^2 + y^2" 

540 
proof  

541 
have "(xy)^2 = x*x  2*x*y + y*y" 

542 
by algebra 

543 
then have "0 \<le> x^2  2*x*y + y^2" 

544 
by (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square) 

545 
then show ?thesis 

546 
by arith 

547 
qed 

22856  548 

55967  549 
lemma arith_geo_mean: 
550 
fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2" 

551 
apply (rule power2_le_imp_le) 

552 
using sum_squares_bound assms 

553 
apply (auto simp: zero_le_mult_iff) 

554 
by (auto simp: algebra_simps power2_eq_square) 

555 

556 
lemma arith_geo_mean_sqrt: 

557 
fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2" 

558 
apply (rule arith_geo_mean) 

559 
using assms 

560 
apply (auto simp: zero_le_mult_iff) 

561 
done 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

562 

22856  563 
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: 
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

564 
"0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))" 
55967  565 
by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero) 
22856  566 

567 
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: 

53076  568 
"(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)" 
44320  569 
by (simp add: zero_le_mult_iff) 
22856  570 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

571 
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0" 
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

572 
by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

573 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

574 
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0" 
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

575 
by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

576 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

577 
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)" 
22856  578 
by (rule power2_le_imp_le, simp_all) 
579 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

580 
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)" 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

581 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

582 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

583 
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)" 
22856  584 
by (rule power2_le_imp_le, simp_all) 
585 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51483
diff
changeset

586 
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)" 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

587 
by (rule power2_le_imp_le, simp_all) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

588 

11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

589 
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)" 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

590 
by (simp add: power2_eq_square [symmetric]) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

591 

22858  592 
lemma real_sqrt_sum_squares_triangle_ineq: 
53015
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wenzelm
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593 
"sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)" 
22858  594 
apply (rule power2_le_imp_le, simp) 
595 
apply (simp add: power2_sum) 

49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49753
diff
changeset

596 
apply (simp only: mult_assoc distrib_left [symmetric]) 
22858  597 
apply (rule mult_left_mono) 
598 
apply (rule power2_le_imp_le) 

599 
apply (simp add: power2_sum power_mult_distrib) 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23475
diff
changeset

600 
apply (simp add: ring_distribs) 
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
changeset

601 
apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2  2 * (a * c) * (b * d)", simp) 
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
changeset

602 
apply (rule_tac b="(a * d  b * c)\<^sup>2" in ord_le_eq_trans) 
22858  603 
apply (rule zero_le_power2) 
604 
apply (simp add: power2_diff power_mult_distrib) 

605 
apply (simp add: mult_nonneg_nonneg) 

606 
apply simp 

607 
apply (simp add: add_increasing) 

608 
done 

609 

23122  610 
lemma real_sqrt_sum_squares_less: 
53015
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wenzelm
parents:
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changeset

611 
"\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u" 
23122  612 
apply (rule power2_less_imp_less, simp) 
613 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

614 
apply (drule power_strict_mono [OF _ abs_ge_zero pos2]) 

615 
apply (simp add: power_divide) 

616 
apply (drule order_le_less_trans [OF abs_ge_zero]) 

617 
apply (simp add: zero_less_divide_iff) 

618 
done 

619 

23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

620 
text{*Needed for the infinitely close relation over the nonstandard 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

621 
complex numbers*} 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
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diff
changeset

622 
lemma lemma_sqrt_hcomplex_capprox: 
53015
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wenzelm
parents:
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changeset

623 
"[ 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y ] ==> sqrt (x\<^sup>2 + y\<^sup>2) < u" 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

624 
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

625 
apply (erule_tac [2] lemma_real_divide_sqrt_less) 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

626 
apply (rule power2_le_imp_le) 
44349
f057535311c5
remove redundant lemma real_0_le_divide_iff in favor or zero_le_divide_iff
huffman
parents:
44320
diff
changeset

627 
apply (auto simp add: zero_le_divide_iff power_divide) 
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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changeset

628 
apply (rule_tac t = "u\<^sup>2" in real_sum_of_halves [THEN subst]) 
23049
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

629 
apply (rule add_mono) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
28952
diff
changeset

630 
apply (auto simp add: four_x_squared intro: power_mono) 
23049
11607c283074
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huffman
parents:
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diff
changeset

631 
done 
11607c283074
moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents:
23047
diff
changeset

632 

22956
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

633 
text "Legacy theorem names:" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset

634 
lemmas real_root_pos2 = real_root_power_cancel 
617140080e6a
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huffman
parents:
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diff
changeset

635 
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le] 
617140080e6a
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huffman
parents:
22943
diff
changeset

636 
lemmas real_root_pos_pos_le = real_root_ge_zero 
617140080e6a
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huffman
parents:
22943
diff
changeset

637 
lemmas real_sqrt_mult_distrib = real_sqrt_mult 
617140080e6a
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huffman
parents:
22943
diff
changeset

638 
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

639 
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

640 

617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

641 
(* needed for CauchysMeanTheorem.het_base from AFP *) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

642 
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x" 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset

643 
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le]) 
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset

644 

14324  645 
end 