src/HOL/NthRoot.thy
author hoelzl
Wed May 07 12:25:35 2014 +0200 (2014-05-07)
changeset 56889 48a745e1bde7
parent 56536 aefb4a8da31f
child 57155 5c59114ff0cb
permissions -rw-r--r--
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
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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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(* Used by Integration/RealRandVar.thy in AFP *)
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lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a"
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by (blast intro: realpow_pos_nth)
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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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    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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  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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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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  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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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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  using real_root_less_mono[of n 0 x] by simp
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lemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x"
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  using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less)
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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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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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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)
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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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lemma odd_real_root_unique:
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  "\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
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by (erule subst, rule odd_real_root_power_cancel)
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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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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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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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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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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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lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)"
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by (insert real_root_less_iff [where x=1], simp)
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lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)"
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by (insert real_root_less_iff [where y=1], simp)
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lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)"
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by (insert real_root_le_iff [where x=1], simp)
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lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)"
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by (insert real_root_le_iff [where y=1], simp)
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lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)"
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by (insert real_root_eq_iff [where y=1], simp)
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text {* Roots of multiplication and division *}
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lemma real_root_mult: "root n (x * y) = root n x * root n y"
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  by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib)
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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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lemma real_root_divide: "root n (x / y) = root n x / root n y"
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  by (simp add: divide_inverse real_root_mult real_root_inverse)
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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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  by (simp add: abs_if real_root_minus)
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lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
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  by (induct k) (simp_all add: real_root_mult)
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text {* Roots of roots *}
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lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
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by (simp add: odd_real_root_unique)
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lemma real_root_mult_exp: "root (m * n) x = root m (root n x)"
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  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)
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lemma real_root_commute: "root m (root n x) = root n (root m x)"
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  by (simp add: real_root_mult_exp [symmetric] mult_commute)
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text {* Monotonicity in first argument *}
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lemma real_root_strict_decreasing:
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  "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x"
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apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
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apply (simp add: real_root_commute power_strict_increasing
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            del: real_root_pow_pos2)
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done
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lemma real_root_strict_increasing:
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  "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x"
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apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
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apply (simp add: real_root_commute power_strict_decreasing
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            del: real_root_pow_pos2)
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done
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lemma real_root_decreasing:
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  "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x"
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by (auto simp add: order_le_less real_root_strict_decreasing)
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lemma real_root_increasing:
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  "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
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by (auto simp add: order_le_less real_root_strict_increasing)
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text {* Continuity and derivatives *}
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lemma isCont_real_root: "isCont (root n) x"
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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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  have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else - ((-x) ^ n) :: real)"
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    using n by (intro continuous_on_If continuous_intros) auto
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  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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  then have [simp]: "\<And>x. isCont ?f x"
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    by (simp add: continuous_on_eq_continuous_at)
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  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
hoelzl@51483
   267
    by (simp add: sgn_power_root n)
hoelzl@51483
   268
qed (simp add: root_def[abs_def])
huffman@23042
   269
hoelzl@51478
   270
lemma tendsto_real_root[tendsto_intros]:
hoelzl@51483
   271
  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) ---> root n x) F"
hoelzl@51483
   272
  using isCont_tendsto_compose[OF isCont_real_root, of f x F] .
hoelzl@51478
   273
hoelzl@51478
   274
lemma continuous_real_root[continuous_intros]:
hoelzl@51483
   275
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
hoelzl@51478
   276
  unfolding continuous_def by (rule tendsto_real_root)
hoelzl@51478
   277
  
hoelzl@56371
   278
lemma continuous_on_real_root[continuous_intros]:
hoelzl@51483
   279
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
hoelzl@51478
   280
  unfolding continuous_on_def by (auto intro: tendsto_real_root)
hoelzl@51478
   281
huffman@23042
   282
lemma DERIV_real_root:
huffman@23042
   283
  assumes n: "0 < n"
huffman@23042
   284
  assumes x: "0 < x"
huffman@23042
   285
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
huffman@23042
   286
proof (rule DERIV_inverse_function)
huffman@23044
   287
  show "0 < x" using x .
huffman@23044
   288
  show "x < x + 1" by simp
huffman@23044
   289
  show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
huffman@23042
   290
    using n by simp
huffman@23042
   291
  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
huffman@23042
   292
    by (rule DERIV_pow)
huffman@23042
   293
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
huffman@23042
   294
    using n x by simp
hoelzl@51483
   295
qed (rule isCont_real_root)
huffman@23042
   296
huffman@23046
   297
lemma DERIV_odd_real_root:
huffman@23046
   298
  assumes n: "odd n"
huffman@23046
   299
  assumes x: "x \<noteq> 0"
huffman@23046
   300
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
huffman@23046
   301
proof (rule DERIV_inverse_function)
huffman@23046
   302
  show "x - 1 < x" by simp
huffman@23046
   303
  show "x < x + 1" by simp
huffman@23046
   304
  show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
huffman@23046
   305
    using n by (simp add: odd_real_root_pow)
huffman@23046
   306
  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
huffman@23046
   307
    by (rule DERIV_pow)
huffman@23046
   308
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
huffman@23046
   309
    using odd_pos [OF n] x by simp
hoelzl@51483
   310
qed (rule isCont_real_root)
huffman@23046
   311
hoelzl@31880
   312
lemma DERIV_even_real_root:
hoelzl@31880
   313
  assumes n: "0 < n" and "even n"
hoelzl@31880
   314
  assumes x: "x < 0"
hoelzl@31880
   315
  shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
hoelzl@31880
   316
proof (rule DERIV_inverse_function)
hoelzl@31880
   317
  show "x - 1 < x" by simp
hoelzl@31880
   318
  show "x < 0" using x .
hoelzl@31880
   319
next
hoelzl@31880
   320
  show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
hoelzl@31880
   321
  proof (rule allI, rule impI, erule conjE)
hoelzl@31880
   322
    fix y assume "x - 1 < y" and "y < 0"
hoelzl@31880
   323
    hence "root n (-y) ^ n = -y" using `0 < n` by simp
hoelzl@51483
   324
    with real_root_minus and `even n`
hoelzl@31880
   325
    show "- (root n y ^ n) = y" by simp
hoelzl@31880
   326
  qed
hoelzl@31880
   327
next
hoelzl@31880
   328
  show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
hoelzl@56381
   329
    by  (auto intro!: derivative_eq_intros simp: real_of_nat_def)
hoelzl@31880
   330
  show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
hoelzl@31880
   331
    using n x by simp
hoelzl@51483
   332
qed (rule isCont_real_root)
hoelzl@31880
   333
hoelzl@31880
   334
lemma DERIV_real_root_generic:
hoelzl@31880
   335
  assumes "0 < n" and "x \<noteq> 0"
wenzelm@49753
   336
    and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
wenzelm@49753
   337
    and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
wenzelm@49753
   338
    and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
hoelzl@31880
   339
  shows "DERIV (root n) x :> D"
hoelzl@31880
   340
using assms by (cases "even n", cases "0 < x",
hoelzl@31880
   341
  auto intro: DERIV_real_root[THEN DERIV_cong]
hoelzl@31880
   342
              DERIV_odd_real_root[THEN DERIV_cong]
hoelzl@31880
   343
              DERIV_even_real_root[THEN DERIV_cong])
hoelzl@31880
   344
huffman@22956
   345
subsection {* Square Root *}
huffman@20687
   346
hoelzl@51483
   347
definition sqrt :: "real \<Rightarrow> real" where
huffman@22956
   348
  "sqrt = root 2"
huffman@20687
   349
huffman@22956
   350
lemma pos2: "0 < (2::nat)" by simp
huffman@22956
   351
wenzelm@53015
   352
lemma real_sqrt_unique: "\<lbrakk>y\<^sup>2 = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
huffman@22956
   353
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
huffman@20687
   354
wenzelm@53015
   355
lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>"
huffman@22956
   356
apply (rule real_sqrt_unique)
huffman@22956
   357
apply (rule power2_abs)
huffman@22956
   358
apply (rule abs_ge_zero)
huffman@22956
   359
done
huffman@20687
   360
wenzelm@53015
   361
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x"
huffman@22956
   362
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
huffman@22856
   363
wenzelm@53015
   364
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
huffman@22856
   365
apply (rule iffI)
huffman@22856
   366
apply (erule subst)
huffman@22856
   367
apply (rule zero_le_power2)
huffman@22856
   368
apply (erule real_sqrt_pow2)
huffman@20687
   369
done
huffman@20687
   370
huffman@22956
   371
lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
huffman@22956
   372
unfolding sqrt_def by (rule real_root_zero)
huffman@22956
   373
huffman@22956
   374
lemma real_sqrt_one [simp]: "sqrt 1 = 1"
huffman@22956
   375
unfolding sqrt_def by (rule real_root_one [OF pos2])
huffman@22956
   376
hoelzl@56889
   377
lemma real_sqrt_four [simp]: "sqrt 4 = 2"
hoelzl@56889
   378
  using real_sqrt_abs[of 2] by simp
hoelzl@56889
   379
huffman@22956
   380
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
hoelzl@51483
   381
unfolding sqrt_def by (rule real_root_minus)
huffman@22956
   382
huffman@22956
   383
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
hoelzl@51483
   384
unfolding sqrt_def by (rule real_root_mult)
huffman@22956
   385
hoelzl@56889
   386
lemma real_sqrt_mult_self[simp]: "sqrt a * sqrt a = \<bar>a\<bar>"
hoelzl@56889
   387
  using real_sqrt_abs[of a] unfolding power2_eq_square real_sqrt_mult .
hoelzl@56889
   388
huffman@22956
   389
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
hoelzl@51483
   390
unfolding sqrt_def by (rule real_root_inverse)
huffman@22956
   391
huffman@22956
   392
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
hoelzl@51483
   393
unfolding sqrt_def by (rule real_root_divide)
huffman@22956
   394
huffman@22956
   395
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
huffman@22956
   396
unfolding sqrt_def by (rule real_root_power [OF pos2])
huffman@22956
   397
huffman@22956
   398
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
huffman@22956
   399
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
huffman@22956
   400
huffman@22956
   401
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
hoelzl@51483
   402
unfolding sqrt_def by (rule real_root_ge_zero)
huffman@20687
   403
huffman@22956
   404
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
huffman@22956
   405
unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
huffman@22956
   406
huffman@22956
   407
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
huffman@22956
   408
unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
huffman@22956
   409
huffman@22956
   410
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
huffman@22956
   411
unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
huffman@22956
   412
huffman@22956
   413
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
huffman@22956
   414
unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
huffman@22956
   415
huffman@22956
   416
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
huffman@22956
   417
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
huffman@22956
   418
hoelzl@54413
   419
lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
hoelzl@54413
   420
  using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
hoelzl@54413
   421
hoelzl@54413
   422
lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
hoelzl@54413
   423
  using real_sqrt_le_mono[of "x\<^sup>2" y] by simp
hoelzl@54413
   424
hoelzl@54413
   425
lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y"
hoelzl@54413
   426
  using real_sqrt_less_mono[of "x\<^sup>2" y] by simp
hoelzl@54413
   427
hoelzl@54413
   428
lemma sqrt_even_pow2:
hoelzl@54413
   429
  assumes n: "even n"
hoelzl@54413
   430
  shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
hoelzl@54413
   431
proof -
hoelzl@54413
   432
  from n obtain m where m: "n = 2 * m"
hoelzl@54413
   433
    unfolding even_mult_two_ex ..
hoelzl@54413
   434
  from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
hoelzl@54413
   435
    by (simp only: power_mult[symmetric] mult_commute)
hoelzl@54413
   436
  then show ?thesis
hoelzl@54413
   437
    using m by simp
hoelzl@54413
   438
qed
hoelzl@54413
   439
huffman@53594
   440
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, unfolded real_sqrt_zero]
huffman@53594
   441
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, unfolded real_sqrt_zero]
huffman@53594
   442
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, unfolded real_sqrt_zero]
huffman@53594
   443
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, unfolded real_sqrt_zero]
huffman@53594
   444
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, unfolded real_sqrt_zero]
huffman@22956
   445
huffman@53594
   446
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, unfolded real_sqrt_one]
huffman@53594
   447
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, unfolded real_sqrt_one]
huffman@53594
   448
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, unfolded real_sqrt_one]
huffman@53594
   449
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, unfolded real_sqrt_one]
huffman@53594
   450
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, unfolded real_sqrt_one]
huffman@20687
   451
huffman@23042
   452
lemma isCont_real_sqrt: "isCont sqrt x"
hoelzl@51483
   453
unfolding sqrt_def by (rule isCont_real_root)
huffman@23042
   454
hoelzl@51478
   455
lemma tendsto_real_sqrt[tendsto_intros]:
hoelzl@51478
   456
  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) ---> sqrt x) F"
hoelzl@51483
   457
  unfolding sqrt_def by (rule tendsto_real_root)
hoelzl@51478
   458
hoelzl@51478
   459
lemma continuous_real_sqrt[continuous_intros]:
hoelzl@51478
   460
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
hoelzl@51483
   461
  unfolding sqrt_def by (rule continuous_real_root)
hoelzl@51478
   462
  
hoelzl@56371
   463
lemma continuous_on_real_sqrt[continuous_intros]:
hoelzl@51478
   464
  "continuous_on s f \<Longrightarrow> 0 < n \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
hoelzl@51483
   465
  unfolding sqrt_def by (rule continuous_on_real_root)
hoelzl@51478
   466
hoelzl@31880
   467
lemma DERIV_real_sqrt_generic:
hoelzl@31880
   468
  assumes "x \<noteq> 0"
hoelzl@31880
   469
  assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
hoelzl@31880
   470
  assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
hoelzl@31880
   471
  shows "DERIV sqrt x :> D"
hoelzl@31880
   472
  using assms unfolding sqrt_def
hoelzl@31880
   473
  by (auto intro!: DERIV_real_root_generic)
hoelzl@31880
   474
huffman@23042
   475
lemma DERIV_real_sqrt:
huffman@23042
   476
  "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
hoelzl@31880
   477
  using DERIV_real_sqrt_generic by simp
hoelzl@31880
   478
hoelzl@31880
   479
declare
hoelzl@56381
   480
  DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
hoelzl@56381
   481
  DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]
huffman@23042
   482
huffman@20687
   483
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
huffman@20687
   484
apply auto
huffman@20687
   485
apply (cut_tac x = x and y = 0 in linorder_less_linear)
huffman@20687
   486
apply (simp add: zero_less_mult_iff)
huffman@20687
   487
done
huffman@20687
   488
huffman@20687
   489
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
huffman@22856
   490
apply (subst power2_eq_square [symmetric])
huffman@20687
   491
apply (rule real_sqrt_abs)
huffman@20687
   492
done
huffman@20687
   493
wenzelm@53076
   494
lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x"
huffman@22856
   495
by (simp add: power_inverse [symmetric])
huffman@20687
   496
huffman@20687
   497
lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
huffman@22956
   498
by simp
huffman@20687
   499
huffman@20687
   500
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
huffman@22956
   501
by simp
huffman@20687
   502
huffman@22443
   503
lemma sqrt_divide_self_eq:
huffman@22443
   504
  assumes nneg: "0 \<le> x"
huffman@22443
   505
  shows "sqrt x / x = inverse (sqrt x)"
huffman@22443
   506
proof cases
huffman@22443
   507
  assume "x=0" thus ?thesis by simp
huffman@22443
   508
next
huffman@22443
   509
  assume nz: "x\<noteq>0" 
huffman@22443
   510
  hence pos: "0<x" using nneg by arith
huffman@22443
   511
  show ?thesis
huffman@22443
   512
  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
huffman@22443
   513
    show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
huffman@22443
   514
    show "inverse (sqrt x) / (sqrt x / x) = 1"
huffman@22443
   515
      by (simp add: divide_inverse mult_assoc [symmetric] 
huffman@22443
   516
                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
huffman@22443
   517
  qed
huffman@22443
   518
qed
huffman@22443
   519
hoelzl@54413
   520
lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
hoelzl@54413
   521
  apply (cases "x = 0")
hoelzl@54413
   522
  apply simp_all
hoelzl@54413
   523
  using sqrt_divide_self_eq[of x]
hoelzl@54413
   524
  apply (simp add: inverse_eq_divide field_simps)
hoelzl@54413
   525
  done
hoelzl@54413
   526
huffman@22721
   527
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
huffman@22721
   528
apply (simp add: divide_inverse)
huffman@22721
   529
apply (case_tac "r=0")
huffman@22721
   530
apply (auto simp add: mult_ac)
huffman@22721
   531
done
huffman@22721
   532
huffman@23049
   533
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
huffman@35216
   534
by (simp add: divide_less_eq)
huffman@23049
   535
huffman@23049
   536
lemma four_x_squared: 
huffman@23049
   537
  fixes x::real
wenzelm@53015
   538
  shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
huffman@23049
   539
by (simp add: power2_eq_square)
huffman@23049
   540
huffman@22856
   541
subsection {* Square Root of Sum of Squares *}
huffman@22856
   542
lp15@55967
   543
lemma sum_squares_bound: 
lp15@55967
   544
  fixes x:: "'a::linordered_field"
lp15@55967
   545
  shows "2*x*y \<le> x^2 + y^2"
lp15@55967
   546
proof -
lp15@55967
   547
  have "(x-y)^2 = x*x - 2*x*y + y*y"
lp15@55967
   548
    by algebra
lp15@55967
   549
  then have "0 \<le> x^2 - 2*x*y + y^2"
lp15@55967
   550
    by (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square)
lp15@55967
   551
  then show ?thesis
lp15@55967
   552
    by arith
lp15@55967
   553
qed
huffman@22856
   554
lp15@55967
   555
lemma arith_geo_mean: 
lp15@55967
   556
  fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2"
lp15@55967
   557
    apply (rule power2_le_imp_le)
lp15@55967
   558
    using sum_squares_bound assms
lp15@55967
   559
    apply (auto simp: zero_le_mult_iff)
lp15@55967
   560
    by (auto simp: algebra_simps power2_eq_square)
lp15@55967
   561
lp15@55967
   562
lemma arith_geo_mean_sqrt: 
lp15@55967
   563
  fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
lp15@55967
   564
  apply (rule arith_geo_mean)
lp15@55967
   565
  using assms
lp15@55967
   566
  apply (auto simp: zero_le_mult_iff)
lp15@55967
   567
  done
huffman@23049
   568
huffman@22856
   569
lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
wenzelm@53015
   570
     "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
lp15@55967
   571
  by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)
huffman@22856
   572
huffman@22856
   573
lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
wenzelm@53076
   574
     "(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)"
huffman@44320
   575
  by (simp add: zero_le_mult_iff)
huffman@22856
   576
wenzelm@53015
   577
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0"
wenzelm@53015
   578
by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
huffman@23049
   579
wenzelm@53015
   580
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0"
wenzelm@53015
   581
by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
huffman@23049
   582
wenzelm@53015
   583
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
huffman@22856
   584
by (rule power2_le_imp_le, simp_all)
huffman@22856
   585
wenzelm@53015
   586
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
huffman@23049
   587
by (rule power2_le_imp_le, simp_all)
huffman@23049
   588
wenzelm@53015
   589
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
huffman@22856
   590
by (rule power2_le_imp_le, simp_all)
huffman@22856
   591
wenzelm@53015
   592
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
huffman@23049
   593
by (rule power2_le_imp_le, simp_all)
huffman@23049
   594
huffman@23049
   595
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
huffman@23049
   596
by (simp add: power2_eq_square [symmetric])
huffman@23049
   597
huffman@22858
   598
lemma real_sqrt_sum_squares_triangle_ineq:
wenzelm@53015
   599
  "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
huffman@22858
   600
apply (rule power2_le_imp_le, simp)
huffman@22858
   601
apply (simp add: power2_sum)
webertj@49962
   602
apply (simp only: mult_assoc distrib_left [symmetric])
huffman@22858
   603
apply (rule mult_left_mono)
huffman@22858
   604
apply (rule power2_le_imp_le)
huffman@22858
   605
apply (simp add: power2_sum power_mult_distrib)
nipkow@23477
   606
apply (simp add: ring_distribs)
wenzelm@53015
   607
apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)", simp)
wenzelm@53015
   608
apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
huffman@22858
   609
apply (rule zero_le_power2)
huffman@22858
   610
apply (simp add: power2_diff power_mult_distrib)
nipkow@56536
   611
apply (simp)
huffman@22858
   612
apply simp
huffman@22858
   613
apply (simp add: add_increasing)
huffman@22858
   614
done
huffman@22858
   615
huffman@23122
   616
lemma real_sqrt_sum_squares_less:
wenzelm@53015
   617
  "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
huffman@23122
   618
apply (rule power2_less_imp_less, simp)
huffman@23122
   619
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
huffman@23122
   620
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
huffman@23122
   621
apply (simp add: power_divide)
huffman@23122
   622
apply (drule order_le_less_trans [OF abs_ge_zero])
huffman@23122
   623
apply (simp add: zero_less_divide_iff)
huffman@23122
   624
done
huffman@23122
   625
huffman@23049
   626
text{*Needed for the infinitely close relation over the nonstandard
huffman@23049
   627
    complex numbers*}
huffman@23049
   628
lemma lemma_sqrt_hcomplex_capprox:
wenzelm@53015
   629
     "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<^sup>2 + y\<^sup>2) < u"
huffman@23049
   630
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
huffman@23049
   631
apply (erule_tac [2] lemma_real_divide_sqrt_less)
huffman@23049
   632
apply (rule power2_le_imp_le)
huffman@44349
   633
apply (auto simp add: zero_le_divide_iff power_divide)
wenzelm@53015
   634
apply (rule_tac t = "u\<^sup>2" in real_sum_of_halves [THEN subst])
huffman@23049
   635
apply (rule add_mono)
huffman@30273
   636
apply (auto simp add: four_x_squared intro: power_mono)
huffman@23049
   637
done
huffman@23049
   638
huffman@22956
   639
text "Legacy theorem names:"
huffman@22956
   640
lemmas real_root_pos2 = real_root_power_cancel
huffman@22956
   641
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
huffman@22956
   642
lemmas real_root_pos_pos_le = real_root_ge_zero
huffman@22956
   643
lemmas real_sqrt_mult_distrib = real_sqrt_mult
huffman@22956
   644
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
huffman@22956
   645
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
huffman@22956
   646
huffman@22956
   647
(* needed for CauchysMeanTheorem.het_base from AFP *)
huffman@22956
   648
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"
huffman@22956
   649
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])
huffman@22956
   650
paulson@14324
   651
end