src/HOL/NthRoot.thy
author hoelzl
Tue, 12 Nov 2013 19:28:54 +0100
changeset 54413 88a036a95967
parent 53594 8a9fb53294f4
child 55967 5dadc93ff3df
permissions -rw-r--r--
add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
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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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   115
    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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parents: 23122
diff changeset
   246
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   247
lemma real_root_increasing:
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   248
  "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
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   249
by (auto simp add: order_le_less real_root_strict_increasing)
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diff changeset
   250
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   251
text {* Continuity and derivatives *}
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   252
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   253
lemma isCont_real_root: "isCont (root n) x"
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   254
proof cases
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   255
  assume n: "0 < n"
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   256
  let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   257
  have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else - ((-x) ^ n) :: real)"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   258
    using n by (intro continuous_on_If continuous_on_intros) auto
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   259
  then have "continuous_on UNIV ?f"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   260
    by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less real_sgn_neg le_less n)
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   261
  then have [simp]: "\<And>x. isCont ?f x"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
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   262
    by (simp add: continuous_on_eq_continuous_at)
23042
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huffman
parents: 23009
diff changeset
   263
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
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   264
  have "isCont (root n) (?f (root n x))"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   265
    by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power n)
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
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   266
  then show ?thesis
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   267
    by (simp add: sgn_power_root n)
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
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   268
qed (simp add: root_def[abs_def])
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   269
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   270
lemma tendsto_real_root[tendsto_intros]:
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   271
  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) ---> root n x) F"
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
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   272
  using isCont_tendsto_compose[OF isCont_real_root, of f x F] .
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)
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parents: 49962
diff changeset
   273
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)
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   274
lemma continuous_real_root[continuous_intros]:
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   275
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
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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)
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parents: 49962
diff changeset
   276
  unfolding continuous_def by (rule tendsto_real_root)
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)
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parents: 49962
diff changeset
   277
  
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)
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parents: 49962
diff changeset
   278
lemma continuous_on_real_root[continuous_on_intros]:
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dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
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   279
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
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
   280
  unfolding continuous_on_def by (auto intro: tendsto_real_root)
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
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diff changeset
   281
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   282
lemma DERIV_real_root:
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   283
  assumes n: "0 < n"
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   284
  assumes x: "0 < x"
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   285
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
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parents: 23009
diff changeset
   286
proof (rule DERIV_inverse_function)
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23042
diff changeset
   287
  show "0 < x" using x .
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23042
diff changeset
   288
  show "x < x + 1" by simp
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huffman
parents: 23042
diff changeset
   289
  show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
23042
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huffman
parents: 23009
diff changeset
   290
    using n by simp
492514b39956 add lemmas about continuity and derivatives of roots
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parents: 23009
diff changeset
   291
  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   292
    by (rule DERIV_pow)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   293
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   294
    using n x by simp
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
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diff changeset
   295
qed (rule isCont_real_root)
23042
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parents: 23009
diff changeset
   296
23046
12f35ece221f add odd_real_root lemmas
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diff changeset
   297
lemma DERIV_odd_real_root:
12f35ece221f add odd_real_root lemmas
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parents: 23044
diff changeset
   298
  assumes n: "odd n"
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diff changeset
   299
  assumes x: "x \<noteq> 0"
12f35ece221f add odd_real_root lemmas
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parents: 23044
diff changeset
   300
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   301
proof (rule DERIV_inverse_function)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   302
  show "x - 1 < x" by simp
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   303
  show "x < x + 1" by simp
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   304
  show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   305
    using n by (simp add: odd_real_root_pow)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   306
  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   307
    by (rule DERIV_pow)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   308
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   309
    using odd_pos [OF n] x by simp
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
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diff changeset
   310
qed (rule isCont_real_root)
23046
12f35ece221f add odd_real_root lemmas
huffman
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diff changeset
   311
31880
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   312
lemma DERIV_even_real_root:
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diff changeset
   313
  assumes n: "0 < n" and "even n"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   314
  assumes x: "x < 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   315
  shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   316
proof (rule DERIV_inverse_function)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   317
  show "x - 1 < x" by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   318
  show "x < 0" using x .
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   319
next
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   320
  show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   321
  proof (rule allI, rule impI, erule conjE)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   322
    fix y assume "x - 1 < y" and "y < 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   323
    hence "root n (-y) ^ n = -y" using `0 < n` by simp
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   324
    with real_root_minus and `even n`
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   325
    show "- (root n y ^ n) = y" by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   326
  qed
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   327
next
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   328
  show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   329
    by  (auto intro!: DERIV_intros)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   330
  show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   331
    using n x by simp
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   332
qed (rule isCont_real_root)
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   333
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   334
lemma DERIV_real_root_generic:
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   335
  assumes "0 < n" and "x \<noteq> 0"
49753
a344f1a21211 eliminated spurious fact duplicates;
wenzelm
parents: 44349
diff changeset
   336
    and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
a344f1a21211 eliminated spurious fact duplicates;
wenzelm
parents: 44349
diff changeset
   337
    and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
a344f1a21211 eliminated spurious fact duplicates;
wenzelm
parents: 44349
diff changeset
   338
    and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   339
  shows "DERIV (root n) x :> D"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   340
using assms by (cases "even n", cases "0 < x",
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   341
  auto intro: DERIV_real_root[THEN DERIV_cong]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   342
              DERIV_odd_real_root[THEN DERIV_cong]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   343
              DERIV_even_real_root[THEN DERIV_cong])
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   344
22956
617140080e6a define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
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diff changeset
   345
subsection {* Square Root *}
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
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parents: 20515
diff changeset
   346
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
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diff changeset
   347
definition sqrt :: "real \<Rightarrow> real" where
22956
617140080e6a define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
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parents: 22943
diff changeset
   348
  "sqrt = root 2"
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
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parents: 20515
diff changeset
   349
22956
617140080e6a define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
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diff changeset
   350
lemma pos2: "0 < (2::nat)" by simp
617140080e6a define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
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diff changeset
   351
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   352
lemma real_sqrt_unique: "\<lbrakk>y\<^sup>2 = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
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
   353
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   354
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   355
lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>"
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
   356
apply (rule real_sqrt_unique)
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
   357
apply (rule power2_abs)
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
   358
apply (rule abs_ge_zero)
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
   359
done
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   360
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   361
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = 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
   362
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   363
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   364
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   365
apply (rule iffI)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   366
apply (erule subst)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   367
apply (rule zero_le_power2)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   368
apply (erule real_sqrt_pow2)
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   369
done
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   370
22956
617140080e6a define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
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parents: 22943
diff changeset
   371
lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
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
   372
unfolding sqrt_def by (rule real_root_zero)
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
   373
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
   374
lemma real_sqrt_one [simp]: "sqrt 1 = 1"
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
   375
unfolding sqrt_def by (rule real_root_one [OF pos2])
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
   376
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
   377
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   378
unfolding sqrt_def by (rule real_root_minus)
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
   379
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
   380
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   381
unfolding sqrt_def by (rule real_root_mult)
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
   382
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
   383
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   384
unfolding sqrt_def by (rule real_root_inverse)
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
   385
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
   386
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   387
unfolding sqrt_def by (rule real_root_divide)
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
   388
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
   389
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
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
   390
unfolding sqrt_def by (rule real_root_power [OF pos2])
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
   391
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
   392
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt 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
   393
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
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
   394
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
   395
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
51483
dc39d69774bb modernized definition of root: use the_inv, handle positive and negative case uniformly, and 0-th root is constant 0
hoelzl
parents: 51478
diff changeset
   396
unfolding sqrt_def by (rule real_root_ge_zero)
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   397
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
   398
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
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
   399
unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
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
   400
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
   401
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
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
   402
unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
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
   403
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
   404
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
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
   405
unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
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
   406
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
   407
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
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
   408
unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
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
   409
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
   410
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
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
   411
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
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
   412
54413
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   413
lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   414
  using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   415
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   416
lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   417
  using real_sqrt_le_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
   418
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
parents: 53594
diff changeset
   419
lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y"
88a036a95967 add finite_select_induct; move generic lemmas from MV_Analysis/Linear_Algebra to the HOL image
hoelzl
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: 53594
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: 53594
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
parents: 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: 53594
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
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   434
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, unfolded real_sqrt_zero]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   435
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, unfolded real_sqrt_zero]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   436
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, unfolded real_sqrt_zero]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   437
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, unfolded real_sqrt_zero]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   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
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   440
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, unfolded real_sqrt_one]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   441
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, unfolded real_sqrt_one]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   442
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, unfolded real_sqrt_one]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   443
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, unfolded real_sqrt_one]
8a9fb53294f4 prefer attribute 'unfolded thm' to 'simplified'
huffman
parents: 53076
diff changeset
   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 0-th 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 0-th 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 0-th 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 0-th 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
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   461
lemma DERIV_real_sqrt_generic:
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   462
  assumes "x \<noteq> 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   463
  assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   464
  assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   465
  shows "DERIV sqrt x :> D"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   466
  using assms unfolding sqrt_def
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   467
  by (auto intro!: DERIV_real_root_generic)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   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
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   471
  using DERIV_real_sqrt_generic by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   472
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   473
declare
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   474
  DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   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
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   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
47c9aff07725 more symbols;
wenzelm
parents: 53015
diff changeset
   488
lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   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
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   497
lemma sqrt_divide_self_eq:
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   498
  assumes nneg: "0 \<le> x"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   499
  shows "sqrt x / x = inverse (sqrt x)"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   500
proof cases
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   501
  assume "x=0" thus ?thesis by simp
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   502
next
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   503
  assume nz: "x\<noteq>0" 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   504
  hence pos: "0<x" using nneg by arith
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   505
  show ?thesis
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   506
  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   507
    show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   508
    show "inverse (sqrt x) / (sqrt x / x) = 1"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   509
      by (simp add: divide_inverse mult_assoc [symmetric] 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   510
                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   511
  qed
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   512
qed
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   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
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 31880
diff changeset
   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
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   535
subsection {* Square Root of Sum of Squares *}
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   536
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   537
lemma real_sqrt_sum_squares_ge_zero: "0 \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
44320
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   538
  by simp (* TODO: delete *)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   539
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   540
declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   541
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   542
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
   543
     "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
44320
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   544
  by (simp add: zero_le_mult_iff)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   545
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   546
lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
53076
47c9aff07725 more symbols;
wenzelm
parents: 53015
diff changeset
   547
     "(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
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   548
  by (simp add: zero_le_mult_iff)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   549
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   550
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
   551
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
   552
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   553
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
   554
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
   555
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   556
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   557
by (rule power2_le_imp_le, simp_all)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   558
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   559
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
   560
by (rule power2_le_imp_le, simp_all)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   561
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   562
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   563
by (rule power2_le_imp_le, simp_all)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   564
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   565
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
   566
by (rule power2_le_imp_le, simp_all)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   567
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   568
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
   569
by (simp add: power2_eq_square [symmetric])
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   570
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   571
lemma real_sqrt_sum_squares_triangle_ineq:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   572
  "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
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   573
apply (rule power2_le_imp_le, simp)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   574
apply (simp add: power2_sum)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 49753
diff changeset
   575
apply (simp only: mult_assoc distrib_left [symmetric])
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   576
apply (rule mult_left_mono)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   577
apply (rule power2_le_imp_le)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   578
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
   579
apply (simp add: ring_distribs)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   580
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: 51483
diff changeset
   581
apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   582
apply (rule zero_le_power2)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   583
apply (simp add: power2_diff power_mult_distrib)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   584
apply (simp add: mult_nonneg_nonneg)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   585
apply simp
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   586
apply (simp add: add_increasing)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   587
done
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   588
23122
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   589
lemma real_sqrt_sum_squares_less:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   590
  "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
23122
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   591
apply (rule power2_less_imp_less, simp)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   592
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   593
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   594
apply (simp add: power_divide)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   595
apply (drule order_le_less_trans [OF abs_ge_zero])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   596
apply (simp add: zero_less_divide_iff)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   597
done
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   598
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   599
text{*Needed for the infinitely close relation over the nonstandard
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   600
    complex numbers*}
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   601
lemma lemma_sqrt_hcomplex_capprox:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   602
     "[| 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
   603
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
   604
apply (erule_tac [2] lemma_real_divide_sqrt_less)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   605
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
   606
apply (auto simp add: zero_le_divide_iff power_divide)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51483
diff changeset
   607
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
   608
apply (rule add_mono)
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 28952
diff changeset
   609
apply (auto simp add: four_x_squared intro: power_mono)
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   610
done
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   611
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
   612
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: 22943
diff changeset
   613
lemmas real_root_pos2 = real_root_power_cancel
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
   614
lemmas real_root_pos_pos = real_root_gt_zero [THEN 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: 22943
diff changeset
   615
lemmas real_root_pos_pos_le = real_root_ge_zero
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
   616
lemmas real_sqrt_mult_distrib = 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
   617
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
   618
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
   619
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
   620
(* 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
   621
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
   622
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: 22943
diff changeset
   623
14324
c9c6832f9b22 converting Hyperreal/NthRoot to Isar
paulson
parents: 14268
diff changeset
   624
end