src/HOL/Hyperreal/NthRoot.thy
author huffman
Sun May 20 09:21:04 2007 +0200 (2007-05-20)
changeset 23047 17f7d831efe2
parent 23046 12f35ece221f
child 23049 11607c283074
permissions -rw-r--r--
add realpow_pos_nth2 back in
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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 SEQ Parity Deriv
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begin
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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 add: isCont_power isCont_Id)
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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:
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  "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)"
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apply (auto intro!: realpow_pos_nth)
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apply (rule_tac n=n in power_eq_imp_eq_base, simp_all)
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done
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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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definition
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  root :: "[nat, real] \<Rightarrow> real" where
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  "root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else
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               if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)"
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lemma real_root_zero [simp]: "root n 0 = 0"
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unfolding root_def by simp
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lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x"
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unfolding root_def by simp
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lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x"
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apply (simp add: root_def)
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apply (drule (1) realpow_pos_nth_unique)
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apply (erule theI' [THEN conjunct1])
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done
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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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apply (simp add: root_def)
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apply (drule (1) realpow_pos_nth_unique)
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apply (erule theI' [THEN conjunct2])
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done
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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 odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n"
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by (cases n, simp_all)
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lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"
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apply (rule_tac x=0 and y=x in linorder_le_cases)
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apply (erule (1) real_root_pow_pos2 [OF odd_pos])
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apply (subgoal_tac "root n (- x) ^ n = - x")
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apply (simp add: real_root_minus odd_pos)
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apply (simp add: odd_pos)
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done
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lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x"
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by (auto simp add: order_le_less real_root_gt_zero)
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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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apply (subgoal_tac "0 \<le> x ^ n")
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apply (subgoal_tac "0 \<le> root n (x ^ n)")
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apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n")
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apply (erule (3) power_eq_imp_eq_base)
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apply (erule (1) real_root_pow_pos2)
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apply (erule (1) real_root_ge_zero)
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apply (erule zero_le_power)
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done
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lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x"
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apply (rule_tac x=0 and y=x in linorder_le_cases)
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apply (erule (1) real_root_power_cancel [OF odd_pos])
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apply (subgoal_tac "root n ((- x) ^ n) = - x")
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apply (simp add: real_root_minus odd_pos)
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apply (erule real_root_power_cancel [OF odd_pos], simp)
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done
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lemma real_root_pos_unique:
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  "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
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by (erule subst, rule real_root_power_cancel)
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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_less_mono_lemma:
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  "\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
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apply (subgoal_tac "0 \<le> y")
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apply (subgoal_tac "root n x ^ n < root n y ^ n")
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apply (erule power_less_imp_less_base)
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apply (erule (1) real_root_ge_zero)
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apply simp
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apply simp
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done
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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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apply (cases "0 \<le> x")
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apply (erule (2) real_root_less_mono_lemma)
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apply (cases "0 \<le> y")
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apply (rule_tac y=0 in order_less_le_trans)
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apply (subgoal_tac "0 < root n (- x)")
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apply (simp add: real_root_minus)
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apply (simp add: real_root_gt_zero)
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apply (simp add: real_root_ge_zero)
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apply (subgoal_tac "root n (- y) < root n (- x)")
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apply (simp add: real_root_minus)
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apply (simp add: real_root_less_mono_lemma)
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done
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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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text {* Roots of multiplication and division *}
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lemma real_root_mult_lemma:
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  "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y"
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by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib)
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lemma real_root_inverse_lemma:
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  "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)"
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by (simp add: real_root_pos_unique power_inverse [symmetric])
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lemma real_root_mult:
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  assumes n: "0 < n"
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  shows "root n (x * y) = root n x * root n y"
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proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases)
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  assume "0 \<le> x" and "0 \<le> y"
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  thus ?thesis by (rule real_root_mult_lemma [OF n])
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next
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  assume "0 \<le> x" and "y \<le> 0"
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  hence "0 \<le> x" and "0 \<le> - y" by simp_all
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  hence "root n (x * - y) = root n x * root n (- y)"
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    by (rule real_root_mult_lemma [OF n])
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  thus ?thesis by (simp add: real_root_minus [OF n])
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next
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  assume "x \<le> 0" and "0 \<le> y"
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  hence "0 \<le> - x" and "0 \<le> y" by simp_all
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  hence "root n (- x * y) = root n (- x) * root n y"
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    by (rule real_root_mult_lemma [OF n])
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  thus ?thesis by (simp add: real_root_minus [OF n])
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next
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  assume "x \<le> 0" and "y \<le> 0"
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  hence "0 \<le> - x" and "0 \<le> - y" by simp_all
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  hence "root n (- x * - y) = root n (- x) * root n (- y)"
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    by (rule real_root_mult_lemma [OF n])
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  thus ?thesis by (simp add: real_root_minus [OF n])
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qed
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lemma real_root_inverse:
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  assumes n: "0 < n"
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  shows "root n (inverse x) = inverse (root n x)"
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proof (rule linorder_le_cases)
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  assume "0 \<le> x"
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  thus ?thesis by (rule real_root_inverse_lemma [OF n])
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next
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  assume "x \<le> 0"
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  hence "0 \<le> - x" by simp
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  hence "root n (inverse (- x)) = inverse (root n (- x))"
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    by (rule real_root_inverse_lemma [OF n])
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  thus ?thesis by (simp add: real_root_minus [OF n])
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qed
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lemma real_root_divide:
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  "0 < n \<Longrightarrow> 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_power:
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  "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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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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text {* Continuity and derivatives *}
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lemma isCont_root_pos:
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  assumes n: "0 < n"
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  assumes x: "0 < x"
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  shows "isCont (root n) x"
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proof -
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  have "isCont (root n) (root n x ^ n)"
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  proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"])
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    show "0 < root n x" using n x by simp
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    show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z"
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      by (simp add: abs_le_iff real_root_power_cancel n)
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    show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z"
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      by (simp add: isCont_power isCont_Id)
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  qed
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  thus ?thesis using n x by simp
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qed
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lemma isCont_root_neg:
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  "\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x"
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apply (subgoal_tac "isCont (\<lambda>x. - root n (- x)) x")
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apply (simp add: real_root_minus)
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apply (rule isCont_o2 [OF isCont_minus [OF isCont_Id]])
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apply (simp add: isCont_minus isCont_root_pos)
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done
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lemma isCont_root_zero:
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  "0 < n \<Longrightarrow> isCont (root n) 0"
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unfolding isCont_def
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apply (rule LIM_I)
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apply (rule_tac x="r ^ n" in exI, safe)
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apply (simp add: zero_less_power)
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apply (simp add: real_root_abs [symmetric])
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apply (rule_tac n="n" in power_less_imp_less_base, simp_all)
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done
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lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x"
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apply (rule_tac x=x and y=0 in linorder_cases)
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apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero)
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done
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lemma DERIV_real_root:
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  assumes n: "0 < n"
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  assumes x: "0 < x"
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  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
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proof (rule DERIV_inverse_function)
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  show "0 < x" using x .
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  show "x < x + 1" by simp
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  show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
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    using n by simp
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  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
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    by (rule DERIV_pow)
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  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
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    using n x by simp
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  show "isCont (root n) x"
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    by (rule isCont_real_root)
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qed
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lemma DERIV_odd_real_root:
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  assumes n: "odd n"
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  assumes x: "x \<noteq> 0"
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   304
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
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proof (rule DERIV_inverse_function)
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   306
  show "x - 1 < x" by simp
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   307
  show "x < x + 1" by simp
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   308
  show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
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   309
    using n by (simp add: odd_real_root_pow)
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   310
  show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
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   311
    by (rule DERIV_pow)
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   312
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
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   313
    using odd_pos [OF n] x by simp
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   314
  show "isCont (root n) x"
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   315
    using odd_pos [OF n] by (rule isCont_real_root)
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   316
qed
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   317
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   318
subsection {* Square Root *}
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   319
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   320
definition
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   321
  sqrt :: "real \<Rightarrow> real" where
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   322
  "sqrt = root 2"
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   323
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   324
lemma pos2: "0 < (2::nat)" by simp
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   325
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   326
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
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   327
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
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   328
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   329
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>"
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   330
apply (rule real_sqrt_unique)
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   331
apply (rule power2_abs)
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   332
apply (rule abs_ge_zero)
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   333
done
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   334
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   335
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x"
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   336
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
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   337
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   338
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
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   339
apply (rule iffI)
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   340
apply (erule subst)
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   341
apply (rule zero_le_power2)
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   342
apply (erule real_sqrt_pow2)
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   343
done
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   344
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   345
lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
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   346
unfolding sqrt_def by (rule real_root_zero)
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   347
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   348
lemma real_sqrt_one [simp]: "sqrt 1 = 1"
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   349
unfolding sqrt_def by (rule real_root_one [OF pos2])
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   350
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   351
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
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   352
unfolding sqrt_def by (rule real_root_minus [OF pos2])
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   353
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   354
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
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   355
unfolding sqrt_def by (rule real_root_mult [OF pos2])
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   356
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   357
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
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   358
unfolding sqrt_def by (rule real_root_inverse [OF pos2])
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   359
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   360
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
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   361
unfolding sqrt_def by (rule real_root_divide [OF pos2])
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   362
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   363
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
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   364
unfolding sqrt_def by (rule real_root_power [OF pos2])
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   365
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   366
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
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   367
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
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   368
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   369
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
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   370
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])
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   371
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   372
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
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   373
unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
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   374
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   375
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
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   376
unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
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   377
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   378
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
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   379
unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
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   380
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   381
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
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   382
unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
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   383
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   384
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
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   385
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
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   386
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   387
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]
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   388
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]
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   389
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]
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   390
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]
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   391
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]
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   392
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   393
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]
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   394
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]
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   395
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]
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   396
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]
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   397
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]
huffman@20687
   398
huffman@23042
   399
lemma isCont_real_sqrt: "isCont sqrt x"
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   400
unfolding sqrt_def by (rule isCont_real_root [OF pos2])
huffman@23042
   401
huffman@23042
   402
lemma DERIV_real_sqrt:
huffman@23042
   403
  "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
huffman@23042
   404
unfolding sqrt_def by (rule DERIV_real_root [OF pos2, simplified])
huffman@23042
   405
huffman@20687
   406
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
huffman@20687
   407
apply auto
huffman@20687
   408
apply (cut_tac x = x and y = 0 in linorder_less_linear)
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   409
apply (simp add: zero_less_mult_iff)
huffman@20687
   410
done
huffman@20687
   411
huffman@20687
   412
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
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   413
apply (subst power2_eq_square [symmetric])
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   414
apply (rule real_sqrt_abs)
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   415
done
huffman@20687
   416
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   417
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
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   418
by simp (* TODO: delete *)
huffman@20687
   419
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   420
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
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   421
by simp (* TODO: delete *)
huffman@20687
   422
huffman@20687
   423
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
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   424
by (simp add: power_inverse [symmetric])
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   425
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   426
lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
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   427
by simp
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   428
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   429
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
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   430
by simp
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   431
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   432
lemma sqrt_divide_self_eq:
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   433
  assumes nneg: "0 \<le> x"
huffman@22443
   434
  shows "sqrt x / x = inverse (sqrt x)"
huffman@22443
   435
proof cases
huffman@22443
   436
  assume "x=0" thus ?thesis by simp
huffman@22443
   437
next
huffman@22443
   438
  assume nz: "x\<noteq>0" 
huffman@22443
   439
  hence pos: "0<x" using nneg by arith
huffman@22443
   440
  show ?thesis
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   441
  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
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   442
    show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
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   443
    show "inverse (sqrt x) / (sqrt x / x) = 1"
huffman@22443
   444
      by (simp add: divide_inverse mult_assoc [symmetric] 
huffman@22443
   445
                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
huffman@22443
   446
  qed
huffman@22443
   447
qed
huffman@22443
   448
huffman@22721
   449
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
huffman@22721
   450
apply (simp add: divide_inverse)
huffman@22721
   451
apply (case_tac "r=0")
huffman@22721
   452
apply (auto simp add: mult_ac)
huffman@22721
   453
done
huffman@22721
   454
huffman@22856
   455
subsection {* Square Root of Sum of Squares *}
huffman@22856
   456
huffman@22856
   457
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
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   458
by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero])
huffman@22856
   459
huffman@22856
   460
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@22961
   461
by simp
huffman@22856
   462
huffman@22856
   463
lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
huffman@22856
   464
     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
huffman@22856
   465
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
huffman@22856
   466
huffman@22856
   467
lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
huffman@22856
   468
     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
huffman@22956
   469
by (auto simp add: zero_le_mult_iff)
huffman@22856
   470
huffman@22856
   471
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
huffman@22856
   472
by (rule power2_le_imp_le, simp_all)
huffman@22856
   473
huffman@22856
   474
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
huffman@22856
   475
by (rule power2_le_imp_le, simp_all)
huffman@22856
   476
huffman@22858
   477
lemma power2_sum:
huffman@22858
   478
  fixes x y :: "'a::{number_ring,recpower}"
huffman@22858
   479
  shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
huffman@22858
   480
by (simp add: left_distrib right_distrib power2_eq_square)
huffman@22858
   481
huffman@22858
   482
lemma power2_diff:
huffman@22858
   483
  fixes x y :: "'a::{number_ring,recpower}"
huffman@22858
   484
  shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
huffman@22858
   485
by (simp add: left_diff_distrib right_diff_distrib power2_eq_square)
huffman@22858
   486
huffman@22858
   487
lemma real_sqrt_sum_squares_triangle_ineq:
huffman@22858
   488
  "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)"
huffman@22858
   489
apply (rule power2_le_imp_le, simp)
huffman@22858
   490
apply (simp add: power2_sum)
huffman@22858
   491
apply (simp only: mult_assoc right_distrib [symmetric])
huffman@22858
   492
apply (rule mult_left_mono)
huffman@22858
   493
apply (rule power2_le_imp_le)
huffman@22858
   494
apply (simp add: power2_sum power_mult_distrib)
huffman@22858
   495
apply (simp add: ring_distrib)
huffman@22858
   496
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp)
huffman@22858
   497
apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans)
huffman@22858
   498
apply (rule zero_le_power2)
huffman@22858
   499
apply (simp add: power2_diff power_mult_distrib)
huffman@22858
   500
apply (simp add: mult_nonneg_nonneg)
huffman@22858
   501
apply simp
huffman@22858
   502
apply (simp add: add_increasing)
huffman@22858
   503
done
huffman@22858
   504
huffman@22956
   505
text "Legacy theorem names:"
huffman@22956
   506
lemmas real_root_pos2 = real_root_power_cancel
huffman@22956
   507
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
huffman@22956
   508
lemmas real_root_pos_pos_le = real_root_ge_zero
huffman@22956
   509
lemmas real_sqrt_mult_distrib = real_sqrt_mult
huffman@22956
   510
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
huffman@22956
   511
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
huffman@22956
   512
huffman@22956
   513
(* needed for CauchysMeanTheorem.het_base from AFP *)
huffman@22956
   514
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"
huffman@22956
   515
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])
huffman@22956
   516
huffman@22956
   517
(* FIXME: the stronger version of real_root_less_iff
huffman@22956
   518
 breaks CauchysMeanTheorem.list_gmean_gt_iff from AFP. *)
huffman@22956
   519
huffman@22956
   520
declare real_root_less_iff [simp del]
huffman@22956
   521
lemma real_root_less_iff_nonneg [simp]:
huffman@22956
   522
  "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (root n x < root n y) = (x < y)"
huffman@22956
   523
by (rule real_root_less_iff)
huffman@22956
   524
paulson@14324
   525
end