src/HOL/NthRoot.thy
author hoelzl
Fri, 22 Mar 2013 10:41:43 +0100
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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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(*  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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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:
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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_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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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 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_pos_mult_exp:
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  "\<lbrakk>0 < m; 0 < n; 0 < x\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)"
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by (rule real_root_pos_unique, simp_all add: power_mult)
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lemma real_root_mult_exp:
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  "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)"
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apply (rule linorder_cases [where x=x and y=0])
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apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))")
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apply (simp add: real_root_minus)
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apply (simp_all add: real_root_pos_mult_exp)
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done
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lemma real_root_commute:
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  "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root m (root n x) = root n (root m x)"
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by (simp add: real_root_mult_exp [symmetric] mult_commute)
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text {* Monotonicity in first argument *}
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lemma real_root_strict_decreasing:
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  "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x"
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apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
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apply (simp add: real_root_commute power_strict_increasing
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            del: real_root_pow_pos2)
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done
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lemma real_root_strict_increasing:
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  "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x"
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apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
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apply (simp add: real_root_commute power_strict_decreasing
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            del: real_root_pow_pos2)
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done
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lemma real_root_decreasing:
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  "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x"
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by (auto simp add: order_le_less real_root_strict_decreasing)
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lemma real_root_increasing:
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  "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
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by (auto simp add: order_le_less real_root_strict_increasing)
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text {* 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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   267
next
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
   268
  assume "x \<le> 0" and "y \<le> 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
   269
  hence "0 \<le> - x" and "0 \<le> - y" by simp_all
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
   270
  hence "root n (- x * - y) = root n (- x) * root n (- 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
   271
    by (rule real_root_mult_lemma [OF n])
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
   272
  thus ?thesis by (simp add: real_root_minus [OF n])
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
   273
qed
22721
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   274
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   275
lemma 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
   276
  assumes n: "0 < n"
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
   277
  shows "root n (inverse x) = inverse (root 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
   278
proof (rule linorder_le_cases)
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
   279
  assume "0 \<le> 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
   280
  thus ?thesis by (rule real_root_inverse_lemma [OF n])
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
   281
next
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
   282
  assume "x \<le> 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
   283
  hence "0 \<le> - x" by simp
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
   284
  hence "root n (inverse (- x)) = inverse (root 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
   285
    by (rule real_root_inverse_lemma [OF n])
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
   286
  thus ?thesis by (simp add: real_root_minus [OF n])
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
   287
qed
22721
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   288
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
   289
lemma real_root_divide:
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
   290
  "0 < n \<Longrightarrow> root n (x / y) = root n x / root n 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
   291
by (simp add: divide_inverse real_root_mult real_root_inverse)
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
   292
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
   293
lemma real_root_power:
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
   294
  "0 < n \<Longrightarrow> root n (x ^ k) = root n 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
   295
by (induct k, simp_all add: real_root_mult)
22721
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   296
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   297
lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   298
by (simp add: abs_if real_root_minus)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   299
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   300
text {* Continuity and derivatives *}
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   301
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   302
lemma isCont_root_pos:
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   303
  assumes n: "0 < n"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   304
  assumes x: "0 < x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   305
  shows "isCont (root n) x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   306
proof -
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   307
  have "isCont (root n) (root n x ^ n)"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   308
  proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"])
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   309
    show "0 < root n x" using n x by simp
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   310
    show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   311
      by (simp add: abs_le_iff real_root_power_cancel n)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   312
    show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z"
44289
d81d09cdab9c optimize some proofs
huffman
parents: 35216
diff changeset
   313
      by simp
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   314
  qed
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   315
  thus ?thesis using n x by simp
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   316
qed
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   317
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   318
lemma isCont_root_neg:
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   319
  "\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   320
apply (subgoal_tac "isCont (\<lambda>x. - root n (- x)) x")
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   321
apply (simp add: real_root_minus)
23069
cdfff0241c12 rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents: 23049
diff changeset
   322
apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]])
44289
d81d09cdab9c optimize some proofs
huffman
parents: 35216
diff changeset
   323
apply (simp add: isCont_root_pos)
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   324
done
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   325
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   326
lemma isCont_root_zero:
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   327
  "0 < n \<Longrightarrow> isCont (root n) 0"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   328
unfolding isCont_def
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   329
apply (rule LIM_I)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   330
apply (rule_tac x="r ^ n" in exI, safe)
25875
536dfdc25e0a added simp attributes/ proofs fixed
nipkow
parents: 25766
diff changeset
   331
apply (simp)
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   332
apply (simp add: real_root_abs [symmetric])
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   333
apply (rule_tac n="n" in power_less_imp_less_base, simp_all)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   334
done
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   335
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   336
lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   337
apply (rule_tac x=x and y=0 in linorder_cases)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   338
apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   339
done
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   340
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
   341
lemma tendsto_real_root[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
   342
  "(f ---> x) F \<Longrightarrow> 0 < n \<Longrightarrow> ((\<lambda>x. root n (f x)) ---> root n x) F"
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
   343
  using isCont_tendsto_compose[OF isCont_real_root, of n f x F] .
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
   344
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
   345
lemma continuous_real_root[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
   346
  "continuous F f \<Longrightarrow> 0 < n \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
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
   347
  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)
hoelzl
parents: 49962
diff changeset
   348
  
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
   349
lemma continuous_on_real_root[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
   350
  "continuous_on s f \<Longrightarrow> 0 < n \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
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
   351
  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
parents: 49962
diff changeset
   352
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   353
lemma DERIV_real_root:
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   354
  assumes n: "0 < n"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   355
  assumes x: "0 < x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   356
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   357
proof (rule DERIV_inverse_function)
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23042
diff changeset
   358
  show "0 < x" using x .
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23042
diff changeset
   359
  show "x < x + 1" by simp
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23042
diff changeset
   360
  show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   361
    using n by simp
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   362
  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
   363
    by (rule DERIV_pow)
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   364
  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
   365
    using n x by simp
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   366
  show "isCont (root n) x"
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23257
diff changeset
   367
    using n by (rule isCont_real_root)
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   368
qed
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   369
23046
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   370
lemma DERIV_odd_real_root:
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   371
  assumes n: "odd n"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   372
  assumes x: "x \<noteq> 0"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   373
  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
   374
proof (rule DERIV_inverse_function)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   375
  show "x - 1 < x" by simp
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   376
  show "x < x + 1" by simp
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   377
  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
   378
    using n by (simp add: odd_real_root_pow)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   379
  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
   380
    by (rule DERIV_pow)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   381
  show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   382
    using odd_pos [OF n] x by simp
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   383
  show "isCont (root n) x"
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   384
    using odd_pos [OF n] by (rule isCont_real_root)
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   385
qed
12f35ece221f add odd_real_root lemmas
huffman
parents: 23044
diff changeset
   386
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   387
lemma DERIV_even_real_root:
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   388
  assumes n: "0 < n" and "even n"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   389
  assumes x: "x < 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   390
  shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   391
proof (rule DERIV_inverse_function)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   392
  show "x - 1 < x" by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   393
  show "x < 0" using x .
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   394
next
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   395
  show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   396
  proof (rule allI, rule impI, erule conjE)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   397
    fix y assume "x - 1 < y" and "y < 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   398
    hence "root n (-y) ^ n = -y" using `0 < n` by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   399
    with real_root_minus[OF `0 < n`] and `even n`
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   400
    show "- (root n y ^ n) = y" by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   401
  qed
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   402
next
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   403
  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
   404
    by  (auto intro!: DERIV_intros)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   405
  show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   406
    using n x by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   407
  show "isCont (root n) x"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   408
    using n by (rule isCont_real_root)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   409
qed
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   410
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   411
lemma DERIV_real_root_generic:
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   412
  assumes "0 < n" and "x \<noteq> 0"
49753
a344f1a21211 eliminated spurious fact duplicates;
wenzelm
parents: 44349
diff changeset
   413
    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
   414
    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
   415
    and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   416
  shows "DERIV (root n) x :> D"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   417
using assms by (cases "even n", cases "0 < x",
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   418
  auto intro: DERIV_real_root[THEN DERIV_cong]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   419
              DERIV_odd_real_root[THEN DERIV_cong]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   420
              DERIV_even_real_root[THEN DERIV_cong])
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   421
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
   422
subsection {* Square Root *}
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   423
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
   424
definition
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
   425
  sqrt :: "real \<Rightarrow> real" where
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
   426
  "sqrt = root 2"
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   427
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
   428
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
parents: 22943
diff changeset
   429
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
   430
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt 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
   431
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
   432
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
   433
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>"
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
   434
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
   435
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
   436
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
   437
done
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   438
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
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = 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
   440
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   441
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
   442
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   443
apply (rule iffI)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   444
apply (erule subst)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   445
apply (rule zero_le_power2)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   446
apply (erule real_sqrt_pow2)
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   447
done
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   448
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
   449
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
   450
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
   451
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
   452
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
   453
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
   454
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
   455
lemma real_sqrt_minus: "sqrt (- x) = - 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
   456
unfolding sqrt_def by (rule real_root_minus [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
   457
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
   458
lemma real_sqrt_mult: "sqrt (x * y) = 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
   459
unfolding sqrt_def by (rule real_root_mult [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
   460
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
   461
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (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
   462
unfolding sqrt_def by (rule real_root_inverse [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
   463
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
   464
lemma real_sqrt_divide: "sqrt (x / y) = 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
   465
unfolding sqrt_def by (rule real_root_divide [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
   466
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
   467
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
   468
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
   469
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
   470
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
   471
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
   472
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
   473
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> 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
   474
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   475
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
   476
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
   477
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
   478
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
   479
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
   480
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
   481
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
   482
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
   483
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
   484
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
   485
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
   486
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
   487
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
   488
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
   489
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
   490
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
   491
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]
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
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]
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
   493
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]
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
   494
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]
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
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]
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
   496
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
   497
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]
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
   498
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]
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
   499
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]
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
   500
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]
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
   501
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   502
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   503
lemma isCont_real_sqrt: "isCont sqrt x"
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   504
unfolding sqrt_def by (rule isCont_real_root [OF pos2])
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   505
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
   506
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
   507
  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) ---> sqrt x) F"
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
   508
  unfolding sqrt_def by (rule tendsto_real_root [OF _ pos2])
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
   509
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
   510
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
   511
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
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
   512
  unfolding sqrt_def by (rule continuous_real_root [OF _ pos2])
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
   513
  
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
   514
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
   515
  "continuous_on s f \<Longrightarrow> 0 < n \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
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
   516
  unfolding sqrt_def by (rule continuous_on_real_root [OF _ pos2])
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
   517
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   518
lemma DERIV_real_sqrt_generic:
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   519
  assumes "x \<noteq> 0"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   520
  assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   521
  assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   522
  shows "DERIV sqrt x :> D"
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   523
  using assms unfolding sqrt_def
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   524
  by (auto intro!: DERIV_real_root_generic)
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   525
23042
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   526
lemma DERIV_real_sqrt:
492514b39956 add lemmas about continuity and derivatives of roots
huffman
parents: 23009
diff changeset
   527
  "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   528
  using DERIV_real_sqrt_generic by simp
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   529
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   530
declare
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   531
  DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31014
diff changeset
   532
  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
   533
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   534
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
   535
apply auto
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   536
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
   537
apply (simp add: zero_less_mult_iff)
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   538
done
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   539
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   540
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   541
apply (subst power2_eq_square [symmetric])
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   542
apply (rule real_sqrt_abs)
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   543
done
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   544
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   545
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   546
by (simp add: power_inverse [symmetric])
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   547
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   548
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
   549
by simp
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   550
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   551
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
   552
by simp
20687
fedb901be392 move root and sqrt stuff from Transcendental to NthRoot
huffman
parents: 20515
diff changeset
   553
22443
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   554
lemma sqrt_divide_self_eq:
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   555
  assumes nneg: "0 \<le> x"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   556
  shows "sqrt x / x = inverse (sqrt x)"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   557
proof cases
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   558
  assume "x=0" thus ?thesis by simp
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   559
next
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   560
  assume nz: "x\<noteq>0" 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   561
  hence pos: "0<x" using nneg by arith
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   562
  show ?thesis
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   563
  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   564
    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
   565
    show "inverse (sqrt x) / (sqrt x / x) = 1"
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   566
      by (simp add: divide_inverse mult_assoc [symmetric] 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   567
                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   568
  qed
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   569
qed
346729a55460 move sqrt_divide_self_eq to NthRoot.thy
huffman
parents: 21865
diff changeset
   570
22721
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   571
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
   572
apply (simp add: divide_inverse)
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   573
apply (case_tac "r=0")
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   574
apply (auto simp add: mult_ac)
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   575
done
d9be18bd7a28 moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 22630
diff changeset
   576
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   577
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
   578
by (simp add: divide_less_eq)
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   579
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   580
lemma four_x_squared: 
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   581
  fixes x::real
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   582
  shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   583
by (simp add: power2_eq_square)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   584
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   585
subsection {* Square Root of Sum of Squares *}
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   586
44320
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   587
lemma real_sqrt_sum_squares_ge_zero: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   588
  by simp (* TODO: delete *)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   589
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   590
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
   591
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   592
lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   593
     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
44320
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   594
  by (simp add: zero_le_mult_iff)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   595
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   596
lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   597
     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
44320
33439faadd67 remove some redundant simp rules about sqrt
huffman
parents: 44289
diff changeset
   598
  by (simp add: zero_le_mult_iff)
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   599
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   600
lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<twosuperior> + y\<twosuperior>) = x \<Longrightarrow> y = 0"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   601
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   602
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   603
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<twosuperior> + y\<twosuperior>) = y \<Longrightarrow> x = 0"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   604
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   605
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   606
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   607
by (rule power2_le_imp_le, simp_all)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   608
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   609
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   610
by (rule power2_le_imp_le, simp_all)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   611
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   612
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
22856
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   613
by (rule power2_le_imp_le, simp_all)
eb0e0582092a cleaned up
huffman
parents: 22721
diff changeset
   614
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   615
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   616
by (rule power2_le_imp_le, simp_all)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   617
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   618
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
   619
by (simp add: power2_eq_square [symmetric])
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   620
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   621
lemma real_sqrt_sum_squares_triangle_ineq:
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   622
  "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)"
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   623
apply (rule power2_le_imp_le, simp)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   624
apply (simp add: power2_sum)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 49753
diff changeset
   625
apply (simp only: mult_assoc distrib_left [symmetric])
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   626
apply (rule mult_left_mono)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   627
apply (rule power2_le_imp_le)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   628
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
   629
apply (simp add: ring_distribs)
22858
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   630
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   631
apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   632
apply (rule zero_le_power2)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   633
apply (simp add: power2_diff power_mult_distrib)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   634
apply (simp add: mult_nonneg_nonneg)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   635
apply simp
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   636
apply (simp add: add_increasing)
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   637
done
5ca5d1cce388 add lemma real_sqrt_sum_squares_triangle_ineq
huffman
parents: 22856
diff changeset
   638
23122
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   639
lemma real_sqrt_sum_squares_less:
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   640
  "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   641
apply (rule power2_less_imp_less, simp)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   642
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   643
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   644
apply (simp add: power_divide)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   645
apply (drule order_le_less_trans [OF abs_ge_zero])
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   646
apply (simp add: zero_less_divide_iff)
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   647
done
3d853d6f2f7d add lemma real_sqrt_sum_squares_less
huffman
parents: 23069
diff changeset
   648
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   649
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
   650
    complex numbers*}
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   651
lemma lemma_sqrt_hcomplex_capprox:
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   652
     "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   653
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
   654
apply (erule_tac [2] lemma_real_divide_sqrt_less)
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   655
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
   656
apply (auto simp add: zero_le_divide_iff power_divide)
23049
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   657
apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   658
apply (rule add_mono)
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 28952
diff changeset
   659
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
   660
done
11607c283074 moved sqrt lemmas from Transcendental.thy to NthRoot.thy
huffman
parents: 23047
diff changeset
   661
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
   662
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
   663
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
   664
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
   665
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
   666
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
   667
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
   668
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
   669
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
   670
(* 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
   671
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
   672
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
   673
14324
c9c6832f9b22 converting Hyperreal/NthRoot to Isar
paulson
parents: 14268
diff changeset
   674
end