paulson@12196: (*  Title       : NthRoot.thy
paulson@12196:     Author      : Jacques D. Fleuriot
paulson@12196:     Copyright   : 1998  University of Cambridge
paulson@14477:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
paulson@12196: *)
paulson@12196: 
huffman@22956: header {* Nth Roots of Real Numbers *}
paulson@14324: 
nipkow@15131: theory NthRoot
haftmann@28952: imports Parity Deriv
nipkow@15131: begin
paulson@14324: 
huffman@22956: subsection {* Existence of Nth Root *}
huffman@20687: 
huffman@23009: text {* Existence follows from the Intermediate Value Theorem *}
paulson@14324: 
huffman@23009: lemma realpow_pos_nth:
huffman@23009:   assumes n: "0 < n"
huffman@23009:   assumes a: "0 < a"
huffman@23009:   shows "\<exists>r>0. r ^ n = (a::real)"
huffman@23009: proof -
huffman@23009:   have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a"
huffman@23009:   proof (rule IVT)
huffman@23009:     show "0 ^ n \<le> a" using n a by (simp add: power_0_left)
huffman@23009:     show "0 \<le> max 1 a" by simp
huffman@23009:     from n have n1: "1 \<le> n" by simp
huffman@23009:     have "a \<le> max 1 a ^ 1" by simp
huffman@23009:     also have "max 1 a ^ 1 \<le> max 1 a ^ n"
huffman@23009:       using n1 by (rule power_increasing, simp)
huffman@23009:     finally show "a \<le> max 1 a ^ n" .
huffman@23009:     show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r"
huffman@23069:       by (simp add: isCont_power)
huffman@23009:   qed
huffman@23009:   then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast
huffman@23009:   with n a have "r \<noteq> 0" by (auto simp add: power_0_left)
huffman@23009:   with r have "0 < r \<and> r ^ n = a" by simp
huffman@23009:   thus ?thesis ..
huffman@23009: qed
paulson@14325: 
huffman@23047: (* Used by Integration/RealRandVar.thy in AFP *)
huffman@23047: lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a"
huffman@23047: by (blast intro: realpow_pos_nth)
huffman@23047: 
huffman@23009: text {* Uniqueness of nth positive root *}
paulson@14324: 
paulson@14324: lemma realpow_pos_nth_unique:
huffman@23009:   "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)"
paulson@14324: apply (auto intro!: realpow_pos_nth)
huffman@23009: apply (rule_tac n=n in power_eq_imp_eq_base, simp_all)
paulson@14324: done
paulson@14324: 
huffman@20687: subsection {* Nth Root *}
huffman@20687: 
huffman@22956: text {* We define roots of negative reals such that
huffman@22956:   @{term "root n (- x) = - root n x"}. This allows
huffman@22956:   us to omit side conditions from many theorems. *}
huffman@20687: 
huffman@22956: definition
huffman@22956:   root :: "[nat, real] \<Rightarrow> real" where
huffman@22956:   "root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else
huffman@22956:                if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)"
huffman@20687: 
huffman@22956: lemma real_root_zero [simp]: "root n 0 = 0"
huffman@22956: unfolding root_def by simp
huffman@22956: 
huffman@22956: lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x"
huffman@22956: unfolding root_def by simp
huffman@22956: 
huffman@22956: lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x"
huffman@20687: apply (simp add: root_def)
huffman@22956: apply (drule (1) realpow_pos_nth_unique)
huffman@20687: apply (erule theI' [THEN conjunct1])
huffman@20687: done
huffman@20687: 
huffman@22956: lemma real_root_pow_pos: (* TODO: rename *)
huffman@22956:   "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
huffman@22956: apply (simp add: root_def)
huffman@22956: apply (drule (1) realpow_pos_nth_unique)
huffman@22956: apply (erule theI' [THEN conjunct2])
huffman@22956: done
huffman@20687: 
huffman@22956: lemma real_root_pow_pos2 [simp]: (* TODO: rename *)
huffman@22956:   "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
huffman@22956: by (auto simp add: order_le_less real_root_pow_pos)
huffman@22956: 
huffman@23046: lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"
huffman@23046: apply (rule_tac x=0 and y=x in linorder_le_cases)
huffman@23046: apply (erule (1) real_root_pow_pos2 [OF odd_pos])
huffman@23046: apply (subgoal_tac "root n (- x) ^ n = - x")
huffman@23046: apply (simp add: real_root_minus odd_pos)
huffman@23046: apply (simp add: odd_pos)
huffman@23046: done
huffman@23046: 
huffman@22956: lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x"
huffman@20687: by (auto simp add: order_le_less real_root_gt_zero)
huffman@20687: 
huffman@22956: lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x"
huffman@22956: apply (subgoal_tac "0 \<le> x ^ n")
huffman@22956: apply (subgoal_tac "0 \<le> root n (x ^ n)")
huffman@22956: apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n")
huffman@22956: apply (erule (3) power_eq_imp_eq_base)
huffman@22956: apply (erule (1) real_root_pow_pos2)
huffman@22956: apply (erule (1) real_root_ge_zero)
huffman@22956: apply (erule zero_le_power)
huffman@20687: done
huffman@20687: 
huffman@23046: lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x"
huffman@23046: apply (rule_tac x=0 and y=x in linorder_le_cases)
huffman@23046: apply (erule (1) real_root_power_cancel [OF odd_pos])
huffman@23046: apply (subgoal_tac "root n ((- x) ^ n) = - x")
huffman@23046: apply (simp add: real_root_minus odd_pos)
huffman@23046: apply (erule real_root_power_cancel [OF odd_pos], simp)
huffman@23046: done
huffman@23046: 
huffman@22956: lemma real_root_pos_unique:
huffman@22956:   "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
huffman@22956: by (erule subst, rule real_root_power_cancel)
huffman@22956: 
huffman@23046: lemma odd_real_root_unique:
huffman@23046:   "\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
huffman@23046: by (erule subst, rule odd_real_root_power_cancel)
huffman@23046: 
huffman@22956: lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1"
huffman@22956: by (simp add: real_root_pos_unique)
huffman@22956: 
huffman@22956: text {* Root function is strictly monotonic, hence injective *}
huffman@22956: 
huffman@22956: lemma real_root_less_mono_lemma:
huffman@22956:   "\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
huffman@22856: apply (subgoal_tac "0 \<le> y")
huffman@22956: apply (subgoal_tac "root n x ^ n < root n y ^ n")
huffman@22956: apply (erule power_less_imp_less_base)
huffman@22956: apply (erule (1) real_root_ge_zero)
huffman@22956: apply simp
huffman@22956: apply simp
huffman@22721: done
huffman@22721: 
huffman@22956: lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
huffman@22956: apply (cases "0 \<le> x")
huffman@22956: apply (erule (2) real_root_less_mono_lemma)
huffman@22956: apply (cases "0 \<le> y")
huffman@22956: apply (rule_tac y=0 in order_less_le_trans)
huffman@22956: apply (subgoal_tac "0 < root n (- x)")
huffman@22956: apply (simp add: real_root_minus)
huffman@22956: apply (simp add: real_root_gt_zero)
huffman@22956: apply (simp add: real_root_ge_zero)
huffman@22956: apply (subgoal_tac "root n (- y) < root n (- x)")
huffman@22956: apply (simp add: real_root_minus)
huffman@22956: apply (simp add: real_root_less_mono_lemma)
huffman@22721: done
huffman@22721: 
huffman@22956: lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y"
huffman@22956: by (auto simp add: order_le_less real_root_less_mono)
huffman@22956: 
huffman@22721: lemma real_root_less_iff [simp]:
huffman@22956:   "0 < n \<Longrightarrow> (root n x < root n y) = (x < y)"
huffman@22956: apply (cases "x < y")
huffman@22956: apply (simp add: real_root_less_mono)
huffman@22956: apply (simp add: linorder_not_less real_root_le_mono)
huffman@22721: done
huffman@22721: 
huffman@22721: lemma real_root_le_iff [simp]:
huffman@22956:   "0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)"
huffman@22956: apply (cases "x \<le> y")
huffman@22956: apply (simp add: real_root_le_mono)
huffman@22956: apply (simp add: linorder_not_le real_root_less_mono)
huffman@22721: done
huffman@22721: 
huffman@22721: lemma real_root_eq_iff [simp]:
huffman@22956:   "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)"
huffman@22956: by (simp add: order_eq_iff)
huffman@22956: 
huffman@22956: lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]
huffman@22956: lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]
huffman@22956: lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified]
huffman@22956: lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]
huffman@22956: lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]
huffman@22721: 
huffman@23257: lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)"
huffman@23257: by (insert real_root_less_iff [where x=1], simp)
huffman@23257: 
huffman@23257: lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)"
huffman@23257: by (insert real_root_less_iff [where y=1], simp)
huffman@23257: 
huffman@23257: lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)"
huffman@23257: by (insert real_root_le_iff [where x=1], simp)
huffman@23257: 
huffman@23257: lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)"
huffman@23257: by (insert real_root_le_iff [where y=1], simp)
huffman@23257: 
huffman@23257: lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)"
huffman@23257: by (insert real_root_eq_iff [where y=1], simp)
huffman@23257: 
huffman@23257: text {* Roots of roots *}
huffman@23257: 
huffman@23257: lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
huffman@23257: by (simp add: odd_real_root_unique)
huffman@23257: 
huffman@23257: lemma real_root_pos_mult_exp:
huffman@23257:   "\<lbrakk>0 < m; 0 < n; 0 < x\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)"
huffman@23257: by (rule real_root_pos_unique, simp_all add: power_mult)
huffman@23257: 
huffman@23257: lemma real_root_mult_exp:
huffman@23257:   "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root (m * n) x = root m (root n x)"
huffman@23257: apply (rule linorder_cases [where x=x and y=0])
huffman@23257: apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))")
huffman@23257: apply (simp add: real_root_minus)
huffman@23257: apply (simp_all add: real_root_pos_mult_exp)
huffman@23257: done
huffman@23257: 
huffman@23257: lemma real_root_commute:
huffman@23257:   "\<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> root m (root n x) = root n (root m x)"
huffman@23257: by (simp add: real_root_mult_exp [symmetric] mult_commute)
huffman@23257: 
huffman@23257: text {* Monotonicity in first argument *}
huffman@23257: 
huffman@23257: lemma real_root_strict_decreasing:
huffman@23257:   "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x"
huffman@23257: apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
huffman@23257: apply (simp add: real_root_commute power_strict_increasing
huffman@23257:             del: real_root_pow_pos2)
huffman@23257: done
huffman@23257: 
huffman@23257: lemma real_root_strict_increasing:
huffman@23257:   "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x"
huffman@23257: apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
huffman@23257: apply (simp add: real_root_commute power_strict_decreasing
huffman@23257:             del: real_root_pow_pos2)
huffman@23257: done
huffman@23257: 
huffman@23257: lemma real_root_decreasing:
huffman@23257:   "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x"
huffman@23257: by (auto simp add: order_le_less real_root_strict_decreasing)
huffman@23257: 
huffman@23257: lemma real_root_increasing:
huffman@23257:   "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
huffman@23257: by (auto simp add: order_le_less real_root_strict_increasing)
huffman@23257: 
huffman@22956: text {* Roots of multiplication and division *}
huffman@22956: 
huffman@22956: lemma real_root_mult_lemma:
huffman@22956:   "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y"
huffman@22956: by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib)
huffman@22956: 
huffman@22956: lemma real_root_inverse_lemma:
huffman@22956:   "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)"
huffman@22956: by (simp add: real_root_pos_unique power_inverse [symmetric])
huffman@22721: 
huffman@22721: lemma real_root_mult:
huffman@22956:   assumes n: "0 < n"
huffman@22956:   shows "root n (x * y) = root n x * root n y"
huffman@22956: proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases)
huffman@22956:   assume "0 \<le> x" and "0 \<le> y"
huffman@22956:   thus ?thesis by (rule real_root_mult_lemma [OF n])
huffman@22956: next
huffman@22956:   assume "0 \<le> x" and "y \<le> 0"
huffman@22956:   hence "0 \<le> x" and "0 \<le> - y" by simp_all
huffman@22956:   hence "root n (x * - y) = root n x * root n (- y)"
huffman@22956:     by (rule real_root_mult_lemma [OF n])
huffman@22956:   thus ?thesis by (simp add: real_root_minus [OF n])
huffman@22956: next
huffman@22956:   assume "x \<le> 0" and "0 \<le> y"
huffman@22956:   hence "0 \<le> - x" and "0 \<le> y" by simp_all
huffman@22956:   hence "root n (- x * y) = root n (- x) * root n y"
huffman@22956:     by (rule real_root_mult_lemma [OF n])
huffman@22956:   thus ?thesis by (simp add: real_root_minus [OF n])
huffman@22956: next
huffman@22956:   assume "x \<le> 0" and "y \<le> 0"
huffman@22956:   hence "0 \<le> - x" and "0 \<le> - y" by simp_all
huffman@22956:   hence "root n (- x * - y) = root n (- x) * root n (- y)"
huffman@22956:     by (rule real_root_mult_lemma [OF n])
huffman@22956:   thus ?thesis by (simp add: real_root_minus [OF n])
huffman@22956: qed
huffman@22721: 
huffman@22721: lemma real_root_inverse:
huffman@22956:   assumes n: "0 < n"
huffman@22956:   shows "root n (inverse x) = inverse (root n x)"
huffman@22956: proof (rule linorder_le_cases)
huffman@22956:   assume "0 \<le> x"
huffman@22956:   thus ?thesis by (rule real_root_inverse_lemma [OF n])
huffman@22956: next
huffman@22956:   assume "x \<le> 0"
huffman@22956:   hence "0 \<le> - x" by simp
huffman@22956:   hence "root n (inverse (- x)) = inverse (root n (- x))"
huffman@22956:     by (rule real_root_inverse_lemma [OF n])
huffman@22956:   thus ?thesis by (simp add: real_root_minus [OF n])
huffman@22956: qed
huffman@22721: 
huffman@22956: lemma real_root_divide:
huffman@22956:   "0 < n \<Longrightarrow> root n (x / y) = root n x / root n y"
huffman@22956: by (simp add: divide_inverse real_root_mult real_root_inverse)
huffman@22956: 
huffman@22956: lemma real_root_power:
huffman@22956:   "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
huffman@22956: by (induct k, simp_all add: real_root_mult)
huffman@22721: 
huffman@23042: lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>"
huffman@23042: by (simp add: abs_if real_root_minus)
huffman@23042: 
huffman@23042: text {* Continuity and derivatives *}
huffman@23042: 
huffman@23042: lemma isCont_root_pos:
huffman@23042:   assumes n: "0 < n"
huffman@23042:   assumes x: "0 < x"
huffman@23042:   shows "isCont (root n) x"
huffman@23042: proof -
huffman@23042:   have "isCont (root n) (root n x ^ n)"
huffman@23042:   proof (rule isCont_inverse_function [where f="\<lambda>a. a ^ n"])
huffman@23042:     show "0 < root n x" using n x by simp
huffman@23042:     show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> root n (z ^ n) = z"
huffman@23042:       by (simp add: abs_le_iff real_root_power_cancel n)
huffman@23042:     show "\<forall>z. \<bar>z - root n x\<bar> \<le> root n x \<longrightarrow> isCont (\<lambda>a. a ^ n) z"
huffman@23069:       by (simp add: isCont_power)
huffman@23042:   qed
huffman@23042:   thus ?thesis using n x by simp
huffman@23042: qed
huffman@23042: 
huffman@23042: lemma isCont_root_neg:
huffman@23042:   "\<lbrakk>0 < n; x < 0\<rbrakk> \<Longrightarrow> isCont (root n) x"
huffman@23042: apply (subgoal_tac "isCont (\<lambda>x. - root n (- x)) x")
huffman@23042: apply (simp add: real_root_minus)
huffman@23069: apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]])
huffman@23042: apply (simp add: isCont_minus isCont_root_pos)
huffman@23042: done
huffman@23042: 
huffman@23042: lemma isCont_root_zero:
huffman@23042:   "0 < n \<Longrightarrow> isCont (root n) 0"
huffman@23042: unfolding isCont_def
huffman@23042: apply (rule LIM_I)
huffman@23042: apply (rule_tac x="r ^ n" in exI, safe)
nipkow@25875: apply (simp)
huffman@23042: apply (simp add: real_root_abs [symmetric])
huffman@23042: apply (rule_tac n="n" in power_less_imp_less_base, simp_all)
huffman@23042: done
huffman@23042: 
huffman@23042: lemma isCont_real_root: "0 < n \<Longrightarrow> isCont (root n) x"
huffman@23042: apply (rule_tac x=x and y=0 in linorder_cases)
huffman@23042: apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero)
huffman@23042: done
huffman@23042: 
huffman@23042: lemma DERIV_real_root:
huffman@23042:   assumes n: "0 < n"
huffman@23042:   assumes x: "0 < x"
huffman@23042:   shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
huffman@23042: proof (rule DERIV_inverse_function)
huffman@23044:   show "0 < x" using x .
huffman@23044:   show "x < x + 1" by simp
huffman@23044:   show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
huffman@23042:     using n by simp
huffman@23042:   show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
huffman@23042:     by (rule DERIV_pow)
huffman@23042:   show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
huffman@23042:     using n x by simp
huffman@23042:   show "isCont (root n) x"
huffman@23441:     using n by (rule isCont_real_root)
huffman@23042: qed
huffman@23042: 
huffman@23046: lemma DERIV_odd_real_root:
huffman@23046:   assumes n: "odd n"
huffman@23046:   assumes x: "x \<noteq> 0"
huffman@23046:   shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
huffman@23046: proof (rule DERIV_inverse_function)
huffman@23046:   show "x - 1 < x" by simp
huffman@23046:   show "x < x + 1" by simp
huffman@23046:   show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
huffman@23046:     using n by (simp add: odd_real_root_pow)
huffman@23046:   show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
huffman@23046:     by (rule DERIV_pow)
huffman@23046:   show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
huffman@23046:     using odd_pos [OF n] x by simp
huffman@23046:   show "isCont (root n) x"
huffman@23046:     using odd_pos [OF n] by (rule isCont_real_root)
huffman@23046: qed
huffman@23046: 
hoelzl@31880: lemma DERIV_even_real_root:
hoelzl@31880:   assumes n: "0 < n" and "even n"
hoelzl@31880:   assumes x: "x < 0"
hoelzl@31880:   shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
hoelzl@31880: proof (rule DERIV_inverse_function)
hoelzl@31880:   show "x - 1 < x" by simp
hoelzl@31880:   show "x < 0" using x .
hoelzl@31880: next
hoelzl@31880:   show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
hoelzl@31880:   proof (rule allI, rule impI, erule conjE)
hoelzl@31880:     fix y assume "x - 1 < y" and "y < 0"
hoelzl@31880:     hence "root n (-y) ^ n = -y" using `0 < n` by simp
hoelzl@31880:     with real_root_minus[OF `0 < n`] and `even n`
hoelzl@31880:     show "- (root n y ^ n) = y" by simp
hoelzl@31880:   qed
hoelzl@31880: next
hoelzl@31880:   show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
hoelzl@31880:     by  (auto intro!: DERIV_intros)
hoelzl@31880:   show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
hoelzl@31880:     using n x by simp
hoelzl@31880:   show "isCont (root n) x"
hoelzl@31880:     using n by (rule isCont_real_root)
hoelzl@31880: qed
hoelzl@31880: 
hoelzl@31880: lemma DERIV_real_root_generic:
hoelzl@31880:   assumes "0 < n" and "x \<noteq> 0"
hoelzl@31880:   and even: "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
hoelzl@31880:   and even: "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
hoelzl@31880:   and odd: "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
hoelzl@31880:   shows "DERIV (root n) x :> D"
hoelzl@31880: using assms by (cases "even n", cases "0 < x",
hoelzl@31880:   auto intro: DERIV_real_root[THEN DERIV_cong]
hoelzl@31880:               DERIV_odd_real_root[THEN DERIV_cong]
hoelzl@31880:               DERIV_even_real_root[THEN DERIV_cong])
hoelzl@31880: 
huffman@22956: subsection {* Square Root *}
huffman@20687: 
huffman@22956: definition
huffman@22956:   sqrt :: "real \<Rightarrow> real" where
huffman@22956:   "sqrt = root 2"
huffman@20687: 
huffman@22956: lemma pos2: "0 < (2::nat)" by simp
huffman@22956: 
huffman@22956: lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
huffman@22956: unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
huffman@20687: 
huffman@22956: lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>"
huffman@22956: apply (rule real_sqrt_unique)
huffman@22956: apply (rule power2_abs)
huffman@22956: apply (rule abs_ge_zero)
huffman@22956: done
huffman@20687: 
huffman@22956: lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x"
huffman@22956: unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
huffman@22856: 
huffman@22956: lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
huffman@22856: apply (rule iffI)
huffman@22856: apply (erule subst)
huffman@22856: apply (rule zero_le_power2)
huffman@22856: apply (erule real_sqrt_pow2)
huffman@20687: done
huffman@20687: 
huffman@22956: lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
huffman@22956: unfolding sqrt_def by (rule real_root_zero)
huffman@22956: 
huffman@22956: lemma real_sqrt_one [simp]: "sqrt 1 = 1"
huffman@22956: unfolding sqrt_def by (rule real_root_one [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
huffman@22956: unfolding sqrt_def by (rule real_root_minus [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
huffman@22956: unfolding sqrt_def by (rule real_root_mult [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
huffman@22956: unfolding sqrt_def by (rule real_root_inverse [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
huffman@22956: unfolding sqrt_def by (rule real_root_divide [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
huffman@22956: unfolding sqrt_def by (rule real_root_power [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
huffman@22956: unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
huffman@22956: unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])
huffman@20687: 
huffman@22956: lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
huffman@22956: unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
huffman@22956: unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
huffman@22956: unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
huffman@22956: unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
huffman@22956: 
huffman@22956: lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
huffman@22956: unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
huffman@22956: 
huffman@22956: lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]
huffman@22956: lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]
huffman@22956: lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]
huffman@22956: lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]
huffman@22956: lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]
huffman@22956: 
huffman@22956: lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]
huffman@22956: lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]
huffman@22956: lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]
huffman@22956: lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]
huffman@22956: lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]
huffman@20687: 
huffman@23042: lemma isCont_real_sqrt: "isCont sqrt x"
huffman@23042: unfolding sqrt_def by (rule isCont_real_root [OF pos2])
huffman@23042: 
hoelzl@31880: lemma DERIV_real_sqrt_generic:
hoelzl@31880:   assumes "x \<noteq> 0"
hoelzl@31880:   assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
hoelzl@31880:   assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
hoelzl@31880:   shows "DERIV sqrt x :> D"
hoelzl@31880:   using assms unfolding sqrt_def
hoelzl@31880:   by (auto intro!: DERIV_real_root_generic)
hoelzl@31880: 
huffman@23042: lemma DERIV_real_sqrt:
huffman@23042:   "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
hoelzl@31880:   using DERIV_real_sqrt_generic by simp
hoelzl@31880: 
hoelzl@31880: declare
hoelzl@31880:   DERIV_real_sqrt_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880:   DERIV_real_root_generic[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
huffman@23042: 
huffman@20687: lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
huffman@20687: apply auto
huffman@20687: apply (cut_tac x = x and y = 0 in linorder_less_linear)
huffman@20687: apply (simp add: zero_less_mult_iff)
huffman@20687: done
huffman@20687: 
huffman@20687: lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
huffman@22856: apply (subst power2_eq_square [symmetric])
huffman@20687: apply (rule real_sqrt_abs)
huffman@20687: done
huffman@20687: 
huffman@20687: lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
huffman@22956: by simp (* TODO: delete *)
huffman@20687: 
huffman@20687: lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
huffman@22956: by simp (* TODO: delete *)
huffman@20687: 
huffman@20687: lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
huffman@22856: by (simp add: power_inverse [symmetric])
huffman@20687: 
huffman@20687: lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
huffman@22956: by simp
huffman@20687: 
huffman@20687: lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
huffman@22956: by simp
huffman@20687: 
huffman@23049: lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
huffman@23049: by simp
huffman@23049: 
huffman@23049: lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
huffman@23049: by simp
huffman@23049: 
huffman@23049: lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
huffman@23049: by simp
huffman@23049: 
huffman@22443: lemma sqrt_divide_self_eq:
huffman@22443:   assumes nneg: "0 \<le> x"
huffman@22443:   shows "sqrt x / x = inverse (sqrt x)"
huffman@22443: proof cases
huffman@22443:   assume "x=0" thus ?thesis by simp
huffman@22443: next
huffman@22443:   assume nz: "x\<noteq>0" 
huffman@22443:   hence pos: "0<x" using nneg by arith
huffman@22443:   show ?thesis
huffman@22443:   proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
huffman@22443:     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
huffman@22443:     show "inverse (sqrt x) / (sqrt x / x) = 1"
huffman@22443:       by (simp add: divide_inverse mult_assoc [symmetric] 
huffman@22443:                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
huffman@22443:   qed
huffman@22443: qed
huffman@22443: 
huffman@22721: lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
huffman@22721: apply (simp add: divide_inverse)
huffman@22721: apply (case_tac "r=0")
huffman@22721: apply (auto simp add: mult_ac)
huffman@22721: done
huffman@22721: 
huffman@23049: lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
huffman@35216: by (simp add: divide_less_eq)
huffman@23049: 
huffman@23049: lemma four_x_squared: 
huffman@23049:   fixes x::real
huffman@23049:   shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
huffman@23049: by (simp add: power2_eq_square)
huffman@23049: 
huffman@22856: subsection {* Square Root of Sum of Squares *}
huffman@22856: 
huffman@22856: lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
huffman@22968: by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero])
huffman@22856: 
huffman@22856: lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@22961: by simp
huffman@22856: 
huffman@23049: declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]
huffman@23049: 
huffman@22856: lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
huffman@22856:      "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
huffman@22856: by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
huffman@22856: 
huffman@22856: lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
huffman@22856:      "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
huffman@22956: by (auto simp add: zero_le_mult_iff)
huffman@22856: 
huffman@23049: lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<twosuperior> + y\<twosuperior>) = x \<Longrightarrow> y = 0"
huffman@23049: by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp)
huffman@23049: 
huffman@23049: lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<twosuperior> + y\<twosuperior>) = y \<Longrightarrow> x = 0"
huffman@23049: by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, simp)
huffman@23049: 
huffman@23049: lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@22856: by (rule power2_le_imp_le, simp_all)
huffman@22856: 
huffman@23049: lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@23049: by (rule power2_le_imp_le, simp_all)
huffman@23049: 
huffman@23049: lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@22856: by (rule power2_le_imp_le, simp_all)
huffman@22856: 
huffman@23049: lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
huffman@23049: by (rule power2_le_imp_le, simp_all)
huffman@23049: 
huffman@23049: lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
huffman@23049: by (simp add: power2_eq_square [symmetric])
huffman@23049: 
huffman@22858: lemma real_sqrt_sum_squares_triangle_ineq:
huffman@22858:   "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)"
huffman@22858: apply (rule power2_le_imp_le, simp)
huffman@22858: apply (simp add: power2_sum)
huffman@22858: apply (simp only: mult_assoc right_distrib [symmetric])
huffman@22858: apply (rule mult_left_mono)
huffman@22858: apply (rule power2_le_imp_le)
huffman@22858: apply (simp add: power2_sum power_mult_distrib)
nipkow@23477: apply (simp add: ring_distribs)
huffman@22858: apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp)
huffman@22858: apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans)
huffman@22858: apply (rule zero_le_power2)
huffman@22858: apply (simp add: power2_diff power_mult_distrib)
huffman@22858: apply (simp add: mult_nonneg_nonneg)
huffman@22858: apply simp
huffman@22858: apply (simp add: add_increasing)
huffman@22858: done
huffman@22858: 
huffman@23122: lemma real_sqrt_sum_squares_less:
huffman@23122:   "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
huffman@23122: apply (rule power2_less_imp_less, simp)
huffman@23122: apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
huffman@23122: apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
huffman@23122: apply (simp add: power_divide)
huffman@23122: apply (drule order_le_less_trans [OF abs_ge_zero])
huffman@23122: apply (simp add: zero_less_divide_iff)
huffman@23122: done
huffman@23122: 
huffman@23049: text{*Needed for the infinitely close relation over the nonstandard
huffman@23049:     complex numbers*}
huffman@23049: lemma lemma_sqrt_hcomplex_capprox:
huffman@23049:      "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
huffman@23049: apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
huffman@23049: apply (erule_tac [2] lemma_real_divide_sqrt_less)
huffman@23049: apply (rule power2_le_imp_le)
huffman@23049: apply (auto simp add: real_0_le_divide_iff power_divide)
huffman@23049: apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
huffman@23049: apply (rule add_mono)
huffman@30273: apply (auto simp add: four_x_squared intro: power_mono)
huffman@23049: done
huffman@23049: 
huffman@22956: text "Legacy theorem names:"
huffman@22956: lemmas real_root_pos2 = real_root_power_cancel
huffman@22956: lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
huffman@22956: lemmas real_root_pos_pos_le = real_root_ge_zero
huffman@22956: lemmas real_sqrt_mult_distrib = real_sqrt_mult
huffman@22956: lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
huffman@22956: lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
huffman@22956: 
huffman@22956: (* needed for CauchysMeanTheorem.het_base from AFP *)
huffman@22956: lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"
huffman@22956: by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])
huffman@22956: 
paulson@14324: end