2 Author : Jacques D. Fleuriot
3 Copyright : 1998 University of Cambridge
4 Conversion to Isar and new proofs by Lawrence C Paulson, 2004
7 section {* Nth Roots of Real Numbers *}
13 lemma abs_sgn_eq: "abs (sgn x :: real) = (if x = 0 then 0 else 1)"
14 by (simp add: sgn_real_def)
16 lemma inverse_sgn: "sgn (inverse a) = inverse (sgn a :: real)"
17 by (simp add: sgn_real_def)
19 lemma power_eq_iff_eq_base:
20 fixes a b :: "_ :: linordered_semidom"
21 shows "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
22 using power_eq_imp_eq_base[of a n b] by auto
24 subsection {* Existence of Nth Root *}
26 text {* Existence follows from the Intermediate Value Theorem *}
28 lemma realpow_pos_nth:
31 shows "\<exists>r>0. r ^ n = (a::real)"
33 have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a"
35 show "0 ^ n \<le> a" using n a by (simp add: power_0_left)
36 show "0 \<le> max 1 a" by simp
37 from n have n1: "1 \<le> n" by simp
38 have "a \<le> max 1 a ^ 1" by simp
39 also have "max 1 a ^ 1 \<le> max 1 a ^ n"
40 using n1 by (rule power_increasing, simp)
41 finally show "a \<le> max 1 a ^ n" .
42 show "\<forall>r. 0 \<le> r \<and> r \<le> max 1 a \<longrightarrow> isCont (\<lambda>x. x ^ n) r"
45 then obtain r where r: "0 \<le> r \<and> r ^ n = a" by fast
46 with n a have "r \<noteq> 0" by (auto simp add: power_0_left)
47 with r have "0 < r \<and> r ^ n = a" by simp
51 (* Used by Integration/RealRandVar.thy in AFP *)
52 lemma realpow_pos_nth2: "(0::real) < a \<Longrightarrow> \<exists>r>0. r ^ Suc n = a"
53 by (blast intro: realpow_pos_nth)
55 text {* Uniqueness of nth positive root *}
57 lemma realpow_pos_nth_unique: "\<lbrakk>0 < n; 0 < a\<rbrakk> \<Longrightarrow> \<exists>!r. 0 < r \<and> r ^ n = (a::real)"
58 by (auto intro!: realpow_pos_nth simp: power_eq_iff_eq_base)
60 subsection {* Nth Root *}
62 text {* We define roots of negative reals such that
63 @{term "root n (- x) = - root n x"}. This allows
64 us to omit side conditions from many theorems. *}
66 lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f")
68 have x: "\<And>a b :: real. (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b) \<Longrightarrow> a \<noteq> b" by auto
69 fix x y assume "?f x = ?f y" with power_eq_iff_eq_base[of n "\<bar>x\<bar>" "\<bar>y\<bar>"] `0<n` show "x = y"
70 by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]])
74 lemma sgn_power_injE: "sgn a * \<bar>a\<bar> ^ n = x \<Longrightarrow> x = sgn b * \<bar>b\<bar> ^ n \<Longrightarrow> 0 < n \<Longrightarrow> a = (b::real)"
75 using inj_sgn_power[THEN injD, of n a b] by simp
77 definition root :: "nat \<Rightarrow> real \<Rightarrow> real" where
78 "root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar>^n) x)"
80 lemma root_0 [simp]: "root 0 x = 0"
81 by (simp add: root_def)
83 lemma root_sgn_power: "0 < n \<Longrightarrow> root n (sgn y * \<bar>y\<bar>^n) = y"
84 using the_inv_f_f[OF inj_sgn_power] by (simp add: root_def)
87 assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x")
90 with realpow_pos_nth[OF `0 < n`, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto
91 with `x \<noteq> 0` have S: "x \<in> range ?f"
92 by (intro image_eqI[of _ _ "sgn x * r"])
93 (auto simp: abs_mult sgn_mult power_mult_distrib abs_sgn_eq mult_sgn_abs)
94 from `0 < n` f_the_inv_into_f[OF inj_sgn_power[OF `0 < n`] this] show ?thesis
95 by (simp add: root_def)
96 qed (insert `0 < n` root_sgn_power[of n 0], simp)
98 lemma split_root: "P (root n x) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar>^n = x \<longrightarrow> P y))"
101 apply (metis root_sgn_power sgn_power_root)
104 lemma real_root_zero [simp]: "root n 0 = 0"
105 by (simp split: split_root add: sgn_zero_iff)
107 lemma real_root_minus: "root n (- x) = - root n x"
108 by (clarsimp split: split_root elim!: sgn_power_injE simp: sgn_minus)
110 lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"
111 proof (clarsimp split: split_root)
112 have x: "\<And>a b :: real. (0 < b \<and> a < 0) \<Longrightarrow> \<not> a > b" by auto
113 fix a b :: real assume "0 < n" "sgn a * \<bar>a\<bar> ^ n < sgn b * \<bar>b\<bar> ^ n" then show "a < b"
114 using power_less_imp_less_base[of a n b] power_less_imp_less_base[of "-b" n "-a"]
115 by (simp add: sgn_real_def power_less_zero_eq x[of "a ^ n" "- ((- b) ^ n)"] split: split_if_asm)
118 lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x"
119 using real_root_less_mono[of n 0 x] by simp
121 lemma real_root_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> root n x"
122 using real_root_gt_zero[of n x] by (cases "n = 0") (auto simp add: le_less)
124 lemma real_root_pow_pos: (* TODO: rename *)
125 "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
126 using sgn_power_root[of n x] real_root_gt_zero[of n x] by simp
128 lemma real_root_pow_pos2 [simp]: (* TODO: rename *)
129 "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x"
130 by (auto simp add: order_le_less real_root_pow_pos)
132 lemma sgn_root: "0 < n \<Longrightarrow> sgn (root n x) = sgn x"
133 by (auto split: split_root simp: sgn_real_def power_less_zero_eq)
135 lemma odd_real_root_pow: "odd n \<Longrightarrow> root n x ^ n = x"
136 using sgn_power_root[of n x] by (simp add: odd_pos sgn_real_def split: split_if_asm)
138 lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x"
139 using root_sgn_power[of n x] by (auto simp add: le_less power_0_left)
141 lemma odd_real_root_power_cancel: "odd n \<Longrightarrow> root n (x ^ n) = x"
142 using root_sgn_power[of n x] by (simp add: odd_pos sgn_real_def power_0_left split: split_if_asm)
144 lemma real_root_pos_unique: "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
145 using root_sgn_power[of n y] by (auto simp add: le_less power_0_left)
147 lemma odd_real_root_unique:
148 "\<lbrakk>odd n; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"
149 by (erule subst, rule odd_real_root_power_cancel)
151 lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1"
152 by (simp add: real_root_pos_unique)
154 text {* Root function is strictly monotonic, hence injective *}
156 lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y"
157 by (auto simp add: order_le_less real_root_less_mono)
159 lemma real_root_less_iff [simp]:
160 "0 < n \<Longrightarrow> (root n x < root n y) = (x < y)"
161 apply (cases "x < y")
162 apply (simp add: real_root_less_mono)
163 apply (simp add: linorder_not_less real_root_le_mono)
166 lemma real_root_le_iff [simp]:
167 "0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)"
168 apply (cases "x \<le> y")
169 apply (simp add: real_root_le_mono)
170 apply (simp add: linorder_not_le real_root_less_mono)
173 lemma real_root_eq_iff [simp]:
174 "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)"
175 by (simp add: order_eq_iff)
177 lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]
178 lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]
179 lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified]
180 lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]
181 lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]
183 lemma real_root_gt_1_iff [simp]: "0 < n \<Longrightarrow> (1 < root n y) = (1 < y)"
184 by (insert real_root_less_iff [where x=1], simp)
186 lemma real_root_lt_1_iff [simp]: "0 < n \<Longrightarrow> (root n x < 1) = (x < 1)"
187 by (insert real_root_less_iff [where y=1], simp)
189 lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> (1 \<le> root n y) = (1 \<le> y)"
190 by (insert real_root_le_iff [where x=1], simp)
192 lemma real_root_le_1_iff [simp]: "0 < n \<Longrightarrow> (root n x \<le> 1) = (x \<le> 1)"
193 by (insert real_root_le_iff [where y=1], simp)
195 lemma real_root_eq_1_iff [simp]: "0 < n \<Longrightarrow> (root n x = 1) = (x = 1)"
196 by (insert real_root_eq_iff [where y=1], simp)
198 text {* Roots of multiplication and division *}
200 lemma real_root_mult: "root n (x * y) = root n x * root n y"
201 by (auto split: split_root elim!: sgn_power_injE simp: sgn_mult abs_mult power_mult_distrib)
203 lemma real_root_inverse: "root n (inverse x) = inverse (root n x)"
204 by (auto split: split_root elim!: sgn_power_injE simp: inverse_sgn power_inverse)
206 lemma real_root_divide: "root n (x / y) = root n x / root n y"
207 by (simp add: divide_inverse real_root_mult real_root_inverse)
209 lemma real_root_abs: "0 < n \<Longrightarrow> root n \<bar>x\<bar> = \<bar>root n x\<bar>"
210 by (simp add: abs_if real_root_minus)
212 lemma real_root_power: "0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k"
213 by (induct k) (simp_all add: real_root_mult)
215 text {* Roots of roots *}
217 lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
218 by (simp add: odd_real_root_unique)
220 lemma real_root_mult_exp: "root (m * n) x = root m (root n x)"
221 by (auto split: split_root elim!: sgn_power_injE
222 simp: sgn_zero_iff sgn_mult power_mult[symmetric] abs_mult power_mult_distrib abs_sgn_eq)
224 lemma real_root_commute: "root m (root n x) = root n (root m x)"
225 by (simp add: real_root_mult_exp [symmetric] mult.commute)
227 text {* Monotonicity in first argument *}
229 lemma real_root_strict_decreasing:
230 "\<lbrakk>0 < n; n < N; 1 < x\<rbrakk> \<Longrightarrow> root N x < root n x"
231 apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
232 apply (simp add: real_root_commute power_strict_increasing
233 del: real_root_pow_pos2)
236 lemma real_root_strict_increasing:
237 "\<lbrakk>0 < n; n < N; 0 < x; x < 1\<rbrakk> \<Longrightarrow> root n x < root N x"
238 apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
239 apply (simp add: real_root_commute power_strict_decreasing
240 del: real_root_pow_pos2)
243 lemma real_root_decreasing:
244 "\<lbrakk>0 < n; n < N; 1 \<le> x\<rbrakk> \<Longrightarrow> root N x \<le> root n x"
245 by (auto simp add: order_le_less real_root_strict_decreasing)
247 lemma real_root_increasing:
248 "\<lbrakk>0 < n; n < N; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> root n x \<le> root N x"
249 by (auto simp add: order_le_less real_root_strict_increasing)
251 text {* Continuity and derivatives *}
253 lemma isCont_real_root: "isCont (root n) x"
256 let ?f = "\<lambda>y::real. sgn y * \<bar>y\<bar>^n"
257 have "continuous_on ({0..} \<union> {.. 0}) (\<lambda>x. if 0 < x then x ^ n else - ((-x) ^ n) :: real)"
258 using n by (intro continuous_on_If continuous_intros) auto
259 then have "continuous_on UNIV ?f"
260 by (rule continuous_on_cong[THEN iffD1, rotated 2]) (auto simp: not_less real_sgn_neg le_less n)
261 then have [simp]: "\<And>x. isCont ?f x"
262 by (simp add: continuous_on_eq_continuous_at)
264 have "isCont (root n) (?f (root n x))"
265 by (rule isCont_inverse_function [where f="?f" and d=1]) (auto simp: root_sgn_power n)
267 by (simp add: sgn_power_root n)
268 qed (simp add: root_def[abs_def])
270 lemma tendsto_real_root[tendsto_intros]:
271 "(f ---> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) ---> root n x) F"
272 using isCont_tendsto_compose[OF isCont_real_root, of f x F] .
274 lemma continuous_real_root[continuous_intros]:
275 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
276 unfolding continuous_def by (rule tendsto_real_root)
278 lemma continuous_on_real_root[continuous_intros]:
279 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
280 unfolding continuous_on_def by (auto intro: tendsto_real_root)
282 lemma DERIV_real_root:
285 shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
286 proof (rule DERIV_inverse_function)
287 show "0 < x" using x .
288 show "x < x + 1" by simp
289 show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
291 show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
293 show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
295 qed (rule isCont_real_root)
297 lemma DERIV_odd_real_root:
299 assumes x: "x \<noteq> 0"
300 shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
301 proof (rule DERIV_inverse_function)
302 show "x - 1 < x" by simp
303 show "x < x + 1" by simp
304 show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
305 using n by (simp add: odd_real_root_pow)
306 show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
308 show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
309 using odd_pos [OF n] x by simp
310 qed (rule isCont_real_root)
312 lemma DERIV_even_real_root:
313 assumes n: "0 < n" and "even n"
315 shows "DERIV (root n) x :> inverse (- real n * root n x ^ (n - Suc 0))"
316 proof (rule DERIV_inverse_function)
317 show "x - 1 < x" by simp
318 show "x < 0" using x .
320 show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
321 proof (rule allI, rule impI, erule conjE)
322 fix y assume "x - 1 < y" and "y < 0"
323 hence "root n (-y) ^ n = -y" using `0 < n` by simp
324 with real_root_minus and `even n`
325 show "- (root n y ^ n) = y" by simp
328 show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
329 by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
330 show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
332 qed (rule isCont_real_root)
334 lemma DERIV_real_root_generic:
335 assumes "0 < n" and "x \<noteq> 0"
336 and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
337 and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
338 and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
339 shows "DERIV (root n) x :> D"
340 using assms by (cases "even n", cases "0 < x",
341 auto intro: DERIV_real_root[THEN DERIV_cong]
342 DERIV_odd_real_root[THEN DERIV_cong]
343 DERIV_even_real_root[THEN DERIV_cong])
345 subsection {* Square Root *}
347 definition sqrt :: "real \<Rightarrow> real" where
350 lemma pos2: "0 < (2::nat)" by simp
352 lemma real_sqrt_unique: "\<lbrakk>y\<^sup>2 = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"
353 unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])
355 lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>"
356 apply (rule real_sqrt_unique)
357 apply (rule power2_abs)
358 apply (rule abs_ge_zero)
361 lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x"
362 unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
364 lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
367 apply (rule zero_le_power2)
368 apply (erule real_sqrt_pow2)
371 lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
372 unfolding sqrt_def by (rule real_root_zero)
374 lemma real_sqrt_one [simp]: "sqrt 1 = 1"
375 unfolding sqrt_def by (rule real_root_one [OF pos2])
377 lemma real_sqrt_four [simp]: "sqrt 4 = 2"
378 using real_sqrt_abs[of 2] by simp
380 lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
381 unfolding sqrt_def by (rule real_root_minus)
383 lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
384 unfolding sqrt_def by (rule real_root_mult)
386 lemma real_sqrt_mult_self[simp]: "sqrt a * sqrt a = \<bar>a\<bar>"
387 using real_sqrt_abs[of a] unfolding power2_eq_square real_sqrt_mult .
389 lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
390 unfolding sqrt_def by (rule real_root_inverse)
392 lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
393 unfolding sqrt_def by (rule real_root_divide)
395 lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
396 unfolding sqrt_def by (rule real_root_power [OF pos2])
398 lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x"
399 unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])
401 lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"
402 unfolding sqrt_def by (rule real_root_ge_zero)
404 lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"
405 unfolding sqrt_def by (rule real_root_less_mono [OF pos2])
407 lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
408 unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
410 lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
411 unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
413 lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
414 unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
416 lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
417 unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
419 lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
420 using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
422 lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
423 using real_sqrt_le_mono[of "x\<^sup>2" y] by simp
425 lemma real_less_rsqrt: "x\<^sup>2 < y \<Longrightarrow> x < sqrt y"
426 using real_sqrt_less_mono[of "x\<^sup>2" y] by simp
428 lemma sqrt_even_pow2:
430 shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
432 from n obtain m where m: "n = 2 * m" ..
433 from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
434 by (simp only: power_mult[symmetric] mult.commute)
439 lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, unfolded real_sqrt_zero]
440 lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, unfolded real_sqrt_zero]
441 lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, unfolded real_sqrt_zero]
442 lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, unfolded real_sqrt_zero]
443 lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, unfolded real_sqrt_zero]
445 lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, unfolded real_sqrt_one]
446 lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, unfolded real_sqrt_one]
447 lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, unfolded real_sqrt_one]
448 lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, unfolded real_sqrt_one]
449 lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, unfolded real_sqrt_one]
451 lemma isCont_real_sqrt: "isCont sqrt x"
452 unfolding sqrt_def by (rule isCont_real_root)
454 lemma tendsto_real_sqrt[tendsto_intros]:
455 "(f ---> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) ---> sqrt x) F"
456 unfolding sqrt_def by (rule tendsto_real_root)
458 lemma continuous_real_sqrt[continuous_intros]:
459 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
460 unfolding sqrt_def by (rule continuous_real_root)
462 lemma continuous_on_real_sqrt[continuous_intros]:
463 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
464 unfolding sqrt_def by (rule continuous_on_real_root)
466 lemma DERIV_real_sqrt_generic:
467 assumes "x \<noteq> 0"
468 assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
469 assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
470 shows "DERIV sqrt x :> D"
471 using assms unfolding sqrt_def
472 by (auto intro!: DERIV_real_root_generic)
474 lemma DERIV_real_sqrt:
475 "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
476 using DERIV_real_sqrt_generic by simp
479 DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
480 DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]
482 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
484 apply (cut_tac x = x and y = 0 in linorder_less_linear)
485 apply (simp add: zero_less_mult_iff)
488 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
489 apply (subst power2_eq_square [symmetric])
490 apply (rule real_sqrt_abs)
493 lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x"
494 by (simp add: power_inverse [symmetric])
496 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
499 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
502 lemma sqrt_divide_self_eq:
503 assumes nneg: "0 \<le> x"
504 shows "sqrt x / x = inverse (sqrt x)"
506 assume "x=0" thus ?thesis by simp
508 assume nz: "x\<noteq>0"
509 hence pos: "0<x" using nneg by arith
511 proof (rule right_inverse_eq [THEN iffD1, THEN sym])
512 show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
513 show "inverse (sqrt x) / (sqrt x / x) = 1"
514 by (simp add: divide_inverse mult.assoc [symmetric]
515 power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
519 lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
520 apply (cases "x = 0")
522 using sqrt_divide_self_eq[of x]
523 apply (simp add: inverse_eq_divide field_simps)
526 lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
527 apply (simp add: divide_inverse)
528 apply (case_tac "r=0")
529 apply (auto simp add: ac_simps)
532 lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
533 by (simp add: divide_less_eq)
535 lemma four_x_squared:
537 shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
538 by (simp add: power2_eq_square)
540 lemma sqrt_at_top: "LIM x at_top. sqrt x :: real :> at_top"
541 by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="power2"])
542 (auto intro: eventually_gt_at_top)
544 subsection {* Square Root of Sum of Squares *}
546 lemma sum_squares_bound:
547 fixes x:: "'a::linordered_field"
548 shows "2*x*y \<le> x^2 + y^2"
550 have "(x-y)^2 = x*x - 2*x*y + y*y"
552 then have "0 \<le> x^2 - 2*x*y + y^2"
553 by (metis sum_power2_ge_zero zero_le_double_add_iff_zero_le_single_add power2_eq_square)
558 lemma arith_geo_mean:
559 fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2"
560 apply (rule power2_le_imp_le)
561 using sum_squares_bound assms
562 apply (auto simp: zero_le_mult_iff)
563 by (auto simp: algebra_simps power2_eq_square)
565 lemma arith_geo_mean_sqrt:
566 fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
567 apply (rule arith_geo_mean)
569 apply (auto simp: zero_le_mult_iff)
572 lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
573 "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
574 by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)
576 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
577 "(sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)))\<^sup>2 = (x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)"
578 by (simp add: zero_le_mult_iff)
580 lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0"
581 by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
583 lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0"
584 by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
586 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
587 by (rule power2_le_imp_le, simp_all)
589 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
590 by (rule power2_le_imp_le, simp_all)
592 lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
593 by (rule power2_le_imp_le, simp_all)
595 lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
596 by (rule power2_le_imp_le, simp_all)
598 lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
599 by (simp add: power2_eq_square [symmetric])
601 lemma real_sqrt_sum_squares_triangle_ineq:
602 "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
603 apply (rule power2_le_imp_le, simp)
604 apply (simp add: power2_sum)
605 apply (simp only: mult.assoc distrib_left [symmetric])
606 apply (rule mult_left_mono)
607 apply (rule power2_le_imp_le)
608 apply (simp add: power2_sum power_mult_distrib)
609 apply (simp add: ring_distribs)
610 apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)", simp)
611 apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
612 apply (rule zero_le_power2)
613 apply (simp add: power2_diff power_mult_distrib)
616 apply (simp add: add_increasing)
619 lemma real_sqrt_sum_squares_less:
620 "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
621 apply (rule power2_less_imp_less, simp)
622 apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
623 apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
624 apply (simp add: power_divide)
625 apply (drule order_le_less_trans [OF abs_ge_zero])
626 apply (simp add: zero_less_divide_iff)
629 text{*Needed for the infinitely close relation over the nonstandard
631 lemma lemma_sqrt_hcomplex_capprox:
632 "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<^sup>2 + y\<^sup>2) < u"
633 apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
634 apply (erule_tac [2] lemma_real_divide_sqrt_less)
635 apply (rule power2_le_imp_le)
636 apply (auto simp add: zero_le_divide_iff power_divide)
637 apply (rule_tac t = "u\<^sup>2" in real_sum_of_halves [THEN subst])
638 apply (rule add_mono)
639 apply (auto simp add: four_x_squared intro: power_mono)
642 text "Legacy theorem names:"
643 lemmas real_root_pos2 = real_root_power_cancel
644 lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
645 lemmas real_root_pos_pos_le = real_root_ge_zero
646 lemmas real_sqrt_mult_distrib = real_sqrt_mult
647 lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
648 lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
650 (* needed for CauchysMeanTheorem.het_base from AFP *)
651 lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"
652 by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])