src/HOL/NthRoot.thy
 author paulson Fri Nov 13 12:27:13 2015 +0000 (2015-11-13) changeset 61649 268d88ec9087 parent 61609 77b453bd616f child 61944 5d06ecfdb472 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
1 (*  Title       : NthRoot.thy
2     Author      : Jacques D. Fleuriot
3     Copyright   : 1998  University of Cambridge
4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
5 *)
7 section \<open>Nth Roots of Real Numbers\<close>
9 theory NthRoot
10 imports Deriv Binomial
11 begin
13 lemma abs_sgn_eq: "abs (sgn x :: real) = (if x = 0 then 0 else 1)"
16 lemma inverse_sgn: "sgn (inverse a) = inverse (sgn a :: real)"
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 \<open>Existence of Nth Root\<close>
26 text \<open>Existence follows from the Intermediate Value Theorem\<close>
28 lemma realpow_pos_nth:
29   assumes n: "0 < n"
30   assumes a: "0 < a"
31   shows "\<exists>r>0. r ^ n = (a::real)"
32 proof -
33   have "\<exists>r\<ge>0. r \<le> (max 1 a) \<and> r ^ n = a"
34   proof (rule IVT)
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"
43       by simp
44   qed
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
48   thus ?thesis ..
49 qed
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 \<open>Uniqueness of nth positive root\<close>
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 \<open>Nth Root\<close>
62 text \<open>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.\<close>
66 lemma inj_sgn_power: assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar>^n :: real)" (is "inj ?f")
67 proof (rule injI)
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>"] \<open>0<n\<close> show "x = y"
70     by (cases rule: linorder_cases[of 0 x, case_product linorder_cases[of 0 y]])
72 qed
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"
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)
86 lemma sgn_power_root:
87   assumes "0 < n" shows "sgn (root n x) * \<bar>(root n x)\<bar>^n = x" (is "?f (root n x) = x")
88 proof cases
89   assume "x \<noteq> 0"
90   with realpow_pos_nth[OF \<open>0 < n\<close>, of "\<bar>x\<bar>"] obtain r where "0 < r" "r ^ n = \<bar>x\<bar>" by auto
91   with \<open>x \<noteq> 0\<close> 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 \<open>0 < n\<close> f_the_inv_into_f[OF inj_sgn_power[OF \<open>0 < n\<close>] this]  show ?thesis
96 qed (insert \<open>0 < n\<close> 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))"
99   apply (cases "n = 0")
100   apply simp_all
101   apply (metis root_sgn_power sgn_power_root)
102   done
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 x [of "a ^ n" "- ((- b) ^ n)"] split: split_if_asm)
116 qed
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)
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"
154 text \<open>Root function is strictly monotonic, hence injective\<close>
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")
163 apply (simp add: linorder_not_less real_root_le_mono)
164 done
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")
170 apply (simp add: linorder_not_le real_root_less_mono)
171 done
173 lemma real_root_eq_iff [simp]:
174   "0 < n \<Longrightarrow> (root n x = root n y) = (x = y)"
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 \<open>Roots of multiplication and division\<close>
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 \<open>Roots of roots\<close>
217 lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
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 \<open>Monotonicity in first argument\<close>
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)
234 done
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)
241 done
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 \<open>Continuity and derivatives\<close>
253 lemma isCont_real_root: "isCont (root n) x"
254 proof cases
255   assume n: "0 < n"
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 sgn_neg le_less n)
261   then have [simp]: "\<And>x. isCont ?f x"
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)
266   then show ?thesis
267     by (simp add: sgn_power_root n)
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:
283   assumes n: "0 < n"
284   assumes x: "0 < x"
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"
290     using n by simp
291   show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
292     by (rule DERIV_pow)
293   show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
294     using n x by simp
295 qed (rule isCont_real_root)
297 lemma DERIV_odd_real_root:
298   assumes n: "odd n"
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)"
307     by (rule DERIV_pow)
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"
314   assumes x: "x < 0"
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 .
319 next
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 \<open>0 < n\<close> by simp
324     with real_root_minus and \<open>even n\<close>
325     show "- (root n y ^ n) = y" by simp
326   qed
327 next
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)
330   show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
331     using n x by simp
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 \<open>Square Root\<close>
347 definition sqrt :: "real \<Rightarrow> real" where
348   "sqrt = root 2"
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)
359 done
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)"
365 apply (rule iffI)
366 apply (erule subst)
367 apply (rule zero_le_power2)
368 apply (erule real_sqrt_pow2)
369 done
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:
429   assumes n: "even n"
430   shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
431 proof -
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)
435   then show ?thesis
436     using m by simp
437 qed
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]
452   assumes "0 \<le> x" "0 \<le> y"
453   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
454 by (rule power2_le_imp_le) (simp_all add: power2_sum assms)
456 lemma isCont_real_sqrt: "isCont sqrt x"
457 unfolding sqrt_def by (rule isCont_real_root)
459 lemma tendsto_real_sqrt[tendsto_intros]:
460   "(f ---> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) ---> sqrt x) F"
461   unfolding sqrt_def by (rule tendsto_real_root)
463 lemma continuous_real_sqrt[continuous_intros]:
464   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
465   unfolding sqrt_def by (rule continuous_real_root)
467 lemma continuous_on_real_sqrt[continuous_intros]:
468   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
469   unfolding sqrt_def by (rule continuous_on_real_root)
471 lemma DERIV_real_sqrt_generic:
472   assumes "x \<noteq> 0"
473   assumes "x > 0 \<Longrightarrow> D = inverse (sqrt x) / 2"
474   assumes "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2"
475   shows "DERIV sqrt x :> D"
476   using assms unfolding sqrt_def
477   by (auto intro!: DERIV_real_root_generic)
479 lemma DERIV_real_sqrt:
480   "0 < x \<Longrightarrow> DERIV sqrt x :> inverse (sqrt x) / 2"
481   using DERIV_real_sqrt_generic by simp
483 declare
484   DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
485   DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]
487 lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
488 apply auto
489 apply (cut_tac x = x and y = 0 in linorder_less_linear)
491 done
493 lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
494 apply (subst power2_eq_square [symmetric])
495 apply (rule real_sqrt_abs)
496 done
498 lemma real_inv_sqrt_pow2: "0 < x ==> (inverse (sqrt x))\<^sup>2 = inverse x"
501 lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
502 by simp
504 lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
505 by simp
507 lemma sqrt_divide_self_eq:
508   assumes nneg: "0 \<le> x"
509   shows "sqrt x / x = inverse (sqrt x)"
510 proof cases
511   assume "x=0" thus ?thesis by simp
512 next
513   assume nz: "x\<noteq>0"
514   hence pos: "0<x" using nneg by arith
515   show ?thesis
516   proof (rule right_inverse_eq [THEN iffD1, THEN sym])
517     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
518     show "inverse (sqrt x) / (sqrt x / x) = 1"
519       by (simp add: divide_inverse mult.assoc [symmetric]
520                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
521   qed
522 qed
524 lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
525   apply (cases "x = 0")
526   apply simp_all
527   using sqrt_divide_self_eq[of x]
529   done
531 lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
533 apply (case_tac "r=0")
534 apply (auto simp add: ac_simps)
535 done
537 lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
540 lemma four_x_squared:
541   fixes x::real
542   shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
545 lemma sqrt_at_top: "LIM x at_top. sqrt x :: real :> at_top"
546   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="power2"])
547      (auto intro: eventually_gt_at_top)
549 subsection \<open>Square Root of Sum of Squares\<close>
551 lemma sum_squares_bound:
552   fixes x:: "'a::linordered_field"
553   shows "2*x*y \<le> x^2 + y^2"
554 proof -
555   have "(x-y)^2 = x*x - 2*x*y + y*y"
556     by algebra
557   then have "0 \<le> x^2 - 2*x*y + y^2"
559   then show ?thesis
560     by arith
561 qed
563 lemma arith_geo_mean:
564   fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2"
565     apply (rule power2_le_imp_le)
566     using sum_squares_bound assms
567     apply (auto simp: zero_le_mult_iff)
568     by (auto simp: algebra_simps power2_eq_square)
570 lemma arith_geo_mean_sqrt:
571   fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
572   apply (rule arith_geo_mean)
573   using assms
574   apply (auto simp: zero_le_mult_iff)
575   done
577 lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
578      "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
579   by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)
581 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
582      "(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)"
585 lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x\<^sup>2 + y\<^sup>2) = x \<Longrightarrow> y = 0"
586 by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
588 lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x\<^sup>2 + y\<^sup>2) = y \<Longrightarrow> x = 0"
589 by (drule_tac f = "%x. x\<^sup>2" in arg_cong, simp)
591 lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
592 by (rule power2_le_imp_le, simp_all)
594 lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
595 by (rule power2_le_imp_le, simp_all)
597 lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
598 by (rule power2_le_imp_le, simp_all)
600 lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
601 by (rule power2_le_imp_le, simp_all)
603 lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
604 by (simp add: power2_eq_square [symmetric])
606 lemma real_sqrt_sum_squares_triangle_ineq:
607   "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
608 apply (rule power2_le_imp_le, simp)
610 apply (simp only: mult.assoc distrib_left [symmetric])
611 apply (rule mult_left_mono)
612 apply (rule power2_le_imp_le)
613 apply (simp add: power2_sum power_mult_distrib)
615 apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)", simp)
616 apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
617 apply (rule zero_le_power2)
618 apply (simp add: power2_diff power_mult_distrib)
619 apply (simp)
620 apply simp
622 done
624 lemma real_sqrt_sum_squares_less:
625   "\<lbrakk>\<bar>x\<bar> < u / sqrt 2; \<bar>y\<bar> < u / sqrt 2\<rbrakk> \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
626 apply (rule power2_less_imp_less, simp)
627 apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
628 apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
630 apply (drule order_le_less_trans [OF abs_ge_zero])
632 done
634 lemma sqrt2_less_2: "sqrt 2 < (2::real)"
635   by (metis not_less not_less_iff_gr_or_eq numeral_less_iff real_sqrt_four real_sqrt_le_iff semiring_norm(75) semiring_norm(78) semiring_norm(85))
638 text\<open>Needed for the infinitely close relation over the nonstandard
639     complex numbers\<close>
640 lemma lemma_sqrt_hcomplex_capprox:
641      "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<^sup>2 + y\<^sup>2) < u"
642   apply (rule real_sqrt_sum_squares_less)
643   apply (auto simp add: abs_if field_simps)
644   apply (rule le_less_trans [where y = "x*2"])
645   using less_eq_real_def sqrt2_less_2 apply force
646   apply assumption
647   apply (rule le_less_trans [where y = "y*2"])
648   using less_eq_real_def sqrt2_less_2 mult_le_cancel_left
649   apply auto
650   done
652 lemma LIMSEQ_root: "(\<lambda>n. root n n) ----> 1"
653 proof -
654   def x \<equiv> "\<lambda>n. root n n - 1"
655   have "x ----> sqrt 0"
656   proof (rule tendsto_sandwich[OF _ _ tendsto_const])
657     show "(\<lambda>x. sqrt (2 / x)) ----> sqrt 0"
658       by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
660     { fix n :: nat assume "2 < n"
661       have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
662         using \<open>2 < n\<close> unfolding gbinomial_def binomial_gbinomial
663         by (simp add: atLeast0AtMost atMost_Suc field_simps of_nat_diff numeral_2_eq_2)
664       also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
666       also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
667         using \<open>2 < n\<close> by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
668       also have "\<dots> = (x n + 1) ^ n"
670       also have "\<dots> = n"
671         using \<open>2 < n\<close> by (simp add: x_def)
672       finally have "real (n - 1) * (real n / 2 * (x n)\<^sup>2) \<le> real (n - 1) * 1"
673         by simp
674       then have "(x n)\<^sup>2 \<le> 2 / real n"
675         using \<open>2 < n\<close> unfolding mult_le_cancel_left by (simp add: field_simps)
676       from real_sqrt_le_mono[OF this] have "x n \<le> sqrt (2 / real n)"
677         by simp }
678     then show "eventually (\<lambda>n. x n \<le> sqrt (2 / real n)) sequentially"
679       by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
680     show "eventually (\<lambda>n. sqrt 0 \<le> x n) sequentially"
681       by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
682   qed
683   from tendsto_add[OF this tendsto_const[of 1]] show ?thesis
685 qed
687 lemma LIMSEQ_root_const:
688   assumes "0 < c"
689   shows "(\<lambda>n. root n c) ----> 1"
690 proof -
691   { fix c :: real assume "1 \<le> c"
692     def x \<equiv> "\<lambda>n. root n c - 1"
693     have "x ----> 0"
694     proof (rule tendsto_sandwich[OF _ _ tendsto_const])
695       show "(\<lambda>n. c / n) ----> 0"
696         by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
698       { fix n :: nat assume "1 < n"
699         have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
700           using \<open>1 < n\<close> unfolding gbinomial_def binomial_gbinomial by simp
701         also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
703         also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
704           using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (intro setsum_mono2) (auto intro!: mult_nonneg_nonneg zero_le_power simp: x_def le_diff_eq)
705         also have "\<dots> = (x n + 1) ^ n"
707         also have "\<dots> = c"
708           using \<open>1 < n\<close> \<open>1 \<le> c\<close> by (simp add: x_def)
709         finally have "x n \<le> c / n"
710           using \<open>1 \<le> c\<close> \<open>1 < n\<close> by (simp add: field_simps) }
711       then show "eventually (\<lambda>n. x n \<le> c / n) sequentially"
712         by (auto intro!: exI[of _ 3] simp: eventually_sequentially)
713       show "eventually (\<lambda>n. 0 \<le> x n) sequentially"
714         using \<open>1 \<le> c\<close> by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
715     qed
716     from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) ----> 1"
717       by (simp add: x_def) }
718   note ge_1 = this
720   show ?thesis
721   proof cases
722     assume "1 \<le> c" with ge_1 show ?thesis by blast
723   next
724     assume "\<not> 1 \<le> c"
725     with \<open>0 < c\<close> have "1 \<le> 1 / c"
726       by simp
727     then have "(\<lambda>n. 1 / root n (1 / c)) ----> 1 / 1"
728       by (intro tendsto_divide tendsto_const ge_1 \<open>1 \<le> 1 / c\<close> one_neq_zero)
729     then show ?thesis
730       by (rule filterlim_cong[THEN iffD1, rotated 3])
731          (auto intro!: exI[of _ 1] simp: eventually_sequentially real_root_divide)
732   qed
733 qed
736 text "Legacy theorem names:"
737 lemmas real_root_pos2 = real_root_power_cancel
738 lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
739 lemmas real_root_pos_pos_le = real_root_ge_zero
740 lemmas real_sqrt_mult_distrib = real_sqrt_mult
741 lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
742 lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff
744 end