2 Author : Jacques D. Fleuriot
3 Copyright : 1998 University of Cambridge
4 Conversion to Isar and new proofs by Lawrence C Paulson, 2004
7 header{* Limits and Continuity *}
13 text{*Standard Definitions*}
16 LIM :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a, 'b] \<Rightarrow> bool"
17 ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
18 "f -- a --> L \<equiv> (f ---> L) (at a)"
21 isCont :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a] \<Rightarrow> bool" where
22 "isCont f a = (f -- a --> (f a))"
25 isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
26 "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
28 subsection {* Limits of Functions *}
30 lemma LIM_def: "f -- a --> L =
31 (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
32 --> dist (f x) L < r)"
33 unfolding tendsto_iff eventually_at ..
36 "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
37 \<Longrightarrow> f -- a --> L"
38 by (simp add: LIM_def)
41 "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
42 \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
43 by (simp add: LIM_def)
46 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
48 (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
49 by (simp add: LIM_def dist_norm)
52 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
53 shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
58 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
59 shows "[| f -- a --> L; 0<r |]
60 ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
64 fixes a :: "'a::real_normed_vector"
65 shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
66 apply (rule topological_tendstoI)
67 apply (drule (2) topological_tendstoD)
68 apply (simp only: eventually_at dist_norm)
69 apply (clarify, rule_tac x=d in exI, safe)
70 apply (drule_tac x="x + k" in spec)
71 apply (simp add: algebra_simps)
74 lemma LIM_offset_zero:
75 fixes a :: "'a::real_normed_vector"
76 shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
77 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
79 lemma LIM_offset_zero_cancel:
80 fixes a :: "'a::real_normed_vector"
81 shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
82 by (drule_tac k="- a" in LIM_offset, simp)
84 lemma LIM_const [simp]: "(%x. k) -- x --> k"
85 by (rule tendsto_const)
87 lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
90 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
91 assumes f: "f -- a --> L" and g: "g -- a --> M"
92 shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
93 using assms by (rule tendsto_add)
96 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
97 shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
98 by (rule tendsto_add_zero)
101 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
102 shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
103 by (rule tendsto_minus)
107 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
108 shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
109 by (intro LIM_add LIM_minus)
112 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
113 shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
114 by (rule tendsto_diff)
117 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
118 shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
119 unfolding tendsto_iff dist_norm by simp
121 lemma LIM_zero_cancel:
122 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
123 shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
124 unfolding tendsto_iff dist_norm by simp
127 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
128 shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
129 unfolding tendsto_iff dist_norm by simp
131 lemma metric_LIM_imp_LIM:
132 assumes f: "f -- a --> l"
133 assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
135 by (rule metric_tendsto_imp_tendsto [OF f],
136 auto simp add: eventually_at_topological le)
139 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
140 fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
141 assumes f: "f -- a --> l"
142 assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
144 by (rule metric_LIM_imp_LIM [OF f],
145 simp add: dist_norm le)
148 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
149 shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
150 by (rule tendsto_norm)
153 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
154 shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
155 by (rule tendsto_norm_zero)
157 lemma LIM_norm_zero_cancel:
158 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
159 shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
160 by (rule tendsto_norm_zero_cancel)
162 lemma LIM_norm_zero_iff:
163 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
164 shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
165 by (rule tendsto_norm_zero_iff)
167 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
168 by (rule tendsto_rabs)
170 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
171 by (rule tendsto_rabs_zero)
173 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
174 by (rule tendsto_rabs_zero_cancel)
176 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
177 by (rule tendsto_rabs_zero_iff)
179 lemma trivial_limit_at:
180 fixes a :: "'a::real_normed_algebra_1"
181 shows "\<not> trivial_limit (at a)" -- {* TODO: find a more appropriate class *}
182 unfolding trivial_limit_def
183 unfolding eventually_at dist_norm
184 by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
186 lemma LIM_const_not_eq:
187 fixes a :: "'a::real_normed_algebra_1"
188 fixes k L :: "'b::t2_space"
189 shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
190 by (simp add: tendsto_const_iff trivial_limit_at)
192 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
195 fixes a :: "'a::real_normed_algebra_1"
196 fixes k L :: "'b::t2_space"
197 shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
198 by (simp add: tendsto_const_iff trivial_limit_at)
201 fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
202 fixes L M :: "'b::t2_space"
203 shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
204 using trivial_limit_at by (rule tendsto_unique)
206 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
207 by (rule tendsto_ident_at)
209 text{*Limits are equal for functions equal except at limit point*}
211 "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
212 unfolding tendsto_def eventually_at_topological by simp
215 "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
216 \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
217 by (simp add: LIM_equal)
219 lemma metric_LIM_equal2:
221 assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
222 shows "g -- a --> l \<Longrightarrow> f -- a --> l"
223 apply (rule topological_tendstoI)
224 apply (drule (2) topological_tendstoD)
225 apply (simp add: eventually_at, safe)
226 apply (rule_tac x="min d R" in exI, safe)
232 fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
234 assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
235 shows "g -- a --> l \<Longrightarrow> f -- a --> l"
236 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
238 text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
240 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
241 shows "[| (%x. f(x) + -g(x)) -- a --> 0; g -- a --> l |] ==> f -- a --> l"
242 apply (drule LIM_add, assumption)
243 apply (auto simp add: add_assoc)
247 assumes g: "g -- l --> g l"
248 assumes f: "f -- a --> l"
249 shows "(\<lambda>x. g (f x)) -- a --> g l"
250 using assms by (rule tendsto_compose)
252 lemma LIM_compose_eventually:
253 assumes f: "f -- a --> b"
254 assumes g: "g -- b --> c"
255 assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
256 shows "(\<lambda>x. g (f x)) -- a --> c"
257 using g f inj by (rule tendsto_compose_eventually)
259 lemma metric_LIM_compose2:
260 assumes f: "f -- a --> b"
261 assumes g: "g -- b --> c"
262 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
263 shows "(\<lambda>x. g (f x)) -- a --> c"
264 using f g inj [folded eventually_at]
265 by (rule LIM_compose_eventually)
268 fixes a :: "'a::real_normed_vector"
269 assumes f: "f -- a --> b"
270 assumes g: "g -- b --> c"
271 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
272 shows "(\<lambda>x. g (f x)) -- a --> c"
273 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
275 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
276 unfolding o_def by (rule LIM_compose)
278 lemma real_LIM_sandwich_zero:
279 fixes f g :: "'a::topological_space \<Rightarrow> real"
280 assumes f: "f -- a --> 0"
281 assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
282 assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
284 proof (rule LIM_imp_LIM [OF f])
285 fix x assume x: "x \<noteq> a"
286 have "norm (g x - 0) = g x" by (simp add: 1 x)
287 also have "g x \<le> f x" by (rule 2 [OF x])
288 also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
289 also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
290 finally show "norm (g x - 0) \<le> norm (f x - 0)" .
293 text {* Bounded Linear Operators *}
295 lemma (in bounded_linear) LIM:
296 "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
299 lemma (in bounded_linear) LIM_zero:
300 "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
301 by (rule tendsto_zero)
303 text {* Bounded Bilinear Operators *}
305 lemma (in bounded_bilinear) LIM:
306 "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
309 lemma (in bounded_bilinear) LIM_prod_zero:
310 fixes a :: "'d::metric_space"
311 assumes f: "f -- a --> 0"
312 assumes g: "g -- a --> 0"
313 shows "(\<lambda>x. f x ** g x) -- a --> 0"
314 using f g by (rule tendsto_zero)
316 lemma (in bounded_bilinear) LIM_left_zero:
317 "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
318 by (rule tendsto_left_zero)
320 lemma (in bounded_bilinear) LIM_right_zero:
321 "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
322 by (rule tendsto_right_zero)
324 lemmas LIM_mult = mult.LIM
326 lemmas LIM_mult_zero = mult.LIM_prod_zero
328 lemmas LIM_mult_left_zero = mult.LIM_left_zero
330 lemmas LIM_mult_right_zero = mult.LIM_right_zero
332 lemmas LIM_scaleR = scaleR.LIM
334 lemmas LIM_of_real = of_real.LIM
337 fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
338 assumes f: "f -- a --> l"
339 shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
340 using assms by (rule tendsto_power)
343 fixes L :: "'a::real_normed_div_algebra"
344 shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
345 by (rule tendsto_inverse)
347 lemma LIM_inverse_fun:
348 assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
349 shows "inverse -- a --> inverse a"
350 by (rule LIM_inverse [OF LIM_ident a])
353 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
354 shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
355 by (rule tendsto_sgn)
358 subsection {* Continuity *}
360 lemma LIM_isCont_iff:
361 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
362 shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
363 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
366 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
367 shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
368 by (simp add: isCont_def LIM_isCont_iff)
370 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
371 unfolding isCont_def by (rule LIM_ident)
373 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
374 unfolding isCont_def by (rule LIM_const)
376 lemma isCont_norm [simp]:
377 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
378 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
379 unfolding isCont_def by (rule LIM_norm)
381 lemma isCont_rabs [simp]:
382 fixes f :: "'a::topological_space \<Rightarrow> real"
383 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
384 unfolding isCont_def by (rule LIM_rabs)
386 lemma isCont_add [simp]:
387 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
388 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
389 unfolding isCont_def by (rule LIM_add)
391 lemma isCont_minus [simp]:
392 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
393 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
394 unfolding isCont_def by (rule LIM_minus)
396 lemma isCont_diff [simp]:
397 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
398 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
399 unfolding isCont_def by (rule LIM_diff)
401 lemma isCont_mult [simp]:
402 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
403 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
404 unfolding isCont_def by (rule LIM_mult)
406 lemma isCont_inverse [simp]:
407 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
408 shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
409 unfolding isCont_def by (rule LIM_inverse)
411 lemma isCont_divide [simp]:
412 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
413 shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
414 unfolding isCont_def by (rule tendsto_divide)
416 lemma isCont_LIM_compose:
417 "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
418 unfolding isCont_def by (rule LIM_compose)
420 lemma metric_isCont_LIM_compose2:
421 assumes f [unfolded isCont_def]: "isCont f a"
422 assumes g: "g -- f a --> l"
423 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
424 shows "(\<lambda>x. g (f x)) -- a --> l"
425 by (rule metric_LIM_compose2 [OF f g inj])
427 lemma isCont_LIM_compose2:
428 fixes a :: "'a::real_normed_vector"
429 assumes f [unfolded isCont_def]: "isCont f a"
430 assumes g: "g -- f a --> l"
431 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
432 shows "(\<lambda>x. g (f x)) -- a --> l"
433 by (rule LIM_compose2 [OF f g inj])
435 lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
436 unfolding isCont_def by (rule LIM_compose)
438 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
439 unfolding o_def by (rule isCont_o2)
441 lemma (in bounded_linear) isCont:
442 "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
443 unfolding isCont_def by (rule LIM)
445 lemma (in bounded_bilinear) isCont:
446 "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
447 unfolding isCont_def by (rule LIM)
449 lemmas isCont_scaleR [simp] = scaleR.isCont
451 lemma isCont_of_real [simp]:
452 "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
453 by (rule of_real.isCont)
455 lemma isCont_power [simp]:
456 fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
457 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
458 unfolding isCont_def by (rule LIM_power)
460 lemma isCont_sgn [simp]:
461 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
462 shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
463 unfolding isCont_def by (rule LIM_sgn)
465 lemma isCont_setsum [simp]:
466 fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
468 shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
469 unfolding isCont_def by (simp add: tendsto_setsum)
471 lemmas isCont_intros =
472 isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
473 isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
474 isCont_of_real isCont_power isCont_sgn isCont_setsum
476 lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
477 and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
480 assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
481 hence "0 < - f x / 2" by auto
482 from isCont[unfolded isCont_def, THEN LIM_D, OF this]
483 obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x' - x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x' - f x \<bar> < - f x / 2" by auto
485 let ?x = "x - min (s / 2) ((x - b) / 2)"
486 have "?x < x" and "\<bar> ?x - x \<bar> < s"
487 using `b < x` and `0 < s` by auto
489 proof (cases "s < x - b")
490 case True thus ?thesis using `0 < s` by auto
492 case False hence "s / 2 \<ge> (x - b) / 2" by auto
493 hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
494 thus ?thesis using `b < x` by auto
496 hence "0 \<le> f ?x" using all_le `?x < x` by auto
497 moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
498 using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
499 hence "f ?x - f x < - f x / 2" by auto
500 hence "f ?x < f x / 2" by auto
501 hence "f ?x < 0" using `f x < 0` by auto
502 thus False using `0 \<le> f ?x` by auto
506 subsection {* Uniform Continuity *}
508 lemma isUCont_isCont: "isUCont f ==> isCont f x"
509 by (simp add: isUCont_def isCont_def LIM_def, force)
511 lemma isUCont_Cauchy:
512 "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
513 unfolding isUCont_def
514 apply (rule metric_CauchyI)
515 apply (drule_tac x=e in spec, safe)
516 apply (drule_tac e=s in metric_CauchyD, safe)
517 apply (rule_tac x=M in exI, simp)
520 lemma (in bounded_linear) isUCont: "isUCont f"
521 unfolding isUCont_def dist_norm
522 proof (intro allI impI)
523 fix r::real assume r: "0 < r"
524 obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
525 using pos_bounded by fast
526 show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
527 proof (rule exI, safe)
528 from r K show "0 < r / K" by (rule divide_pos_pos)
531 assume xy: "norm (x - y) < r / K"
532 have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
533 also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
534 also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
535 finally show "norm (f x - f y) < r" .
539 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
540 by (rule isUCont [THEN isUCont_Cauchy])
543 subsection {* Relation of LIM and LIMSEQ *}
545 lemma LIMSEQ_SEQ_conv1:
546 assumes X: "X -- a --> L"
547 shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
548 using tendsto_compose_eventually [OF X, where F=sequentially] by simp
550 lemma LIMSEQ_SEQ_conv2:
551 fixes a :: real and L :: "'a::metric_space"
552 assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
555 assume "\<not> (X -- a --> L)"
556 hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
557 unfolding LIM_def dist_norm .
558 hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> dist (X x) L < r)" by simp
559 hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less)
560 then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto
562 let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
563 have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
565 hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
567 hence F1: "\<And>n. ?F n \<noteq> a"
568 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
569 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
573 proof (rule LIMSEQ_I, unfold real_norm_def)
576 (* choose no such that inverse (real (Suc n)) < e *)
577 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
578 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
579 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
580 proof (intro exI allI impI)
582 assume mlen: "m \<le> n"
583 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
585 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
588 "inverse (real (Suc m)) < e" .
589 finally show "\<bar>?F n - a\<bar> < e" .
593 moreover have "\<forall>n. ?F n \<noteq> a"
594 by (rule allI) (rule F1)
596 moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
597 ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
599 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
603 obtain n where "n = no + 1" by simp
604 then have nolen: "no \<le> n" by simp
605 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
606 have "dist (X (?F n)) L \<ge> r"
608 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
610 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
611 with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
612 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
614 ultimately show False by simp
617 lemma LIMSEQ_SEQ_conv:
618 "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
619 (X -- a --> (L::'a::metric_space))"
620 using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..