src/HOL/Limits.thy
 author hoelzl Tue Mar 26 12:20:58 2013 +0100 (2013-03-26) changeset 51526 155263089e7b parent 51524 7cb5ac44ca9e child 51529 2d2f59e6055a permissions -rw-r--r--
move SEQ.thy and Lim.thy to Limits.thy
1 (*  Title:      Limits.thy
2     Author:     Brian Huffman
3     Author:     Jacques D. Fleuriot, University of Cambridge
4     Author:     Lawrence C Paulson
7 *)
9 header {* Limits on Real Vector Spaces *}
11 theory Limits
12 imports Real_Vector_Spaces
13 begin
15 (* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
16    Hence it was references as Limits.eventually_within, but now it is Basic_Topology.eventually_within *)
17 lemmas eventually_within = eventually_within
19 subsection {* Filter going to infinity norm *}
21 definition at_infinity :: "'a::real_normed_vector filter" where
22   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
24 lemma eventually_at_infinity:
25   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
26 unfolding at_infinity_def
27 proof (rule eventually_Abs_filter, rule is_filter.intro)
28   fix P Q :: "'a \<Rightarrow> bool"
29   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
30   then obtain r s where
31     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
32   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
33   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
34 qed auto
36 lemma at_infinity_eq_at_top_bot:
37   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
38   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
39 proof (intro arg_cong[where f=Abs_filter] ext iffI)
40   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
41   then guess r ..
42   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
43   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
44 next
45   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
46   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
47   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
48     by (intro exI[of _ "max p (-q)"])
49        (auto simp: abs_real_def)
50 qed
52 lemma at_top_le_at_infinity:
53   "at_top \<le> (at_infinity :: real filter)"
54   unfolding at_infinity_eq_at_top_bot by simp
56 lemma at_bot_le_at_infinity:
57   "at_bot \<le> (at_infinity :: real filter)"
58   unfolding at_infinity_eq_at_top_bot by simp
60 subsection {* Boundedness *}
62 lemma Bfun_def:
63   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
64   unfolding Bfun_metric_def norm_conv_dist
65 proof safe
66   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
67   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
68     by (intro always_eventually) (metis dist_commute dist_triangle)
69   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
70     by eventually_elim auto
71   with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
72     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
73 qed auto
75 lemma BfunI:
76   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
77 unfolding Bfun_def
78 proof (intro exI conjI allI)
79   show "0 < max K 1" by simp
80 next
81   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
82     using K by (rule eventually_elim1, simp)
83 qed
85 lemma BfunE:
86   assumes "Bfun f F"
87   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
88 using assms unfolding Bfun_def by fast
90 subsection {* Convergence to Zero *}
92 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
93   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
95 lemma ZfunI:
96   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
97   unfolding Zfun_def by simp
99 lemma ZfunD:
100   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
101   unfolding Zfun_def by simp
103 lemma Zfun_ssubst:
104   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
105   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
107 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
108   unfolding Zfun_def by simp
110 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
111   unfolding Zfun_def by simp
113 lemma Zfun_imp_Zfun:
114   assumes f: "Zfun f F"
115   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
116   shows "Zfun (\<lambda>x. g x) F"
117 proof (cases)
118   assume K: "0 < K"
119   show ?thesis
120   proof (rule ZfunI)
121     fix r::real assume "0 < r"
122     hence "0 < r / K"
123       using K by (rule divide_pos_pos)
124     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
125       using ZfunD [OF f] by fast
126     with g show "eventually (\<lambda>x. norm (g x) < r) F"
127     proof eventually_elim
128       case (elim x)
129       hence "norm (f x) * K < r"
130         by (simp add: pos_less_divide_eq K)
131       thus ?case
132         by (simp add: order_le_less_trans [OF elim(1)])
133     qed
134   qed
135 next
136   assume "\<not> 0 < K"
137   hence K: "K \<le> 0" by (simp only: not_less)
138   show ?thesis
139   proof (rule ZfunI)
140     fix r :: real
141     assume "0 < r"
142     from g show "eventually (\<lambda>x. norm (g x) < r) F"
143     proof eventually_elim
144       case (elim x)
145       also have "norm (f x) * K \<le> norm (f x) * 0"
146         using K norm_ge_zero by (rule mult_left_mono)
147       finally show ?case
148         using `0 < r` by simp
149     qed
150   qed
151 qed
153 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
154   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
157   assumes f: "Zfun f F" and g: "Zfun g F"
158   shows "Zfun (\<lambda>x. f x + g x) F"
159 proof (rule ZfunI)
160   fix r::real assume "0 < r"
161   hence r: "0 < r / 2" by simp
162   have "eventually (\<lambda>x. norm (f x) < r/2) F"
163     using f r by (rule ZfunD)
164   moreover
165   have "eventually (\<lambda>x. norm (g x) < r/2) F"
166     using g r by (rule ZfunD)
167   ultimately
168   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
169   proof eventually_elim
170     case (elim x)
171     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
172       by (rule norm_triangle_ineq)
173     also have "\<dots> < r/2 + r/2"
174       using elim by (rule add_strict_mono)
175     finally show ?case
176       by simp
177   qed
178 qed
180 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
181   unfolding Zfun_def by simp
183 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
184   by (simp only: diff_minus Zfun_add Zfun_minus)
186 lemma (in bounded_linear) Zfun:
187   assumes g: "Zfun g F"
188   shows "Zfun (\<lambda>x. f (g x)) F"
189 proof -
190   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
191     using bounded by fast
192   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
193     by simp
194   with g show ?thesis
195     by (rule Zfun_imp_Zfun)
196 qed
198 lemma (in bounded_bilinear) Zfun:
199   assumes f: "Zfun f F"
200   assumes g: "Zfun g F"
201   shows "Zfun (\<lambda>x. f x ** g x) F"
202 proof (rule ZfunI)
203   fix r::real assume r: "0 < r"
204   obtain K where K: "0 < K"
205     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
206     using pos_bounded by fast
207   from K have K': "0 < inverse K"
208     by (rule positive_imp_inverse_positive)
209   have "eventually (\<lambda>x. norm (f x) < r) F"
210     using f r by (rule ZfunD)
211   moreover
212   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
213     using g K' by (rule ZfunD)
214   ultimately
215   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
216   proof eventually_elim
217     case (elim x)
218     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
219       by (rule norm_le)
220     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
221       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
222     also from K have "r * inverse K * K = r"
223       by simp
224     finally show ?case .
225   qed
226 qed
228 lemma (in bounded_bilinear) Zfun_left:
229   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
230   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
232 lemma (in bounded_bilinear) Zfun_right:
233   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
234   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
236 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
237 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
238 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
240 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
241   by (simp only: tendsto_iff Zfun_def dist_norm)
243 subsubsection {* Distance and norms *}
245 lemma tendsto_norm [tendsto_intros]:
246   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
247   unfolding norm_conv_dist by (intro tendsto_intros)
249 lemma continuous_norm [continuous_intros]:
250   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
251   unfolding continuous_def by (rule tendsto_norm)
253 lemma continuous_on_norm [continuous_on_intros]:
254   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
255   unfolding continuous_on_def by (auto intro: tendsto_norm)
257 lemma tendsto_norm_zero:
258   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
259   by (drule tendsto_norm, simp)
261 lemma tendsto_norm_zero_cancel:
262   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
263   unfolding tendsto_iff dist_norm by simp
265 lemma tendsto_norm_zero_iff:
266   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
267   unfolding tendsto_iff dist_norm by simp
269 lemma tendsto_rabs [tendsto_intros]:
270   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
271   by (fold real_norm_def, rule tendsto_norm)
273 lemma continuous_rabs [continuous_intros]:
274   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
275   unfolding real_norm_def[symmetric] by (rule continuous_norm)
277 lemma continuous_on_rabs [continuous_on_intros]:
278   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
279   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
281 lemma tendsto_rabs_zero:
282   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
283   by (fold real_norm_def, rule tendsto_norm_zero)
285 lemma tendsto_rabs_zero_cancel:
286   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
287   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
289 lemma tendsto_rabs_zero_iff:
290   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
291   by (fold real_norm_def, rule tendsto_norm_zero_iff)
293 subsubsection {* Addition and subtraction *}
296   fixes a b :: "'a::real_normed_vector"
297   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
301   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
302   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
303   unfolding continuous_def by (rule tendsto_add)
306   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
307   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
308   unfolding continuous_on_def by (auto intro: tendsto_add)
311   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
312   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
313   by (drule (1) tendsto_add, simp)
315 lemma tendsto_minus [tendsto_intros]:
316   fixes a :: "'a::real_normed_vector"
317   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
318   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
320 lemma continuous_minus [continuous_intros]:
321   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
322   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
323   unfolding continuous_def by (rule tendsto_minus)
325 lemma continuous_on_minus [continuous_on_intros]:
326   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
327   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
328   unfolding continuous_on_def by (auto intro: tendsto_minus)
330 lemma tendsto_minus_cancel:
331   fixes a :: "'a::real_normed_vector"
332   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
333   by (drule tendsto_minus, simp)
335 lemma tendsto_minus_cancel_left:
336     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
337   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
338   by auto
340 lemma tendsto_diff [tendsto_intros]:
341   fixes a b :: "'a::real_normed_vector"
342   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
345 lemma continuous_diff [continuous_intros]:
346   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
347   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
348   unfolding continuous_def by (rule tendsto_diff)
350 lemma continuous_on_diff [continuous_on_intros]:
351   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
352   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
353   unfolding continuous_on_def by (auto intro: tendsto_diff)
355 lemma tendsto_setsum [tendsto_intros]:
356   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
357   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
358   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
359 proof (cases "finite S")
360   assume "finite S" thus ?thesis using assms
362 next
363   assume "\<not> finite S" thus ?thesis
365 qed
367 lemma continuous_setsum [continuous_intros]:
368   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
369   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
370   unfolding continuous_def by (rule tendsto_setsum)
372 lemma continuous_on_setsum [continuous_intros]:
373   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
374   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
375   unfolding continuous_on_def by (auto intro: tendsto_setsum)
377 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
379 subsubsection {* Linear operators and multiplication *}
381 lemma (in bounded_linear) tendsto:
382   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
383   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
385 lemma (in bounded_linear) continuous:
386   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
387   using tendsto[of g _ F] by (auto simp: continuous_def)
389 lemma (in bounded_linear) continuous_on:
390   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
391   using tendsto[of g] by (auto simp: continuous_on_def)
393 lemma (in bounded_linear) tendsto_zero:
394   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
395   by (drule tendsto, simp only: zero)
397 lemma (in bounded_bilinear) tendsto:
398   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
399   by (simp only: tendsto_Zfun_iff prod_diff_prod
402 lemma (in bounded_bilinear) continuous:
403   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
404   using tendsto[of f _ F g] by (auto simp: continuous_def)
406 lemma (in bounded_bilinear) continuous_on:
407   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
408   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
410 lemma (in bounded_bilinear) tendsto_zero:
411   assumes f: "(f ---> 0) F"
412   assumes g: "(g ---> 0) F"
413   shows "((\<lambda>x. f x ** g x) ---> 0) F"
414   using tendsto [OF f g] by (simp add: zero_left)
416 lemma (in bounded_bilinear) tendsto_left_zero:
417   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
418   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
420 lemma (in bounded_bilinear) tendsto_right_zero:
421   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
422   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
424 lemmas tendsto_of_real [tendsto_intros] =
425   bounded_linear.tendsto [OF bounded_linear_of_real]
427 lemmas tendsto_scaleR [tendsto_intros] =
428   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
430 lemmas tendsto_mult [tendsto_intros] =
431   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
433 lemmas continuous_of_real [continuous_intros] =
434   bounded_linear.continuous [OF bounded_linear_of_real]
436 lemmas continuous_scaleR [continuous_intros] =
437   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
439 lemmas continuous_mult [continuous_intros] =
440   bounded_bilinear.continuous [OF bounded_bilinear_mult]
442 lemmas continuous_on_of_real [continuous_on_intros] =
443   bounded_linear.continuous_on [OF bounded_linear_of_real]
445 lemmas continuous_on_scaleR [continuous_on_intros] =
446   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
448 lemmas continuous_on_mult [continuous_on_intros] =
449   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
451 lemmas tendsto_mult_zero =
452   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
454 lemmas tendsto_mult_left_zero =
455   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
457 lemmas tendsto_mult_right_zero =
458   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
460 lemma tendsto_power [tendsto_intros]:
461   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
462   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
463   by (induct n) (simp_all add: tendsto_const tendsto_mult)
465 lemma continuous_power [continuous_intros]:
466   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
467   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
468   unfolding continuous_def by (rule tendsto_power)
470 lemma continuous_on_power [continuous_on_intros]:
471   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
472   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
473   unfolding continuous_on_def by (auto intro: tendsto_power)
475 lemma tendsto_setprod [tendsto_intros]:
476   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
477   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
478   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
479 proof (cases "finite S")
480   assume "finite S" thus ?thesis using assms
482 next
483   assume "\<not> finite S" thus ?thesis
485 qed
487 lemma continuous_setprod [continuous_intros]:
488   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
489   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
490   unfolding continuous_def by (rule tendsto_setprod)
492 lemma continuous_on_setprod [continuous_intros]:
493   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
494   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
495   unfolding continuous_on_def by (auto intro: tendsto_setprod)
497 subsubsection {* Inverse and division *}
499 lemma (in bounded_bilinear) Zfun_prod_Bfun:
500   assumes f: "Zfun f F"
501   assumes g: "Bfun g F"
502   shows "Zfun (\<lambda>x. f x ** g x) F"
503 proof -
504   obtain K where K: "0 \<le> K"
505     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
506     using nonneg_bounded by fast
507   obtain B where B: "0 < B"
508     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
509     using g by (rule BfunE)
510   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
511   using norm_g proof eventually_elim
512     case (elim x)
513     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
514       by (rule norm_le)
515     also have "\<dots> \<le> norm (f x) * B * K"
516       by (intro mult_mono' order_refl norm_g norm_ge_zero
517                 mult_nonneg_nonneg K elim)
518     also have "\<dots> = norm (f x) * (B * K)"
519       by (rule mult_assoc)
520     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
521   qed
522   with f show ?thesis
523     by (rule Zfun_imp_Zfun)
524 qed
526 lemma (in bounded_bilinear) flip:
527   "bounded_bilinear (\<lambda>x y. y ** x)"
528   apply default
531   apply (rule scaleR_right)
532   apply (rule scaleR_left)
533   apply (subst mult_commute)
534   using bounded by fast
536 lemma (in bounded_bilinear) Bfun_prod_Zfun:
537   assumes f: "Bfun f F"
538   assumes g: "Zfun g F"
539   shows "Zfun (\<lambda>x. f x ** g x) F"
540   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
542 lemma Bfun_inverse_lemma:
543   fixes x :: "'a::real_normed_div_algebra"
544   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
545   apply (subst nonzero_norm_inverse, clarsimp)
546   apply (erule (1) le_imp_inverse_le)
547   done
549 lemma Bfun_inverse:
550   fixes a :: "'a::real_normed_div_algebra"
551   assumes f: "(f ---> a) F"
552   assumes a: "a \<noteq> 0"
553   shows "Bfun (\<lambda>x. inverse (f x)) F"
554 proof -
555   from a have "0 < norm a" by simp
556   hence "\<exists>r>0. r < norm a" by (rule dense)
557   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
558   have "eventually (\<lambda>x. dist (f x) a < r) F"
559     using tendstoD [OF f r1] by fast
560   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
561   proof eventually_elim
562     case (elim x)
563     hence 1: "norm (f x - a) < r"
565     hence 2: "f x \<noteq> 0" using r2 by auto
566     hence "norm (inverse (f x)) = inverse (norm (f x))"
567       by (rule nonzero_norm_inverse)
568     also have "\<dots> \<le> inverse (norm a - r)"
569     proof (rule le_imp_inverse_le)
570       show "0 < norm a - r" using r2 by simp
571     next
572       have "norm a - norm (f x) \<le> norm (a - f x)"
573         by (rule norm_triangle_ineq2)
574       also have "\<dots> = norm (f x - a)"
575         by (rule norm_minus_commute)
576       also have "\<dots> < r" using 1 .
577       finally show "norm a - r \<le> norm (f x)" by simp
578     qed
579     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
580   qed
581   thus ?thesis by (rule BfunI)
582 qed
584 lemma tendsto_inverse [tendsto_intros]:
585   fixes a :: "'a::real_normed_div_algebra"
586   assumes f: "(f ---> a) F"
587   assumes a: "a \<noteq> 0"
588   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
589 proof -
590   from a have "0 < norm a" by simp
591   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
592     by (rule tendstoD)
593   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
594     unfolding dist_norm by (auto elim!: eventually_elim1)
595   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
596     - (inverse (f x) * (f x - a) * inverse a)) F"
597     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
598   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
599     by (intro Zfun_minus Zfun_mult_left
600       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
601       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
602   ultimately show ?thesis
603     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
604 qed
606 lemma continuous_inverse:
607   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
608   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
609   shows "continuous F (\<lambda>x. inverse (f x))"
610   using assms unfolding continuous_def by (rule tendsto_inverse)
612 lemma continuous_at_within_inverse[continuous_intros]:
613   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
614   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
615   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
616   using assms unfolding continuous_within by (rule tendsto_inverse)
618 lemma isCont_inverse[continuous_intros, simp]:
619   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
620   assumes "isCont f a" and "f a \<noteq> 0"
621   shows "isCont (\<lambda>x. inverse (f x)) a"
622   using assms unfolding continuous_at by (rule tendsto_inverse)
624 lemma continuous_on_inverse[continuous_on_intros]:
625   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
626   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
627   shows "continuous_on s (\<lambda>x. inverse (f x))"
628   using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
630 lemma tendsto_divide [tendsto_intros]:
631   fixes a b :: "'a::real_normed_field"
632   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
633     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
634   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
636 lemma continuous_divide:
637   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
638   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
639   shows "continuous F (\<lambda>x. (f x) / (g x))"
640   using assms unfolding continuous_def by (rule tendsto_divide)
642 lemma continuous_at_within_divide[continuous_intros]:
643   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
644   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
645   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
646   using assms unfolding continuous_within by (rule tendsto_divide)
648 lemma isCont_divide[continuous_intros, simp]:
649   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
650   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
651   shows "isCont (\<lambda>x. (f x) / g x) a"
652   using assms unfolding continuous_at by (rule tendsto_divide)
654 lemma continuous_on_divide[continuous_on_intros]:
655   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
656   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
657   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
658   using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
660 lemma tendsto_sgn [tendsto_intros]:
661   fixes l :: "'a::real_normed_vector"
662   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
663   unfolding sgn_div_norm by (simp add: tendsto_intros)
665 lemma continuous_sgn:
666   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
667   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
668   shows "continuous F (\<lambda>x. sgn (f x))"
669   using assms unfolding continuous_def by (rule tendsto_sgn)
671 lemma continuous_at_within_sgn[continuous_intros]:
672   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
673   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
674   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
675   using assms unfolding continuous_within by (rule tendsto_sgn)
677 lemma isCont_sgn[continuous_intros]:
678   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
679   assumes "isCont f a" and "f a \<noteq> 0"
680   shows "isCont (\<lambda>x. sgn (f x)) a"
681   using assms unfolding continuous_at by (rule tendsto_sgn)
683 lemma continuous_on_sgn[continuous_on_intros]:
684   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
685   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
686   shows "continuous_on s (\<lambda>x. sgn (f x))"
687   using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
689 lemma filterlim_at_infinity:
690   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
691   assumes "0 \<le> c"
692   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
693   unfolding filterlim_iff eventually_at_infinity
694 proof safe
695   fix P :: "'a \<Rightarrow> bool" and b
696   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
697     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
698   have "max b (c + 1) > c" by auto
699   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
700     by auto
701   then show "eventually (\<lambda>x. P (f x)) F"
702   proof eventually_elim
703     fix x assume "max b (c + 1) \<le> norm (f x)"
704     with P show "P (f x)" by auto
705   qed
706 qed force
708 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
710 text {*
712 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
713 @{term "at_right x"} and also @{term "at_right 0"}.
715 *}
717 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
719 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
720   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
721   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
723 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
724   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
725   apply (intro allI ex_cong)
726   apply (auto simp: dist_real_def field_simps)
727   apply (erule_tac x="-x" in allE)
728   apply simp
729   done
731 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
732   unfolding at_def filtermap_nhds_shift[symmetric]
733   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
735 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
736   unfolding filtermap_at_shift[symmetric]
737   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
739 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
740   using filtermap_at_right_shift[of "-a" 0] by simp
742 lemma filterlim_at_right_to_0:
743   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
744   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
746 lemma eventually_at_right_to_0:
747   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
748   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
750 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
751   unfolding at_def filtermap_nhds_minus[symmetric]
752   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
754 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
755   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
757 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
758   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
760 lemma filterlim_at_left_to_right:
761   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
762   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
764 lemma eventually_at_left_to_right:
765   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
766   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
768 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
769   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
770   by (metis le_minus_iff minus_minus)
772 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
773   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
775 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
776   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
778 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
779   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
781 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
782   unfolding filterlim_at_top eventually_at_bot_dense
783   by (metis leI minus_less_iff order_less_asym)
785 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
786   unfolding filterlim_at_bot eventually_at_top_dense
787   by (metis leI less_minus_iff order_less_asym)
789 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
790   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
791   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
792   by auto
794 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
795   unfolding filterlim_uminus_at_top by simp
797 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
798   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
799 proof safe
800   fix Z :: real assume [arith]: "0 < Z"
801   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
802     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
803   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
804     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
805 qed
807 lemma filterlim_inverse_at_top:
808   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
809   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
810      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
812 lemma filterlim_inverse_at_bot_neg:
813   "LIM x (at_left (0::real)). inverse x :> at_bot"
814   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
816 lemma filterlim_inverse_at_bot:
817   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
818   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
819   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
821 lemma tendsto_inverse_0:
822   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
823   shows "(inverse ---> (0::'a)) at_infinity"
824   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
825 proof safe
826   fix r :: real assume "0 < r"
827   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
828   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
829     fix x :: 'a
830     from `0 < r` have "0 < inverse (r / 2)" by simp
831     also assume *: "inverse (r / 2) \<le> norm x"
832     finally show "norm (inverse x) < r"
833       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
834   qed
835 qed
837 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
838 proof (rule antisym)
839   have "(inverse ---> (0::real)) at_top"
840     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
841   then show "filtermap inverse at_top \<le> at_right (0::real)"
842     unfolding at_within_eq
843     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
844 next
845   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
846     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
847   then show "at_right (0::real) \<le> filtermap inverse at_top"
848     by (simp add: filtermap_ident filtermap_filtermap)
849 qed
851 lemma eventually_at_right_to_top:
852   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
853   unfolding at_right_to_top eventually_filtermap ..
855 lemma filterlim_at_right_to_top:
856   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
857   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
859 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
860   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
862 lemma eventually_at_top_to_right:
863   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
864   unfolding at_top_to_right eventually_filtermap ..
866 lemma filterlim_at_top_to_right:
867   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
868   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
870 lemma filterlim_inverse_at_infinity:
871   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
872   shows "filterlim inverse at_infinity (at (0::'a))"
873   unfolding filterlim_at_infinity[OF order_refl]
874 proof safe
875   fix r :: real assume "0 < r"
876   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
877     unfolding eventually_at norm_inverse
878     by (intro exI[of _ "inverse r"])
879        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
880 qed
882 lemma filterlim_inverse_at_iff:
883   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
884   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
885   unfolding filterlim_def filtermap_filtermap[symmetric]
886 proof
887   assume "filtermap g F \<le> at_infinity"
888   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
889     by (rule filtermap_mono)
890   also have "\<dots> \<le> at 0"
891     using tendsto_inverse_0
892     by (auto intro!: le_withinI exI[of _ 1]
893              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
894   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
895 next
896   assume "filtermap inverse (filtermap g F) \<le> at 0"
897   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
898     by (rule filtermap_mono)
899   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
900     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
901 qed
903 lemma tendsto_inverse_0_at_top:
904   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
905  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
907 text {*
909 We only show rules for multiplication and addition when the functions are either against a real
910 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
912 *}
914 lemma filterlim_tendsto_pos_mult_at_top:
915   assumes f: "(f ---> c) F" and c: "0 < c"
916   assumes g: "LIM x F. g x :> at_top"
917   shows "LIM x F. (f x * g x :: real) :> at_top"
918   unfolding filterlim_at_top_gt[where c=0]
919 proof safe
920   fix Z :: real assume "0 < Z"
921   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
922     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
923              simp: dist_real_def abs_real_def split: split_if_asm)
924   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
925     unfolding filterlim_at_top by auto
926   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
927   proof eventually_elim
928     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
929     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
930       by (intro mult_mono) (auto simp: zero_le_divide_iff)
931     with `0 < c` show "Z \<le> f x * g x"
932        by simp
933   qed
934 qed
936 lemma filterlim_at_top_mult_at_top:
937   assumes f: "LIM x F. f x :> at_top"
938   assumes g: "LIM x F. g x :> at_top"
939   shows "LIM x F. (f x * g x :: real) :> at_top"
940   unfolding filterlim_at_top_gt[where c=0]
941 proof safe
942   fix Z :: real assume "0 < Z"
943   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
944     unfolding filterlim_at_top by auto
945   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
946     unfolding filterlim_at_top by auto
947   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
948   proof eventually_elim
949     fix x assume "1 \<le> f x" "Z \<le> g x"
950     with `0 < Z` have "1 * Z \<le> f x * g x"
951       by (intro mult_mono) (auto simp: zero_le_divide_iff)
952     then show "Z \<le> f x * g x"
953        by simp
954   qed
955 qed
957 lemma filterlim_tendsto_pos_mult_at_bot:
958   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
959   shows "LIM x F. f x * g x :> at_bot"
960   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
961   unfolding filterlim_uminus_at_bot by simp
964   assumes f: "(f ---> c) F"
965   assumes g: "LIM x F. g x :> at_top"
966   shows "LIM x F. (f x + g x :: real) :> at_top"
967   unfolding filterlim_at_top_gt[where c=0]
968 proof safe
969   fix Z :: real assume "0 < Z"
970   from f have "eventually (\<lambda>x. c - 1 < f x) F"
971     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
972   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
973     unfolding filterlim_at_top by auto
974   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
975     by eventually_elim simp
976 qed
978 lemma LIM_at_top_divide:
979   fixes f g :: "'a \<Rightarrow> real"
980   assumes f: "(f ---> a) F" "0 < a"
981   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
982   shows "LIM x F. f x / g x :> at_top"
983   unfolding divide_inverse
984   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
987   assumes f: "LIM x F. f x :> at_top"
988   assumes g: "LIM x F. g x :> at_top"
989   shows "LIM x F. (f x + g x :: real) :> at_top"
990   unfolding filterlim_at_top_gt[where c=0]
991 proof safe
992   fix Z :: real assume "0 < Z"
993   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
994     unfolding filterlim_at_top by auto
995   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
996     unfolding filterlim_at_top by auto
997   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
998     by eventually_elim simp
999 qed
1001 lemma tendsto_divide_0:
1002   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
1003   assumes f: "(f ---> c) F"
1004   assumes g: "LIM x F. g x :> at_infinity"
1005   shows "((\<lambda>x. f x / g x) ---> 0) F"
1006   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
1008 lemma linear_plus_1_le_power:
1009   fixes x :: real
1010   assumes x: "0 \<le> x"
1011   shows "real n * x + 1 \<le> (x + 1) ^ n"
1012 proof (induct n)
1013   case (Suc n)
1014   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
1015     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
1016   also have "\<dots> \<le> (x + 1)^Suc n"
1017     using Suc x by (simp add: mult_left_mono)
1018   finally show ?case .
1019 qed simp
1021 lemma filterlim_realpow_sequentially_gt1:
1022   fixes x :: "'a :: real_normed_div_algebra"
1023   assumes x[arith]: "1 < norm x"
1024   shows "LIM n sequentially. x ^ n :> at_infinity"
1025 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
1026   fix y :: real assume "0 < y"
1027   have "0 < norm x - 1" by simp
1028   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
1029   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
1030   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
1031   also have "\<dots> = norm x ^ N" by simp
1032   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
1033     by (metis order_less_le_trans power_increasing order_less_imp_le x)
1034   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
1035     unfolding eventually_sequentially
1036     by (auto simp: norm_power)
1037 qed simp
1040 subsection {* Limits of Sequences *}
1042 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
1043   by simp
1045 lemma LIMSEQ_iff:
1046   fixes L :: "'a::real_normed_vector"
1047   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
1048 unfolding LIMSEQ_def dist_norm ..
1050 lemma LIMSEQ_I:
1051   fixes L :: "'a::real_normed_vector"
1052   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
1055 lemma LIMSEQ_D:
1056   fixes L :: "'a::real_normed_vector"
1057   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
1060 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
1061   unfolding tendsto_def eventually_sequentially
1062   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
1064 lemma Bseq_inverse_lemma:
1065   fixes x :: "'a::real_normed_div_algebra"
1066   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
1067 apply (subst nonzero_norm_inverse, clarsimp)
1068 apply (erule (1) le_imp_inverse_le)
1069 done
1071 lemma Bseq_inverse:
1072   fixes a :: "'a::real_normed_div_algebra"
1073   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
1074   by (rule Bfun_inverse)
1076 lemma LIMSEQ_diff_approach_zero:
1077   fixes L :: "'a::real_normed_vector"
1078   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
1079   by (drule (1) tendsto_add, simp)
1081 lemma LIMSEQ_diff_approach_zero2:
1082   fixes L :: "'a::real_normed_vector"
1083   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
1084   by (drule (1) tendsto_diff, simp)
1086 text{*An unbounded sequence's inverse tends to 0*}
1088 lemma LIMSEQ_inverse_zero:
1089   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
1090   apply (rule filterlim_compose[OF tendsto_inverse_0])
1091   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
1092   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
1093   done
1095 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
1097 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
1098   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
1099             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
1101 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
1102 infinity is now easily proved*}
1105      "(%n. r + inverse(real(Suc n))) ----> r"
1106   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
1109      "(%n. r + -inverse(real(Suc n))) ----> r"
1110   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
1111   by auto
1114      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
1115   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
1116   by auto
1118 subsection {* Convergence on sequences *}
1121   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
1122   assumes "convergent (\<lambda>n. X n)"
1123   assumes "convergent (\<lambda>n. Y n)"
1124   shows "convergent (\<lambda>n. X n + Y n)"
1125   using assms unfolding convergent_def by (fast intro: tendsto_add)
1127 lemma convergent_setsum:
1128   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
1129   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
1130   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
1131 proof (cases "finite A")
1132   case True from this and assms show ?thesis
1136 lemma (in bounded_linear) convergent:
1137   assumes "convergent (\<lambda>n. X n)"
1138   shows "convergent (\<lambda>n. f (X n))"
1139   using assms unfolding convergent_def by (fast intro: tendsto)
1141 lemma (in bounded_bilinear) convergent:
1142   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
1143   shows "convergent (\<lambda>n. X n ** Y n)"
1144   using assms unfolding convergent_def by (fast intro: tendsto)
1146 lemma convergent_minus_iff:
1147   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1148   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
1150 apply (auto dest: tendsto_minus)
1151 apply (drule tendsto_minus, auto)
1152 done
1154 subsection {* Bounded Monotonic Sequences *}
1156 subsubsection {* Bounded Sequences *}
1158 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
1159   by (intro BfunI) (auto simp: eventually_sequentially)
1161 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
1162   by (intro BfunI) (auto simp: eventually_sequentially)
1164 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
1165   unfolding Bfun_def eventually_sequentially
1166 proof safe
1167   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
1168   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
1169     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
1170        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
1171 qed auto
1173 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
1174 unfolding Bseq_def by auto
1176 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
1179 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
1180 by (auto simp add: Bseq_def)
1182 lemma lemma_NBseq_def:
1183   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
1184 proof safe
1185   fix K :: real
1186   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
1187   then have "K \<le> real (Suc n)" by auto
1188   moreover assume "\<forall>m. norm (X m) \<le> K"
1189   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
1190     by (blast intro: order_trans)
1191   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
1192 qed (force simp add: real_of_nat_Suc)
1194 text{* alternative definition for Bseq *}
1195 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
1197 apply (simp (no_asm) add: lemma_NBseq_def)
1198 done
1200 lemma lemma_NBseq_def2:
1201      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
1202 apply (subst lemma_NBseq_def, auto)
1203 apply (rule_tac x = "Suc N" in exI)
1204 apply (rule_tac [2] x = N in exI)
1205 apply (auto simp add: real_of_nat_Suc)
1206  prefer 2 apply (blast intro: order_less_imp_le)
1207 apply (drule_tac x = n in spec, simp)
1208 done
1210 (* yet another definition for Bseq *)
1211 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
1212 by (simp add: Bseq_def lemma_NBseq_def2)
1214 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
1216 text{*alternative formulation for boundedness*}
1217 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
1218 apply (unfold Bseq_def, safe)
1219 apply (rule_tac [2] x = "k + norm x" in exI)
1220 apply (rule_tac x = K in exI, simp)
1221 apply (rule exI [where x = 0], auto)
1222 apply (erule order_less_le_trans, simp)
1223 apply (drule_tac x=n in spec, fold diff_minus)
1224 apply (drule order_trans [OF norm_triangle_ineq2])
1225 apply simp
1226 done
1228 text{*alternative formulation for boundedness*}
1229 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
1230 apply safe
1231 apply (simp add: Bseq_def, safe)
1232 apply (rule_tac x = "K + norm (X N)" in exI)
1233 apply auto
1234 apply (erule order_less_le_trans, simp)
1235 apply (rule_tac x = N in exI, safe)
1236 apply (drule_tac x = n in spec)
1237 apply (rule order_trans [OF norm_triangle_ineq], simp)
1238 apply (auto simp add: Bseq_iff2)
1239 done
1241 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
1243 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
1244 apply (drule_tac x = n in spec, arith)
1245 done
1247 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
1249 lemma Bseq_isUb:
1250   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
1251 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
1253 text{* Use completeness of reals (supremum property)
1254    to show that any bounded sequence has a least upper bound*}
1256 lemma Bseq_isLub:
1257   "!!(X::nat=>real). Bseq X ==>
1258    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
1259 by (blast intro: reals_complete Bseq_isUb)
1261 subsubsection{*A Bounded and Monotonic Sequence Converges*}
1263 (* TODO: delete *)
1264 (* FIXME: one use in NSA/HSEQ.thy *)
1265 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
1266   apply (rule_tac x="X m" in exI)
1267   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
1268   unfolding eventually_sequentially
1269   apply blast
1270   done
1272 text {* A monotone sequence converges to its least upper bound. *}
1274 lemma isLub_mono_imp_LIMSEQ:
1275   fixes X :: "nat \<Rightarrow> real"
1276   assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
1277   assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
1278   shows "X ----> u"
1279 proof (rule LIMSEQ_I)
1280   have 1: "\<forall>n. X n \<le> u"
1281     using isLubD2 [OF u] by auto
1282   have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
1283     using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
1284   hence 2: "\<forall>y<u. \<exists>n. y < X n"
1285     by (metis not_le)
1286   fix r :: real assume "0 < r"
1287   hence "u - r < u" by simp
1288   hence "\<exists>m. u - r < X m" using 2 by simp
1289   then obtain m where "u - r < X m" ..
1290   with X have "\<forall>n\<ge>m. u - r < X n"
1291     by (fast intro: less_le_trans)
1292   hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
1293   thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
1295 qed
1297 text{*A standard proof of the theorem for monotone increasing sequence*}
1299 lemma Bseq_mono_convergent:
1300    "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
1301   by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
1303 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
1306 text{*Main monotonicity theorem*}
1308 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
1309   by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
1310             Bseq_mono_convergent)
1312 lemma Cauchy_iff:
1313   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1314   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
1315   unfolding Cauchy_def dist_norm ..
1317 lemma CauchyI:
1318   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1319   shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
1322 lemma CauchyD:
1323   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1324   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
1327 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
1329   apply (rule BseqI'[where K="max (norm a) (norm b)"])
1330   apply (erule_tac x=n in allE)
1331   apply auto
1332   done
1334 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
1335   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
1337 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
1338   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
1340 lemma incseq_convergent:
1341   fixes X :: "nat \<Rightarrow> real"
1342   assumes "incseq X" and "\<forall>i. X i \<le> B"
1343   obtains L where "X ----> L" "\<forall>i. X i \<le> L"
1344 proof atomize_elim
1345   from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
1346   obtain L where "X ----> L"
1347     by (auto simp: convergent_def monoseq_def incseq_def)
1348   with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
1349     by (auto intro!: exI[of _ L] incseq_le)
1350 qed
1352 lemma decseq_convergent:
1353   fixes X :: "nat \<Rightarrow> real"
1354   assumes "decseq X" and "\<forall>i. B \<le> X i"
1355   obtains L where "X ----> L" "\<forall>i. L \<le> X i"
1356 proof atomize_elim
1357   from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
1358   obtain L where "X ----> L"
1359     by (auto simp: convergent_def monoseq_def decseq_def)
1360   with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
1361     by (auto intro!: exI[of _ L] decseq_le)
1362 qed
1364 subsubsection {* Cauchy Sequences are Bounded *}
1366 text{*A Cauchy sequence is bounded -- this is the standard
1367   proof mechanization rather than the nonstandard proof*}
1369 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
1370           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
1371 apply (clarify, drule spec, drule (1) mp)
1372 apply (simp only: norm_minus_commute)
1373 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
1374 apply simp
1375 done
1377 class banach = real_normed_vector + complete_space
1379 instance real :: banach by default
1381 subsection {* Power Sequences *}
1383 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
1384 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
1385   also fact that bounded and monotonic sequence converges.*}
1387 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
1389 apply (rule_tac x = 1 in exI)
1391 apply (auto dest: power_mono)
1392 done
1394 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
1395 apply (clarify intro!: mono_SucI2)
1396 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
1397 done
1399 lemma convergent_realpow:
1400   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
1401 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
1403 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
1404   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
1406 lemma LIMSEQ_realpow_zero:
1407   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1408 proof cases
1409   assume "0 \<le> x" and "x \<noteq> 0"
1410   hence x0: "0 < x" by simp
1411   assume x1: "x < 1"
1412   from x0 x1 have "1 < inverse x"
1413     by (rule one_less_inverse)
1414   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
1415     by (rule LIMSEQ_inverse_realpow_zero)
1416   thus ?thesis by (simp add: power_inverse)
1417 qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
1419 lemma LIMSEQ_power_zero:
1420   fixes x :: "'a::{real_normed_algebra_1}"
1421   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1422 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
1423 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
1424 apply (simp add: power_abs norm_power_ineq)
1425 done
1427 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
1428   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
1430 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
1432 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
1433   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
1435 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
1436   by (rule LIMSEQ_power_zero) simp
1439 subsection {* Limits of Functions *}
1441 lemma LIM_eq:
1442   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1443   shows "f -- a --> L =
1444      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
1445 by (simp add: LIM_def dist_norm)
1447 lemma LIM_I:
1448   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1449   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
1450       ==> f -- a --> L"
1453 lemma LIM_D:
1454   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1455   shows "[| f -- a --> L; 0<r |]
1456       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
1459 lemma LIM_offset:
1460   fixes a :: "'a::real_normed_vector"
1461   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
1462 apply (rule topological_tendstoI)
1463 apply (drule (2) topological_tendstoD)
1464 apply (simp only: eventually_at dist_norm)
1465 apply (clarify, rule_tac x=d in exI, safe)
1466 apply (drule_tac x="x + k" in spec)
1468 done
1470 lemma LIM_offset_zero:
1471   fixes a :: "'a::real_normed_vector"
1472   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
1475 lemma LIM_offset_zero_cancel:
1476   fixes a :: "'a::real_normed_vector"
1477   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
1478 by (drule_tac k="- a" in LIM_offset, simp)
1480 lemma LIM_zero:
1481   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1482   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
1483 unfolding tendsto_iff dist_norm by simp
1485 lemma LIM_zero_cancel:
1486   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1487   shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
1488 unfolding tendsto_iff dist_norm by simp
1490 lemma LIM_zero_iff:
1491   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1492   shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
1493 unfolding tendsto_iff dist_norm by simp
1495 lemma LIM_imp_LIM:
1496   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1497   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
1498   assumes f: "f -- a --> l"
1499   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
1500   shows "g -- a --> m"
1501   by (rule metric_LIM_imp_LIM [OF f],
1504 lemma LIM_equal2:
1505   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1506   assumes 1: "0 < R"
1507   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
1508   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
1509 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
1511 lemma LIM_compose2:
1512   fixes a :: "'a::real_normed_vector"
1513   assumes f: "f -- a --> b"
1514   assumes g: "g -- b --> c"
1515   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
1516   shows "(\<lambda>x. g (f x)) -- a --> c"
1517 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
1519 lemma real_LIM_sandwich_zero:
1520   fixes f g :: "'a::topological_space \<Rightarrow> real"
1521   assumes f: "f -- a --> 0"
1522   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
1523   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
1524   shows "g -- a --> 0"
1525 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
1526   fix x assume x: "x \<noteq> a"
1527   have "norm (g x - 0) = g x" by (simp add: 1 x)
1528   also have "g x \<le> f x" by (rule 2 [OF x])
1529   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
1530   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
1531   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
1532 qed
1535 subsection {* Continuity *}
1537 lemma LIM_isCont_iff:
1538   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1539   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
1540 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
1542 lemma isCont_iff:
1543   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1544   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
1545 by (simp add: isCont_def LIM_isCont_iff)
1547 lemma isCont_LIM_compose2:
1548   fixes a :: "'a::real_normed_vector"
1549   assumes f [unfolded isCont_def]: "isCont f a"
1550   assumes g: "g -- f a --> l"
1551   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
1552   shows "(\<lambda>x. g (f x)) -- a --> l"
1553 by (rule LIM_compose2 [OF f g inj])
1556 lemma isCont_norm [simp]:
1557   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1558   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
1559   by (fact continuous_norm)
1561 lemma isCont_rabs [simp]:
1562   fixes f :: "'a::t2_space \<Rightarrow> real"
1563   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
1564   by (fact continuous_rabs)
1567   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1568   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
1571 lemma isCont_minus [simp]:
1572   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1573   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
1574   by (fact continuous_minus)
1576 lemma isCont_diff [simp]:
1577   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1578   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
1579   by (fact continuous_diff)
1581 lemma isCont_mult [simp]:
1582   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
1583   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
1584   by (fact continuous_mult)
1586 lemma (in bounded_linear) isCont:
1587   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
1588   by (fact continuous)
1590 lemma (in bounded_bilinear) isCont:
1591   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
1592   by (fact continuous)
1594 lemmas isCont_scaleR [simp] =
1595   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
1597 lemmas isCont_of_real [simp] =
1598   bounded_linear.isCont [OF bounded_linear_of_real]
1600 lemma isCont_power [simp]:
1601   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
1602   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
1603   by (fact continuous_power)
1605 lemma isCont_setsum [simp]:
1606   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
1607   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
1608   by (auto intro: continuous_setsum)
1610 lemmas isCont_intros =
1611   isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
1612   isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
1613   isCont_of_real isCont_power isCont_sgn isCont_setsum
1615 subsection {* Uniform Continuity *}
1617 lemma (in bounded_linear) isUCont: "isUCont f"
1618 unfolding isUCont_def dist_norm
1619 proof (intro allI impI)
1620   fix r::real assume r: "0 < r"
1621   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
1622     using pos_bounded by fast
1623   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
1624   proof (rule exI, safe)
1625     from r K show "0 < r / K" by (rule divide_pos_pos)
1626   next
1627     fix x y :: 'a
1628     assume xy: "norm (x - y) < r / K"
1629     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
1630     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
1631     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
1632     finally show "norm (f x - f y) < r" .
1633   qed
1634 qed
1636 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
1637 by (rule isUCont [THEN isUCont_Cauchy])
1640 lemma LIM_less_bound:
1641   fixes f :: "real \<Rightarrow> real"
1642   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
1643   shows "0 \<le> f x"
1644 proof (rule tendsto_le_const)
1645   show "(f ---> f x) (at_left x)"
1646     using `isCont f x` by (simp add: filterlim_at_split isCont_def)
1647   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
1648     using ev by (auto simp: eventually_within_less dist_real_def intro!: exI[of _ "x - b"])
1649 qed simp
1651 end