src/HOL/Limits.thy
 author huffman Sun Apr 25 11:58:39 2010 -0700 (2010-04-25) changeset 36358 246493d61204 parent 31902 862ae16a799d child 36360 9d8f7efd9289 permissions -rw-r--r--
define nets directly as filters, instead of as filter bases
1 (*  Title       : Limits.thy
2     Author      : Brian Huffman
3 *)
5 header {* Filters and Limits *}
7 theory Limits
8 imports RealVector RComplete
9 begin
11 subsection {* Nets *}
13 text {*
14   A net is now defined simply as a filter.
15   The definition also allows non-proper filters.
16 *}
18 locale is_filter =
19   fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
20   assumes True: "net (\<lambda>x. True)"
21   assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
24 typedef (open) 'a net =
25   "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
26 proof
27   show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
28 qed
30 lemma is_filter_Rep_net: "is_filter (Rep_net net)"
31 using Rep_net [of net] by simp
33 lemma Abs_net_inverse':
34   assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
35 using assms by (simp add: Abs_net_inverse)
38 subsection {* Eventually *}
40 definition
41   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
42   [code del]: "eventually P net \<longleftrightarrow> Rep_net net P"
44 lemma eventually_Abs_net:
45   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
46 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
48 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
49 unfolding eventually_def
50 by (rule is_filter.True [OF is_filter_Rep_net])
52 lemma eventually_mono:
53   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
54 unfolding eventually_def
55 by (rule is_filter.mono [OF is_filter_Rep_net])
57 lemma eventually_conj:
58   assumes P: "eventually (\<lambda>x. P x) net"
59   assumes Q: "eventually (\<lambda>x. Q x) net"
60   shows "eventually (\<lambda>x. P x \<and> Q x) net"
61 using assms unfolding eventually_def
62 by (rule is_filter.conj [OF is_filter_Rep_net])
64 lemma eventually_mp:
65   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
66   assumes "eventually (\<lambda>x. P x) net"
67   shows "eventually (\<lambda>x. Q x) net"
68 proof (rule eventually_mono)
69   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
70   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
71     using assms by (rule eventually_conj)
72 qed
74 lemma eventually_rev_mp:
75   assumes "eventually (\<lambda>x. P x) net"
76   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
77   shows "eventually (\<lambda>x. Q x) net"
78 using assms(2) assms(1) by (rule eventually_mp)
80 lemma eventually_conj_iff:
81   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
82 by (auto intro: eventually_conj elim: eventually_rev_mp)
84 lemma eventually_elim1:
85   assumes "eventually (\<lambda>i. P i) net"
86   assumes "\<And>i. P i \<Longrightarrow> Q i"
87   shows "eventually (\<lambda>i. Q i) net"
88 using assms by (auto elim!: eventually_rev_mp)
90 lemma eventually_elim2:
91   assumes "eventually (\<lambda>i. P i) net"
92   assumes "eventually (\<lambda>i. Q i) net"
93   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
94   shows "eventually (\<lambda>i. R i) net"
95 using assms by (auto elim!: eventually_rev_mp)
98 subsection {* Standard Nets *}
100 definition
101   sequentially :: "nat net"
102 where [code del]:
103   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
105 definition
106   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
107 where [code del]:
108   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
110 definition
111   at :: "'a::topological_space \<Rightarrow> 'a net"
112 where [code del]:
113   "at a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
115 lemma eventually_sequentially:
116   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
117 unfolding sequentially_def
118 proof (rule eventually_Abs_net, rule is_filter.intro)
119   fix P Q :: "nat \<Rightarrow> bool"
120   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
121   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
122   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
123   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
124 qed auto
126 lemma eventually_within:
127   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
128 unfolding within_def
129 by (rule eventually_Abs_net, rule is_filter.intro)
130    (auto elim!: eventually_rev_mp)
132 lemma eventually_at_topological:
133   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
134 unfolding at_def
135 proof (rule eventually_Abs_net, rule is_filter.intro)
136   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. x \<noteq> a \<longrightarrow> True)" by simp
137   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> True)" by - rule
138 next
139   fix P Q
140   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
141      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)"
142   then obtain S T where
143     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
144     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)" by auto
145   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). x \<noteq> a \<longrightarrow> P x \<and> Q x)"
146     by (simp add: open_Int)
147   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x \<and> Q x)" by - rule
148 qed auto
150 lemma eventually_at:
151   fixes a :: "'a::metric_space"
152   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
153 unfolding eventually_at_topological open_dist
154 apply safe
155 apply fast
156 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
157 apply clarsimp
158 apply (rule_tac x="d - dist x a" in exI, clarsimp)
159 apply (simp only: less_diff_eq)
160 apply (erule le_less_trans [OF dist_triangle])
161 done
164 subsection {* Boundedness *}
166 definition
167   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
168   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
170 lemma BfunI:
171   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
172 unfolding Bfun_def
173 proof (intro exI conjI allI)
174   show "0 < max K 1" by simp
175 next
176   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
177     using K by (rule eventually_elim1, simp)
178 qed
180 lemma BfunE:
181   assumes "Bfun f net"
182   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
183 using assms unfolding Bfun_def by fast
186 subsection {* Convergence to Zero *}
188 definition
189   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
190   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
192 lemma ZfunI:
193   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
194 unfolding Zfun_def by simp
196 lemma ZfunD:
197   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
198 unfolding Zfun_def by simp
200 lemma Zfun_ssubst:
201   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
202 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
204 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
205 unfolding Zfun_def by simp
207 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
208 unfolding Zfun_def by simp
210 lemma Zfun_imp_Zfun:
211   assumes f: "Zfun f net"
212   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
213   shows "Zfun (\<lambda>x. g x) net"
214 proof (cases)
215   assume K: "0 < K"
216   show ?thesis
217   proof (rule ZfunI)
218     fix r::real assume "0 < r"
219     hence "0 < r / K"
220       using K by (rule divide_pos_pos)
221     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
222       using ZfunD [OF f] by fast
223     with g show "eventually (\<lambda>x. norm (g x) < r) net"
224     proof (rule eventually_elim2)
225       fix x
226       assume *: "norm (g x) \<le> norm (f x) * K"
227       assume "norm (f x) < r / K"
228       hence "norm (f x) * K < r"
229         by (simp add: pos_less_divide_eq K)
230       thus "norm (g x) < r"
231         by (simp add: order_le_less_trans [OF *])
232     qed
233   qed
234 next
235   assume "\<not> 0 < K"
236   hence K: "K \<le> 0" by (simp only: not_less)
237   show ?thesis
238   proof (rule ZfunI)
239     fix r :: real
240     assume "0 < r"
241     from g show "eventually (\<lambda>x. norm (g x) < r) net"
242     proof (rule eventually_elim1)
243       fix x
244       assume "norm (g x) \<le> norm (f x) * K"
245       also have "\<dots> \<le> norm (f x) * 0"
246         using K norm_ge_zero by (rule mult_left_mono)
247       finally show "norm (g x) < r"
248         using `0 < r` by simp
249     qed
250   qed
251 qed
253 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
254 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
257   assumes f: "Zfun f net" and g: "Zfun g net"
258   shows "Zfun (\<lambda>x. f x + g x) net"
259 proof (rule ZfunI)
260   fix r::real assume "0 < r"
261   hence r: "0 < r / 2" by simp
262   have "eventually (\<lambda>x. norm (f x) < r/2) net"
263     using f r by (rule ZfunD)
264   moreover
265   have "eventually (\<lambda>x. norm (g x) < r/2) net"
266     using g r by (rule ZfunD)
267   ultimately
268   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
269   proof (rule eventually_elim2)
270     fix x
271     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
272     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
273       by (rule norm_triangle_ineq)
274     also have "\<dots> < r/2 + r/2"
275       using * by (rule add_strict_mono)
276     finally show "norm (f x + g x) < r"
277       by simp
278   qed
279 qed
281 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
282 unfolding Zfun_def by simp
284 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
285 by (simp only: diff_minus Zfun_add Zfun_minus)
287 lemma (in bounded_linear) Zfun:
288   assumes g: "Zfun g net"
289   shows "Zfun (\<lambda>x. f (g x)) net"
290 proof -
291   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
292     using bounded by fast
293   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
294     by simp
295   with g show ?thesis
296     by (rule Zfun_imp_Zfun)
297 qed
299 lemma (in bounded_bilinear) Zfun:
300   assumes f: "Zfun f net"
301   assumes g: "Zfun g net"
302   shows "Zfun (\<lambda>x. f x ** g x) net"
303 proof (rule ZfunI)
304   fix r::real assume r: "0 < r"
305   obtain K where K: "0 < K"
306     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
307     using pos_bounded by fast
308   from K have K': "0 < inverse K"
309     by (rule positive_imp_inverse_positive)
310   have "eventually (\<lambda>x. norm (f x) < r) net"
311     using f r by (rule ZfunD)
312   moreover
313   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
314     using g K' by (rule ZfunD)
315   ultimately
316   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
317   proof (rule eventually_elim2)
318     fix x
319     assume *: "norm (f x) < r" "norm (g x) < inverse K"
320     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
321       by (rule norm_le)
322     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
323       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
324     also from K have "r * inverse K * K = r"
325       by simp
326     finally show "norm (f x ** g x) < r" .
327   qed
328 qed
330 lemma (in bounded_bilinear) Zfun_left:
331   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
332 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
334 lemma (in bounded_bilinear) Zfun_right:
335   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
336 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
338 lemmas Zfun_mult = mult.Zfun
339 lemmas Zfun_mult_right = mult.Zfun_right
340 lemmas Zfun_mult_left = mult.Zfun_left
343 subsection {* Limits *}
345 definition
346   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
347     (infixr "--->" 55)
348 where [code del]:
349   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
351 ML {*
352 structure Tendsto_Intros = Named_Thms
353 (
354   val name = "tendsto_intros"
355   val description = "introduction rules for tendsto"
356 )
357 *}
359 setup Tendsto_Intros.setup
361 lemma topological_tendstoI:
362   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
363     \<Longrightarrow> (f ---> l) net"
364   unfolding tendsto_def by auto
366 lemma topological_tendstoD:
367   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
368   unfolding tendsto_def by auto
370 lemma tendstoI:
371   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
372   shows "(f ---> l) net"
373 apply (rule topological_tendstoI)
374 apply (simp add: open_dist)
375 apply (drule (1) bspec, clarify)
376 apply (drule assms)
377 apply (erule eventually_elim1, simp)
378 done
380 lemma tendstoD:
381   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
382 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
383 apply (clarsimp simp add: open_dist)
384 apply (rule_tac x="e - dist x l" in exI, clarsimp)
385 apply (simp only: less_diff_eq)
386 apply (erule le_less_trans [OF dist_triangle])
387 apply simp
388 apply simp
389 done
391 lemma tendsto_iff:
392   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
393 using tendstoI tendstoD by fast
395 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
396 by (simp only: tendsto_iff Zfun_def dist_norm)
398 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
399 unfolding tendsto_def eventually_at_topological by auto
401 lemma tendsto_ident_at_within [tendsto_intros]:
402   "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
403 unfolding tendsto_def eventually_within eventually_at_topological by auto
405 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
406 by (simp add: tendsto_def)
408 lemma tendsto_dist [tendsto_intros]:
409   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
410   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
411 proof (rule tendstoI)
412   fix e :: real assume "0 < e"
413   hence e2: "0 < e/2" by simp
414   from tendstoD [OF f e2] tendstoD [OF g e2]
415   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
416   proof (rule eventually_elim2)
417     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
418     then show "dist (dist (f x) (g x)) (dist l m) < e"
419       unfolding dist_real_def
420       using dist_triangle2 [of "f x" "g x" "l"]
421       using dist_triangle2 [of "g x" "l" "m"]
422       using dist_triangle3 [of "l" "m" "f x"]
423       using dist_triangle [of "f x" "m" "g x"]
424       by arith
425   qed
426 qed
428 lemma tendsto_norm [tendsto_intros]:
429   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
430 apply (simp add: tendsto_iff dist_norm, safe)
431 apply (drule_tac x="e" in spec, safe)
432 apply (erule eventually_elim1)
433 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
434 done
437   fixes a b c d :: "'a::ab_group_add"
438   shows "(a + c) - (b + d) = (a - b) + (c - d)"
439 by simp
441 lemma minus_diff_minus:
442   fixes a b :: "'a::ab_group_add"
443   shows "(- a) - (- b) = - (a - b)"
444 by simp
446 lemma tendsto_add [tendsto_intros]:
447   fixes a b :: "'a::real_normed_vector"
448   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
451 lemma tendsto_minus [tendsto_intros]:
452   fixes a :: "'a::real_normed_vector"
453   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
454 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
456 lemma tendsto_minus_cancel:
457   fixes a :: "'a::real_normed_vector"
458   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
459 by (drule tendsto_minus, simp)
461 lemma tendsto_diff [tendsto_intros]:
462   fixes a b :: "'a::real_normed_vector"
463   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
464 by (simp add: diff_minus tendsto_add tendsto_minus)
466 lemma tendsto_setsum [tendsto_intros]:
467   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
468   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
469   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
470 proof (cases "finite S")
471   assume "finite S" thus ?thesis using assms
472   proof (induct set: finite)
473     case empty show ?case
474       by (simp add: tendsto_const)
475   next
476     case (insert i F) thus ?case
478   qed
479 next
480   assume "\<not> finite S" thus ?thesis
481     by (simp add: tendsto_const)
482 qed
484 lemma (in bounded_linear) tendsto [tendsto_intros]:
485   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
486 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
488 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
489   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
490 by (simp only: tendsto_Zfun_iff prod_diff_prod
491                Zfun_add Zfun Zfun_left Zfun_right)
494 subsection {* Continuity of Inverse *}
496 lemma (in bounded_bilinear) Zfun_prod_Bfun:
497   assumes f: "Zfun f net"
498   assumes g: "Bfun g net"
499   shows "Zfun (\<lambda>x. f x ** g x) net"
500 proof -
501   obtain K where K: "0 \<le> K"
502     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
503     using nonneg_bounded by fast
504   obtain B where B: "0 < B"
505     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
506     using g by (rule BfunE)
507   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
508   using norm_g proof (rule eventually_elim1)
509     fix x
510     assume *: "norm (g x) \<le> B"
511     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
512       by (rule norm_le)
513     also have "\<dots> \<le> norm (f x) * B * K"
514       by (intro mult_mono' order_refl norm_g norm_ge_zero
515                 mult_nonneg_nonneg K *)
516     also have "\<dots> = norm (f x) * (B * K)"
517       by (rule mult_assoc)
518     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
519   qed
520   with f show ?thesis
521     by (rule Zfun_imp_Zfun)
522 qed
524 lemma (in bounded_bilinear) flip:
525   "bounded_bilinear (\<lambda>x y. y ** x)"
526 apply default
527 apply (rule add_right)
528 apply (rule add_left)
529 apply (rule scaleR_right)
530 apply (rule scaleR_left)
531 apply (subst mult_commute)
532 using bounded by fast
534 lemma (in bounded_bilinear) Bfun_prod_Zfun:
535   assumes f: "Bfun f net"
536   assumes g: "Zfun g net"
537   shows "Zfun (\<lambda>x. f x ** g x) net"
538 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
540 lemma inverse_diff_inverse:
541   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
542    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
543 by (simp add: algebra_simps)
545 lemma Bfun_inverse_lemma:
546   fixes x :: "'a::real_normed_div_algebra"
547   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
548 apply (subst nonzero_norm_inverse, clarsimp)
549 apply (erule (1) le_imp_inverse_le)
550 done
552 lemma Bfun_inverse:
553   fixes a :: "'a::real_normed_div_algebra"
554   assumes f: "(f ---> a) net"
555   assumes a: "a \<noteq> 0"
556   shows "Bfun (\<lambda>x. inverse (f x)) net"
557 proof -
558   from a have "0 < norm a" by simp
559   hence "\<exists>r>0. r < norm a" by (rule dense)
560   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
561   have "eventually (\<lambda>x. dist (f x) a < r) net"
562     using tendstoD [OF f r1] by fast
563   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
564   proof (rule eventually_elim1)
565     fix x
566     assume "dist (f x) a < r"
567     hence 1: "norm (f x - a) < r"
568       by (simp add: dist_norm)
569     hence 2: "f x \<noteq> 0" using r2 by auto
570     hence "norm (inverse (f x)) = inverse (norm (f x))"
571       by (rule nonzero_norm_inverse)
572     also have "\<dots> \<le> inverse (norm a - r)"
573     proof (rule le_imp_inverse_le)
574       show "0 < norm a - r" using r2 by simp
575     next
576       have "norm a - norm (f x) \<le> norm (a - f x)"
577         by (rule norm_triangle_ineq2)
578       also have "\<dots> = norm (f x - a)"
579         by (rule norm_minus_commute)
580       also have "\<dots> < r" using 1 .
581       finally show "norm a - r \<le> norm (f x)" by simp
582     qed
583     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
584   qed
585   thus ?thesis by (rule BfunI)
586 qed
588 lemma tendsto_inverse_lemma:
589   fixes a :: "'a::real_normed_div_algebra"
590   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
591          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
592 apply (subst tendsto_Zfun_iff)
593 apply (rule Zfun_ssubst)
594 apply (erule eventually_elim1)
595 apply (erule (1) inverse_diff_inverse)
596 apply (rule Zfun_minus)
597 apply (rule Zfun_mult_left)
598 apply (rule mult.Bfun_prod_Zfun)
599 apply (erule (1) Bfun_inverse)
600 apply (simp add: tendsto_Zfun_iff)
601 done
603 lemma tendsto_inverse [tendsto_intros]:
604   fixes a :: "'a::real_normed_div_algebra"
605   assumes f: "(f ---> a) net"
606   assumes a: "a \<noteq> 0"
607   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
608 proof -
609   from a have "0 < norm a" by simp
610   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
611     by (rule tendstoD)
612   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
613     unfolding dist_norm by (auto elim!: eventually_elim1)
614   with f a show ?thesis
615     by (rule tendsto_inverse_lemma)
616 qed
618 lemma tendsto_divide [tendsto_intros]:
619   fixes a b :: "'a::real_normed_field"
620   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
621     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
622 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
624 end