src/HOL/Limits.thy
 author huffman Sat Jun 06 09:11:12 2009 -0700 (2009-06-06) changeset 31488 5691ccb8d6b5 parent 31487 93938cafc0e6 child 31492 5400beeddb55 permissions -rw-r--r--
generalize tendsto to class topological_space
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 as a filter base.
15   The definition also allows non-proper filter bases.
16 *}
18 typedef (open) 'a net =
19   "{net :: 'a set set. (\<exists>A. A \<in> net)
20     \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
21 proof
22   show "UNIV \<in> ?net" by auto
23 qed
25 lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
26 using Rep_net [of net] by simp
28 lemma Rep_net_directed:
29   "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
30 using Rep_net [of net] by simp
32 lemma Abs_net_inverse':
33   assumes "\<exists>A. A \<in> net"
34   assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B"
35   shows "Rep_net (Abs_net net) = net"
36 using assms by (simp add: Abs_net_inverse)
38 lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
39 by auto
42 subsection {* Eventually *}
44 definition
45   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
46   [code del]: "eventually P net \<longleftrightarrow> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
48 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
49 unfolding eventually_def using Rep_net_nonempty [of net] by fast
51 lemma eventually_mono:
52   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
53 unfolding eventually_def by blast
55 lemma eventually_conj:
56   assumes P: "eventually (\<lambda>x. P x) net"
57   assumes Q: "eventually (\<lambda>x. Q x) net"
58   shows "eventually (\<lambda>x. P x \<and> Q x) net"
59 proof -
60   obtain A where A: "A \<in> Rep_net net" "\<forall>x\<in>A. P x"
61     using P unfolding eventually_def by fast
62   obtain B where B: "B \<in> Rep_net net" "\<forall>x\<in>B. Q x"
63     using Q unfolding eventually_def by fast
64   obtain C where C: "C \<in> Rep_net net" "C \<subseteq> A" "C \<subseteq> B"
65     using Rep_net_directed [OF A(1) B(1)] by fast
66   then have "\<forall>x\<in>C. P x \<and> Q x" "C \<in> Rep_net net"
67     using A(2) B(2) by auto
68   then show ?thesis unfolding eventually_def ..
69 qed
71 lemma eventually_mp:
72   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
73   assumes "eventually (\<lambda>x. P x) net"
74   shows "eventually (\<lambda>x. Q x) net"
75 proof (rule eventually_mono)
76   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
77   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
78     using assms by (rule eventually_conj)
79 qed
81 lemma eventually_rev_mp:
82   assumes "eventually (\<lambda>x. P x) net"
83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
84   shows "eventually (\<lambda>x. Q x) net"
85 using assms(2) assms(1) by (rule eventually_mp)
87 lemma eventually_conj_iff:
88   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
89 by (auto intro: eventually_conj elim: eventually_rev_mp)
91 lemma eventually_elim1:
92   assumes "eventually (\<lambda>i. P i) net"
93   assumes "\<And>i. P i \<Longrightarrow> Q i"
94   shows "eventually (\<lambda>i. Q i) net"
95 using assms by (auto elim!: eventually_rev_mp)
97 lemma eventually_elim2:
98   assumes "eventually (\<lambda>i. P i) net"
99   assumes "eventually (\<lambda>i. Q i) net"
100   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
101   shows "eventually (\<lambda>i. R i) net"
102 using assms by (auto elim!: eventually_rev_mp)
105 subsection {* Standard Nets *}
107 definition
108   sequentially :: "nat net" where
109   [code del]: "sequentially = Abs_net (range (\<lambda>n. {n..}))"
111 definition
112   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
113   [code del]: "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
115 definition
116   at :: "'a::topological_space \<Rightarrow> 'a net" where
117   [code del]: "at a = Abs_net ((\<lambda>S. S - {a}) ` {S\<in>topo. a \<in> S})"
119 lemma Rep_net_sequentially:
120   "Rep_net sequentially = range (\<lambda>n. {n..})"
121 unfolding sequentially_def
122 apply (rule Abs_net_inverse')
123 apply (rule image_nonempty, simp)
124 apply (clarsimp, rename_tac m n)
125 apply (rule_tac x="max m n" in exI, auto)
126 done
128 lemma Rep_net_within:
129   "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
130 unfolding within_def
131 apply (rule Abs_net_inverse')
132 apply (rule image_nonempty, rule Rep_net_nonempty)
133 apply (clarsimp, rename_tac A B)
134 apply (drule (1) Rep_net_directed)
135 apply (clarify, rule_tac x=C in bexI, auto)
136 done
138 lemma Rep_net_at:
139   "Rep_net (at a) = ((\<lambda>S. S - {a}) ` {S\<in>topo. a \<in> S})"
140 unfolding at_def
141 apply (rule Abs_net_inverse')
142 apply (rule image_nonempty)
143 apply (rule_tac x="UNIV" in exI, simp add: topo_UNIV)
144 apply (clarsimp, rename_tac S T)
145 apply (rule_tac x="S \<inter> T" in exI, auto simp add: topo_Int)
146 done
148 lemma eventually_sequentially:
149   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
150 unfolding eventually_def Rep_net_sequentially by auto
152 lemma eventually_within:
153   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
154 unfolding eventually_def Rep_net_within by auto
156 lemma eventually_at_topological:
157   "eventually P (at a) \<longleftrightarrow> (\<exists>S\<in>topo. a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
158 unfolding eventually_def Rep_net_at by auto
160 lemma eventually_at:
161   fixes a :: "'a::metric_space"
162   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
163 unfolding eventually_at_topological topo_dist
164 apply safe
165 apply fast
166 apply (rule_tac x="{x. dist x a < d}" in bexI, simp)
167 apply clarsimp
168 apply (rule_tac x="d - dist x a" in exI, clarsimp)
169 apply (simp only: less_diff_eq)
170 apply (erule le_less_trans [OF dist_triangle])
171 done
174 subsection {* Boundedness *}
176 definition
177   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
178   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
180 lemma BfunI:
181   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
182 unfolding Bfun_def
183 proof (intro exI conjI allI)
184   show "0 < max K 1" by simp
185 next
186   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
187     using K by (rule eventually_elim1, simp)
188 qed
190 lemma BfunE:
191   assumes "Bfun f net"
192   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
193 using assms unfolding Bfun_def by fast
196 subsection {* Convergence to Zero *}
198 definition
199   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
200   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
202 lemma ZfunI:
203   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
204 unfolding Zfun_def by simp
206 lemma ZfunD:
207   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
208 unfolding Zfun_def by simp
210 lemma Zfun_ssubst:
211   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
212 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
214 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
215 unfolding Zfun_def by simp
217 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
218 unfolding Zfun_def by simp
220 lemma Zfun_imp_Zfun:
221   assumes f: "Zfun f net"
222   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
223   shows "Zfun (\<lambda>x. g x) net"
224 proof (cases)
225   assume K: "0 < K"
226   show ?thesis
227   proof (rule ZfunI)
228     fix r::real assume "0 < r"
229     hence "0 < r / K"
230       using K by (rule divide_pos_pos)
231     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
232       using ZfunD [OF f] by fast
233     with g show "eventually (\<lambda>x. norm (g x) < r) net"
234     proof (rule eventually_elim2)
235       fix x
236       assume *: "norm (g x) \<le> norm (f x) * K"
237       assume "norm (f x) < r / K"
238       hence "norm (f x) * K < r"
239         by (simp add: pos_less_divide_eq K)
240       thus "norm (g x) < r"
241         by (simp add: order_le_less_trans [OF *])
242     qed
243   qed
244 next
245   assume "\<not> 0 < K"
246   hence K: "K \<le> 0" by (simp only: not_less)
247   show ?thesis
248   proof (rule ZfunI)
249     fix r :: real
250     assume "0 < r"
251     from g show "eventually (\<lambda>x. norm (g x) < r) net"
252     proof (rule eventually_elim1)
253       fix x
254       assume "norm (g x) \<le> norm (f x) * K"
255       also have "\<dots> \<le> norm (f x) * 0"
256         using K norm_ge_zero by (rule mult_left_mono)
257       finally show "norm (g x) < r"
258         using `0 < r` by simp
259     qed
260   qed
261 qed
263 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
264 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
267   assumes f: "Zfun f net" and g: "Zfun g net"
268   shows "Zfun (\<lambda>x. f x + g x) net"
269 proof (rule ZfunI)
270   fix r::real assume "0 < r"
271   hence r: "0 < r / 2" by simp
272   have "eventually (\<lambda>x. norm (f x) < r/2) net"
273     using f r by (rule ZfunD)
274   moreover
275   have "eventually (\<lambda>x. norm (g x) < r/2) net"
276     using g r by (rule ZfunD)
277   ultimately
278   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
279   proof (rule eventually_elim2)
280     fix x
281     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
282     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
283       by (rule norm_triangle_ineq)
284     also have "\<dots> < r/2 + r/2"
285       using * by (rule add_strict_mono)
286     finally show "norm (f x + g x) < r"
287       by simp
288   qed
289 qed
291 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
292 unfolding Zfun_def by simp
294 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
295 by (simp only: diff_minus Zfun_add Zfun_minus)
297 lemma (in bounded_linear) Zfun:
298   assumes g: "Zfun g net"
299   shows "Zfun (\<lambda>x. f (g x)) net"
300 proof -
301   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
302     using bounded by fast
303   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
304     by simp
305   with g show ?thesis
306     by (rule Zfun_imp_Zfun)
307 qed
309 lemma (in bounded_bilinear) Zfun:
310   assumes f: "Zfun f net"
311   assumes g: "Zfun g net"
312   shows "Zfun (\<lambda>x. f x ** g x) net"
313 proof (rule ZfunI)
314   fix r::real assume r: "0 < r"
315   obtain K where K: "0 < K"
316     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
317     using pos_bounded by fast
318   from K have K': "0 < inverse K"
319     by (rule positive_imp_inverse_positive)
320   have "eventually (\<lambda>x. norm (f x) < r) net"
321     using f r by (rule ZfunD)
322   moreover
323   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
324     using g K' by (rule ZfunD)
325   ultimately
326   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
327   proof (rule eventually_elim2)
328     fix x
329     assume *: "norm (f x) < r" "norm (g x) < inverse K"
330     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
331       by (rule norm_le)
332     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
333       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
334     also from K have "r * inverse K * K = r"
335       by simp
336     finally show "norm (f x ** g x) < r" .
337   qed
338 qed
340 lemma (in bounded_bilinear) Zfun_left:
341   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
342 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
344 lemma (in bounded_bilinear) Zfun_right:
345   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
346 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
348 lemmas Zfun_mult = mult.Zfun
349 lemmas Zfun_mult_right = mult.Zfun_right
350 lemmas Zfun_mult_left = mult.Zfun_left
353 subsection{* Limits *}
355 definition
356   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
357     (infixr "--->" 55)
358 where [code del]:
359   "(f ---> l) net \<longleftrightarrow> (\<forall>S\<in>topo. l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
361 lemma topological_tendstoI:
362   "(\<And>S. S \<in> topo \<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> S \<in> topo \<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: topo_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: topo_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_const: "((\<lambda>x. k) ---> k) net"
399 by (simp add: tendsto_def)
401 lemma tendsto_norm:
402   fixes a :: "'a::real_normed_vector"
403   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
404 apply (simp add: tendsto_iff dist_norm, safe)
405 apply (drule_tac x="e" in spec, safe)
406 apply (erule eventually_elim1)
407 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
408 done
411   fixes a b c d :: "'a::ab_group_add"
412   shows "(a + c) - (b + d) = (a - b) + (c - d)"
413 by simp
415 lemma minus_diff_minus:
416   fixes a b :: "'a::ab_group_add"
417   shows "(- a) - (- b) = - (a - b)"
418 by simp
421   fixes a b :: "'a::real_normed_vector"
422   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
425 lemma tendsto_minus:
426   fixes a :: "'a::real_normed_vector"
427   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
428 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
430 lemma tendsto_minus_cancel:
431   fixes a :: "'a::real_normed_vector"
432   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
433 by (drule tendsto_minus, simp)
435 lemma tendsto_diff:
436   fixes a b :: "'a::real_normed_vector"
437   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
438 by (simp add: diff_minus tendsto_add tendsto_minus)
440 lemma (in bounded_linear) tendsto:
441   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
442 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
444 lemma (in bounded_bilinear) tendsto:
445   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
446 by (simp only: tendsto_Zfun_iff prod_diff_prod
447                Zfun_add Zfun Zfun_left Zfun_right)
450 subsection {* Continuity of Inverse *}
452 lemma (in bounded_bilinear) Zfun_prod_Bfun:
453   assumes f: "Zfun f net"
454   assumes g: "Bfun g net"
455   shows "Zfun (\<lambda>x. f x ** g x) net"
456 proof -
457   obtain K where K: "0 \<le> K"
458     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
459     using nonneg_bounded by fast
460   obtain B where B: "0 < B"
461     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
462     using g by (rule BfunE)
463   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
464   using norm_g proof (rule eventually_elim1)
465     fix x
466     assume *: "norm (g x) \<le> B"
467     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
468       by (rule norm_le)
469     also have "\<dots> \<le> norm (f x) * B * K"
470       by (intro mult_mono' order_refl norm_g norm_ge_zero
471                 mult_nonneg_nonneg K *)
472     also have "\<dots> = norm (f x) * (B * K)"
473       by (rule mult_assoc)
474     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
475   qed
476   with f show ?thesis
477     by (rule Zfun_imp_Zfun)
478 qed
480 lemma (in bounded_bilinear) flip:
481   "bounded_bilinear (\<lambda>x y. y ** x)"
482 apply default
483 apply (rule add_right)
484 apply (rule add_left)
485 apply (rule scaleR_right)
486 apply (rule scaleR_left)
487 apply (subst mult_commute)
488 using bounded by fast
490 lemma (in bounded_bilinear) Bfun_prod_Zfun:
491   assumes f: "Bfun f net"
492   assumes g: "Zfun g net"
493   shows "Zfun (\<lambda>x. f x ** g x) net"
494 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
496 lemma inverse_diff_inverse:
497   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
498    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
499 by (simp add: algebra_simps)
501 lemma Bfun_inverse_lemma:
502   fixes x :: "'a::real_normed_div_algebra"
503   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
504 apply (subst nonzero_norm_inverse, clarsimp)
505 apply (erule (1) le_imp_inverse_le)
506 done
508 lemma Bfun_inverse:
509   fixes a :: "'a::real_normed_div_algebra"
510   assumes f: "(f ---> a) net"
511   assumes a: "a \<noteq> 0"
512   shows "Bfun (\<lambda>x. inverse (f x)) net"
513 proof -
514   from a have "0 < norm a" by simp
515   hence "\<exists>r>0. r < norm a" by (rule dense)
516   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
517   have "eventually (\<lambda>x. dist (f x) a < r) net"
518     using tendstoD [OF f r1] by fast
519   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
520   proof (rule eventually_elim1)
521     fix x
522     assume "dist (f x) a < r"
523     hence 1: "norm (f x - a) < r"
524       by (simp add: dist_norm)
525     hence 2: "f x \<noteq> 0" using r2 by auto
526     hence "norm (inverse (f x)) = inverse (norm (f x))"
527       by (rule nonzero_norm_inverse)
528     also have "\<dots> \<le> inverse (norm a - r)"
529     proof (rule le_imp_inverse_le)
530       show "0 < norm a - r" using r2 by simp
531     next
532       have "norm a - norm (f x) \<le> norm (a - f x)"
533         by (rule norm_triangle_ineq2)
534       also have "\<dots> = norm (f x - a)"
535         by (rule norm_minus_commute)
536       also have "\<dots> < r" using 1 .
537       finally show "norm a - r \<le> norm (f x)" by simp
538     qed
539     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
540   qed
541   thus ?thesis by (rule BfunI)
542 qed
544 lemma tendsto_inverse_lemma:
545   fixes a :: "'a::real_normed_div_algebra"
546   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
547          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
548 apply (subst tendsto_Zfun_iff)
549 apply (rule Zfun_ssubst)
550 apply (erule eventually_elim1)
551 apply (erule (1) inverse_diff_inverse)
552 apply (rule Zfun_minus)
553 apply (rule Zfun_mult_left)
554 apply (rule mult.Bfun_prod_Zfun)
555 apply (erule (1) Bfun_inverse)
556 apply (simp add: tendsto_Zfun_iff)
557 done
559 lemma tendsto_inverse:
560   fixes a :: "'a::real_normed_div_algebra"
561   assumes f: "(f ---> a) net"
562   assumes a: "a \<noteq> 0"
563   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
564 proof -
565   from a have "0 < norm a" by simp
566   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
567     by (rule tendstoD)
568   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
569     unfolding dist_norm by (auto elim!: eventually_elim1)
570   with f a show ?thesis
571     by (rule tendsto_inverse_lemma)
572 qed
574 lemma tendsto_divide:
575   fixes a b :: "'a::real_normed_field"
576   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
577     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
578 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
580 end