src/HOL/Limits.thy
 author hoelzl Fri Mar 22 10:41:42 2013 +0100 (2013-03-22) changeset 51471 cad22a3cc09c parent 51360 c4367ed99b5e child 51472 adb441e4b9e9 permissions -rw-r--r--
move topological_space to its own theory
1 (*  Title       : Limits.thy
2     Author      : Brian Huffman
3 *)
5 header {* Filters and Limits *}
7 theory Limits
8 imports RealVector
9 begin
11 definition at_infinity :: "'a::real_normed_vector filter" where
12   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
15 lemma eventually_nhds_metric:
16   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
17 unfolding eventually_nhds open_dist
18 apply safe
19 apply fast
20 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
21 apply clarsimp
22 apply (rule_tac x="d - dist x a" in exI, clarsimp)
23 apply (simp only: less_diff_eq)
24 apply (erule le_less_trans [OF dist_triangle])
25 done
27 lemma eventually_at:
28   fixes a :: "'a::metric_space"
29   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
30 unfolding at_def eventually_within eventually_nhds_metric by auto
31 lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
32   "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
33   unfolding eventually_within eventually_at dist_nz by auto
35 lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
36   "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
37   unfolding eventually_within_less by auto (metis dense order_le_less_trans)
39 lemma eventually_at_infinity:
40   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
41 unfolding at_infinity_def
42 proof (rule eventually_Abs_filter, rule is_filter.intro)
43   fix P Q :: "'a \<Rightarrow> bool"
44   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
45   then obtain r s where
46     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
47   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
48   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
49 qed auto
51 lemma at_infinity_eq_at_top_bot:
52   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
53   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
54 proof (intro arg_cong[where f=Abs_filter] ext iffI)
55   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
56   then guess r ..
57   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
58   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
59 next
60   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
61   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
62   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
63     by (intro exI[of _ "max p (-q)"])
64        (auto simp: abs_real_def)
65 qed
67 lemma at_top_le_at_infinity:
68   "at_top \<le> (at_infinity :: real filter)"
69   unfolding at_infinity_eq_at_top_bot by simp
71 lemma at_bot_le_at_infinity:
72   "at_bot \<le> (at_infinity :: real filter)"
73   unfolding at_infinity_eq_at_top_bot by simp
75 subsection {* Boundedness *}
77 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
78   where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
80 lemma BfunI:
81   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
82 unfolding Bfun_def
83 proof (intro exI conjI allI)
84   show "0 < max K 1" by simp
85 next
86   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
87     using K by (rule eventually_elim1, simp)
88 qed
90 lemma BfunE:
91   assumes "Bfun f F"
92   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
93 using assms unfolding Bfun_def by fast
96 subsection {* Convergence to Zero *}
98 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
99   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
101 lemma ZfunI:
102   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
103   unfolding Zfun_def by simp
105 lemma ZfunD:
106   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
107   unfolding Zfun_def by simp
109 lemma Zfun_ssubst:
110   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
111   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
113 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
114   unfolding Zfun_def by simp
116 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
117   unfolding Zfun_def by simp
119 lemma Zfun_imp_Zfun:
120   assumes f: "Zfun f F"
121   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
122   shows "Zfun (\<lambda>x. g x) F"
123 proof (cases)
124   assume K: "0 < K"
125   show ?thesis
126   proof (rule ZfunI)
127     fix r::real assume "0 < r"
128     hence "0 < r / K"
129       using K by (rule divide_pos_pos)
130     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
131       using ZfunD [OF f] by fast
132     with g show "eventually (\<lambda>x. norm (g x) < r) F"
133     proof eventually_elim
134       case (elim x)
135       hence "norm (f x) * K < r"
136         by (simp add: pos_less_divide_eq K)
137       thus ?case
138         by (simp add: order_le_less_trans [OF elim(1)])
139     qed
140   qed
141 next
142   assume "\<not> 0 < K"
143   hence K: "K \<le> 0" by (simp only: not_less)
144   show ?thesis
145   proof (rule ZfunI)
146     fix r :: real
147     assume "0 < r"
148     from g show "eventually (\<lambda>x. norm (g x) < r) F"
149     proof eventually_elim
150       case (elim x)
151       also have "norm (f x) * K \<le> norm (f x) * 0"
152         using K norm_ge_zero by (rule mult_left_mono)
153       finally show ?case
154         using `0 < r` by simp
155     qed
156   qed
157 qed
159 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
160   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
163   assumes f: "Zfun f F" and g: "Zfun g F"
164   shows "Zfun (\<lambda>x. f x + g x) F"
165 proof (rule ZfunI)
166   fix r::real assume "0 < r"
167   hence r: "0 < r / 2" by simp
168   have "eventually (\<lambda>x. norm (f x) < r/2) F"
169     using f r by (rule ZfunD)
170   moreover
171   have "eventually (\<lambda>x. norm (g x) < r/2) F"
172     using g r by (rule ZfunD)
173   ultimately
174   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
175   proof eventually_elim
176     case (elim x)
177     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
178       by (rule norm_triangle_ineq)
179     also have "\<dots> < r/2 + r/2"
180       using elim by (rule add_strict_mono)
181     finally show ?case
182       by simp
183   qed
184 qed
186 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
187   unfolding Zfun_def by simp
189 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
190   by (simp only: diff_minus Zfun_add Zfun_minus)
192 lemma (in bounded_linear) Zfun:
193   assumes g: "Zfun g F"
194   shows "Zfun (\<lambda>x. f (g x)) F"
195 proof -
196   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
197     using bounded by fast
198   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
199     by simp
200   with g show ?thesis
201     by (rule Zfun_imp_Zfun)
202 qed
204 lemma (in bounded_bilinear) Zfun:
205   assumes f: "Zfun f F"
206   assumes g: "Zfun g F"
207   shows "Zfun (\<lambda>x. f x ** g x) F"
208 proof (rule ZfunI)
209   fix r::real assume r: "0 < r"
210   obtain K where K: "0 < K"
211     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
212     using pos_bounded by fast
213   from K have K': "0 < inverse K"
214     by (rule positive_imp_inverse_positive)
215   have "eventually (\<lambda>x. norm (f x) < r) F"
216     using f r by (rule ZfunD)
217   moreover
218   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
219     using g K' by (rule ZfunD)
220   ultimately
221   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
222   proof eventually_elim
223     case (elim x)
224     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
225       by (rule norm_le)
226     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
227       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
228     also from K have "r * inverse K * K = r"
229       by simp
230     finally show ?case .
231   qed
232 qed
234 lemma (in bounded_bilinear) Zfun_left:
235   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
236   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
238 lemma (in bounded_bilinear) Zfun_right:
239   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
240   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
242 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
243 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
244 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
246 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
247   by (simp only: tendsto_iff Zfun_def dist_norm)
250 lemma metric_tendsto_imp_tendsto:
251   assumes f: "(f ---> a) F"
252   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
253   shows "(g ---> b) F"
254 proof (rule tendstoI)
255   fix e :: real assume "0 < e"
256   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
257   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
258     using le_less_trans by (rule eventually_elim2)
259 qed
260 subsubsection {* Distance and norms *}
262 lemma tendsto_dist [tendsto_intros]:
263   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
264   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
265 proof (rule tendstoI)
266   fix e :: real assume "0 < e"
267   hence e2: "0 < e/2" by simp
268   from tendstoD [OF f e2] tendstoD [OF g e2]
269   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
270   proof (eventually_elim)
271     case (elim x)
272     then show "dist (dist (f x) (g x)) (dist l m) < e"
273       unfolding dist_real_def
274       using dist_triangle2 [of "f x" "g x" "l"]
275       using dist_triangle2 [of "g x" "l" "m"]
276       using dist_triangle3 [of "l" "m" "f x"]
277       using dist_triangle [of "f x" "m" "g x"]
278       by arith
279   qed
280 qed
282 lemma norm_conv_dist: "norm x = dist x 0"
283   unfolding dist_norm by simp
285 lemma tendsto_norm [tendsto_intros]:
286   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
287   unfolding norm_conv_dist by (intro tendsto_intros)
289 lemma tendsto_norm_zero:
290   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
291   by (drule tendsto_norm, simp)
293 lemma tendsto_norm_zero_cancel:
294   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
295   unfolding tendsto_iff dist_norm by simp
297 lemma tendsto_norm_zero_iff:
298   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
299   unfolding tendsto_iff dist_norm by simp
301 lemma tendsto_rabs [tendsto_intros]:
302   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
303   by (fold real_norm_def, rule tendsto_norm)
305 lemma tendsto_rabs_zero:
306   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
307   by (fold real_norm_def, rule tendsto_norm_zero)
309 lemma tendsto_rabs_zero_cancel:
310   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
311   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
313 lemma tendsto_rabs_zero_iff:
314   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
315   by (fold real_norm_def, rule tendsto_norm_zero_iff)
317 subsubsection {* Addition and subtraction *}
319 lemma tendsto_add [tendsto_intros]:
320   fixes a b :: "'a::real_normed_vector"
321   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
325   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
326   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
327   by (drule (1) tendsto_add, simp)
329 lemma tendsto_minus [tendsto_intros]:
330   fixes a :: "'a::real_normed_vector"
331   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
332   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
334 lemma tendsto_minus_cancel:
335   fixes a :: "'a::real_normed_vector"
336   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
337   by (drule tendsto_minus, simp)
339 lemma tendsto_minus_cancel_left:
340     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
341   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
342   by auto
344 lemma tendsto_diff [tendsto_intros]:
345   fixes a b :: "'a::real_normed_vector"
346   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
347   by (simp add: diff_minus tendsto_add tendsto_minus)
349 lemma tendsto_setsum [tendsto_intros]:
350   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
351   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
352   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
353 proof (cases "finite S")
354   assume "finite S" thus ?thesis using assms
356 next
357   assume "\<not> finite S" thus ?thesis
358     by (simp add: tendsto_const)
359 qed
361 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
363 subsubsection {* Linear operators and multiplication *}
365 lemma (in bounded_linear) tendsto:
366   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
367   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
369 lemma (in bounded_linear) tendsto_zero:
370   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
371   by (drule tendsto, simp only: zero)
373 lemma (in bounded_bilinear) tendsto:
374   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
375   by (simp only: tendsto_Zfun_iff prod_diff_prod
376                  Zfun_add Zfun Zfun_left Zfun_right)
378 lemma (in bounded_bilinear) tendsto_zero:
379   assumes f: "(f ---> 0) F"
380   assumes g: "(g ---> 0) F"
381   shows "((\<lambda>x. f x ** g x) ---> 0) F"
382   using tendsto [OF f g] by (simp add: zero_left)
384 lemma (in bounded_bilinear) tendsto_left_zero:
385   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
386   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
388 lemma (in bounded_bilinear) tendsto_right_zero:
389   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
390   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
392 lemmas tendsto_of_real [tendsto_intros] =
393   bounded_linear.tendsto [OF bounded_linear_of_real]
395 lemmas tendsto_scaleR [tendsto_intros] =
396   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
398 lemmas tendsto_mult [tendsto_intros] =
399   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
401 lemmas tendsto_mult_zero =
402   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
404 lemmas tendsto_mult_left_zero =
405   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
407 lemmas tendsto_mult_right_zero =
408   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
410 lemma tendsto_power [tendsto_intros]:
411   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
412   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
413   by (induct n) (simp_all add: tendsto_const tendsto_mult)
415 lemma tendsto_setprod [tendsto_intros]:
416   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
417   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
418   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
419 proof (cases "finite S")
420   assume "finite S" thus ?thesis using assms
421     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
422 next
423   assume "\<not> finite S" thus ?thesis
424     by (simp add: tendsto_const)
425 qed
427 subsubsection {* Inverse and division *}
429 lemma (in bounded_bilinear) Zfun_prod_Bfun:
430   assumes f: "Zfun f F"
431   assumes g: "Bfun g F"
432   shows "Zfun (\<lambda>x. f x ** g x) F"
433 proof -
434   obtain K where K: "0 \<le> K"
435     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
436     using nonneg_bounded by fast
437   obtain B where B: "0 < B"
438     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
439     using g by (rule BfunE)
440   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
441   using norm_g proof eventually_elim
442     case (elim x)
443     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
444       by (rule norm_le)
445     also have "\<dots> \<le> norm (f x) * B * K"
446       by (intro mult_mono' order_refl norm_g norm_ge_zero
447                 mult_nonneg_nonneg K elim)
448     also have "\<dots> = norm (f x) * (B * K)"
449       by (rule mult_assoc)
450     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
451   qed
452   with f show ?thesis
453     by (rule Zfun_imp_Zfun)
454 qed
456 lemma (in bounded_bilinear) flip:
457   "bounded_bilinear (\<lambda>x y. y ** x)"
458   apply default
459   apply (rule add_right)
460   apply (rule add_left)
461   apply (rule scaleR_right)
462   apply (rule scaleR_left)
463   apply (subst mult_commute)
464   using bounded by fast
466 lemma (in bounded_bilinear) Bfun_prod_Zfun:
467   assumes f: "Bfun f F"
468   assumes g: "Zfun g F"
469   shows "Zfun (\<lambda>x. f x ** g x) F"
470   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
472 lemma Bfun_inverse_lemma:
473   fixes x :: "'a::real_normed_div_algebra"
474   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
475   apply (subst nonzero_norm_inverse, clarsimp)
476   apply (erule (1) le_imp_inverse_le)
477   done
479 lemma Bfun_inverse:
480   fixes a :: "'a::real_normed_div_algebra"
481   assumes f: "(f ---> a) F"
482   assumes a: "a \<noteq> 0"
483   shows "Bfun (\<lambda>x. inverse (f x)) F"
484 proof -
485   from a have "0 < norm a" by simp
486   hence "\<exists>r>0. r < norm a" by (rule dense)
487   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
488   have "eventually (\<lambda>x. dist (f x) a < r) F"
489     using tendstoD [OF f r1] by fast
490   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
491   proof eventually_elim
492     case (elim x)
493     hence 1: "norm (f x - a) < r"
494       by (simp add: dist_norm)
495     hence 2: "f x \<noteq> 0" using r2 by auto
496     hence "norm (inverse (f x)) = inverse (norm (f x))"
497       by (rule nonzero_norm_inverse)
498     also have "\<dots> \<le> inverse (norm a - r)"
499     proof (rule le_imp_inverse_le)
500       show "0 < norm a - r" using r2 by simp
501     next
502       have "norm a - norm (f x) \<le> norm (a - f x)"
503         by (rule norm_triangle_ineq2)
504       also have "\<dots> = norm (f x - a)"
505         by (rule norm_minus_commute)
506       also have "\<dots> < r" using 1 .
507       finally show "norm a - r \<le> norm (f x)" by simp
508     qed
509     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
510   qed
511   thus ?thesis by (rule BfunI)
512 qed
514 lemma tendsto_inverse [tendsto_intros]:
515   fixes a :: "'a::real_normed_div_algebra"
516   assumes f: "(f ---> a) F"
517   assumes a: "a \<noteq> 0"
518   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
519 proof -
520   from a have "0 < norm a" by simp
521   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
522     by (rule tendstoD)
523   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
524     unfolding dist_norm by (auto elim!: eventually_elim1)
525   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
526     - (inverse (f x) * (f x - a) * inverse a)) F"
527     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
528   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
529     by (intro Zfun_minus Zfun_mult_left
530       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
531       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
532   ultimately show ?thesis
533     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
534 qed
536 lemma tendsto_divide [tendsto_intros]:
537   fixes a b :: "'a::real_normed_field"
538   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
539     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
540   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
542 lemma tendsto_sgn [tendsto_intros]:
543   fixes l :: "'a::real_normed_vector"
544   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
545   unfolding sgn_div_norm by (simp add: tendsto_intros)
547 lemma filterlim_at_bot_at_right:
548   fixes f :: "real \<Rightarrow> 'b::linorder"
549   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
550   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
551   assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
552   assumes P: "eventually P at_bot"
553   shows "filterlim f at_bot (at_right a)"
554 proof -
555   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
556     unfolding eventually_at_bot_linorder by auto
557   show ?thesis
558   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
559     fix z assume "z \<le> x"
560     with x have "P z" by auto
561     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
562       using bound[OF bij(2)[OF `P z`]]
563       by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "g z - a"])
564     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
565       by eventually_elim (metis bij `P z` mono)
566   qed
567 qed
569 lemma filterlim_at_top_at_left:
570   fixes f :: "real \<Rightarrow> 'b::linorder"
571   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
572   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
573   assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
574   assumes P: "eventually P at_top"
575   shows "filterlim f at_top (at_left a)"
576 proof -
577   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
578     unfolding eventually_at_top_linorder by auto
579   show ?thesis
580   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
581     fix z assume "x \<le> z"
582     with x have "P z" by auto
583     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
584       using bound[OF bij(2)[OF `P z`]]
585       by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "a - g z"])
586     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
587       by eventually_elim (metis bij `P z` mono)
588   qed
589 qed
591 lemma filterlim_at_infinity:
592   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
593   assumes "0 \<le> c"
594   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
595   unfolding filterlim_iff eventually_at_infinity
596 proof safe
597   fix P :: "'a \<Rightarrow> bool" and b
598   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
599     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
600   have "max b (c + 1) > c" by auto
601   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
602     by auto
603   then show "eventually (\<lambda>x. P (f x)) F"
604   proof eventually_elim
605     fix x assume "max b (c + 1) \<le> norm (f x)"
606     with P show "P (f x)" by auto
607   qed
608 qed force
610 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
611   unfolding filterlim_at_top
612   apply (intro allI)
613   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
614   apply (auto simp: natceiling_le_eq)
615   done
617 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
619 text {*
621 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
622 @{term "at_right x"} and also @{term "at_right 0"}.
624 *}
626 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
628 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
629   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
630   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
632 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
633   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
634   apply (intro allI ex_cong)
635   apply (auto simp: dist_real_def field_simps)
636   apply (erule_tac x="-x" in allE)
637   apply simp
638   done
640 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
641   unfolding at_def filtermap_nhds_shift[symmetric]
642   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
644 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
645   unfolding filtermap_at_shift[symmetric]
646   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
648 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
649   using filtermap_at_right_shift[of "-a" 0] by simp
651 lemma filterlim_at_right_to_0:
652   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
653   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
655 lemma eventually_at_right_to_0:
656   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
657   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
659 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
660   unfolding at_def filtermap_nhds_minus[symmetric]
661   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
663 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
664   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
666 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
667   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
669 lemma filterlim_at_left_to_right:
670   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
671   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
673 lemma eventually_at_left_to_right:
674   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
675   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
677 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
678   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
679   by (metis le_minus_iff minus_minus)
681 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
682   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
684 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
685   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
687 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
688   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
690 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
691   unfolding filterlim_at_top eventually_at_bot_dense
692   by (metis leI minus_less_iff order_less_asym)
694 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
695   unfolding filterlim_at_bot eventually_at_top_dense
696   by (metis leI less_minus_iff order_less_asym)
698 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
699   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
700   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
701   by auto
703 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
704   unfolding filterlim_uminus_at_top by simp
706 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
707   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
708 proof safe
709   fix Z :: real assume [arith]: "0 < Z"
710   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
711     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
712   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
713     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
714 qed
716 lemma filterlim_inverse_at_top:
717   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
718   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
719      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
721 lemma filterlim_inverse_at_bot_neg:
722   "LIM x (at_left (0::real)). inverse x :> at_bot"
723   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
725 lemma filterlim_inverse_at_bot:
726   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
727   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
728   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
730 lemma tendsto_inverse_0:
731   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
732   shows "(inverse ---> (0::'a)) at_infinity"
733   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
734 proof safe
735   fix r :: real assume "0 < r"
736   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
737   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
738     fix x :: 'a
739     from `0 < r` have "0 < inverse (r / 2)" by simp
740     also assume *: "inverse (r / 2) \<le> norm x"
741     finally show "norm (inverse x) < r"
742       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
743   qed
744 qed
746 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
747 proof (rule antisym)
748   have "(inverse ---> (0::real)) at_top"
749     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
750   then show "filtermap inverse at_top \<le> at_right (0::real)"
751     unfolding at_within_eq
752     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
753 next
754   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
755     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
756   then show "at_right (0::real) \<le> filtermap inverse at_top"
757     by (simp add: filtermap_ident filtermap_filtermap)
758 qed
760 lemma eventually_at_right_to_top:
761   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
762   unfolding at_right_to_top eventually_filtermap ..
764 lemma filterlim_at_right_to_top:
765   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
766   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
768 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
769   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
771 lemma eventually_at_top_to_right:
772   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
773   unfolding at_top_to_right eventually_filtermap ..
775 lemma filterlim_at_top_to_right:
776   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
777   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
779 lemma filterlim_inverse_at_infinity:
780   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
781   shows "filterlim inverse at_infinity (at (0::'a))"
782   unfolding filterlim_at_infinity[OF order_refl]
783 proof safe
784   fix r :: real assume "0 < r"
785   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
786     unfolding eventually_at norm_inverse
787     by (intro exI[of _ "inverse r"])
788        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
789 qed
791 lemma filterlim_inverse_at_iff:
792   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
793   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
794   unfolding filterlim_def filtermap_filtermap[symmetric]
795 proof
796   assume "filtermap g F \<le> at_infinity"
797   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
798     by (rule filtermap_mono)
799   also have "\<dots> \<le> at 0"
800     using tendsto_inverse_0
801     by (auto intro!: le_withinI exI[of _ 1]
802              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
803   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
804 next
805   assume "filtermap inverse (filtermap g F) \<le> at 0"
806   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
807     by (rule filtermap_mono)
808   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
809     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
810 qed
812 lemma tendsto_inverse_0_at_top:
813   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
814  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
816 text {*
818 We only show rules for multiplication and addition when the functions are either against a real
819 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
821 *}
823 lemma filterlim_tendsto_pos_mult_at_top:
824   assumes f: "(f ---> c) F" and c: "0 < c"
825   assumes g: "LIM x F. g x :> at_top"
826   shows "LIM x F. (f x * g x :: real) :> at_top"
827   unfolding filterlim_at_top_gt[where c=0]
828 proof safe
829   fix Z :: real assume "0 < Z"
830   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
831     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
832              simp: dist_real_def abs_real_def split: split_if_asm)
833   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
834     unfolding filterlim_at_top by auto
835   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
836   proof eventually_elim
837     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
838     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
839       by (intro mult_mono) (auto simp: zero_le_divide_iff)
840     with `0 < c` show "Z \<le> f x * g x"
841        by simp
842   qed
843 qed
845 lemma filterlim_at_top_mult_at_top:
846   assumes f: "LIM x F. f x :> at_top"
847   assumes g: "LIM x F. g x :> at_top"
848   shows "LIM x F. (f x * g x :: real) :> at_top"
849   unfolding filterlim_at_top_gt[where c=0]
850 proof safe
851   fix Z :: real assume "0 < Z"
852   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
853     unfolding filterlim_at_top by auto
854   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
855     unfolding filterlim_at_top by auto
856   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
857   proof eventually_elim
858     fix x assume "1 \<le> f x" "Z \<le> g x"
859     with `0 < Z` have "1 * Z \<le> f x * g x"
860       by (intro mult_mono) (auto simp: zero_le_divide_iff)
861     then show "Z \<le> f x * g x"
862        by simp
863   qed
864 qed
866 lemma filterlim_tendsto_pos_mult_at_bot:
867   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
868   shows "LIM x F. f x * g x :> at_bot"
869   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
870   unfolding filterlim_uminus_at_bot by simp
873   assumes f: "(f ---> c) F"
874   assumes g: "LIM x F. g x :> at_top"
875   shows "LIM x F. (f x + g x :: real) :> at_top"
876   unfolding filterlim_at_top_gt[where c=0]
877 proof safe
878   fix Z :: real assume "0 < Z"
879   from f have "eventually (\<lambda>x. c - 1 < f x) F"
880     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
881   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
882     unfolding filterlim_at_top by auto
883   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
884     by eventually_elim simp
885 qed
887 lemma LIM_at_top_divide:
888   fixes f g :: "'a \<Rightarrow> real"
889   assumes f: "(f ---> a) F" "0 < a"
890   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
891   shows "LIM x F. f x / g x :> at_top"
892   unfolding divide_inverse
893   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
896   assumes f: "LIM x F. f x :> at_top"
897   assumes g: "LIM x F. g x :> at_top"
898   shows "LIM x F. (f x + g x :: real) :> at_top"
899   unfolding filterlim_at_top_gt[where c=0]
900 proof safe
901   fix Z :: real assume "0 < Z"
902   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
903     unfolding filterlim_at_top by auto
904   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
905     unfolding filterlim_at_top by auto
906   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
907     by eventually_elim simp
908 qed
910 lemma tendsto_divide_0:
911   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
912   assumes f: "(f ---> c) F"
913   assumes g: "LIM x F. g x :> at_infinity"
914   shows "((\<lambda>x. f x / g x) ---> 0) F"
915   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
917 lemma linear_plus_1_le_power:
918   fixes x :: real
919   assumes x: "0 \<le> x"
920   shows "real n * x + 1 \<le> (x + 1) ^ n"
921 proof (induct n)
922   case (Suc n)
923   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
924     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
925   also have "\<dots> \<le> (x + 1)^Suc n"
926     using Suc x by (simp add: mult_left_mono)
927   finally show ?case .
928 qed simp
930 lemma filterlim_realpow_sequentially_gt1:
931   fixes x :: "'a :: real_normed_div_algebra"
932   assumes x[arith]: "1 < norm x"
933   shows "LIM n sequentially. x ^ n :> at_infinity"
934 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
935   fix y :: real assume "0 < y"
936   have "0 < norm x - 1" by simp
937   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
938   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
939   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
940   also have "\<dots> = norm x ^ N" by simp
941   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
942     by (metis order_less_le_trans power_increasing order_less_imp_le x)
943   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
944     unfolding eventually_sequentially
945     by (auto simp: norm_power)
946 qed simp
949 (* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
950    Hence it was references as Limits.within, but now it is Basic_Topology.eventually_within *)
951 lemmas eventually_within = eventually_within
953 end