src/HOL/Limits.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51474 1e9e68247ad1 parent 51472 adb441e4b9e9 child 51478 270b21f3ae0a permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
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)"
14 lemma eventually_at_infinity:
15   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
16 unfolding at_infinity_def
17 proof (rule eventually_Abs_filter, rule is_filter.intro)
18   fix P Q :: "'a \<Rightarrow> bool"
19   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
20   then obtain r s where
21     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
22   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
23   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
24 qed auto
26 lemma at_infinity_eq_at_top_bot:
27   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
28   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
29 proof (intro arg_cong[where f=Abs_filter] ext iffI)
30   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
31   then guess r ..
32   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
33   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
34 next
35   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
36   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
37   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
38     by (intro exI[of _ "max p (-q)"])
39        (auto simp: abs_real_def)
40 qed
42 lemma at_top_le_at_infinity:
43   "at_top \<le> (at_infinity :: real filter)"
44   unfolding at_infinity_eq_at_top_bot by simp
46 lemma at_bot_le_at_infinity:
47   "at_bot \<le> (at_infinity :: real filter)"
48   unfolding at_infinity_eq_at_top_bot by simp
50 subsection {* Boundedness *}
52 lemma Bfun_def:
53   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
54   unfolding Bfun_metric_def norm_conv_dist
55 proof safe
56   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
57   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
58     by (intro always_eventually) (metis dist_commute dist_triangle)
59   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
60     by eventually_elim auto
61   with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
62     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
63 qed auto
65 lemma BfunI:
66   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
67 unfolding Bfun_def
68 proof (intro exI conjI allI)
69   show "0 < max K 1" by simp
70 next
71   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
72     using K by (rule eventually_elim1, simp)
73 qed
75 lemma BfunE:
76   assumes "Bfun f F"
77   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
78 using assms unfolding Bfun_def by fast
80 subsection {* Convergence to Zero *}
82 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
83   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
85 lemma ZfunI:
86   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
87   unfolding Zfun_def by simp
89 lemma ZfunD:
90   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
91   unfolding Zfun_def by simp
93 lemma Zfun_ssubst:
94   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
95   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
97 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
98   unfolding Zfun_def by simp
100 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
101   unfolding Zfun_def by simp
103 lemma Zfun_imp_Zfun:
104   assumes f: "Zfun f F"
105   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
106   shows "Zfun (\<lambda>x. g x) F"
107 proof (cases)
108   assume K: "0 < K"
109   show ?thesis
110   proof (rule ZfunI)
111     fix r::real assume "0 < r"
112     hence "0 < r / K"
113       using K by (rule divide_pos_pos)
114     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
115       using ZfunD [OF f] by fast
116     with g show "eventually (\<lambda>x. norm (g x) < r) F"
117     proof eventually_elim
118       case (elim x)
119       hence "norm (f x) * K < r"
120         by (simp add: pos_less_divide_eq K)
121       thus ?case
122         by (simp add: order_le_less_trans [OF elim(1)])
123     qed
124   qed
125 next
126   assume "\<not> 0 < K"
127   hence K: "K \<le> 0" by (simp only: not_less)
128   show ?thesis
129   proof (rule ZfunI)
130     fix r :: real
131     assume "0 < r"
132     from g show "eventually (\<lambda>x. norm (g x) < r) F"
133     proof eventually_elim
134       case (elim x)
135       also have "norm (f x) * K \<le> norm (f x) * 0"
136         using K norm_ge_zero by (rule mult_left_mono)
137       finally show ?case
138         using `0 < r` by simp
139     qed
140   qed
141 qed
143 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
144   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
147   assumes f: "Zfun f F" and g: "Zfun g F"
148   shows "Zfun (\<lambda>x. f x + g x) F"
149 proof (rule ZfunI)
150   fix r::real assume "0 < r"
151   hence r: "0 < r / 2" by simp
152   have "eventually (\<lambda>x. norm (f x) < r/2) F"
153     using f r by (rule ZfunD)
154   moreover
155   have "eventually (\<lambda>x. norm (g x) < r/2) F"
156     using g r by (rule ZfunD)
157   ultimately
158   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
159   proof eventually_elim
160     case (elim x)
161     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
162       by (rule norm_triangle_ineq)
163     also have "\<dots> < r/2 + r/2"
164       using elim by (rule add_strict_mono)
165     finally show ?case
166       by simp
167   qed
168 qed
170 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
171   unfolding Zfun_def by simp
173 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
174   by (simp only: diff_minus Zfun_add Zfun_minus)
176 lemma (in bounded_linear) Zfun:
177   assumes g: "Zfun g F"
178   shows "Zfun (\<lambda>x. f (g x)) F"
179 proof -
180   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
181     using bounded by fast
182   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
183     by simp
184   with g show ?thesis
185     by (rule Zfun_imp_Zfun)
186 qed
188 lemma (in bounded_bilinear) Zfun:
189   assumes f: "Zfun f F"
190   assumes g: "Zfun g F"
191   shows "Zfun (\<lambda>x. f x ** g x) F"
192 proof (rule ZfunI)
193   fix r::real assume r: "0 < r"
194   obtain K where K: "0 < K"
195     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
196     using pos_bounded by fast
197   from K have K': "0 < inverse K"
198     by (rule positive_imp_inverse_positive)
199   have "eventually (\<lambda>x. norm (f x) < r) F"
200     using f r by (rule ZfunD)
201   moreover
202   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
203     using g K' by (rule ZfunD)
204   ultimately
205   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
206   proof eventually_elim
207     case (elim x)
208     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
209       by (rule norm_le)
210     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
211       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
212     also from K have "r * inverse K * K = r"
213       by simp
214     finally show ?case .
215   qed
216 qed
218 lemma (in bounded_bilinear) Zfun_left:
219   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
220   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
222 lemma (in bounded_bilinear) Zfun_right:
223   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
224   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
226 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
227 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
228 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
230 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
231   by (simp only: tendsto_iff Zfun_def dist_norm)
233 subsubsection {* Distance and norms *}
235 lemma tendsto_norm [tendsto_intros]:
236   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
237   unfolding norm_conv_dist by (intro tendsto_intros)
239 lemma tendsto_norm_zero:
240   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
241   by (drule tendsto_norm, simp)
243 lemma tendsto_norm_zero_cancel:
244   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
245   unfolding tendsto_iff dist_norm by simp
247 lemma tendsto_norm_zero_iff:
248   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
249   unfolding tendsto_iff dist_norm by simp
251 lemma tendsto_rabs [tendsto_intros]:
252   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
253   by (fold real_norm_def, rule tendsto_norm)
255 lemma tendsto_rabs_zero:
256   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
257   by (fold real_norm_def, rule tendsto_norm_zero)
259 lemma tendsto_rabs_zero_cancel:
260   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
261   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
263 lemma tendsto_rabs_zero_iff:
264   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
265   by (fold real_norm_def, rule tendsto_norm_zero_iff)
267 subsubsection {* Addition and subtraction *}
269 lemma tendsto_add [tendsto_intros]:
270   fixes a b :: "'a::real_normed_vector"
271   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
275   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
276   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
277   by (drule (1) tendsto_add, simp)
279 lemma tendsto_minus [tendsto_intros]:
280   fixes a :: "'a::real_normed_vector"
281   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
282   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
284 lemma tendsto_minus_cancel:
285   fixes a :: "'a::real_normed_vector"
286   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
287   by (drule tendsto_minus, simp)
289 lemma tendsto_minus_cancel_left:
290     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
291   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
292   by auto
294 lemma tendsto_diff [tendsto_intros]:
295   fixes a b :: "'a::real_normed_vector"
296   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
297   by (simp add: diff_minus tendsto_add tendsto_minus)
299 lemma tendsto_setsum [tendsto_intros]:
300   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
301   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
302   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
303 proof (cases "finite S")
304   assume "finite S" thus ?thesis using assms
306 next
307   assume "\<not> finite S" thus ?thesis
308     by (simp add: tendsto_const)
309 qed
311 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
313 subsubsection {* Linear operators and multiplication *}
315 lemma (in bounded_linear) tendsto:
316   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
317   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
319 lemma (in bounded_linear) tendsto_zero:
320   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
321   by (drule tendsto, simp only: zero)
323 lemma (in bounded_bilinear) tendsto:
324   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
325   by (simp only: tendsto_Zfun_iff prod_diff_prod
326                  Zfun_add Zfun Zfun_left Zfun_right)
328 lemma (in bounded_bilinear) tendsto_zero:
329   assumes f: "(f ---> 0) F"
330   assumes g: "(g ---> 0) F"
331   shows "((\<lambda>x. f x ** g x) ---> 0) F"
332   using tendsto [OF f g] by (simp add: zero_left)
334 lemma (in bounded_bilinear) tendsto_left_zero:
335   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
336   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
338 lemma (in bounded_bilinear) tendsto_right_zero:
339   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
340   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
342 lemmas tendsto_of_real [tendsto_intros] =
343   bounded_linear.tendsto [OF bounded_linear_of_real]
345 lemmas tendsto_scaleR [tendsto_intros] =
346   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
348 lemmas tendsto_mult [tendsto_intros] =
349   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
351 lemmas tendsto_mult_zero =
352   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
354 lemmas tendsto_mult_left_zero =
355   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
357 lemmas tendsto_mult_right_zero =
358   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
360 lemma tendsto_power [tendsto_intros]:
361   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
362   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
363   by (induct n) (simp_all add: tendsto_const tendsto_mult)
365 lemma tendsto_setprod [tendsto_intros]:
366   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
367   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
368   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
369 proof (cases "finite S")
370   assume "finite S" thus ?thesis using assms
371     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
372 next
373   assume "\<not> finite S" thus ?thesis
374     by (simp add: tendsto_const)
375 qed
377 subsubsection {* Inverse and division *}
379 lemma (in bounded_bilinear) Zfun_prod_Bfun:
380   assumes f: "Zfun f F"
381   assumes g: "Bfun g F"
382   shows "Zfun (\<lambda>x. f x ** g x) F"
383 proof -
384   obtain K where K: "0 \<le> K"
385     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
386     using nonneg_bounded by fast
387   obtain B where B: "0 < B"
388     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
389     using g by (rule BfunE)
390   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
391   using norm_g proof eventually_elim
392     case (elim x)
393     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
394       by (rule norm_le)
395     also have "\<dots> \<le> norm (f x) * B * K"
396       by (intro mult_mono' order_refl norm_g norm_ge_zero
397                 mult_nonneg_nonneg K elim)
398     also have "\<dots> = norm (f x) * (B * K)"
399       by (rule mult_assoc)
400     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
401   qed
402   with f show ?thesis
403     by (rule Zfun_imp_Zfun)
404 qed
406 lemma (in bounded_bilinear) flip:
407   "bounded_bilinear (\<lambda>x y. y ** x)"
408   apply default
409   apply (rule add_right)
410   apply (rule add_left)
411   apply (rule scaleR_right)
412   apply (rule scaleR_left)
413   apply (subst mult_commute)
414   using bounded by fast
416 lemma (in bounded_bilinear) Bfun_prod_Zfun:
417   assumes f: "Bfun f F"
418   assumes g: "Zfun g F"
419   shows "Zfun (\<lambda>x. f x ** g x) F"
420   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
422 lemma Bfun_inverse_lemma:
423   fixes x :: "'a::real_normed_div_algebra"
424   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
425   apply (subst nonzero_norm_inverse, clarsimp)
426   apply (erule (1) le_imp_inverse_le)
427   done
429 lemma Bfun_inverse:
430   fixes a :: "'a::real_normed_div_algebra"
431   assumes f: "(f ---> a) F"
432   assumes a: "a \<noteq> 0"
433   shows "Bfun (\<lambda>x. inverse (f x)) F"
434 proof -
435   from a have "0 < norm a" by simp
436   hence "\<exists>r>0. r < norm a" by (rule dense)
437   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
438   have "eventually (\<lambda>x. dist (f x) a < r) F"
439     using tendstoD [OF f r1] by fast
440   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
441   proof eventually_elim
442     case (elim x)
443     hence 1: "norm (f x - a) < r"
444       by (simp add: dist_norm)
445     hence 2: "f x \<noteq> 0" using r2 by auto
446     hence "norm (inverse (f x)) = inverse (norm (f x))"
447       by (rule nonzero_norm_inverse)
448     also have "\<dots> \<le> inverse (norm a - r)"
449     proof (rule le_imp_inverse_le)
450       show "0 < norm a - r" using r2 by simp
451     next
452       have "norm a - norm (f x) \<le> norm (a - f x)"
453         by (rule norm_triangle_ineq2)
454       also have "\<dots> = norm (f x - a)"
455         by (rule norm_minus_commute)
456       also have "\<dots> < r" using 1 .
457       finally show "norm a - r \<le> norm (f x)" by simp
458     qed
459     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
460   qed
461   thus ?thesis by (rule BfunI)
462 qed
464 lemma tendsto_inverse [tendsto_intros]:
465   fixes a :: "'a::real_normed_div_algebra"
466   assumes f: "(f ---> a) F"
467   assumes a: "a \<noteq> 0"
468   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
469 proof -
470   from a have "0 < norm a" by simp
471   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
472     by (rule tendstoD)
473   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
474     unfolding dist_norm by (auto elim!: eventually_elim1)
475   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
476     - (inverse (f x) * (f x - a) * inverse a)) F"
477     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
478   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
479     by (intro Zfun_minus Zfun_mult_left
480       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
481       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
482   ultimately show ?thesis
483     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
484 qed
486 lemma tendsto_divide [tendsto_intros]:
487   fixes a b :: "'a::real_normed_field"
488   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
489     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
490   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
492 lemma tendsto_sgn [tendsto_intros]:
493   fixes l :: "'a::real_normed_vector"
494   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
495   unfolding sgn_div_norm by (simp add: tendsto_intros)
497 lemma filterlim_at_infinity:
498   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
499   assumes "0 \<le> c"
500   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
501   unfolding filterlim_iff eventually_at_infinity
502 proof safe
503   fix P :: "'a \<Rightarrow> bool" and b
504   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
505     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
506   have "max b (c + 1) > c" by auto
507   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
508     by auto
509   then show "eventually (\<lambda>x. P (f x)) F"
510   proof eventually_elim
511     fix x assume "max b (c + 1) \<le> norm (f x)"
512     with P show "P (f x)" by auto
513   qed
514 qed force
516 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
518 text {*
520 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
521 @{term "at_right x"} and also @{term "at_right 0"}.
523 *}
525 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
527 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
528   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
529   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
531 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
532   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
533   apply (intro allI ex_cong)
534   apply (auto simp: dist_real_def field_simps)
535   apply (erule_tac x="-x" in allE)
536   apply simp
537   done
539 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
540   unfolding at_def filtermap_nhds_shift[symmetric]
541   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
543 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
544   unfolding filtermap_at_shift[symmetric]
545   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
547 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
548   using filtermap_at_right_shift[of "-a" 0] by simp
550 lemma filterlim_at_right_to_0:
551   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
552   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
554 lemma eventually_at_right_to_0:
555   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
556   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
558 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
559   unfolding at_def filtermap_nhds_minus[symmetric]
560   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
562 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
563   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
565 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
566   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
568 lemma filterlim_at_left_to_right:
569   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
570   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
572 lemma eventually_at_left_to_right:
573   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
574   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
576 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
577   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
578   by (metis le_minus_iff minus_minus)
580 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
581   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
583 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
584   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
586 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
587   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
589 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
590   unfolding filterlim_at_top eventually_at_bot_dense
591   by (metis leI minus_less_iff order_less_asym)
593 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
594   unfolding filterlim_at_bot eventually_at_top_dense
595   by (metis leI less_minus_iff order_less_asym)
597 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
598   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
599   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
600   by auto
602 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
603   unfolding filterlim_uminus_at_top by simp
605 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
606   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
607 proof safe
608   fix Z :: real assume [arith]: "0 < Z"
609   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
610     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
611   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
612     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
613 qed
615 lemma filterlim_inverse_at_top:
616   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
617   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
618      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
620 lemma filterlim_inverse_at_bot_neg:
621   "LIM x (at_left (0::real)). inverse x :> at_bot"
622   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
624 lemma filterlim_inverse_at_bot:
625   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
626   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
627   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
629 lemma tendsto_inverse_0:
630   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
631   shows "(inverse ---> (0::'a)) at_infinity"
632   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
633 proof safe
634   fix r :: real assume "0 < r"
635   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
636   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
637     fix x :: 'a
638     from `0 < r` have "0 < inverse (r / 2)" by simp
639     also assume *: "inverse (r / 2) \<le> norm x"
640     finally show "norm (inverse x) < r"
641       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
642   qed
643 qed
645 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
646 proof (rule antisym)
647   have "(inverse ---> (0::real)) at_top"
648     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
649   then show "filtermap inverse at_top \<le> at_right (0::real)"
650     unfolding at_within_eq
651     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
652 next
653   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
654     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
655   then show "at_right (0::real) \<le> filtermap inverse at_top"
656     by (simp add: filtermap_ident filtermap_filtermap)
657 qed
659 lemma eventually_at_right_to_top:
660   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
661   unfolding at_right_to_top eventually_filtermap ..
663 lemma filterlim_at_right_to_top:
664   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
665   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
667 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
668   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
670 lemma eventually_at_top_to_right:
671   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
672   unfolding at_top_to_right eventually_filtermap ..
674 lemma filterlim_at_top_to_right:
675   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
676   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
678 lemma filterlim_inverse_at_infinity:
679   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
680   shows "filterlim inverse at_infinity (at (0::'a))"
681   unfolding filterlim_at_infinity[OF order_refl]
682 proof safe
683   fix r :: real assume "0 < r"
684   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
685     unfolding eventually_at norm_inverse
686     by (intro exI[of _ "inverse r"])
687        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
688 qed
690 lemma filterlim_inverse_at_iff:
691   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
692   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
693   unfolding filterlim_def filtermap_filtermap[symmetric]
694 proof
695   assume "filtermap g F \<le> at_infinity"
696   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
697     by (rule filtermap_mono)
698   also have "\<dots> \<le> at 0"
699     using tendsto_inverse_0
700     by (auto intro!: le_withinI exI[of _ 1]
701              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
702   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
703 next
704   assume "filtermap inverse (filtermap g F) \<le> at 0"
705   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
706     by (rule filtermap_mono)
707   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
708     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
709 qed
711 lemma tendsto_inverse_0_at_top:
712   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
713  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
715 text {*
717 We only show rules for multiplication and addition when the functions are either against a real
718 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
720 *}
722 lemma filterlim_tendsto_pos_mult_at_top:
723   assumes f: "(f ---> c) F" and c: "0 < c"
724   assumes g: "LIM x F. g x :> at_top"
725   shows "LIM x F. (f x * g x :: real) :> at_top"
726   unfolding filterlim_at_top_gt[where c=0]
727 proof safe
728   fix Z :: real assume "0 < Z"
729   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
730     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
731              simp: dist_real_def abs_real_def split: split_if_asm)
732   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
733     unfolding filterlim_at_top by auto
734   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
735   proof eventually_elim
736     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
737     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
738       by (intro mult_mono) (auto simp: zero_le_divide_iff)
739     with `0 < c` show "Z \<le> f x * g x"
740        by simp
741   qed
742 qed
744 lemma filterlim_at_top_mult_at_top:
745   assumes f: "LIM x F. f x :> at_top"
746   assumes g: "LIM x F. g x :> at_top"
747   shows "LIM x F. (f x * g x :: real) :> at_top"
748   unfolding filterlim_at_top_gt[where c=0]
749 proof safe
750   fix Z :: real assume "0 < Z"
751   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
752     unfolding filterlim_at_top by auto
753   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
754     unfolding filterlim_at_top by auto
755   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
756   proof eventually_elim
757     fix x assume "1 \<le> f x" "Z \<le> g x"
758     with `0 < Z` have "1 * Z \<le> f x * g x"
759       by (intro mult_mono) (auto simp: zero_le_divide_iff)
760     then show "Z \<le> f x * g x"
761        by simp
762   qed
763 qed
765 lemma filterlim_tendsto_pos_mult_at_bot:
766   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
767   shows "LIM x F. f x * g x :> at_bot"
768   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
769   unfolding filterlim_uminus_at_bot by simp
772   assumes f: "(f ---> c) F"
773   assumes g: "LIM x F. g x :> at_top"
774   shows "LIM x F. (f x + g x :: real) :> at_top"
775   unfolding filterlim_at_top_gt[where c=0]
776 proof safe
777   fix Z :: real assume "0 < Z"
778   from f have "eventually (\<lambda>x. c - 1 < f x) F"
779     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
780   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
781     unfolding filterlim_at_top by auto
782   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
783     by eventually_elim simp
784 qed
786 lemma LIM_at_top_divide:
787   fixes f g :: "'a \<Rightarrow> real"
788   assumes f: "(f ---> a) F" "0 < a"
789   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
790   shows "LIM x F. f x / g x :> at_top"
791   unfolding divide_inverse
792   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
795   assumes f: "LIM x F. f x :> at_top"
796   assumes g: "LIM x F. g x :> at_top"
797   shows "LIM x F. (f x + g x :: real) :> at_top"
798   unfolding filterlim_at_top_gt[where c=0]
799 proof safe
800   fix Z :: real assume "0 < Z"
801   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
802     unfolding filterlim_at_top by auto
803   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
804     unfolding filterlim_at_top by auto
805   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
806     by eventually_elim simp
807 qed
809 lemma tendsto_divide_0:
810   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
811   assumes f: "(f ---> c) F"
812   assumes g: "LIM x F. g x :> at_infinity"
813   shows "((\<lambda>x. f x / g x) ---> 0) F"
814   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
816 lemma linear_plus_1_le_power:
817   fixes x :: real
818   assumes x: "0 \<le> x"
819   shows "real n * x + 1 \<le> (x + 1) ^ n"
820 proof (induct n)
821   case (Suc n)
822   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
823     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
824   also have "\<dots> \<le> (x + 1)^Suc n"
825     using Suc x by (simp add: mult_left_mono)
826   finally show ?case .
827 qed simp
829 lemma filterlim_realpow_sequentially_gt1:
830   fixes x :: "'a :: real_normed_div_algebra"
831   assumes x[arith]: "1 < norm x"
832   shows "LIM n sequentially. x ^ n :> at_infinity"
833 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
834   fix y :: real assume "0 < y"
835   have "0 < norm x - 1" by simp
836   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
837   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
838   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
839   also have "\<dots> = norm x ^ N" by simp
840   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
841     by (metis order_less_le_trans power_increasing order_less_imp_le x)
842   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
843     unfolding eventually_sequentially
844     by (auto simp: norm_power)
845 qed simp
848 (* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
849    Hence it was references as Limits.within, but now it is Basic_Topology.eventually_within *)
850 lemmas eventually_within = eventually_within
852 end