move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
1 (* Title: HOL/Limits.thy
3 Author: Jacques D. Fleuriot, University of Cambridge
4 Author: Lawrence C Paulson
9 header {* Limits on Real Vector Spaces *}
12 imports Real_Vector_Spaces
15 subsection {* Filter going to infinity norm *}
17 definition at_infinity :: "'a::real_normed_vector filter" where
18 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
20 lemma eventually_at_infinity:
21 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
22 unfolding at_infinity_def
23 proof (rule eventually_Abs_filter, rule is_filter.intro)
24 fix P Q :: "'a \<Rightarrow> bool"
25 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
27 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
28 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
29 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
32 lemma at_infinity_eq_at_top_bot:
33 "(at_infinity \<Colon> real filter) = sup at_top at_bot"
34 unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
35 proof (intro arg_cong[where f=Abs_filter] ext iffI)
36 fix P :: "real \<Rightarrow> bool"
37 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
38 then obtain r where "\<forall>x. r \<le> norm x \<longrightarrow> P x" ..
39 then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
40 then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
42 fix P :: "real \<Rightarrow> bool"
43 assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
44 then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
45 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
46 by (intro exI[of _ "max p (-q)"]) (auto simp: abs_real_def)
49 lemma at_top_le_at_infinity:
50 "at_top \<le> (at_infinity :: real filter)"
51 unfolding at_infinity_eq_at_top_bot by simp
53 lemma at_bot_le_at_infinity:
54 "at_bot \<le> (at_infinity :: real filter)"
55 unfolding at_infinity_eq_at_top_bot by simp
57 subsubsection {* Boundedness *}
59 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
60 Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
62 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
63 "Bseq X \<equiv> Bfun X sequentially"
65 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
67 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
68 unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
70 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
71 unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
74 "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
75 unfolding Bfun_metric_def norm_conv_dist
77 fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
78 moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
79 by (intro always_eventually) (metis dist_commute dist_triangle)
80 with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
81 by eventually_elim auto
82 with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
83 by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
87 assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
89 proof (intro exI conjI allI)
90 show "0 < max K 1" by simp
92 show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
93 using K by (rule eventually_elim1, simp)
98 obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
99 using assms unfolding Bfun_def by fast
101 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
102 unfolding Cauchy_def Bfun_metric_def eventually_sequentially
103 apply (erule_tac x=1 in allE)
106 apply (rule_tac x="X M" in exI)
107 apply (rule_tac x=1 in exI)
108 apply (erule_tac x=M in allE)
110 apply (rule_tac x=M in exI)
111 apply (auto simp: dist_commute)
115 subsubsection {* Bounded Sequences *}
117 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
118 by (intro BfunI) (auto simp: eventually_sequentially)
120 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
121 by (intro BfunI) (auto simp: eventually_sequentially)
123 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
124 unfolding Bfun_def eventually_sequentially
126 fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
127 then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
128 by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
129 (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
132 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
133 unfolding Bseq_def by auto
135 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
136 by (simp add: Bseq_def)
138 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
139 by (auto simp add: Bseq_def)
141 lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
142 proof (elim BseqE, intro bdd_aboveI2)
143 fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
144 by (auto elim!: allE[of _ n])
147 lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
148 proof (elim BseqE, intro bdd_belowI2)
149 fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
150 by (auto elim!: allE[of _ n])
153 lemma lemma_NBseq_def:
154 "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
157 from reals_Archimedean2 obtain n :: nat where "K < real n" ..
158 then have "K \<le> real (Suc n)" by auto
159 moreover assume "\<forall>m. norm (X m) \<le> K"
160 ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
161 by (blast intro: order_trans)
162 then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
163 qed (force simp add: real_of_nat_Suc)
165 text{* alternative definition for Bseq *}
166 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
167 apply (simp add: Bseq_def)
168 apply (simp (no_asm) add: lemma_NBseq_def)
171 lemma lemma_NBseq_def2:
172 "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
173 apply (subst lemma_NBseq_def, auto)
174 apply (rule_tac x = "Suc N" in exI)
175 apply (rule_tac [2] x = N in exI)
176 apply (auto simp add: real_of_nat_Suc)
177 prefer 2 apply (blast intro: order_less_imp_le)
178 apply (drule_tac x = n in spec, simp)
181 (* yet another definition for Bseq *)
182 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
183 by (simp add: Bseq_def lemma_NBseq_def2)
185 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
187 text{*alternative formulation for boundedness*}
188 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
189 apply (unfold Bseq_def, safe)
190 apply (rule_tac [2] x = "k + norm x" in exI)
191 apply (rule_tac x = K in exI, simp)
192 apply (rule exI [where x = 0], auto)
193 apply (erule order_less_le_trans, simp)
194 apply (drule_tac x=n in spec)
195 apply (drule order_trans [OF norm_triangle_ineq2])
199 text{*alternative formulation for boundedness*}
201 "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
205 where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
206 from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
207 from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
208 by (auto intro: order_trans norm_triangle_ineq4)
209 then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
211 with `0 < K + norm (X 0)` show ?Q by blast
213 assume ?Q then show ?P by (auto simp add: Bseq_iff2)
216 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
217 apply (simp add: Bseq_def)
218 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
219 apply (drule_tac x = n in spec, arith)
223 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
225 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
226 by (simp add: Bseq_def)
228 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
229 apply (simp add: subset_eq)
230 apply (rule BseqI'[where K="max (norm a) (norm b)"])
231 apply (erule_tac x=n in allE)
235 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
236 by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
238 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
239 by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
241 subsection {* Bounded Monotonic Sequences *}
243 subsubsection{*A Bounded and Monotonic Sequence Converges*}
246 (* FIXME: one use in NSA/HSEQ.thy *)
247 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
248 apply (rule_tac x="X m" in exI)
249 apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
250 unfolding eventually_sequentially
254 subsection {* Convergence to Zero *}
256 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
257 where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
260 "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
261 unfolding Zfun_def by simp
264 "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
265 unfolding Zfun_def by simp
268 "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
269 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
271 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
272 unfolding Zfun_def by simp
274 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
275 unfolding Zfun_def by simp
278 assumes f: "Zfun f F"
279 assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
280 shows "Zfun (\<lambda>x. g x) F"
285 fix r::real assume "0 < r"
287 using K by (rule divide_pos_pos)
288 then have "eventually (\<lambda>x. norm (f x) < r / K) F"
289 using ZfunD [OF f] by fast
290 with g show "eventually (\<lambda>x. norm (g x) < r) F"
291 proof eventually_elim
293 hence "norm (f x) * K < r"
294 by (simp add: pos_less_divide_eq K)
296 by (simp add: order_le_less_trans [OF elim(1)])
300 assume "\<not> 0 < K"
301 hence K: "K \<le> 0" by (simp only: not_less)
306 from g show "eventually (\<lambda>x. norm (g x) < r) F"
307 proof eventually_elim
309 also have "norm (f x) * K \<le> norm (f x) * 0"
310 using K norm_ge_zero by (rule mult_left_mono)
312 using `0 < r` by simp
317 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
318 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
321 assumes f: "Zfun f F" and g: "Zfun g F"
322 shows "Zfun (\<lambda>x. f x + g x) F"
324 fix r::real assume "0 < r"
325 hence r: "0 < r / 2" by simp
326 have "eventually (\<lambda>x. norm (f x) < r/2) F"
327 using f r by (rule ZfunD)
329 have "eventually (\<lambda>x. norm (g x) < r/2) F"
330 using g r by (rule ZfunD)
332 show "eventually (\<lambda>x. norm (f x + g x) < r) F"
333 proof eventually_elim
335 have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
336 by (rule norm_triangle_ineq)
337 also have "\<dots> < r/2 + r/2"
338 using elim by (rule add_strict_mono)
344 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
345 unfolding Zfun_def by simp
347 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
348 using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
350 lemma (in bounded_linear) Zfun:
351 assumes g: "Zfun g F"
352 shows "Zfun (\<lambda>x. f (g x)) F"
354 obtain K where "\<And>x. norm (f x) \<le> norm x * K"
355 using bounded by fast
356 then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
359 by (rule Zfun_imp_Zfun)
362 lemma (in bounded_bilinear) Zfun:
363 assumes f: "Zfun f F"
364 assumes g: "Zfun g F"
365 shows "Zfun (\<lambda>x. f x ** g x) F"
367 fix r::real assume r: "0 < r"
368 obtain K where K: "0 < K"
369 and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
370 using pos_bounded by fast
371 from K have K': "0 < inverse K"
372 by (rule positive_imp_inverse_positive)
373 have "eventually (\<lambda>x. norm (f x) < r) F"
374 using f r by (rule ZfunD)
376 have "eventually (\<lambda>x. norm (g x) < inverse K) F"
377 using g K' by (rule ZfunD)
379 show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
380 proof eventually_elim
382 have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
384 also have "norm (f x) * norm (g x) * K < r * inverse K * K"
385 by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
386 also from K have "r * inverse K * K = r"
392 lemma (in bounded_bilinear) Zfun_left:
393 "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
394 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
396 lemma (in bounded_bilinear) Zfun_right:
397 "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
398 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
400 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
401 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
402 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
404 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
405 by (simp only: tendsto_iff Zfun_def dist_norm)
407 subsubsection {* Distance and norms *}
409 lemma tendsto_dist [tendsto_intros]:
410 fixes l m :: "'a :: metric_space"
411 assumes f: "(f ---> l) F" and g: "(g ---> m) F"
412 shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
413 proof (rule tendstoI)
414 fix e :: real assume "0 < e"
415 hence e2: "0 < e/2" by simp
416 from tendstoD [OF f e2] tendstoD [OF g e2]
417 show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
418 proof (eventually_elim)
420 then show "dist (dist (f x) (g x)) (dist l m) < e"
421 unfolding dist_real_def
422 using dist_triangle2 [of "f x" "g x" "l"]
423 using dist_triangle2 [of "g x" "l" "m"]
424 using dist_triangle3 [of "l" "m" "f x"]
425 using dist_triangle [of "f x" "m" "g x"]
430 lemma continuous_dist[continuous_intros]:
431 fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
432 shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
433 unfolding continuous_def by (rule tendsto_dist)
435 lemma continuous_on_dist[continuous_on_intros]:
436 fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
437 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
438 unfolding continuous_on_def by (auto intro: tendsto_dist)
440 lemma tendsto_norm [tendsto_intros]:
441 "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
442 unfolding norm_conv_dist by (intro tendsto_intros)
444 lemma continuous_norm [continuous_intros]:
445 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
446 unfolding continuous_def by (rule tendsto_norm)
448 lemma continuous_on_norm [continuous_on_intros]:
449 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
450 unfolding continuous_on_def by (auto intro: tendsto_norm)
452 lemma tendsto_norm_zero:
453 "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
454 by (drule tendsto_norm, simp)
456 lemma tendsto_norm_zero_cancel:
457 "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
458 unfolding tendsto_iff dist_norm by simp
460 lemma tendsto_norm_zero_iff:
461 "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
462 unfolding tendsto_iff dist_norm by simp
464 lemma tendsto_rabs [tendsto_intros]:
465 "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
466 by (fold real_norm_def, rule tendsto_norm)
468 lemma continuous_rabs [continuous_intros]:
469 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
470 unfolding real_norm_def[symmetric] by (rule continuous_norm)
472 lemma continuous_on_rabs [continuous_on_intros]:
473 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
474 unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
476 lemma tendsto_rabs_zero:
477 "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
478 by (fold real_norm_def, rule tendsto_norm_zero)
480 lemma tendsto_rabs_zero_cancel:
481 "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
482 by (fold real_norm_def, rule tendsto_norm_zero_cancel)
484 lemma tendsto_rabs_zero_iff:
485 "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
486 by (fold real_norm_def, rule tendsto_norm_zero_iff)
488 subsubsection {* Addition and subtraction *}
490 lemma tendsto_add [tendsto_intros]:
491 fixes a b :: "'a::real_normed_vector"
492 shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
493 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
495 lemma continuous_add [continuous_intros]:
496 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
497 shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
498 unfolding continuous_def by (rule tendsto_add)
500 lemma continuous_on_add [continuous_on_intros]:
501 fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
502 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
503 unfolding continuous_on_def by (auto intro: tendsto_add)
505 lemma tendsto_add_zero:
506 fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
507 shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
508 by (drule (1) tendsto_add, simp)
510 lemma tendsto_minus [tendsto_intros]:
511 fixes a :: "'a::real_normed_vector"
512 shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
513 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
515 lemma continuous_minus [continuous_intros]:
516 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
517 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
518 unfolding continuous_def by (rule tendsto_minus)
520 lemma continuous_on_minus [continuous_on_intros]:
521 fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
522 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
523 unfolding continuous_on_def by (auto intro: tendsto_minus)
525 lemma tendsto_minus_cancel:
526 fixes a :: "'a::real_normed_vector"
527 shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
528 by (drule tendsto_minus, simp)
530 lemma tendsto_minus_cancel_left:
531 "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
532 using tendsto_minus_cancel[of f "- y" F] tendsto_minus[of f "- y" F]
535 lemma tendsto_diff [tendsto_intros]:
536 fixes a b :: "'a::real_normed_vector"
537 shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
538 using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
540 lemma continuous_diff [continuous_intros]:
541 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
542 shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
543 unfolding continuous_def by (rule tendsto_diff)
545 lemma continuous_on_diff [continuous_on_intros]:
546 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
547 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
548 unfolding continuous_on_def by (auto intro: tendsto_diff)
550 lemma tendsto_setsum [tendsto_intros]:
551 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
552 assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
553 shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
554 proof (cases "finite S")
555 assume "finite S" thus ?thesis using assms
556 by (induct, simp add: tendsto_const, simp add: tendsto_add)
558 assume "\<not> finite S" thus ?thesis
559 by (simp add: tendsto_const)
562 lemma continuous_setsum [continuous_intros]:
563 fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
564 shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
565 unfolding continuous_def by (rule tendsto_setsum)
567 lemma continuous_on_setsum [continuous_intros]:
568 fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
569 shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
570 unfolding continuous_on_def by (auto intro: tendsto_setsum)
572 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
574 subsubsection {* Linear operators and multiplication *}
576 lemma (in bounded_linear) tendsto:
577 "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
578 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
580 lemma (in bounded_linear) continuous:
581 "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
582 using tendsto[of g _ F] by (auto simp: continuous_def)
584 lemma (in bounded_linear) continuous_on:
585 "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
586 using tendsto[of g] by (auto simp: continuous_on_def)
588 lemma (in bounded_linear) tendsto_zero:
589 "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
590 by (drule tendsto, simp only: zero)
592 lemma (in bounded_bilinear) tendsto:
593 "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
594 by (simp only: tendsto_Zfun_iff prod_diff_prod
595 Zfun_add Zfun Zfun_left Zfun_right)
597 lemma (in bounded_bilinear) continuous:
598 "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
599 using tendsto[of f _ F g] by (auto simp: continuous_def)
601 lemma (in bounded_bilinear) continuous_on:
602 "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
603 using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
605 lemma (in bounded_bilinear) tendsto_zero:
606 assumes f: "(f ---> 0) F"
607 assumes g: "(g ---> 0) F"
608 shows "((\<lambda>x. f x ** g x) ---> 0) F"
609 using tendsto [OF f g] by (simp add: zero_left)
611 lemma (in bounded_bilinear) tendsto_left_zero:
612 "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
613 by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
615 lemma (in bounded_bilinear) tendsto_right_zero:
616 "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
617 by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
619 lemmas tendsto_of_real [tendsto_intros] =
620 bounded_linear.tendsto [OF bounded_linear_of_real]
622 lemmas tendsto_scaleR [tendsto_intros] =
623 bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
625 lemmas tendsto_mult [tendsto_intros] =
626 bounded_bilinear.tendsto [OF bounded_bilinear_mult]
628 lemmas continuous_of_real [continuous_intros] =
629 bounded_linear.continuous [OF bounded_linear_of_real]
631 lemmas continuous_scaleR [continuous_intros] =
632 bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
634 lemmas continuous_mult [continuous_intros] =
635 bounded_bilinear.continuous [OF bounded_bilinear_mult]
637 lemmas continuous_on_of_real [continuous_on_intros] =
638 bounded_linear.continuous_on [OF bounded_linear_of_real]
640 lemmas continuous_on_scaleR [continuous_on_intros] =
641 bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
643 lemmas continuous_on_mult [continuous_on_intros] =
644 bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
646 lemmas tendsto_mult_zero =
647 bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
649 lemmas tendsto_mult_left_zero =
650 bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
652 lemmas tendsto_mult_right_zero =
653 bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
655 lemma tendsto_power [tendsto_intros]:
656 fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
657 shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
658 by (induct n) (simp_all add: tendsto_const tendsto_mult)
660 lemma continuous_power [continuous_intros]:
661 fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
662 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
663 unfolding continuous_def by (rule tendsto_power)
665 lemma continuous_on_power [continuous_on_intros]:
666 fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
667 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
668 unfolding continuous_on_def by (auto intro: tendsto_power)
670 lemma tendsto_setprod [tendsto_intros]:
671 fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
672 assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
673 shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
674 proof (cases "finite S")
675 assume "finite S" thus ?thesis using assms
676 by (induct, simp add: tendsto_const, simp add: tendsto_mult)
678 assume "\<not> finite S" thus ?thesis
679 by (simp add: tendsto_const)
682 lemma continuous_setprod [continuous_intros]:
683 fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
684 shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
685 unfolding continuous_def by (rule tendsto_setprod)
687 lemma continuous_on_setprod [continuous_intros]:
688 fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
689 shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
690 unfolding continuous_on_def by (auto intro: tendsto_setprod)
692 subsubsection {* Inverse and division *}
694 lemma (in bounded_bilinear) Zfun_prod_Bfun:
695 assumes f: "Zfun f F"
696 assumes g: "Bfun g F"
697 shows "Zfun (\<lambda>x. f x ** g x) F"
699 obtain K where K: "0 \<le> K"
700 and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
701 using nonneg_bounded by fast
702 obtain B where B: "0 < B"
703 and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
704 using g by (rule BfunE)
705 have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
706 using norm_g proof eventually_elim
708 have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
710 also have "\<dots> \<le> norm (f x) * B * K"
711 by (intro mult_mono' order_refl norm_g norm_ge_zero
712 mult_nonneg_nonneg K elim)
713 also have "\<dots> = norm (f x) * (B * K)"
715 finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
718 by (rule Zfun_imp_Zfun)
721 lemma (in bounded_bilinear) flip:
722 "bounded_bilinear (\<lambda>x y. y ** x)"
724 apply (rule add_right)
725 apply (rule add_left)
726 apply (rule scaleR_right)
727 apply (rule scaleR_left)
728 apply (subst mult_commute)
729 using bounded by fast
731 lemma (in bounded_bilinear) Bfun_prod_Zfun:
732 assumes f: "Bfun f F"
733 assumes g: "Zfun g F"
734 shows "Zfun (\<lambda>x. f x ** g x) F"
735 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
737 lemma Bfun_inverse_lemma:
738 fixes x :: "'a::real_normed_div_algebra"
739 shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
740 apply (subst nonzero_norm_inverse, clarsimp)
741 apply (erule (1) le_imp_inverse_le)
745 fixes a :: "'a::real_normed_div_algebra"
746 assumes f: "(f ---> a) F"
747 assumes a: "a \<noteq> 0"
748 shows "Bfun (\<lambda>x. inverse (f x)) F"
750 from a have "0 < norm a" by simp
751 hence "\<exists>r>0. r < norm a" by (rule dense)
752 then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
753 have "eventually (\<lambda>x. dist (f x) a < r) F"
754 using tendstoD [OF f r1] by fast
755 hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
756 proof eventually_elim
758 hence 1: "norm (f x - a) < r"
759 by (simp add: dist_norm)
760 hence 2: "f x \<noteq> 0" using r2 by auto
761 hence "norm (inverse (f x)) = inverse (norm (f x))"
762 by (rule nonzero_norm_inverse)
763 also have "\<dots> \<le> inverse (norm a - r)"
764 proof (rule le_imp_inverse_le)
765 show "0 < norm a - r" using r2 by simp
767 have "norm a - norm (f x) \<le> norm (a - f x)"
768 by (rule norm_triangle_ineq2)
769 also have "\<dots> = norm (f x - a)"
770 by (rule norm_minus_commute)
771 also have "\<dots> < r" using 1 .
772 finally show "norm a - r \<le> norm (f x)" by simp
774 finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
776 thus ?thesis by (rule BfunI)
779 lemma tendsto_inverse [tendsto_intros]:
780 fixes a :: "'a::real_normed_div_algebra"
781 assumes f: "(f ---> a) F"
782 assumes a: "a \<noteq> 0"
783 shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
785 from a have "0 < norm a" by simp
786 with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
788 then have "eventually (\<lambda>x. f x \<noteq> 0) F"
789 unfolding dist_norm by (auto elim!: eventually_elim1)
790 with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
791 - (inverse (f x) * (f x - a) * inverse a)) F"
792 by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
793 moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
794 by (intro Zfun_minus Zfun_mult_left
795 bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
796 Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
797 ultimately show ?thesis
798 unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
801 lemma continuous_inverse:
802 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
803 assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
804 shows "continuous F (\<lambda>x. inverse (f x))"
805 using assms unfolding continuous_def by (rule tendsto_inverse)
807 lemma continuous_at_within_inverse[continuous_intros]:
808 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
809 assumes "continuous (at a within s) f" and "f a \<noteq> 0"
810 shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
811 using assms unfolding continuous_within by (rule tendsto_inverse)
813 lemma isCont_inverse[continuous_intros, simp]:
814 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
815 assumes "isCont f a" and "f a \<noteq> 0"
816 shows "isCont (\<lambda>x. inverse (f x)) a"
817 using assms unfolding continuous_at by (rule tendsto_inverse)
819 lemma continuous_on_inverse[continuous_on_intros]:
820 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
821 assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
822 shows "continuous_on s (\<lambda>x. inverse (f x))"
823 using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
825 lemma tendsto_divide [tendsto_intros]:
826 fixes a b :: "'a::real_normed_field"
827 shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
828 \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
829 by (simp add: tendsto_mult tendsto_inverse divide_inverse)
831 lemma continuous_divide:
832 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
833 assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
834 shows "continuous F (\<lambda>x. (f x) / (g x))"
835 using assms unfolding continuous_def by (rule tendsto_divide)
837 lemma continuous_at_within_divide[continuous_intros]:
838 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
839 assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
840 shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
841 using assms unfolding continuous_within by (rule tendsto_divide)
843 lemma isCont_divide[continuous_intros, simp]:
844 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
845 assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
846 shows "isCont (\<lambda>x. (f x) / g x) a"
847 using assms unfolding continuous_at by (rule tendsto_divide)
849 lemma continuous_on_divide[continuous_on_intros]:
850 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
851 assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
852 shows "continuous_on s (\<lambda>x. (f x) / (g x))"
853 using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
855 lemma tendsto_sgn [tendsto_intros]:
856 fixes l :: "'a::real_normed_vector"
857 shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
858 unfolding sgn_div_norm by (simp add: tendsto_intros)
860 lemma continuous_sgn:
861 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
862 assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
863 shows "continuous F (\<lambda>x. sgn (f x))"
864 using assms unfolding continuous_def by (rule tendsto_sgn)
866 lemma continuous_at_within_sgn[continuous_intros]:
867 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
868 assumes "continuous (at a within s) f" and "f a \<noteq> 0"
869 shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
870 using assms unfolding continuous_within by (rule tendsto_sgn)
872 lemma isCont_sgn[continuous_intros]:
873 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
874 assumes "isCont f a" and "f a \<noteq> 0"
875 shows "isCont (\<lambda>x. sgn (f x)) a"
876 using assms unfolding continuous_at by (rule tendsto_sgn)
878 lemma continuous_on_sgn[continuous_on_intros]:
879 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
880 assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
881 shows "continuous_on s (\<lambda>x. sgn (f x))"
882 using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
884 lemma filterlim_at_infinity:
885 fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
887 shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
888 unfolding filterlim_iff eventually_at_infinity
890 fix P :: "'a \<Rightarrow> bool" and b
891 assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
892 and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
893 have "max b (c + 1) > c" by auto
894 with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
896 then show "eventually (\<lambda>x. P (f x)) F"
897 proof eventually_elim
898 fix x assume "max b (c + 1) \<le> norm (f x)"
899 with P show "P (f x)" by auto
904 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
908 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
909 @{term "at_right x"} and also @{term "at_right 0"}.
913 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
915 lemma filtermap_homeomorph:
916 assumes f: "continuous (at a) f"
917 assumes g: "continuous (at (f a)) g"
918 assumes bij1: "\<forall>x. f (g x) = x" and bij2: "\<forall>x. g (f x) = x"
919 shows "filtermap f (nhds a) = nhds (f a)"
920 unfolding filter_eq_iff eventually_filtermap eventually_nhds
922 fix P S assume S: "open S" "f a \<in> S" and P: "\<forall>x\<in>S. P x"
923 from continuous_within_topological[THEN iffD1, rule_format, OF f S] P
924 show "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P (f x))" by auto
926 fix P S assume S: "open S" "a \<in> S" and P: "\<forall>x\<in>S. P (f x)"
927 with continuous_within_topological[THEN iffD1, rule_format, OF g, of S] bij2
928 obtain A where "open A" "f a \<in> A" "(\<forall>y\<in>A. g y \<in> S)"
930 with P bij1 show "\<exists>S. open S \<and> f a \<in> S \<and> (\<forall>x\<in>S. P x)"
931 by (force intro!: exI[of _ A])
934 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
935 by (rule filtermap_homeomorph[where g="\<lambda>x. x + d"]) (auto intro: continuous_intros)
937 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
938 by (rule filtermap_homeomorph[where g=uminus]) (auto intro: continuous_minus)
940 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
941 by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
943 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
944 by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
946 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
947 using filtermap_at_right_shift[of "-a" 0] by simp
949 lemma filterlim_at_right_to_0:
950 "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
951 unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
953 lemma eventually_at_right_to_0:
954 "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
955 unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
957 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
958 by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
960 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
961 by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
963 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
964 by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
966 lemma filterlim_at_left_to_right:
967 "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
968 unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
970 lemma eventually_at_left_to_right:
971 "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
972 unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
974 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
975 unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
976 by (metis le_minus_iff minus_minus)
978 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
979 unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
981 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
982 unfolding filterlim_def at_top_mirror filtermap_filtermap ..
984 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
985 unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
987 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
988 unfolding filterlim_at_top eventually_at_bot_dense
989 by (metis leI minus_less_iff order_less_asym)
991 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
992 unfolding filterlim_at_bot eventually_at_top_dense
993 by (metis leI less_minus_iff order_less_asym)
995 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
996 using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
997 using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
1000 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
1001 unfolding filterlim_uminus_at_top by simp
1003 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
1004 unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
1006 fix Z :: real assume [arith]: "0 < Z"
1007 then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
1008 by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
1009 then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
1010 by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
1013 lemma filterlim_inverse_at_top:
1014 "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
1015 by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
1016 (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
1018 lemma filterlim_inverse_at_bot_neg:
1019 "LIM x (at_left (0::real)). inverse x :> at_bot"
1020 by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
1022 lemma filterlim_inverse_at_bot:
1023 "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
1024 unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
1025 by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
1027 lemma tendsto_inverse_0:
1028 fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
1029 shows "(inverse ---> (0::'a)) at_infinity"
1030 unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
1032 fix r :: real assume "0 < r"
1033 show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
1034 proof (intro exI[of _ "inverse (r / 2)"] allI impI)
1036 from `0 < r` have "0 < inverse (r / 2)" by simp
1037 also assume *: "inverse (r / 2) \<le> norm x"
1038 finally show "norm (inverse x) < r"
1039 using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
1043 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
1044 proof (rule antisym)
1045 have "(inverse ---> (0::real)) at_top"
1046 by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
1047 then show "filtermap inverse at_top \<le> at_right (0::real)"
1048 by (simp add: le_principal eventually_filtermap eventually_gt_at_top filterlim_def at_within_def)
1050 have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
1051 using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
1052 then show "at_right (0::real) \<le> filtermap inverse at_top"
1053 by (simp add: filtermap_ident filtermap_filtermap)
1056 lemma eventually_at_right_to_top:
1057 "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
1058 unfolding at_right_to_top eventually_filtermap ..
1060 lemma filterlim_at_right_to_top:
1061 "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
1062 unfolding filterlim_def at_right_to_top filtermap_filtermap ..
1064 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
1065 unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
1067 lemma eventually_at_top_to_right:
1068 "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
1069 unfolding at_top_to_right eventually_filtermap ..
1071 lemma filterlim_at_top_to_right:
1072 "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
1073 unfolding filterlim_def at_top_to_right filtermap_filtermap ..
1075 lemma filterlim_inverse_at_infinity:
1076 fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
1077 shows "filterlim inverse at_infinity (at (0::'a))"
1078 unfolding filterlim_at_infinity[OF order_refl]
1080 fix r :: real assume "0 < r"
1081 then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
1082 unfolding eventually_at norm_inverse
1083 by (intro exI[of _ "inverse r"])
1084 (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
1087 lemma filterlim_inverse_at_iff:
1088 fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
1089 shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
1090 unfolding filterlim_def filtermap_filtermap[symmetric]
1092 assume "filtermap g F \<le> at_infinity"
1093 then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
1094 by (rule filtermap_mono)
1095 also have "\<dots> \<le> at 0"
1096 using tendsto_inverse_0[where 'a='b]
1097 by (auto intro!: exI[of _ 1]
1098 simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
1099 finally show "filtermap inverse (filtermap g F) \<le> at 0" .
1101 assume "filtermap inverse (filtermap g F) \<le> at 0"
1102 then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
1103 by (rule filtermap_mono)
1104 with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
1105 by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
1108 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
1109 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
1113 We only show rules for multiplication and addition when the functions are either against a real
1114 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
1118 lemma filterlim_tendsto_pos_mult_at_top:
1119 assumes f: "(f ---> c) F" and c: "0 < c"
1120 assumes g: "LIM x F. g x :> at_top"
1121 shows "LIM x F. (f x * g x :: real) :> at_top"
1122 unfolding filterlim_at_top_gt[where c=0]
1124 fix Z :: real assume "0 < Z"
1125 from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
1126 by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
1127 simp: dist_real_def abs_real_def split: split_if_asm)
1128 moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
1129 unfolding filterlim_at_top by auto
1130 ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
1131 proof eventually_elim
1132 fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
1133 with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
1134 by (intro mult_mono) (auto simp: zero_le_divide_iff)
1135 with `0 < c` show "Z \<le> f x * g x"
1140 lemma filterlim_at_top_mult_at_top:
1141 assumes f: "LIM x F. f x :> at_top"
1142 assumes g: "LIM x F. g x :> at_top"
1143 shows "LIM x F. (f x * g x :: real) :> at_top"
1144 unfolding filterlim_at_top_gt[where c=0]
1146 fix Z :: real assume "0 < Z"
1147 from f have "eventually (\<lambda>x. 1 \<le> f x) F"
1148 unfolding filterlim_at_top by auto
1149 moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
1150 unfolding filterlim_at_top by auto
1151 ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
1152 proof eventually_elim
1153 fix x assume "1 \<le> f x" "Z \<le> g x"
1154 with `0 < Z` have "1 * Z \<le> f x * g x"
1155 by (intro mult_mono) (auto simp: zero_le_divide_iff)
1156 then show "Z \<le> f x * g x"
1161 lemma filterlim_tendsto_pos_mult_at_bot:
1162 assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
1163 shows "LIM x F. f x * g x :> at_bot"
1164 using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
1165 unfolding filterlim_uminus_at_bot by simp
1167 lemma filterlim_tendsto_add_at_top:
1168 assumes f: "(f ---> c) F"
1169 assumes g: "LIM x F. g x :> at_top"
1170 shows "LIM x F. (f x + g x :: real) :> at_top"
1171 unfolding filterlim_at_top_gt[where c=0]
1173 fix Z :: real assume "0 < Z"
1174 from f have "eventually (\<lambda>x. c - 1 < f x) F"
1175 by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
1176 moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
1177 unfolding filterlim_at_top by auto
1178 ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
1179 by eventually_elim simp
1182 lemma LIM_at_top_divide:
1183 fixes f g :: "'a \<Rightarrow> real"
1184 assumes f: "(f ---> a) F" "0 < a"
1185 assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
1186 shows "LIM x F. f x / g x :> at_top"
1187 unfolding divide_inverse
1188 by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
1190 lemma filterlim_at_top_add_at_top:
1191 assumes f: "LIM x F. f x :> at_top"
1192 assumes g: "LIM x F. g x :> at_top"
1193 shows "LIM x F. (f x + g x :: real) :> at_top"
1194 unfolding filterlim_at_top_gt[where c=0]
1196 fix Z :: real assume "0 < Z"
1197 from f have "eventually (\<lambda>x. 0 \<le> f x) F"
1198 unfolding filterlim_at_top by auto
1199 moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
1200 unfolding filterlim_at_top by auto
1201 ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
1202 by eventually_elim simp
1205 lemma tendsto_divide_0:
1206 fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
1207 assumes f: "(f ---> c) F"
1208 assumes g: "LIM x F. g x :> at_infinity"
1209 shows "((\<lambda>x. f x / g x) ---> 0) F"
1210 using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
1212 lemma linear_plus_1_le_power:
1214 assumes x: "0 \<le> x"
1215 shows "real n * x + 1 \<le> (x + 1) ^ n"
1218 have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
1219 by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
1220 also have "\<dots> \<le> (x + 1)^Suc n"
1221 using Suc x by (simp add: mult_left_mono)
1222 finally show ?case .
1225 lemma filterlim_realpow_sequentially_gt1:
1226 fixes x :: "'a :: real_normed_div_algebra"
1227 assumes x[arith]: "1 < norm x"
1228 shows "LIM n sequentially. x ^ n :> at_infinity"
1229 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
1230 fix y :: real assume "0 < y"
1231 have "0 < norm x - 1" by simp
1232 then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
1233 also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
1234 also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
1235 also have "\<dots> = norm x ^ N" by simp
1236 finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
1237 by (metis order_less_le_trans power_increasing order_less_imp_le x)
1238 then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
1239 unfolding eventually_sequentially
1240 by (auto simp: norm_power)
1244 subsection {* Limits of Sequences *}
1246 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
1250 fixes L :: "'a::real_normed_vector"
1251 shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
1252 unfolding LIMSEQ_def dist_norm ..
1255 fixes L :: "'a::real_normed_vector"
1256 shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
1257 by (simp add: LIMSEQ_iff)
1260 fixes L :: "'a::real_normed_vector"
1261 shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
1262 by (simp add: LIMSEQ_iff)
1264 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
1265 unfolding tendsto_def eventually_sequentially
1266 by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
1268 lemma Bseq_inverse_lemma:
1269 fixes x :: "'a::real_normed_div_algebra"
1270 shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
1271 apply (subst nonzero_norm_inverse, clarsimp)
1272 apply (erule (1) le_imp_inverse_le)
1276 fixes a :: "'a::real_normed_div_algebra"
1277 shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
1278 by (rule Bfun_inverse)
1280 lemma LIMSEQ_diff_approach_zero:
1281 fixes L :: "'a::real_normed_vector"
1282 shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
1283 by (drule (1) tendsto_add, simp)
1285 lemma LIMSEQ_diff_approach_zero2:
1286 fixes L :: "'a::real_normed_vector"
1287 shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
1288 by (drule (1) tendsto_diff, simp)
1290 text{*An unbounded sequence's inverse tends to 0*}
1292 lemma LIMSEQ_inverse_zero:
1293 "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
1294 apply (rule filterlim_compose[OF tendsto_inverse_0])
1295 apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
1296 apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
1299 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
1301 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
1302 by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
1303 filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
1305 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
1306 infinity is now easily proved*}
1308 lemma LIMSEQ_inverse_real_of_nat_add:
1309 "(%n. r + inverse(real(Suc n))) ----> r"
1310 using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
1312 lemma LIMSEQ_inverse_real_of_nat_add_minus:
1313 "(%n. r + -inverse(real(Suc n))) ----> r"
1314 using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
1317 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
1318 "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
1319 using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
1322 subsection {* Convergence on sequences *}
1324 lemma convergent_add:
1325 fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
1326 assumes "convergent (\<lambda>n. X n)"
1327 assumes "convergent (\<lambda>n. Y n)"
1328 shows "convergent (\<lambda>n. X n + Y n)"
1329 using assms unfolding convergent_def by (fast intro: tendsto_add)
1331 lemma convergent_setsum:
1332 fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
1333 assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
1334 shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
1335 proof (cases "finite A")
1336 case True from this and assms show ?thesis
1337 by (induct A set: finite) (simp_all add: convergent_const convergent_add)
1338 qed (simp add: convergent_const)
1340 lemma (in bounded_linear) convergent:
1341 assumes "convergent (\<lambda>n. X n)"
1342 shows "convergent (\<lambda>n. f (X n))"
1343 using assms unfolding convergent_def by (fast intro: tendsto)
1345 lemma (in bounded_bilinear) convergent:
1346 assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
1347 shows "convergent (\<lambda>n. X n ** Y n)"
1348 using assms unfolding convergent_def by (fast intro: tendsto)
1350 lemma convergent_minus_iff:
1351 fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1352 shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
1353 apply (simp add: convergent_def)
1354 apply (auto dest: tendsto_minus)
1355 apply (drule tendsto_minus, auto)
1359 text {* A monotone sequence converges to its least upper bound. *}
1361 lemma LIMSEQ_incseq_SUP:
1362 fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
1363 assumes u: "bdd_above (range X)"
1364 assumes X: "incseq X"
1365 shows "X ----> (SUP i. X i)"
1366 by (rule order_tendstoI)
1367 (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
1369 lemma LIMSEQ_decseq_INF:
1370 fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
1371 assumes u: "bdd_below (range X)"
1372 assumes X: "decseq X"
1373 shows "X ----> (INF i. X i)"
1374 by (rule order_tendstoI)
1375 (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
1377 text{*Main monotonicity theorem*}
1379 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
1380 by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
1382 lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
1383 by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
1386 fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1387 shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
1388 unfolding Cauchy_def dist_norm ..
1391 fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1392 shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
1393 by (simp add: Cauchy_iff)
1396 fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1397 shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
1398 by (simp add: Cauchy_iff)
1400 lemma incseq_convergent:
1401 fixes X :: "nat \<Rightarrow> real"
1402 assumes "incseq X" and "\<forall>i. X i \<le> B"
1403 obtains L where "X ----> L" "\<forall>i. X i \<le> L"
1405 from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
1406 obtain L where "X ----> L"
1407 by (auto simp: convergent_def monoseq_def incseq_def)
1408 with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
1409 by (auto intro!: exI[of _ L] incseq_le)
1412 lemma decseq_convergent:
1413 fixes X :: "nat \<Rightarrow> real"
1414 assumes "decseq X" and "\<forall>i. B \<le> X i"
1415 obtains L where "X ----> L" "\<forall>i. L \<le> X i"
1417 from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
1418 obtain L where "X ----> L"
1419 by (auto simp: convergent_def monoseq_def decseq_def)
1420 with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
1421 by (auto intro!: exI[of _ L] decseq_le)
1424 subsubsection {* Cauchy Sequences are Bounded *}
1426 text{*A Cauchy sequence is bounded -- this is the standard
1427 proof mechanization rather than the nonstandard proof*}
1429 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
1430 ==> \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
1431 apply (clarify, drule spec, drule (1) mp)
1432 apply (simp only: norm_minus_commute)
1433 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
1437 subsection {* Power Sequences *}
1439 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
1440 "x<1"}. Proof will use (NS) Cauchy equivalence for convergence and
1441 also fact that bounded and monotonic sequence converges.*}
1443 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
1444 apply (simp add: Bseq_def)
1445 apply (rule_tac x = 1 in exI)
1446 apply (simp add: power_abs)
1447 apply (auto dest: power_mono)
1450 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
1451 apply (clarify intro!: mono_SucI2)
1452 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
1455 lemma convergent_realpow:
1456 "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
1457 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
1459 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
1460 by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
1462 lemma LIMSEQ_realpow_zero:
1463 "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1465 assume "0 \<le> x" and "x \<noteq> 0"
1466 hence x0: "0 < x" by simp
1468 from x0 x1 have "1 < inverse x"
1469 by (rule one_less_inverse)
1470 hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
1471 by (rule LIMSEQ_inverse_realpow_zero)
1472 thus ?thesis by (simp add: power_inverse)
1473 qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
1475 lemma LIMSEQ_power_zero:
1476 fixes x :: "'a::{real_normed_algebra_1}"
1477 shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1478 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
1479 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
1480 apply (simp add: power_abs norm_power_ineq)
1483 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
1484 by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
1486 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
1488 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
1489 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
1491 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
1492 by (rule LIMSEQ_power_zero) simp
1495 subsection {* Limits of Functions *}
1498 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1499 shows "f -- a --> L =
1500 (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
1501 by (simp add: LIM_def dist_norm)
1504 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1505 shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
1507 by (simp add: LIM_eq)
1510 fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1511 shows "[| f -- a --> L; 0<r |]
1512 ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
1513 by (simp add: LIM_eq)
1516 fixes a :: "'a::real_normed_vector"
1517 shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
1518 unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
1520 lemma LIM_offset_zero:
1521 fixes a :: "'a::real_normed_vector"
1522 shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
1523 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
1525 lemma LIM_offset_zero_cancel:
1526 fixes a :: "'a::real_normed_vector"
1527 shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
1528 by (drule_tac k="- a" in LIM_offset, simp)
1530 lemma LIM_offset_zero_iff:
1531 fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
1532 shows "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
1533 using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
1536 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1537 shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
1538 unfolding tendsto_iff dist_norm by simp
1540 lemma LIM_zero_cancel:
1541 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1542 shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
1543 unfolding tendsto_iff dist_norm by simp
1546 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1547 shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
1548 unfolding tendsto_iff dist_norm by simp
1551 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1552 fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
1553 assumes f: "f -- a --> l"
1554 assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
1555 shows "g -- a --> m"
1556 by (rule metric_LIM_imp_LIM [OF f],
1557 simp add: dist_norm le)
1560 fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1562 assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
1563 shows "g -- a --> l \<Longrightarrow> f -- a --> l"
1564 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
1567 fixes a :: "'a::real_normed_vector"
1568 assumes f: "f -- a --> b"
1569 assumes g: "g -- b --> c"
1570 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
1571 shows "(\<lambda>x. g (f x)) -- a --> c"
1572 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
1574 lemma real_LIM_sandwich_zero:
1575 fixes f g :: "'a::topological_space \<Rightarrow> real"
1576 assumes f: "f -- a --> 0"
1577 assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
1578 assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
1579 shows "g -- a --> 0"
1580 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
1581 fix x assume x: "x \<noteq> a"
1582 have "norm (g x - 0) = g x" by (simp add: 1 x)
1583 also have "g x \<le> f x" by (rule 2 [OF x])
1584 also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
1585 also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
1586 finally show "norm (g x - 0) \<le> norm (f x - 0)" .
1590 subsection {* Continuity *}
1592 lemma LIM_isCont_iff:
1593 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1594 shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
1595 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
1598 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
1599 shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
1600 by (simp add: isCont_def LIM_isCont_iff)
1602 lemma isCont_LIM_compose2:
1603 fixes a :: "'a::real_normed_vector"
1604 assumes f [unfolded isCont_def]: "isCont f a"
1605 assumes g: "g -- f a --> l"
1606 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
1607 shows "(\<lambda>x. g (f x)) -- a --> l"
1608 by (rule LIM_compose2 [OF f g inj])
1611 lemma isCont_norm [simp]:
1612 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1613 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
1614 by (fact continuous_norm)
1616 lemma isCont_rabs [simp]:
1617 fixes f :: "'a::t2_space \<Rightarrow> real"
1618 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
1619 by (fact continuous_rabs)
1621 lemma isCont_add [simp]:
1622 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1623 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
1624 by (fact continuous_add)
1626 lemma isCont_minus [simp]:
1627 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1628 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
1629 by (fact continuous_minus)
1631 lemma isCont_diff [simp]:
1632 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1633 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
1634 by (fact continuous_diff)
1636 lemma isCont_mult [simp]:
1637 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
1638 shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
1639 by (fact continuous_mult)
1641 lemma (in bounded_linear) isCont:
1642 "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
1643 by (fact continuous)
1645 lemma (in bounded_bilinear) isCont:
1646 "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
1647 by (fact continuous)
1649 lemmas isCont_scaleR [simp] =
1650 bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
1652 lemmas isCont_of_real [simp] =
1653 bounded_linear.isCont [OF bounded_linear_of_real]
1655 lemma isCont_power [simp]:
1656 fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
1657 shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
1658 by (fact continuous_power)
1660 lemma isCont_setsum [simp]:
1661 fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
1662 shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
1663 by (auto intro: continuous_setsum)
1665 lemmas isCont_intros =
1666 isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
1667 isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
1668 isCont_of_real isCont_power isCont_sgn isCont_setsum
1670 subsection {* Uniform Continuity *}
1673 isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
1674 "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
1676 lemma isUCont_isCont: "isUCont f ==> isCont f x"
1677 by (simp add: isUCont_def isCont_def LIM_def, force)
1679 lemma isUCont_Cauchy:
1680 "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
1681 unfolding isUCont_def
1682 apply (rule metric_CauchyI)
1683 apply (drule_tac x=e in spec, safe)
1684 apply (drule_tac e=s in metric_CauchyD, safe)
1685 apply (rule_tac x=M in exI, simp)
1688 lemma (in bounded_linear) isUCont: "isUCont f"
1689 unfolding isUCont_def dist_norm
1690 proof (intro allI impI)
1691 fix r::real assume r: "0 < r"
1692 obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
1693 using pos_bounded by fast
1694 show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
1695 proof (rule exI, safe)
1696 from r K show "0 < r / K" by (rule divide_pos_pos)
1699 assume xy: "norm (x - y) < r / K"
1700 have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
1701 also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
1702 also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
1703 finally show "norm (f x - f y) < r" .
1707 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
1708 by (rule isUCont [THEN isUCont_Cauchy])
1710 lemma LIM_less_bound:
1711 fixes f :: "real \<Rightarrow> real"
1712 assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
1714 proof (rule tendsto_le_const)
1715 show "(f ---> f x) (at_left x)"
1716 using `isCont f x` by (simp add: filterlim_at_split isCont_def)
1717 show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
1718 using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
1722 subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
1724 lemma nested_sequence_unique:
1725 assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) ----> 0"
1726 shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
1728 have "incseq f" unfolding incseq_Suc_iff by fact
1729 have "decseq g" unfolding decseq_Suc_iff by fact
1732 from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
1733 with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
1734 then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
1735 using incseq_convergent[OF `incseq f`] by auto
1738 from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
1739 with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
1740 then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
1741 using decseq_convergent[OF `decseq g`] by auto
1742 moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
1743 ultimately show ?thesis by auto
1746 lemma Bolzano[consumes 1, case_names trans local]:
1747 fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
1748 assumes [arith]: "a \<le> b"
1749 assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
1750 assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
1753 def bisect \<equiv> "nat_rec (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
1754 def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
1755 have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
1756 and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
1757 by (simp_all add: l_def u_def bisect_def split: prod.split)
1759 { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
1761 have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
1762 proof (safe intro!: nested_sequence_unique)
1763 fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
1765 { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
1766 then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
1768 then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
1769 obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
1770 using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
1774 assume "\<not> P a b"
1775 { fix n have "\<not> P (l n) (u n)"
1777 case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
1778 qed (simp add: `\<not> P a b`) }
1780 { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
1781 using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
1782 moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
1783 using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
1784 ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
1785 proof eventually_elim
1786 fix n assume "x - d / 2 < l n" "u n < x + d / 2"
1787 from add_strict_mono[OF this] have "u n - l n < d" by simp
1788 with x show "P (l n) (u n)" by (rule d)
1790 ultimately show False by simp
1794 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
1795 proof (cases "a \<le> b", rule compactI)
1796 fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
1798 from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
1799 proof (induct rule: Bolzano)
1801 then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
1802 from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
1804 with trans show ?case
1805 unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
1808 then have "x \<in> \<Union>C" using C by auto
1809 with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
1810 then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
1811 by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
1812 with `c \<in> C` show ?case
1813 by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
1818 subsection {* Boundedness of continuous functions *}
1820 text{*By bisection, function continuous on closed interval is bounded above*}
1823 fixes f :: "real \<Rightarrow> 'a::linorder_topology"
1824 shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
1825 \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
1826 using continuous_attains_sup[of "{a .. b}" f]
1827 by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
1830 fixes f :: "real \<Rightarrow> 'a::linorder_topology"
1831 shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
1832 \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
1833 using continuous_attains_inf[of "{a .. b}" f]
1834 by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
1836 lemma isCont_bounded:
1837 fixes f :: "real \<Rightarrow> 'a::linorder_topology"
1838 shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
1839 using isCont_eq_Ub[of a b f] by auto
1841 lemma isCont_has_Ub:
1842 fixes f :: "real \<Rightarrow> 'a::linorder_topology"
1843 shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
1844 \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
1845 using isCont_eq_Ub[of a b f] by auto
1847 (*HOL style here: object-level formulations*)
1848 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
1849 (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
1850 --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
1851 by (blast intro: IVT)
1853 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
1854 (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
1855 --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
1856 by (blast intro: IVT2)
1859 fixes f :: "real \<Rightarrow> real"
1860 assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
1861 shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
1862 (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
1864 obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
1865 using isCont_eq_Ub[OF assms] by auto
1866 obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
1867 using isCont_eq_Lb[OF assms] by auto
1869 using IVT[of f L _ M] IVT2[of f L _ M] M L assms
1870 apply (rule_tac x="f L" in exI)
1871 apply (rule_tac x="f M" in exI)
1872 apply (cases "L \<le> M")
1873 apply (simp, metis order_trans)
1874 apply (simp, metis order_trans)
1879 text{*Continuity of inverse function*}
1881 lemma isCont_inverse_function:
1882 fixes f g :: "real \<Rightarrow> real"
1884 and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
1885 and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
1886 shows "isCont g (f x)"
1888 let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
1890 have f: "continuous_on ?D f"
1891 using cont by (intro continuous_at_imp_continuous_on ballI) auto
1892 then have g: "continuous_on (f`?D) g"
1893 using inj by (intro continuous_on_inv) auto
1895 from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
1896 by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
1897 with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
1898 by (rule continuous_on_subset)
1900 have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
1901 using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
1902 then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
1906 by (simp add: continuous_on_eq_continuous_at)
1909 lemma isCont_inverse_function2:
1910 fixes f g :: "real \<Rightarrow> real" shows
1911 "\<lbrakk>a < x; x < b;
1912 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
1913 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
1914 \<Longrightarrow> isCont g (f x)"
1915 apply (rule isCont_inverse_function
1916 [where f=f and d="min (x - a) (b - x)"])
1917 apply (simp_all add: abs_le_iff)
1920 (* need to rename second isCont_inverse *)
1922 lemma isCont_inv_fun:
1923 fixes f g :: "real \<Rightarrow> real"
1924 shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
1925 \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
1927 by (rule isCont_inverse_function)
1929 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
1930 lemma LIM_fun_gt_zero:
1931 fixes f :: "real \<Rightarrow> real"
1932 shows "f -- c --> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
1933 apply (drule (1) LIM_D, clarify)
1934 apply (rule_tac x = s in exI)
1935 apply (simp add: abs_less_iff)
1938 lemma LIM_fun_less_zero:
1939 fixes f :: "real \<Rightarrow> real"
1940 shows "f -- c --> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
1941 apply (drule LIM_D [where r="-l"], simp, clarify)
1942 apply (rule_tac x = s in exI)
1943 apply (simp add: abs_less_iff)
1946 lemma LIM_fun_not_zero:
1947 fixes f :: "real \<Rightarrow> real"
1948 shows "f -- c --> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
1949 using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)