src/HOL/SEQ.thy
 author blanchet Wed Mar 04 11:05:29 2009 +0100 (2009-03-04) changeset 30242 aea5d7fa7ef5 parent 30240 5b25fee0362c parent 30196 6ffaa79c352c child 30273 ecd6f0ca62ea permissions -rw-r--r--
Merge.
1 (*  Title       : SEQ.thy
2     Author      : Jacques D. Fleuriot
3     Copyright   : 1998  University of Cambridge
4     Description : Convergence of sequences and series
5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
6     Additional contributions by Jeremy Avigad and Brian Huffman
7 *)
9 header {* Sequences and Convergence *}
11 theory SEQ
12 imports RealVector RComplete
13 begin
15 definition
16   Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
17     --{*Standard definition of sequence converging to zero*}
18   [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
20 definition
21   LIMSEQ :: "[nat => 'a::real_normed_vector, 'a] => bool"
22     ("((_)/ ----> (_))" [60, 60] 60) where
23     --{*Standard definition of convergence of sequence*}
24   [code del]: "X ----> L = (\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X n - L) < r))"
26 definition
27   lim :: "(nat => 'a::real_normed_vector) => 'a" where
28     --{*Standard definition of limit using choice operator*}
29   "lim X = (THE L. X ----> L)"
31 definition
32   convergent :: "(nat => 'a::real_normed_vector) => bool" where
33     --{*Standard definition of convergence*}
34   "convergent X = (\<exists>L. X ----> L)"
36 definition
37   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
38     --{*Standard definition for bounded sequence*}
39   [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
41 definition
42   monoseq :: "(nat=>real)=>bool" where
43     --{*Definition for monotonicity*}
44   [code del]: "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
46 definition
47   subseq :: "(nat => nat) => bool" where
48     --{*Definition of subsequence*}
49   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
51 definition
52   Cauchy :: "(nat => 'a::real_normed_vector) => bool" where
53     --{*Standard definition of the Cauchy condition*}
54   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. norm (X m - X n) < e)"
57 subsection {* Bounded Sequences *}
59 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
60 unfolding Bseq_def
61 proof (intro exI conjI allI)
62   show "0 < max K 1" by simp
63 next
64   fix n::nat
65   have "norm (X n) \<le> K" by (rule K)
66   thus "norm (X n) \<le> max K 1" by simp
67 qed
69 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
70 unfolding Bseq_def by auto
72 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
73 proof (rule BseqI')
74   let ?A = "norm ` X ` {..N}"
75   have 1: "finite ?A" by simp
76   fix n::nat
77   show "norm (X n) \<le> max K (Max ?A)"
78   proof (cases rule: linorder_le_cases)
79     assume "n \<ge> N"
80     hence "norm (X n) \<le> K" using K by simp
81     thus "norm (X n) \<le> max K (Max ?A)" by simp
82   next
83     assume "n \<le> N"
84     hence "norm (X n) \<in> ?A" by simp
85     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
86     thus "norm (X n) \<le> max K (Max ?A)" by simp
87   qed
88 qed
90 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
91 unfolding Bseq_def by auto
93 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
94 apply (erule BseqE)
95 apply (rule_tac N="k" and K="K" in BseqI2')
96 apply clarify
97 apply (drule_tac x="n - k" in spec, simp)
98 done
101 subsection {* Sequences That Converge to Zero *}
103 lemma ZseqI:
104   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
105 unfolding Zseq_def by simp
107 lemma ZseqD:
108   "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
109 unfolding Zseq_def by simp
111 lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
112 unfolding Zseq_def by simp
114 lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
115 unfolding Zseq_def by force
117 lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
118 unfolding Zseq_def by simp
120 lemma Zseq_imp_Zseq:
121   assumes X: "Zseq X"
122   assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
123   shows "Zseq (\<lambda>n. Y n)"
124 proof (cases)
125   assume K: "0 < K"
126   show ?thesis
127   proof (rule ZseqI)
128     fix r::real assume "0 < r"
129     hence "0 < r / K"
130       using K by (rule divide_pos_pos)
131     then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
132       using ZseqD [OF X] by fast
133     hence "\<forall>n\<ge>N. norm (X n) * K < r"
134       by (simp add: pos_less_divide_eq K)
135     hence "\<forall>n\<ge>N. norm (Y n) < r"
136       by (simp add: order_le_less_trans [OF Y])
137     thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
138   qed
139 next
140   assume "\<not> 0 < K"
141   hence K: "K \<le> 0" by (simp only: linorder_not_less)
142   {
143     fix n::nat
144     have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
145     also have "\<dots> \<le> norm (X n) * 0"
146       using K norm_ge_zero by (rule mult_left_mono)
147     finally have "norm (Y n) = 0" by simp
148   }
149   thus ?thesis by (simp add: Zseq_zero)
150 qed
152 lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
153 by (erule_tac K="1" in Zseq_imp_Zseq, simp)
156   assumes X: "Zseq X"
157   assumes Y: "Zseq Y"
158   shows "Zseq (\<lambda>n. X n + Y n)"
159 proof (rule ZseqI)
160   fix r::real assume "0 < r"
161   hence r: "0 < r / 2" by simp
162   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
163     using ZseqD [OF X r] by fast
164   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
165     using ZseqD [OF Y r] by fast
166   show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
167   proof (intro exI allI impI)
168     fix n assume n: "max M N \<le> n"
169     have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
170       by (rule norm_triangle_ineq)
171     also have "\<dots> < r/2 + r/2"
172     proof (rule add_strict_mono)
173       from M n show "norm (X n) < r/2" by simp
174       from N n show "norm (Y n) < r/2" by simp
175     qed
176     finally show "norm (X n + Y n) < r" by simp
177   qed
178 qed
180 lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
181 unfolding Zseq_def by simp
183 lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
184 by (simp only: diff_minus Zseq_add Zseq_minus)
186 lemma (in bounded_linear) Zseq:
187   assumes X: "Zseq X"
188   shows "Zseq (\<lambda>n. f (X n))"
189 proof -
190   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
191     using bounded by fast
192   with X show ?thesis
193     by (rule Zseq_imp_Zseq)
194 qed
196 lemma (in bounded_bilinear) Zseq:
197   assumes X: "Zseq X"
198   assumes Y: "Zseq Y"
199   shows "Zseq (\<lambda>n. X n ** Y n)"
200 proof (rule ZseqI)
201   fix r::real assume r: "0 < r"
202   obtain K where K: "0 < K"
203     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
204     using pos_bounded by fast
205   from K have K': "0 < inverse K"
206     by (rule positive_imp_inverse_positive)
207   obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
208     using ZseqD [OF X r] by fast
209   obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
210     using ZseqD [OF Y K'] by fast
211   show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
212   proof (intro exI allI impI)
213     fix n assume n: "max M N \<le> n"
214     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
215       by (rule norm_le)
216     also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
217     proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
218       from M n show Xn: "norm (X n) < r" by simp
219       from N n show Yn: "norm (Y n) < inverse K" by simp
220     qed
221     also from K have "r * inverse K * K = r" by simp
222     finally show "norm (X n ** Y n) < r" .
223   qed
224 qed
226 lemma (in bounded_bilinear) Zseq_prod_Bseq:
227   assumes X: "Zseq X"
228   assumes Y: "Bseq Y"
229   shows "Zseq (\<lambda>n. X n ** Y n)"
230 proof -
231   obtain K where K: "0 \<le> K"
232     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
233     using nonneg_bounded by fast
234   obtain B where B: "0 < B"
235     and norm_Y: "\<And>n. norm (Y n) \<le> B"
236     using Y [unfolded Bseq_def] by fast
237   from X show ?thesis
238   proof (rule Zseq_imp_Zseq)
239     fix n::nat
240     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
241       by (rule norm_le)
242     also have "\<dots> \<le> norm (X n) * B * K"
243       by (intro mult_mono' order_refl norm_Y norm_ge_zero
244                 mult_nonneg_nonneg K)
245     also have "\<dots> = norm (X n) * (B * K)"
246       by (rule mult_assoc)
247     finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
248   qed
249 qed
251 lemma (in bounded_bilinear) Bseq_prod_Zseq:
252   assumes X: "Bseq X"
253   assumes Y: "Zseq Y"
254   shows "Zseq (\<lambda>n. X n ** Y n)"
255 proof -
256   obtain K where K: "0 \<le> K"
257     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
258     using nonneg_bounded by fast
259   obtain B where B: "0 < B"
260     and norm_X: "\<And>n. norm (X n) \<le> B"
261     using X [unfolded Bseq_def] by fast
262   from Y show ?thesis
263   proof (rule Zseq_imp_Zseq)
264     fix n::nat
265     have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
266       by (rule norm_le)
267     also have "\<dots> \<le> B * norm (Y n) * K"
268       by (intro mult_mono' order_refl norm_X norm_ge_zero
269                 mult_nonneg_nonneg K)
270     also have "\<dots> = norm (Y n) * (B * K)"
271       by (simp only: mult_ac)
272     finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
273   qed
274 qed
276 lemma (in bounded_bilinear) Zseq_left:
277   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
278 by (rule bounded_linear_left [THEN bounded_linear.Zseq])
280 lemma (in bounded_bilinear) Zseq_right:
281   "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
282 by (rule bounded_linear_right [THEN bounded_linear.Zseq])
284 lemmas Zseq_mult = mult.Zseq
285 lemmas Zseq_mult_right = mult.Zseq_right
286 lemmas Zseq_mult_left = mult.Zseq_left
289 subsection {* Limits of Sequences *}
291 lemma LIMSEQ_iff:
292       "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
293 by (rule LIMSEQ_def)
295 lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
296 by (simp only: LIMSEQ_def Zseq_def)
298 lemma LIMSEQ_I:
299   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
300 by (simp add: LIMSEQ_def)
302 lemma LIMSEQ_D:
303   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
304 by (simp add: LIMSEQ_def)
306 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
307 by (simp add: LIMSEQ_def)
309 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l = (k = l)"
310 by (simp add: LIMSEQ_Zseq_iff Zseq_const_iff)
312 lemma LIMSEQ_norm: "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
313 apply (simp add: LIMSEQ_def, safe)
314 apply (drule_tac x="r" in spec, safe)
315 apply (rule_tac x="no" in exI, safe)
316 apply (drule_tac x="n" in spec, safe)
317 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
318 done
320 lemma LIMSEQ_ignore_initial_segment:
321   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
322 apply (rule LIMSEQ_I)
323 apply (drule (1) LIMSEQ_D)
324 apply (erule exE, rename_tac N)
325 apply (rule_tac x=N in exI)
326 apply simp
327 done
329 lemma LIMSEQ_offset:
330   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
331 apply (rule LIMSEQ_I)
332 apply (drule (1) LIMSEQ_D)
333 apply (erule exE, rename_tac N)
334 apply (rule_tac x="N + k" in exI)
335 apply clarify
336 apply (drule_tac x="n - k" in spec)
337 apply (simp add: le_diff_conv2)
338 done
340 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
341 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
343 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
344 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
346 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
347 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
349 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
350   unfolding LIMSEQ_def
351   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
355   fixes a b c d :: "'a::ab_group_add"
356   shows "(a + c) - (b + d) = (a - b) + (c - d)"
357 by simp
359 lemma minus_diff_minus:
360   fixes a b :: "'a::ab_group_add"
361   shows "(- a) - (- b) = - (a - b)"
362 by simp
364 lemma LIMSEQ_add: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
367 lemma LIMSEQ_minus: "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
368 by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
370 lemma LIMSEQ_minus_cancel: "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
371 by (drule LIMSEQ_minus, simp)
373 lemma LIMSEQ_diff: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
374 by (simp add: diff_minus LIMSEQ_add LIMSEQ_minus)
376 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
377 by (drule (1) LIMSEQ_diff, simp add: LIMSEQ_const_iff)
379 lemma (in bounded_linear) LIMSEQ:
380   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
381 by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
383 lemma (in bounded_bilinear) LIMSEQ:
384   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
385 by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
386                Zseq_add Zseq Zseq_left Zseq_right)
388 lemma LIMSEQ_mult:
389   fixes a b :: "'a::real_normed_algebra"
390   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
391 by (rule mult.LIMSEQ)
393 lemma inverse_diff_inverse:
394   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
395    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
396 by (simp add: algebra_simps)
398 lemma Bseq_inverse_lemma:
399   fixes x :: "'a::real_normed_div_algebra"
400   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
401 apply (subst nonzero_norm_inverse, clarsimp)
402 apply (erule (1) le_imp_inverse_le)
403 done
405 lemma Bseq_inverse:
406   fixes a :: "'a::real_normed_div_algebra"
407   assumes X: "X ----> a"
408   assumes a: "a \<noteq> 0"
409   shows "Bseq (\<lambda>n. inverse (X n))"
410 proof -
411   from a have "0 < norm a" by simp
412   hence "\<exists>r>0. r < norm a" by (rule dense)
413   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
414   obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
415     using LIMSEQ_D [OF X r1] by fast
416   show ?thesis
417   proof (rule BseqI2' [rule_format])
418     fix n assume n: "N \<le> n"
419     hence 1: "norm (X n - a) < r" by (rule N)
420     hence 2: "X n \<noteq> 0" using r2 by auto
421     hence "norm (inverse (X n)) = inverse (norm (X n))"
422       by (rule nonzero_norm_inverse)
423     also have "\<dots> \<le> inverse (norm a - r)"
424     proof (rule le_imp_inverse_le)
425       show "0 < norm a - r" using r2 by simp
426     next
427       have "norm a - norm (X n) \<le> norm (a - X n)"
428         by (rule norm_triangle_ineq2)
429       also have "\<dots> = norm (X n - a)"
430         by (rule norm_minus_commute)
431       also have "\<dots> < r" using 1 .
432       finally show "norm a - r \<le> norm (X n)" by simp
433     qed
434     finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
435   qed
436 qed
438 lemma LIMSEQ_inverse_lemma:
439   fixes a :: "'a::real_normed_div_algebra"
440   shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
441          \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
442 apply (subst LIMSEQ_Zseq_iff)
443 apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
444 apply (rule Zseq_minus)
445 apply (rule Zseq_mult_left)
446 apply (rule mult.Bseq_prod_Zseq)
447 apply (erule (1) Bseq_inverse)
448 apply (simp add: LIMSEQ_Zseq_iff)
449 done
451 lemma LIMSEQ_inverse:
452   fixes a :: "'a::real_normed_div_algebra"
453   assumes X: "X ----> a"
454   assumes a: "a \<noteq> 0"
455   shows "(\<lambda>n. inverse (X n)) ----> inverse a"
456 proof -
457   from a have "0 < norm a" by simp
458   then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
459     using LIMSEQ_D [OF X] by fast
460   hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
461   hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
463   from X have "(\<lambda>n. X (n + k)) ----> a"
464     by (rule LIMSEQ_ignore_initial_segment)
465   hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
466     using a k by (rule LIMSEQ_inverse_lemma)
467   thus "(\<lambda>n. inverse (X n)) ----> inverse a"
468     by (rule LIMSEQ_offset)
469 qed
471 lemma LIMSEQ_divide:
472   fixes a b :: "'a::real_normed_field"
473   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
474 by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
476 lemma LIMSEQ_pow:
477   fixes a :: "'a::{real_normed_algebra,recpower}"
478   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
479 by (induct m) (simp_all add: power_Suc LIMSEQ_const LIMSEQ_mult)
481 lemma LIMSEQ_setsum:
482   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
483   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
484 proof (cases "finite S")
485   case True
486   thus ?thesis using n
487   proof (induct)
488     case empty
489     show ?case
490       by (simp add: LIMSEQ_const)
491   next
492     case insert
493     thus ?case
495   qed
496 next
497   case False
498   thus ?thesis
499     by (simp add: LIMSEQ_const)
500 qed
502 lemma LIMSEQ_setprod:
503   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
504   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
505   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
506 proof (cases "finite S")
507   case True
508   thus ?thesis using n
509   proof (induct)
510     case empty
511     show ?case
512       by (simp add: LIMSEQ_const)
513   next
514     case insert
515     thus ?case
516       by (simp add: LIMSEQ_mult)
517   qed
518 next
519   case False
520   thus ?thesis
521     by (simp add: setprod_def LIMSEQ_const)
522 qed
524 lemma LIMSEQ_add_const: "f ----> a ==> (%n.(f n + b)) ----> a + b"
527 (* FIXME: delete *)
529      "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
530 by (simp only: LIMSEQ_add LIMSEQ_minus)
532 lemma LIMSEQ_diff_const: "f ----> a ==> (%n.(f n  - b)) ----> a - b"
533 by (simp add: LIMSEQ_diff LIMSEQ_const)
535 lemma LIMSEQ_diff_approach_zero:
536   "g ----> L ==> (%x. f x - g x) ----> 0  ==>
537      f ----> L"
538   apply (drule LIMSEQ_add)
539   apply assumption
540   apply simp
541 done
543 lemma LIMSEQ_diff_approach_zero2:
544   "f ----> L ==> (%x. f x - g x) ----> 0  ==>
545      g ----> L";
546   apply (drule LIMSEQ_diff)
547   apply assumption
548   apply simp
549 done
551 text{*A sequence tends to zero iff its abs does*}
552 lemma LIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----> 0) = (X ----> 0)"
553 by (simp add: LIMSEQ_def)
555 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
556 by (simp add: LIMSEQ_def)
558 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
559 by (drule LIMSEQ_norm, simp)
561 text{*An unbounded sequence's inverse tends to 0*}
563 lemma LIMSEQ_inverse_zero:
564   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
565 apply (rule LIMSEQ_I)
566 apply (drule_tac x="inverse r" in spec, safe)
567 apply (rule_tac x="N" in exI, safe)
568 apply (drule_tac x="n" in spec, safe)
569 apply (frule positive_imp_inverse_positive)
570 apply (frule (1) less_imp_inverse_less)
571 apply (subgoal_tac "0 < X n", simp)
572 apply (erule (1) order_less_trans)
573 done
575 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
577 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
578 apply (rule LIMSEQ_inverse_zero, safe)
579 apply (cut_tac x = r in reals_Archimedean2)
580 apply (safe, rule_tac x = n in exI)
581 apply (auto simp add: real_of_nat_Suc)
582 done
584 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
585 infinity is now easily proved*}
588      "(%n. r + inverse(real(Suc n))) ----> r"
589 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
592      "(%n. r + -inverse(real(Suc n))) ----> r"
593 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
596      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
597 by (cut_tac b=1 in
598         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
600 lemma LIMSEQ_le_const:
601   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
602 apply (rule ccontr, simp only: linorder_not_le)
603 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
604 apply clarsimp
605 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
606 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
607 apply simp
608 done
610 lemma LIMSEQ_le_const2:
611   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
612 apply (subgoal_tac "- a \<le> - x", simp)
613 apply (rule LIMSEQ_le_const)
614 apply (erule LIMSEQ_minus)
615 apply simp
616 done
618 lemma LIMSEQ_le:
619   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
620 apply (subgoal_tac "0 \<le> y - x", simp)
621 apply (rule LIMSEQ_le_const)
622 apply (erule (1) LIMSEQ_diff)
623 apply (simp add: le_diff_eq)
624 done
627 subsection {* Convergence *}
629 lemma limI: "X ----> L ==> lim X = L"
630 apply (simp add: lim_def)
631 apply (blast intro: LIMSEQ_unique)
632 done
634 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
635 by (simp add: convergent_def)
637 lemma convergentI: "(X ----> L) ==> convergent X"
638 by (auto simp add: convergent_def)
640 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
641 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
643 lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
644 apply (simp add: convergent_def)
645 apply (auto dest: LIMSEQ_minus)
646 apply (drule LIMSEQ_minus, auto)
647 done
649 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
651 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
652   unfolding Ex1_def
653   apply (rule_tac x="nat_rec e f" in exI)
654   apply (rule conjI)+
655 apply (rule def_nat_rec_0, simp)
656 apply (rule allI, rule def_nat_rec_Suc, simp)
657 apply (rule allI, rule impI, rule ext)
658 apply (erule conjE)
659 apply (induct_tac x)
660 apply (simp add: nat_rec_0)
661 apply (erule_tac x="n" in allE)
662 apply (simp)
663 done
665 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
667 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
668 apply (simp add: subseq_def)
669 apply (auto dest!: less_imp_Suc_add)
670 apply (induct_tac k)
671 apply (auto intro: less_trans)
672 done
674 lemma monoseq_Suc:
675    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
676                  | (\<forall>n. X (Suc n) \<le> X n))"
677 apply (simp add: monoseq_def)
678 apply (auto dest!: le_imp_less_or_eq)
679 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
680 apply (induct_tac "ka")
681 apply (auto intro: order_trans)
682 apply (erule contrapos_np)
683 apply (induct_tac "k")
684 apply (auto intro: order_trans)
685 done
687 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
688 by (simp add: monoseq_def)
690 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
691 by (simp add: monoseq_def)
693 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
694 by (simp add: monoseq_Suc)
696 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
697 by (simp add: monoseq_Suc)
699 lemma monoseq_minus: assumes "monoseq a"
700   shows "monoseq (\<lambda> n. - a n)"
701 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
702   case True
703   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
704   thus ?thesis by (rule monoI2)
705 next
706   case False
707   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
708   thus ?thesis by (rule monoI1)
709 qed
711 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
712   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or>
713          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
714 proof -
715   { fix x n fix a :: "nat \<Rightarrow> real"
716     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
717     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
718     have "a n \<le> x"
719     proof (rule ccontr)
720       assume "\<not> a n \<le> x" hence "x < a n" by auto
721       hence "0 < a n - x" by auto
722       from `a ----> x`[THEN LIMSEQ_D, OF this]
723       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
724       hence "norm (a (max no n) - x) < a n - x" by auto
725       moreover
726       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
727       hence "x < a (max no n)" by auto
728       ultimately
729       have "a (max no n) < a n" by auto
730       with monotone[where m=n and n="max no n"]
731       show False by auto
732     qed
733   } note top_down = this
734   { fix x n m fix a :: "nat \<Rightarrow> real"
735     assume "a ----> x" and "monoseq a" and "a m < x"
736     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
737     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
738       case True with top_down and `a ----> x` show ?thesis by auto
739     next
740       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
741       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
742       hence False using `a m < x` by auto
743       thus ?thesis ..
744     qed
745   } note when_decided = this
747   show ?thesis
748   proof (cases "\<exists> m. a m \<noteq> x")
749     case True then obtain m where "a m \<noteq> x" by auto
750     show ?thesis
751     proof (cases "a m < x")
752       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
753       show ?thesis by blast
754     next
755       case False hence "- a m < - x" using `a m \<noteq> x` by auto
756       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
757       show ?thesis by auto
758     qed
759   qed auto
760 qed
762 text{* for any sequence, there is a mootonic subsequence *}
763 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
764 proof-
765   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
766     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
767     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
768     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
769     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
770       using H apply -
771       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI)
772       unfolding order_le_less by blast
773     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
774     {fix n
775       have "?P (f (Suc n)) (f n)"
776 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
777 	using H apply -
778       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI)
779       unfolding order_le_less by blast
780     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
781   note fSuc = this
782     {fix p q assume pq: "p \<ge> f q"
783       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
784 	by (cases q, simp_all) }
785     note pqth = this
786     {fix q
787       have "f (Suc q) > f q" apply (induct q)
788 	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
789     note fss = this
790     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
791     {fix a b
792       have "f a \<le> f (a + b)"
793       proof(induct b)
794 	case 0 thus ?case by simp
795       next
796 	case (Suc b)
797 	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
798       qed}
799     note fmon0 = this
800     have "monoseq (\<lambda>n. s (f n))"
801     proof-
802       {fix n
803 	have "s (f n) \<ge> s (f (Suc n))"
804 	proof(cases n)
805 	  case 0
806 	  assume n0: "n = 0"
807 	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
808 	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
809 	next
810 	  case (Suc m)
811 	  assume m: "n = Suc m"
812 	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
813 	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp
814 	qed}
815       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast
816     qed
817     with th1 have ?thesis by blast}
818   moreover
819   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
820     {fix p assume p: "p \<ge> Suc N"
821       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
822       have "m \<noteq> p" using m(2) by auto
823       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
824     note th0 = this
825     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
826     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
827     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))"
828       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
829     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
830       using N apply -
831       apply (erule allE[where x="Suc N"], clarsimp)
832       apply (rule_tac x="m" in exI)
833       apply auto
834       apply (subgoal_tac "Suc N \<noteq> m")
835       apply simp
836       apply (rule ccontr, simp)
837       done
838     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
839     {fix n
840       have "f n > N \<and> ?P (f (Suc n)) (f n)"
841 	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
842       proof (induct n)
843 	case 0 thus ?case
844 	  using f0 N apply auto
845 	  apply (erule allE[where x="f 0"], clarsimp)
846 	  apply (rule_tac x="m" in exI, simp)
847 	  by (subgoal_tac "f 0 \<noteq> m", auto)
848       next
849 	case (Suc n)
850 	from Suc.hyps have Nfn: "N < f n" by blast
851 	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
852 	with Nfn have mN: "m > N" by arith
853 	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
855 	from key have th0: "f (Suc n) > N" by simp
856 	from N[rule_format, OF th0]
857 	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
858 	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
859 	hence "m' > f (Suc n)" using m'(1) by simp
860 	with key m'(2) show ?case by auto
861       qed}
862     note fSuc = this
863     {fix n
864       have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto
865       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
866     note thf = this
867     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
868     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
869       apply -
870       apply (rule disjI1)
871       apply auto
872       apply (rule order_less_imp_le)
873       apply blast
874       done
875     then have ?thesis  using sqf by blast}
876   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
877 qed
879 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
880 proof(induct n)
881   case 0 thus ?case by simp
882 next
883   case (Suc n)
884   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
885   have "n < f (Suc n)" by arith
886   thus ?case by arith
887 qed
889 subsection {* Bounded Monotonic Sequences *}
892 text{*Bounded Sequence*}
894 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
895 by (simp add: Bseq_def)
897 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
898 by (auto simp add: Bseq_def)
900 lemma lemma_NBseq_def:
901      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
902       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
903 apply auto
904  prefer 2 apply force
905 apply (cut_tac x = K in reals_Archimedean2, clarify)
906 apply (rule_tac x = n in exI, clarify)
907 apply (drule_tac x = na in spec)
908 apply (auto simp add: real_of_nat_Suc)
909 done
911 text{* alternative definition for Bseq *}
912 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
913 apply (simp add: Bseq_def)
914 apply (simp (no_asm) add: lemma_NBseq_def)
915 done
917 lemma lemma_NBseq_def2:
918      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
919 apply (subst lemma_NBseq_def, auto)
920 apply (rule_tac x = "Suc N" in exI)
921 apply (rule_tac  x = N in exI)
922 apply (auto simp add: real_of_nat_Suc)
923  prefer 2 apply (blast intro: order_less_imp_le)
924 apply (drule_tac x = n in spec, simp)
925 done
927 (* yet another definition for Bseq *)
928 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
929 by (simp add: Bseq_def lemma_NBseq_def2)
931 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
933 lemma Bseq_isUb:
934   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
935 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
938 text{* Use completeness of reals (supremum property)
939    to show that any bounded sequence has a least upper bound*}
941 lemma Bseq_isLub:
942   "!!(X::nat=>real). Bseq X ==>
943    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
944 by (blast intro: reals_complete Bseq_isUb)
946 subsubsection{*A Bounded and Monotonic Sequence Converges*}
948 lemma lemma_converg1:
949      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
950                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
951                |] ==> \<forall>n \<ge> ma. X n = X ma"
952 apply safe
953 apply (drule_tac y = "X n" in isLubD2)
954 apply (blast dest: order_antisym)+
955 done
957 text{* The best of both worlds: Easier to prove this result as a standard
958    theorem and then use equivalence to "transfer" it into the
959    equivalent nonstandard form if needed!*}
961 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
962 apply (simp add: LIMSEQ_def)
963 apply (rule_tac x = "X m" in exI, safe)
964 apply (rule_tac x = m in exI, safe)
965 apply (drule spec, erule impE, auto)
966 done
968 lemma lemma_converg2:
969    "!!(X::nat=>real).
970     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
971 apply safe
972 apply (drule_tac y = "X m" in isLubD2)
973 apply (auto dest!: order_le_imp_less_or_eq)
974 done
976 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
977 by (rule setleI [THEN isUbI], auto)
979 text{* FIXME: @{term "U - T < U"} is redundant *}
980 lemma lemma_converg4: "!!(X::nat=> real).
981                [| \<forall>m. X m ~= U;
982                   isLub UNIV {x. \<exists>n. X n = x} U;
983                   0 < T;
984                   U + - T < U
985                |] ==> \<exists>m. U + -T < X m & X m < U"
986 apply (drule lemma_converg2, assumption)
987 apply (rule ccontr, simp)
988 apply (simp add: linorder_not_less)
989 apply (drule lemma_converg3)
990 apply (drule isLub_le_isUb, assumption)
991 apply (auto dest: order_less_le_trans)
992 done
994 text{*A standard proof of the theorem for monotone increasing sequence*}
996 lemma Bseq_mono_convergent:
997      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
998 apply (simp add: convergent_def)
999 apply (frule Bseq_isLub, safe)
1000 apply (case_tac "\<exists>m. X m = U", auto)
1001 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
1002 (* second case *)
1003 apply (rule_tac x = U in exI)
1004 apply (subst LIMSEQ_iff, safe)
1005 apply (frule lemma_converg2, assumption)
1006 apply (drule lemma_converg4, auto)
1007 apply (rule_tac x = m in exI, safe)
1008 apply (subgoal_tac "X m \<le> X n")
1009  prefer 2 apply blast
1010 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
1011 done
1013 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
1014 by (simp add: Bseq_def)
1016 text{*Main monotonicity theorem*}
1017 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
1018 apply (simp add: monoseq_def, safe)
1019 apply (rule_tac  convergent_minus_iff [THEN ssubst])
1020 apply (drule_tac  Bseq_minus_iff [THEN ssubst])
1021 apply (auto intro!: Bseq_mono_convergent)
1022 done
1024 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
1026 text{*alternative formulation for boundedness*}
1027 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
1028 apply (unfold Bseq_def, safe)
1029 apply (rule_tac  x = "k + norm x" in exI)
1030 apply (rule_tac x = K in exI, simp)
1031 apply (rule exI [where x = 0], auto)
1032 apply (erule order_less_le_trans, simp)
1033 apply (drule_tac x=n in spec, fold diff_def)
1034 apply (drule order_trans [OF norm_triangle_ineq2])
1035 apply simp
1036 done
1038 text{*alternative formulation for boundedness*}
1039 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
1040 apply safe
1041 apply (simp add: Bseq_def, safe)
1042 apply (rule_tac x = "K + norm (X N)" in exI)
1043 apply auto
1044 apply (erule order_less_le_trans, simp)
1045 apply (rule_tac x = N in exI, safe)
1046 apply (drule_tac x = n in spec)
1047 apply (rule order_trans [OF norm_triangle_ineq], simp)
1048 apply (auto simp add: Bseq_iff2)
1049 done
1051 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
1052 apply (simp add: Bseq_def)
1053 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
1054 apply (drule_tac x = n in spec, arith)
1055 done
1058 subsection {* Cauchy Sequences *}
1060 lemma CauchyI:
1061   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
1062 by (simp add: Cauchy_def)
1064 lemma CauchyD:
1065   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
1066 by (simp add: Cauchy_def)
1068 subsubsection {* Cauchy Sequences are Bounded *}
1070 text{*A Cauchy sequence is bounded -- this is the standard
1071   proof mechanization rather than the nonstandard proof*}
1073 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
1074           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
1075 apply (clarify, drule spec, drule (1) mp)
1076 apply (simp only: norm_minus_commute)
1077 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
1078 apply simp
1079 done
1081 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
1082 apply (simp add: Cauchy_def)
1083 apply (drule spec, drule mp, rule zero_less_one, safe)
1084 apply (drule_tac x="M" in spec, simp)
1085 apply (drule lemmaCauchy)
1086 apply (rule_tac k="M" in Bseq_offset)
1087 apply (simp add: Bseq_def)
1088 apply (rule_tac x="1 + norm (X M)" in exI)
1089 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
1090 apply (simp add: order_less_imp_le)
1091 done
1093 subsubsection {* Cauchy Sequences are Convergent *}
1095 axclass banach \<subseteq> real_normed_vector
1096   Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
1098 theorem LIMSEQ_imp_Cauchy:
1099   assumes X: "X ----> a" shows "Cauchy X"
1100 proof (rule CauchyI)
1101   fix e::real assume "0 < e"
1102   hence "0 < e/2" by simp
1103   with X have "\<exists>N. \<forall>n\<ge>N. norm (X n - a) < e/2" by (rule LIMSEQ_D)
1104   then obtain N where N: "\<forall>n\<ge>N. norm (X n - a) < e/2" ..
1105   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < e"
1106   proof (intro exI allI impI)
1107     fix m assume "N \<le> m"
1108     hence m: "norm (X m - a) < e/2" using N by fast
1109     fix n assume "N \<le> n"
1110     hence n: "norm (X n - a) < e/2" using N by fast
1111     have "norm (X m - X n) = norm ((X m - a) - (X n - a))" by simp
1112     also have "\<dots> \<le> norm (X m - a) + norm (X n - a)"
1113       by (rule norm_triangle_ineq4)
1114     also from m n have "\<dots> < e" by(simp add:field_simps)
1115     finally show "norm (X m - X n) < e" .
1116   qed
1117 qed
1119 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
1120 unfolding convergent_def
1121 by (erule exE, erule LIMSEQ_imp_Cauchy)
1123 text {*
1124 Proof that Cauchy sequences converge based on the one from
1125 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
1126 *}
1128 text {*
1129   If sequence @{term "X"} is Cauchy, then its limit is the lub of
1130   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
1131 *}
1133 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
1134 by (simp add: isUbI setleI)
1136 lemma real_abs_diff_less_iff:
1137   "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
1138 by auto
1140 locale real_Cauchy =
1141   fixes X :: "nat \<Rightarrow> real"
1142   assumes X: "Cauchy X"
1143   fixes S :: "real set"
1144   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
1146 lemma real_CauchyI:
1147   assumes "Cauchy X"
1148   shows "real_Cauchy X"
1149   proof qed (fact assms)
1151 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
1152 by (unfold S_def, auto)
1154 lemma (in real_Cauchy) bound_isUb:
1155   assumes N: "\<forall>n\<ge>N. X n < x"
1156   shows "isUb UNIV S x"
1157 proof (rule isUb_UNIV_I)
1158   fix y::real assume "y \<in> S"
1159   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
1160     by (simp add: S_def)
1161   then obtain M where "\<forall>n\<ge>M. y < X n" ..
1162   hence "y < X (max M N)" by simp
1163   also have "\<dots> < x" using N by simp
1164   finally show "y \<le> x"
1165     by (rule order_less_imp_le)
1166 qed
1168 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
1169 proof (rule reals_complete)
1170   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
1171     using CauchyD [OF X zero_less_one] by fast
1172   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
1173   show "\<exists>x. x \<in> S"
1174   proof
1175     from N have "\<forall>n\<ge>N. X N - 1 < X n"
1176       by (simp add: real_abs_diff_less_iff)
1177     thus "X N - 1 \<in> S" by (rule mem_S)
1178   qed
1179   show "\<exists>u. isUb UNIV S u"
1180   proof
1181     from N have "\<forall>n\<ge>N. X n < X N + 1"
1182       by (simp add: real_abs_diff_less_iff)
1183     thus "isUb UNIV S (X N + 1)"
1184       by (rule bound_isUb)
1185   qed
1186 qed
1188 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
1189   assumes x: "isLub UNIV S x"
1190   shows "X ----> x"
1191 proof (rule LIMSEQ_I)
1192   fix r::real assume "0 < r"
1193   hence r: "0 < r/2" by simp
1194   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
1195     using CauchyD [OF X r] by fast
1196   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
1197   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
1198     by (simp only: real_norm_def real_abs_diff_less_iff)
1200   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
1201   hence "X N - r/2 \<in> S" by (rule mem_S)
1202   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
1204   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
1205   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
1206   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
1208   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
1209   proof (intro exI allI impI)
1210     fix n assume n: "N \<le> n"
1211     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
1212     thus "norm (X n - x) < r" using 1 2
1213       by (simp add: real_abs_diff_less_iff)
1214   qed
1215 qed
1217 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
1218 proof -
1219   obtain x where "isLub UNIV S x"
1220     using isLub_ex by fast
1221   hence "X ----> x"
1222     by (rule isLub_imp_LIMSEQ)
1223   thus ?thesis ..
1224 qed
1226 lemma real_Cauchy_convergent:
1227   fixes X :: "nat \<Rightarrow> real"
1228   shows "Cauchy X \<Longrightarrow> convergent X"
1229 unfolding convergent_def
1230 by (rule real_Cauchy.LIMSEQ_ex)
1231  (rule real_CauchyI)
1233 instance real :: banach
1234 by intro_classes (rule real_Cauchy_convergent)
1236 lemma Cauchy_convergent_iff:
1237   fixes X :: "nat \<Rightarrow> 'a::banach"
1238   shows "Cauchy X = convergent X"
1239 by (fast intro: Cauchy_convergent convergent_Cauchy)
1242 subsection {* Power Sequences *}
1244 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
1245 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
1246   also fact that bounded and monotonic sequence converges.*}
1248 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
1249 apply (simp add: Bseq_def)
1250 apply (rule_tac x = 1 in exI)
1251 apply (simp add: power_abs)
1252 apply (auto dest: power_mono)
1253 done
1255 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
1256 apply (clarify intro!: mono_SucI2)
1257 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
1258 done
1260 lemma convergent_realpow:
1261   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
1262 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
1264 lemma LIMSEQ_inverse_realpow_zero_lemma:
1265   fixes x :: real
1266   assumes x: "0 \<le> x"
1267   shows "real n * x + 1 \<le> (x + 1) ^ n"
1268 apply (induct n)
1269 apply simp
1270 apply simp
1271 apply (rule order_trans)
1272 prefer 2
1273 apply (erule mult_left_mono)
1274 apply (rule add_increasing [OF x], simp)
1275 apply (simp add: real_of_nat_Suc)
1276 apply (simp add: ring_distribs)
1277 apply (simp add: mult_nonneg_nonneg x)
1278 done
1280 lemma LIMSEQ_inverse_realpow_zero:
1281   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
1282 proof (rule LIMSEQ_inverse_zero [rule_format])
1283   fix y :: real
1284   assume x: "1 < x"
1285   hence "0 < x - 1" by simp
1286   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
1287     by (rule reals_Archimedean3)
1288   hence "\<exists>N::nat. y < real N * (x - 1)" ..
1289   then obtain N::nat where "y < real N * (x - 1)" ..
1290   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
1291   also have "\<dots> \<le> (x - 1 + 1) ^ N"
1292     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
1293   also have "\<dots> = x ^ N" by simp
1294   finally have "y < x ^ N" .
1295   hence "\<forall>n\<ge>N. y < x ^ n"
1296     apply clarify
1297     apply (erule order_less_le_trans)
1298     apply (erule power_increasing)
1299     apply (rule order_less_imp_le [OF x])
1300     done
1301   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
1302 qed
1304 lemma LIMSEQ_realpow_zero:
1305   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1306 proof (cases)
1307   assume "x = 0"
1308   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
1309   thus ?thesis by (rule LIMSEQ_imp_Suc)
1310 next
1311   assume "0 \<le> x" and "x \<noteq> 0"
1312   hence x0: "0 < x" by simp
1313   assume x1: "x < 1"
1314   from x0 x1 have "1 < inverse x"
1315     by (rule real_inverse_gt_one)
1316   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
1317     by (rule LIMSEQ_inverse_realpow_zero)
1318   thus ?thesis by (simp add: power_inverse)
1319 qed
1321 lemma LIMSEQ_power_zero:
1322   fixes x :: "'a::{real_normed_algebra_1,recpower}"
1323   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
1324 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
1325 apply (simp only: LIMSEQ_Zseq_iff, erule Zseq_le)
1326 apply (simp add: power_abs norm_power_ineq)
1327 done
1329 lemma LIMSEQ_divide_realpow_zero:
1330   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
1331 apply (cut_tac a = a and x1 = "inverse x" in
1332         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
1333 apply (auto simp add: divide_inverse power_inverse)
1334 apply (simp add: inverse_eq_divide pos_divide_less_eq)
1335 done
1337 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
1339 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
1340 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
1342 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
1343 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
1344 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
1345 done
1347 end