move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
1 (* Title: HOL/Real_Vector_Spaces.thy
6 header {* Vector Spaces and Algebras over the Reals *}
8 theory Real_Vector_Spaces
9 imports Real Topological_Spaces
12 subsection {* Locale for additive functions *}
15 fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
16 assumes add: "f (x + y) = f x + f y"
21 have "f 0 = f (0 + 0)" by simp
22 also have "\<dots> = f 0 + f 0" by (rule add)
23 finally show "f 0 = 0" by simp
26 lemma minus: "f (- x) = - f x"
28 have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
29 also have "\<dots> = - f x + f x" by (simp add: zero)
30 finally show "f (- x) = - f x" by (rule add_right_imp_eq)
33 lemma diff: "f (x - y) = f x - f y"
34 using add [of x "- y"] by (simp add: minus)
36 lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
37 apply (cases "finite A")
38 apply (induct set: finite)
39 apply (simp add: zero)
41 apply (simp add: zero)
46 subsection {* Vector spaces *}
49 fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
50 assumes scale_right_distrib [algebra_simps]:
51 "scale a (x + y) = scale a x + scale a y"
52 and scale_left_distrib [algebra_simps]:
53 "scale (a + b) x = scale a x + scale b x"
54 and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
55 and scale_one [simp]: "scale 1 x = x"
58 lemma scale_left_commute:
59 "scale a (scale b x) = scale b (scale a x)"
60 by (simp add: mult_commute)
62 lemma scale_zero_left [simp]: "scale 0 x = 0"
63 and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
64 and scale_left_diff_distrib [algebra_simps]:
65 "scale (a - b) x = scale a x - scale b x"
66 and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
68 interpret s: additive "\<lambda>a. scale a x"
69 proof qed (rule scale_left_distrib)
70 show "scale 0 x = 0" by (rule s.zero)
71 show "scale (- a) x = - (scale a x)" by (rule s.minus)
72 show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
73 show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)
76 lemma scale_zero_right [simp]: "scale a 0 = 0"
77 and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
78 and scale_right_diff_distrib [algebra_simps]:
79 "scale a (x - y) = scale a x - scale a y"
80 and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
82 interpret s: additive "\<lambda>x. scale a x"
83 proof qed (rule scale_right_distrib)
84 show "scale a 0 = 0" by (rule s.zero)
85 show "scale a (- x) = - (scale a x)" by (rule s.minus)
86 show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
87 show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
90 lemma scale_eq_0_iff [simp]:
91 "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
93 assume "a = 0" thus ?thesis by simp
95 assume anz [simp]: "a \<noteq> 0"
96 { assume "scale a x = 0"
97 hence "scale (inverse a) (scale a x) = 0" by simp
98 hence "x = 0" by simp }
102 lemma scale_left_imp_eq:
103 "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
105 assume nonzero: "a \<noteq> 0"
106 assume "scale a x = scale a y"
107 hence "scale a (x - y) = 0"
108 by (simp add: scale_right_diff_distrib)
109 hence "x - y = 0" by (simp add: nonzero)
110 thus "x = y" by (simp only: right_minus_eq)
113 lemma scale_right_imp_eq:
114 "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
116 assume nonzero: "x \<noteq> 0"
117 assume "scale a x = scale b x"
118 hence "scale (a - b) x = 0"
119 by (simp add: scale_left_diff_distrib)
120 hence "a - b = 0" by (simp add: nonzero)
121 thus "a = b" by (simp only: right_minus_eq)
124 lemma scale_cancel_left [simp]:
125 "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
126 by (auto intro: scale_left_imp_eq)
128 lemma scale_cancel_right [simp]:
129 "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
130 by (auto intro: scale_right_imp_eq)
134 subsection {* Real vector spaces *}
137 fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
141 divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
143 "x /\<^sub>R r == scaleR (inverse r) x"
147 class real_vector = scaleR + ab_group_add +
148 assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"
149 and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"
150 and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
151 and scaleR_one: "scaleR 1 x = x"
153 interpretation real_vector:
154 vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
156 apply (rule scaleR_add_right)
157 apply (rule scaleR_add_left)
158 apply (rule scaleR_scaleR)
159 apply (rule scaleR_one)
162 text {* Recover original theorem names *}
164 lemmas scaleR_left_commute = real_vector.scale_left_commute
165 lemmas scaleR_zero_left = real_vector.scale_zero_left
166 lemmas scaleR_minus_left = real_vector.scale_minus_left
167 lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib
168 lemmas scaleR_setsum_left = real_vector.scale_setsum_left
169 lemmas scaleR_zero_right = real_vector.scale_zero_right
170 lemmas scaleR_minus_right = real_vector.scale_minus_right
171 lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib
172 lemmas scaleR_setsum_right = real_vector.scale_setsum_right
173 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
174 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
175 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
176 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
177 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
179 text {* Legacy names *}
181 lemmas scaleR_left_distrib = scaleR_add_left
182 lemmas scaleR_right_distrib = scaleR_add_right
183 lemmas scaleR_left_diff_distrib = scaleR_diff_left
184 lemmas scaleR_right_diff_distrib = scaleR_diff_right
186 lemma scaleR_minus1_left [simp]:
187 fixes x :: "'a::real_vector"
188 shows "scaleR (-1) x = - x"
189 using scaleR_minus_left [of 1 x] by simp
191 class real_algebra = real_vector + ring +
192 assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
193 and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
195 class real_algebra_1 = real_algebra + ring_1
197 class real_div_algebra = real_algebra_1 + division_ring
199 class real_field = real_div_algebra + field
201 instantiation real :: real_field
205 real_scaleR_def [simp]: "scaleR a x = a * x"
208 qed (simp_all add: algebra_simps)
212 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
213 proof qed (rule scaleR_left_distrib)
215 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
216 proof qed (rule scaleR_right_distrib)
218 lemma nonzero_inverse_scaleR_distrib:
219 fixes x :: "'a::real_div_algebra" shows
220 "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
221 by (rule inverse_unique, simp)
223 lemma inverse_scaleR_distrib:
224 fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"
225 shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
226 apply (case_tac "a = 0", simp)
227 apply (case_tac "x = 0", simp)
228 apply (erule (1) nonzero_inverse_scaleR_distrib)
232 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
236 of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
237 "of_real r = scaleR r 1"
239 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
240 by (simp add: of_real_def)
242 lemma of_real_0 [simp]: "of_real 0 = 0"
243 by (simp add: of_real_def)
245 lemma of_real_1 [simp]: "of_real 1 = 1"
246 by (simp add: of_real_def)
248 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
249 by (simp add: of_real_def scaleR_left_distrib)
251 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
252 by (simp add: of_real_def)
254 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
255 by (simp add: of_real_def scaleR_left_diff_distrib)
257 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
258 by (simp add: of_real_def mult_commute)
260 lemma nonzero_of_real_inverse:
261 "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
262 inverse (of_real x :: 'a::real_div_algebra)"
263 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
265 lemma of_real_inverse [simp]:
266 "of_real (inverse x) =
267 inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
268 by (simp add: of_real_def inverse_scaleR_distrib)
270 lemma nonzero_of_real_divide:
271 "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
272 (of_real x / of_real y :: 'a::real_field)"
273 by (simp add: divide_inverse nonzero_of_real_inverse)
275 lemma of_real_divide [simp]:
277 (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
278 by (simp add: divide_inverse)
280 lemma of_real_power [simp]:
281 "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
282 by (induct n) simp_all
284 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
285 by (simp add: of_real_def)
289 by (auto intro: injI)
291 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
293 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
296 show "of_real r = id r"
297 by (simp add: of_real_def)
300 text{*Collapse nested embeddings*}
301 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
304 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
305 by (cases z rule: int_diff_cases, simp)
307 lemma of_real_numeral: "of_real (numeral w) = numeral w"
308 using of_real_of_int_eq [of "numeral w"] by simp
310 lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"
311 using of_real_of_int_eq [of "neg_numeral w"] by simp
313 text{*Every real algebra has characteristic zero*}
315 instance real_algebra_1 < ring_char_0
317 from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
318 then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
321 instance real_field < field_char_0 ..
324 subsection {* The Set of Real Numbers *}
326 definition Reals :: "'a::real_algebra_1 set" where
327 "Reals = range of_real"
332 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
333 by (simp add: Reals_def)
335 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
336 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
338 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
339 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
341 lemma Reals_numeral [simp]: "numeral w \<in> Reals"
342 by (subst of_real_numeral [symmetric], rule Reals_of_real)
344 lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals"
345 by (subst of_real_neg_numeral [symmetric], rule Reals_of_real)
347 lemma Reals_0 [simp]: "0 \<in> Reals"
348 apply (unfold Reals_def)
349 apply (rule range_eqI)
350 apply (rule of_real_0 [symmetric])
353 lemma Reals_1 [simp]: "1 \<in> Reals"
354 apply (unfold Reals_def)
355 apply (rule range_eqI)
356 apply (rule of_real_1 [symmetric])
359 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
360 apply (auto simp add: Reals_def)
361 apply (rule range_eqI)
362 apply (rule of_real_add [symmetric])
365 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
366 apply (auto simp add: Reals_def)
367 apply (rule range_eqI)
368 apply (rule of_real_minus [symmetric])
371 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
372 apply (auto simp add: Reals_def)
373 apply (rule range_eqI)
374 apply (rule of_real_diff [symmetric])
377 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
378 apply (auto simp add: Reals_def)
379 apply (rule range_eqI)
380 apply (rule of_real_mult [symmetric])
383 lemma nonzero_Reals_inverse:
384 fixes a :: "'a::real_div_algebra"
385 shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
386 apply (auto simp add: Reals_def)
387 apply (rule range_eqI)
388 apply (erule nonzero_of_real_inverse [symmetric])
391 lemma Reals_inverse [simp]:
392 fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
393 shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
394 apply (auto simp add: Reals_def)
395 apply (rule range_eqI)
396 apply (rule of_real_inverse [symmetric])
399 lemma nonzero_Reals_divide:
400 fixes a b :: "'a::real_field"
401 shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
402 apply (auto simp add: Reals_def)
403 apply (rule range_eqI)
404 apply (erule nonzero_of_real_divide [symmetric])
407 lemma Reals_divide [simp]:
408 fixes a b :: "'a::{real_field, field_inverse_zero}"
409 shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
410 apply (auto simp add: Reals_def)
411 apply (rule range_eqI)
412 apply (rule of_real_divide [symmetric])
415 lemma Reals_power [simp]:
416 fixes a :: "'a::{real_algebra_1}"
417 shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
418 apply (auto simp add: Reals_def)
419 apply (rule range_eqI)
420 apply (rule of_real_power [symmetric])
423 lemma Reals_cases [cases set: Reals]:
424 assumes "q \<in> \<real>"
425 obtains (of_real) r where "q = of_real r"
428 from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
429 then obtain r where "q = of_real r" ..
433 lemma Reals_induct [case_names of_real, induct set: Reals]:
434 "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
435 by (rule Reals_cases) auto
438 subsection {* Real normed vector spaces *}
441 fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
444 fixes norm :: "'a \<Rightarrow> real"
446 class sgn_div_norm = scaleR + norm + sgn +
447 assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
449 class dist_norm = dist + norm + minus +
450 assumes dist_norm: "dist x y = norm (x - y)"
452 class open_dist = "open" + dist +
453 assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
455 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
456 assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
457 and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
458 and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
461 lemma norm_ge_zero [simp]: "0 \<le> norm x"
463 have "0 = norm (x + -1 *\<^sub>R x)"
464 using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
465 also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
466 finally show ?thesis by simp
471 class real_normed_algebra = real_algebra + real_normed_vector +
472 assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
474 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
475 assumes norm_one [simp]: "norm 1 = 1"
477 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
478 assumes norm_mult: "norm (x * y) = norm x * norm y"
480 class real_normed_field = real_field + real_normed_div_algebra
482 instance real_normed_div_algebra < real_normed_algebra_1
485 show "norm (x * y) \<le> norm x * norm y"
486 by (simp add: norm_mult)
488 have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
490 thus "norm (1::'a) = 1" by simp
493 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
496 lemma zero_less_norm_iff [simp]:
497 fixes x :: "'a::real_normed_vector"
498 shows "(0 < norm x) = (x \<noteq> 0)"
499 by (simp add: order_less_le)
501 lemma norm_not_less_zero [simp]:
502 fixes x :: "'a::real_normed_vector"
503 shows "\<not> norm x < 0"
504 by (simp add: linorder_not_less)
506 lemma norm_le_zero_iff [simp]:
507 fixes x :: "'a::real_normed_vector"
508 shows "(norm x \<le> 0) = (x = 0)"
509 by (simp add: order_le_less)
511 lemma norm_minus_cancel [simp]:
512 fixes x :: "'a::real_normed_vector"
513 shows "norm (- x) = norm x"
515 have "norm (- x) = norm (scaleR (- 1) x)"
516 by (simp only: scaleR_minus_left scaleR_one)
517 also have "\<dots> = \<bar>- 1\<bar> * norm x"
518 by (rule norm_scaleR)
519 finally show ?thesis by simp
522 lemma norm_minus_commute:
523 fixes a b :: "'a::real_normed_vector"
524 shows "norm (a - b) = norm (b - a)"
526 have "norm (- (b - a)) = norm (b - a)"
527 by (rule norm_minus_cancel)
531 lemma norm_triangle_ineq2:
532 fixes a b :: "'a::real_normed_vector"
533 shows "norm a - norm b \<le> norm (a - b)"
535 have "norm (a - b + b) \<le> norm (a - b) + norm b"
536 by (rule norm_triangle_ineq)
540 lemma norm_triangle_ineq3:
541 fixes a b :: "'a::real_normed_vector"
542 shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
543 apply (subst abs_le_iff)
545 apply (rule norm_triangle_ineq2)
546 apply (subst norm_minus_commute)
547 apply (rule norm_triangle_ineq2)
550 lemma norm_triangle_ineq4:
551 fixes a b :: "'a::real_normed_vector"
552 shows "norm (a - b) \<le> norm a + norm b"
554 have "norm (a + - b) \<le> norm a + norm (- b)"
555 by (rule norm_triangle_ineq)
556 then show ?thesis by simp
559 lemma norm_diff_ineq:
560 fixes a b :: "'a::real_normed_vector"
561 shows "norm a - norm b \<le> norm (a + b)"
563 have "norm a - norm (- b) \<le> norm (a - - b)"
564 by (rule norm_triangle_ineq2)
568 lemma norm_diff_triangle_ineq:
569 fixes a b c d :: "'a::real_normed_vector"
570 shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
572 have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
573 by (simp add: algebra_simps)
574 also have "\<dots> \<le> norm (a - c) + norm (b - d)"
575 by (rule norm_triangle_ineq)
576 finally show ?thesis .
579 lemma abs_norm_cancel [simp]:
580 fixes a :: "'a::real_normed_vector"
581 shows "\<bar>norm a\<bar> = norm a"
582 by (rule abs_of_nonneg [OF norm_ge_zero])
585 fixes x y :: "'a::real_normed_vector"
586 shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
587 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
589 lemma norm_mult_less:
590 fixes x y :: "'a::real_normed_algebra"
591 shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
592 apply (rule order_le_less_trans [OF norm_mult_ineq])
593 apply (simp add: mult_strict_mono')
596 lemma norm_of_real [simp]:
597 "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
598 unfolding of_real_def by simp
600 lemma norm_numeral [simp]:
601 "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
602 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
604 lemma norm_neg_numeral [simp]:
605 "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w"
606 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
608 lemma norm_of_int [simp]:
609 "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
610 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
612 lemma norm_of_nat [simp]:
613 "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
614 apply (subst of_real_of_nat_eq [symmetric])
615 apply (subst norm_of_real, simp)
618 lemma nonzero_norm_inverse:
619 fixes a :: "'a::real_normed_div_algebra"
620 shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
621 apply (rule inverse_unique [symmetric])
622 apply (simp add: norm_mult [symmetric])
626 fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
627 shows "norm (inverse a) = inverse (norm a)"
628 apply (case_tac "a = 0", simp)
629 apply (erule nonzero_norm_inverse)
632 lemma nonzero_norm_divide:
633 fixes a b :: "'a::real_normed_field"
634 shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
635 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
638 fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
639 shows "norm (a / b) = norm a / norm b"
640 by (simp add: divide_inverse norm_mult norm_inverse)
642 lemma norm_power_ineq:
643 fixes x :: "'a::{real_normed_algebra_1}"
644 shows "norm (x ^ n) \<le> norm x ^ n"
646 case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
649 have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
650 by (rule norm_mult_ineq)
651 also from Suc have "\<dots> \<le> norm x * norm x ^ n"
652 using norm_ge_zero by (rule mult_left_mono)
653 finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
658 fixes x :: "'a::{real_normed_div_algebra}"
659 shows "norm (x ^ n) = norm x ^ n"
660 by (induct n) (simp_all add: norm_mult)
663 subsection {* Metric spaces *}
665 class metric_space = open_dist +
666 assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
667 assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
670 lemma dist_self [simp]: "dist x x = 0"
673 lemma zero_le_dist [simp]: "0 \<le> dist x y"
674 using dist_triangle2 [of x x y] by simp
676 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
677 by (simp add: less_le)
679 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
680 by (simp add: not_less)
682 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
683 by (simp add: le_less)
685 lemma dist_commute: "dist x y = dist y x"
686 proof (rule order_antisym)
687 show "dist x y \<le> dist y x"
688 using dist_triangle2 [of x y x] by simp
689 show "dist y x \<le> dist x y"
690 using dist_triangle2 [of y x y] by simp
693 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
694 using dist_triangle2 [of x z y] by (simp add: dist_commute)
696 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
697 using dist_triangle2 [of x y a] by (simp add: dist_commute)
699 lemma dist_triangle_alt:
700 shows "dist y z <= dist x y + dist x z"
701 by (rule dist_triangle3)
704 shows "x \<noteq> y ==> 0 < dist x y"
705 by (simp add: zero_less_dist_iff)
708 shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
709 by (simp add: zero_less_dist_iff)
711 lemma dist_triangle_le:
712 shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
713 by (rule order_trans [OF dist_triangle2])
715 lemma dist_triangle_lt:
716 shows "dist x z + dist y z < e ==> dist x y < e"
717 by (rule le_less_trans [OF dist_triangle2])
719 lemma dist_triangle_half_l:
720 shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
721 by (rule dist_triangle_lt [where z=y], simp)
723 lemma dist_triangle_half_r:
724 shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
725 by (rule dist_triangle_half_l, simp_all add: dist_commute)
727 subclass topological_space
729 have "\<exists>e::real. 0 < e"
730 by (fast intro: zero_less_one)
731 then show "open UNIV"
732 unfolding open_dist by simp
734 fix S T assume "open S" "open T"
735 then show "open (S \<inter> T)"
738 apply (drule (1) bspec)+
739 apply (clarify, rename_tac r s)
740 apply (rule_tac x="min r s" in exI, simp)
743 fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
744 unfolding open_dist by fast
747 lemma open_ball: "open {y. dist x y < d}"
748 proof (unfold open_dist, intro ballI)
749 fix y assume *: "y \<in> {y. dist x y < d}"
750 then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
751 by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
754 subclass first_countable_topology
757 show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
758 proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
759 fix S assume "open S" "x \<in> S"
760 then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
761 by (auto simp: open_dist subset_eq dist_commute)
763 from e obtain i where "inverse (Suc i) < e"
764 by (auto dest!: reals_Archimedean)
765 then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
767 ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
769 qed (auto intro: open_ball)
774 instance metric_space \<subseteq> t2_space
776 fix x y :: "'a::metric_space"
777 assume xy: "x \<noteq> y"
778 let ?U = "{y'. dist x y' < dist x y / 2}"
779 let ?V = "{x'. dist y x' < dist x y / 2}"
780 have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
781 \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
782 have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
783 using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
784 using open_ball[of _ "dist x y / 2"] by auto
785 then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
789 text {* Every normed vector space is a metric space. *}
791 instance real_normed_vector < metric_space
793 fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
794 unfolding dist_norm by simp
796 fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
798 using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
801 subsection {* Class instances for real numbers *}
803 instantiation real :: real_normed_field
806 definition dist_real_def:
807 "dist x y = \<bar>x - y\<bar>"
809 definition open_real_def [code del]:
810 "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
812 definition real_norm_def [simp]:
813 "norm r = \<bar>r\<bar>"
816 apply (intro_classes, unfold real_norm_def real_scaleR_def)
817 apply (rule dist_real_def)
818 apply (rule open_real_def)
819 apply (simp add: sgn_real_def)
820 apply (rule abs_eq_0)
821 apply (rule abs_triangle_ineq)
822 apply (rule abs_mult)
823 apply (rule abs_mult)
828 code_abort "open :: real set \<Rightarrow> bool"
830 instance real :: linorder_topology
832 show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
833 proof (rule ext, safe)
834 fix S :: "real set" assume "open S"
835 then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
836 unfolding open_real_def bchoice_iff ..
837 then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
838 by (fastforce simp: dist_real_def)
839 show "generate_topology (range lessThan \<union> range greaterThan) S"
841 apply (intro generate_topology_Union generate_topology.Int)
842 apply (auto intro: generate_topology.Basis)
845 fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
846 moreover have "\<And>a::real. open {..<a}"
847 unfolding open_real_def dist_real_def
849 fix x a :: real assume "x < a"
850 hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
851 thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
853 moreover have "\<And>a::real. open {a <..}"
854 unfolding open_real_def dist_real_def
856 fix x a :: real assume "a < x"
857 hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
858 thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
860 ultimately show "open S"
865 instance real :: linear_continuum_topology ..
867 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
868 lemmas open_real_lessThan = open_lessThan[where 'a=real]
869 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
870 lemmas closed_real_atMost = closed_atMost[where 'a=real]
871 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
872 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
874 subsection {* Extra type constraints *}
876 text {* Only allow @{term "open"} in class @{text topological_space}. *}
878 setup {* Sign.add_const_constraint
879 (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
881 text {* Only allow @{term dist} in class @{text metric_space}. *}
883 setup {* Sign.add_const_constraint
884 (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
886 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
888 setup {* Sign.add_const_constraint
889 (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
891 subsection {* Sign function *}
894 "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
895 by (simp add: sgn_div_norm)
897 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
898 by (simp add: sgn_div_norm)
900 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
901 by (simp add: sgn_div_norm)
903 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
904 by (simp add: sgn_div_norm)
907 "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
908 by (simp add: sgn_div_norm mult_ac)
910 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
911 by (simp add: sgn_div_norm)
914 "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
915 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
918 fixes x y :: "'a::real_normed_div_algebra"
919 shows "sgn (x * y) = sgn x * sgn y"
920 by (simp add: sgn_div_norm norm_mult mult_commute)
922 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
923 by (simp add: sgn_div_norm divide_inverse)
925 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
926 unfolding real_sgn_eq by simp
928 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
929 unfolding real_sgn_eq by simp
931 lemma norm_conv_dist: "norm x = dist x 0"
932 unfolding dist_norm by simp
934 subsection {* Bounded Linear and Bilinear Operators *}
936 locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
937 assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
940 assumes "\<And>x y. f (x + y) = f x + f y"
941 assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
943 by default (rule assms)+
945 locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
946 assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
950 "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
952 obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
953 using bounded by fast
955 proof (intro exI impI conjI allI)
957 by (rule order_less_le_trans [OF zero_less_one le_maxI1])
960 have "norm (f x) \<le> norm x * K" using K .
961 also have "\<dots> \<le> norm x * max 1 K"
962 by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
963 finally show "norm (f x) \<le> norm x * max 1 K" .
967 lemma nonneg_bounded:
968 "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
971 show ?thesis by (auto intro: order_less_imp_le)
976 lemma bounded_linear_intro:
977 assumes "\<And>x y. f (x + y) = f x + f y"
978 assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
979 assumes "\<And>x. norm (f x) \<le> norm x * K"
980 shows "bounded_linear f"
981 by default (fast intro: assms)+
983 locale bounded_bilinear =
984 fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
985 \<Rightarrow> 'c::real_normed_vector"
987 assumes add_left: "prod (a + a') b = prod a b + prod a' b"
988 assumes add_right: "prod a (b + b') = prod a b + prod a b'"
989 assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
990 assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
991 assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
995 "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
996 apply (cut_tac bounded, erule exE)
997 apply (rule_tac x="max 1 K" in exI, safe)
998 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
999 apply (drule spec, drule spec, erule order_trans)
1000 apply (rule mult_left_mono [OF le_maxI2])
1001 apply (intro mult_nonneg_nonneg norm_ge_zero)
1004 lemma nonneg_bounded:
1005 "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
1008 show ?thesis by (auto intro: order_less_imp_le)
1011 lemma additive_right: "additive (\<lambda>b. prod a b)"
1012 by (rule additive.intro, rule add_right)
1014 lemma additive_left: "additive (\<lambda>a. prod a b)"
1015 by (rule additive.intro, rule add_left)
1017 lemma zero_left: "prod 0 b = 0"
1018 by (rule additive.zero [OF additive_left])
1020 lemma zero_right: "prod a 0 = 0"
1021 by (rule additive.zero [OF additive_right])
1023 lemma minus_left: "prod (- a) b = - prod a b"
1024 by (rule additive.minus [OF additive_left])
1026 lemma minus_right: "prod a (- b) = - prod a b"
1027 by (rule additive.minus [OF additive_right])
1030 "prod (a - a') b = prod a b - prod a' b"
1031 by (rule additive.diff [OF additive_left])
1034 "prod a (b - b') = prod a b - prod a b'"
1035 by (rule additive.diff [OF additive_right])
1037 lemma bounded_linear_left:
1038 "bounded_linear (\<lambda>a. a ** b)"
1039 apply (cut_tac bounded, safe)
1040 apply (rule_tac K="norm b * K" in bounded_linear_intro)
1041 apply (rule add_left)
1042 apply (rule scaleR_left)
1043 apply (simp add: mult_ac)
1046 lemma bounded_linear_right:
1047 "bounded_linear (\<lambda>b. a ** b)"
1048 apply (cut_tac bounded, safe)
1049 apply (rule_tac K="norm a * K" in bounded_linear_intro)
1050 apply (rule add_right)
1051 apply (rule scaleR_right)
1052 apply (simp add: mult_ac)
1055 lemma prod_diff_prod:
1056 "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
1057 by (simp add: diff_left diff_right)
1061 lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
1062 by default (auto intro!: exI[of _ 1])
1064 lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
1065 by default (auto intro!: exI[of _ 1])
1067 lemma bounded_linear_add:
1068 assumes "bounded_linear f"
1069 assumes "bounded_linear g"
1070 shows "bounded_linear (\<lambda>x. f x + g x)"
1072 interpret f: bounded_linear f by fact
1073 interpret g: bounded_linear g by fact
1076 from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
1077 from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
1078 show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
1079 using add_mono[OF Kf Kg]
1080 by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
1081 qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
1084 lemma bounded_linear_minus:
1085 assumes "bounded_linear f"
1086 shows "bounded_linear (\<lambda>x. - f x)"
1088 interpret f: bounded_linear f by fact
1089 show ?thesis apply (unfold_locales)
1090 apply (simp add: f.add)
1091 apply (simp add: f.scaleR)
1092 apply (simp add: f.bounded)
1096 lemma bounded_linear_compose:
1097 assumes "bounded_linear f"
1098 assumes "bounded_linear g"
1099 shows "bounded_linear (\<lambda>x. f (g x))"
1101 interpret f: bounded_linear f by fact
1102 interpret g: bounded_linear g by fact
1103 show ?thesis proof (unfold_locales)
1104 fix x y show "f (g (x + y)) = f (g x) + f (g y)"
1105 by (simp only: f.add g.add)
1107 fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
1108 by (simp only: f.scaleR g.scaleR)
1111 obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
1113 obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
1114 show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
1115 proof (intro exI allI)
1117 have "norm (f (g x)) \<le> norm (g x) * Kf"
1119 also have "\<dots> \<le> (norm x * Kg) * Kf"
1120 using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
1121 also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
1122 by (rule mult_assoc)
1123 finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
1128 lemma bounded_bilinear_mult:
1129 "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
1130 apply (rule bounded_bilinear.intro)
1131 apply (rule distrib_right)
1132 apply (rule distrib_left)
1133 apply (rule mult_scaleR_left)
1134 apply (rule mult_scaleR_right)
1135 apply (rule_tac x="1" in exI)
1136 apply (simp add: norm_mult_ineq)
1139 lemma bounded_linear_mult_left:
1140 "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
1141 using bounded_bilinear_mult
1142 by (rule bounded_bilinear.bounded_linear_left)
1144 lemma bounded_linear_mult_right:
1145 "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
1146 using bounded_bilinear_mult
1147 by (rule bounded_bilinear.bounded_linear_right)
1149 lemmas bounded_linear_mult_const =
1150 bounded_linear_mult_left [THEN bounded_linear_compose]
1152 lemmas bounded_linear_const_mult =
1153 bounded_linear_mult_right [THEN bounded_linear_compose]
1155 lemma bounded_linear_divide:
1156 "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
1157 unfolding divide_inverse by (rule bounded_linear_mult_left)
1159 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
1160 apply (rule bounded_bilinear.intro)
1161 apply (rule scaleR_left_distrib)
1162 apply (rule scaleR_right_distrib)
1164 apply (rule scaleR_left_commute)
1165 apply (rule_tac x="1" in exI, simp)
1168 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
1169 using bounded_bilinear_scaleR
1170 by (rule bounded_bilinear.bounded_linear_left)
1172 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
1173 using bounded_bilinear_scaleR
1174 by (rule bounded_bilinear.bounded_linear_right)
1176 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
1177 unfolding of_real_def by (rule bounded_linear_scaleR_left)
1179 lemma real_bounded_linear:
1180 fixes f :: "real \<Rightarrow> real"
1181 shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
1183 { fix x assume "bounded_linear f"
1184 then interpret bounded_linear f .
1185 from scaleR[of x 1] have "f x = x * f 1"
1188 by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
1191 instance real_normed_algebra_1 \<subseteq> perfect_space
1194 show "\<not> open {x}"
1195 unfolding open_dist dist_norm
1196 by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
1199 subsection {* Filters and Limits on Metric Space *}
1201 lemma eventually_nhds_metric:
1202 fixes a :: "'a :: metric_space"
1203 shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
1204 unfolding eventually_nhds open_dist
1207 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
1209 apply (rule_tac x="d - dist x a" in exI, clarsimp)
1210 apply (simp only: less_diff_eq)
1211 apply (erule le_less_trans [OF dist_triangle])
1214 lemma eventually_at:
1215 fixes a :: "'a :: metric_space"
1216 shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
1217 unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
1219 lemma eventually_at_le:
1220 fixes a :: "'a::metric_space"
1221 shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
1222 unfolding eventually_at_filter eventually_nhds_metric
1224 apply (rule_tac x="d / 2" in exI)
1229 fixes l :: "'a :: metric_space"
1230 assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
1231 shows "(f ---> l) F"
1232 apply (rule topological_tendstoI)
1233 apply (simp add: open_dist)
1234 apply (drule (1) bspec, clarify)
1236 apply (erule eventually_elim1, simp)
1240 fixes l :: "'a :: metric_space"
1241 shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
1242 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
1243 apply (clarsimp simp add: open_dist)
1244 apply (rule_tac x="e - dist x l" in exI, clarsimp)
1245 apply (simp only: less_diff_eq)
1246 apply (erule le_less_trans [OF dist_triangle])
1252 fixes l :: "'a :: metric_space"
1253 shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
1254 using tendstoI tendstoD by fast
1256 lemma metric_tendsto_imp_tendsto:
1257 fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
1258 assumes f: "(f ---> a) F"
1259 assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
1260 shows "(g ---> b) F"
1261 proof (rule tendstoI)
1262 fix e :: real assume "0 < e"
1263 with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
1264 with le show "eventually (\<lambda>x. dist (g x) b < e) F"
1265 using le_less_trans by (rule eventually_elim2)
1268 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
1269 unfolding filterlim_at_top
1271 apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
1272 apply (auto simp: natceiling_le_eq)
1275 subsubsection {* Limits of Sequences *}
1277 lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
1278 unfolding tendsto_iff eventually_sequentially ..
1280 lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
1281 unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
1283 lemma metric_LIMSEQ_I:
1284 "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
1285 by (simp add: LIMSEQ_def)
1287 lemma metric_LIMSEQ_D:
1288 "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
1289 by (simp add: LIMSEQ_def)
1292 subsubsection {* Limits of Functions *}
1294 lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
1295 (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
1296 --> dist (f x) L < r)"
1297 unfolding tendsto_iff eventually_at by simp
1300 "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
1301 \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
1302 by (simp add: LIM_def)
1305 "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
1306 \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
1307 by (simp add: LIM_def)
1309 lemma metric_LIM_imp_LIM:
1310 assumes f: "f -- a --> (l::'a::metric_space)"
1311 assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
1312 shows "g -- a --> (m::'b::metric_space)"
1313 by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
1315 lemma metric_LIM_equal2:
1317 assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
1318 shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
1319 apply (rule topological_tendstoI)
1320 apply (drule (2) topological_tendstoD)
1321 apply (simp add: eventually_at, safe)
1322 apply (rule_tac x="min d R" in exI, safe)
1327 lemma metric_LIM_compose2:
1328 assumes f: "f -- (a::'a::metric_space) --> b"
1329 assumes g: "g -- b --> c"
1330 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
1331 shows "(\<lambda>x. g (f x)) -- a --> c"
1333 by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
1335 lemma metric_isCont_LIM_compose2:
1336 fixes f :: "'a :: metric_space \<Rightarrow> _"
1337 assumes f [unfolded isCont_def]: "isCont f a"
1338 assumes g: "g -- f a --> l"
1339 assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
1340 shows "(\<lambda>x. g (f x)) -- a --> l"
1341 by (rule metric_LIM_compose2 [OF f g inj])
1343 subsection {* Complete metric spaces *}
1345 subsection {* Cauchy sequences *}
1347 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
1348 "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
1350 subsection {* Cauchy Sequences *}
1352 lemma metric_CauchyI:
1353 "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
1354 by (simp add: Cauchy_def)
1356 lemma metric_CauchyD:
1357 "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
1358 by (simp add: Cauchy_def)
1360 lemma metric_Cauchy_iff2:
1361 "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
1362 apply (simp add: Cauchy_def, auto)
1363 apply (drule reals_Archimedean, safe)
1364 apply (drule_tac x = n in spec, auto)
1365 apply (rule_tac x = M in exI, auto)
1366 apply (drule_tac x = m in spec, simp)
1367 apply (drule_tac x = na in spec, auto)
1371 "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
1372 unfolding metric_Cauchy_iff2 dist_real_def ..
1374 lemma Cauchy_subseq_Cauchy:
1375 "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
1376 apply (auto simp add: Cauchy_def)
1377 apply (drule_tac x=e in spec, clarify)
1378 apply (rule_tac x=M in exI, clarify)
1379 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
1382 theorem LIMSEQ_imp_Cauchy:
1383 assumes X: "X ----> a" shows "Cauchy X"
1384 proof (rule metric_CauchyI)
1385 fix e::real assume "0 < e"
1386 hence "0 < e/2" by simp
1387 with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
1388 then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
1389 show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
1390 proof (intro exI allI impI)
1391 fix m assume "N \<le> m"
1392 hence m: "dist (X m) a < e/2" using N by fast
1393 fix n assume "N \<le> n"
1394 hence n: "dist (X n) a < e/2" using N by fast
1395 have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
1396 by (rule dist_triangle2)
1397 also from m n have "\<dots> < e" by simp
1398 finally show "dist (X m) (X n) < e" .
1402 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
1403 unfolding convergent_def
1404 by (erule exE, erule LIMSEQ_imp_Cauchy)
1406 subsubsection {* Cauchy Sequences are Convergent *}
1408 class complete_space = metric_space +
1409 assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
1411 lemma Cauchy_convergent_iff:
1412 fixes X :: "nat \<Rightarrow> 'a::complete_space"
1413 shows "Cauchy X = convergent X"
1414 by (fast intro: Cauchy_convergent convergent_Cauchy)
1416 subsection {* The set of real numbers is a complete metric space *}
1419 Proof that Cauchy sequences converge based on the one from
1420 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
1424 If sequence @{term "X"} is Cauchy, then its limit is the lub of
1425 @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
1428 lemma increasing_LIMSEQ:
1429 fixes f :: "nat \<Rightarrow> real"
1430 assumes inc: "\<And>n. f n \<le> f (Suc n)"
1431 and bdd: "\<And>n. f n \<le> l"
1432 and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
1434 proof (rule increasing_tendsto)
1435 fix x assume "x < l"
1436 with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
1438 from en[OF `0 < e`] obtain n where "l - e \<le> f n"
1439 by (auto simp: field_simps)
1440 with `e < l - x` `0 < e` have "x < f n" by simp
1441 with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
1442 by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
1443 qed (insert bdd, auto)
1445 lemma real_Cauchy_convergent:
1446 fixes X :: "nat \<Rightarrow> real"
1447 assumes X: "Cauchy X"
1448 shows "convergent X"
1450 def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
1451 then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
1453 { fix N x assume N: "\<forall>n\<ge>N. X n < x"
1454 fix y::real assume "y \<in> S"
1455 hence "\<exists>M. \<forall>n\<ge>M. y < X n"
1456 by (simp add: S_def)
1457 then obtain M where "\<forall>n\<ge>M. y < X n" ..
1458 hence "y < X (max M N)" by simp
1459 also have "\<dots> < x" using N by simp
1460 finally have "y \<le> x"
1461 by (rule order_less_imp_le) }
1462 note bound_isUb = this
1464 obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
1465 using X[THEN metric_CauchyD, OF zero_less_one] by auto
1466 hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
1467 have [simp]: "S \<noteq> {}"
1468 proof (intro exI ex_in_conv[THEN iffD1])
1469 from N have "\<forall>n\<ge>N. X N - 1 < X n"
1470 by (simp add: abs_diff_less_iff dist_real_def)
1471 thus "X N - 1 \<in> S" by (rule mem_S)
1473 have [simp]: "bdd_above S"
1475 from N have "\<forall>n\<ge>N. X n < X N + 1"
1476 by (simp add: abs_diff_less_iff dist_real_def)
1477 thus "\<And>s. s \<in> S \<Longrightarrow> s \<le> X N + 1"
1478 by (rule bound_isUb)
1480 have "X ----> Sup S"
1481 proof (rule metric_LIMSEQ_I)
1482 fix r::real assume "0 < r"
1483 hence r: "0 < r/2" by simp
1484 obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
1485 using metric_CauchyD [OF X r] by auto
1486 hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
1487 hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
1488 by (simp only: dist_real_def abs_diff_less_iff)
1490 from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
1491 hence "X N - r/2 \<in> S" by (rule mem_S)
1492 hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
1494 from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
1495 from bound_isUb[OF this]
1496 have 2: "Sup S \<le> X N + r/2"
1497 by (intro cSup_least) simp_all
1499 show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
1500 proof (intro exI allI impI)
1501 fix n assume n: "N \<le> n"
1502 from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
1503 thus "dist (X n) (Sup S) < r" using 1 2
1504 by (simp add: abs_diff_less_iff dist_real_def)
1507 then show ?thesis unfolding convergent_def by auto
1510 instance real :: complete_space
1511 by intro_classes (rule real_Cauchy_convergent)
1513 class banach = real_normed_vector + complete_space
1515 instance real :: banach by default
1517 lemma tendsto_at_topI_sequentially:
1518 fixes f :: "real \<Rightarrow> real"
1519 assumes mono: "mono f"
1520 assumes limseq: "(\<lambda>n. f (real n)) ----> y"
1521 shows "(f ---> y) at_top"
1522 proof (rule tendstoI)
1523 fix e :: real assume "0 < e"
1524 with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
1525 by (auto simp: LIMSEQ_def dist_real_def)
1527 obtain n where "x \<le> real_of_nat n"
1528 using ex_le_of_nat[of x] ..
1529 note monoD[OF mono this]
1530 also have "f (real_of_nat n) \<le> y"
1531 by (rule LIMSEQ_le_const[OF limseq])
1532 (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
1533 finally have "f x \<le> y" . }
1535 have "eventually (\<lambda>x. real N \<le> x) at_top"
1536 by (rule eventually_ge_at_top)
1537 then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
1538 proof eventually_elim
1539 fix x assume N': "real N \<le> x"
1540 with N[of N] le have "y - f (real N) < e" by auto
1541 moreover note monoD[OF mono N']
1542 ultimately show "dist (f x) y < e"
1543 using le[of x] by (auto simp: dist_real_def field_simps)