src/HOL/RealVector.thy
 author wenzelm Mon Mar 22 20:58:52 2010 +0100 (2010-03-22) changeset 35898 c890a3835d15 parent 35216 7641e8d831d2 child 36349 39be26d1bc28 permissions -rw-r--r--
1 (*  Title:      HOL/RealVector.thy
2     Author:     Brian Huffman
3 *)
5 header {* Vector Spaces and Algebras over the Reals *}
7 theory RealVector
8 imports RealPow
9 begin
11 subsection {* Locale for additive functions *}
13 locale additive =
14   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
15   assumes add: "f (x + y) = f x + f y"
16 begin
18 lemma zero: "f 0 = 0"
19 proof -
20   have "f 0 = f (0 + 0)" by simp
21   also have "\<dots> = f 0 + f 0" by (rule add)
22   finally show "f 0 = 0" by simp
23 qed
25 lemma minus: "f (- x) = - f x"
26 proof -
27   have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
28   also have "\<dots> = - f x + f x" by (simp add: zero)
29   finally show "f (- x) = - f x" by (rule add_right_imp_eq)
30 qed
32 lemma diff: "f (x - y) = f x - f y"
33 by (simp add: diff_def add minus)
35 lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
36 apply (cases "finite A")
37 apply (induct set: finite)
38 apply (simp add: zero)
40 apply (simp add: zero)
41 done
43 end
45 subsection {* Vector spaces *}
47 locale vector_space =
48   fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
49   assumes scale_right_distrib [algebra_simps]:
50     "scale a (x + y) = scale a x + scale a y"
51   and scale_left_distrib [algebra_simps]:
52     "scale (a + b) x = scale a x + scale b x"
53   and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
54   and scale_one [simp]: "scale 1 x = x"
55 begin
57 lemma scale_left_commute:
58   "scale a (scale b x) = scale b (scale a x)"
59 by (simp add: mult_commute)
61 lemma scale_zero_left [simp]: "scale 0 x = 0"
62   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
63   and scale_left_diff_distrib [algebra_simps]:
64         "scale (a - b) x = scale a x - scale b x"
65 proof -
66   interpret s: additive "\<lambda>a. scale a x"
67     proof qed (rule scale_left_distrib)
68   show "scale 0 x = 0" by (rule s.zero)
69   show "scale (- a) x = - (scale a x)" by (rule s.minus)
70   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
71 qed
73 lemma scale_zero_right [simp]: "scale a 0 = 0"
74   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
75   and scale_right_diff_distrib [algebra_simps]:
76         "scale a (x - y) = scale a x - scale a y"
77 proof -
78   interpret s: additive "\<lambda>x. scale a x"
79     proof qed (rule scale_right_distrib)
80   show "scale a 0 = 0" by (rule s.zero)
81   show "scale a (- x) = - (scale a x)" by (rule s.minus)
82   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
83 qed
85 lemma scale_eq_0_iff [simp]:
86   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
87 proof cases
88   assume "a = 0" thus ?thesis by simp
89 next
90   assume anz [simp]: "a \<noteq> 0"
91   { assume "scale a x = 0"
92     hence "scale (inverse a) (scale a x) = 0" by simp
93     hence "x = 0" by simp }
94   thus ?thesis by force
95 qed
97 lemma scale_left_imp_eq:
98   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
99 proof -
100   assume nonzero: "a \<noteq> 0"
101   assume "scale a x = scale a y"
102   hence "scale a (x - y) = 0"
103      by (simp add: scale_right_diff_distrib)
104   hence "x - y = 0" by (simp add: nonzero)
105   thus "x = y" by (simp only: right_minus_eq)
106 qed
108 lemma scale_right_imp_eq:
109   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
110 proof -
111   assume nonzero: "x \<noteq> 0"
112   assume "scale a x = scale b x"
113   hence "scale (a - b) x = 0"
114      by (simp add: scale_left_diff_distrib)
115   hence "a - b = 0" by (simp add: nonzero)
116   thus "a = b" by (simp only: right_minus_eq)
117 qed
119 lemma scale_cancel_left [simp]:
120   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
121 by (auto intro: scale_left_imp_eq)
123 lemma scale_cancel_right [simp]:
124   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
125 by (auto intro: scale_right_imp_eq)
127 end
129 subsection {* Real vector spaces *}
131 class scaleR =
132   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
133 begin
135 abbreviation
136   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
137 where
138   "x /\<^sub>R r == scaleR (inverse r) x"
140 end
142 class real_vector = scaleR + ab_group_add +
143   assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
144   and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
145   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
146   and scaleR_one: "scaleR 1 x = x"
148 interpretation real_vector:
149   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
150 apply unfold_locales
151 apply (rule scaleR_right_distrib)
152 apply (rule scaleR_left_distrib)
153 apply (rule scaleR_scaleR)
154 apply (rule scaleR_one)
155 done
157 text {* Recover original theorem names *}
159 lemmas scaleR_left_commute = real_vector.scale_left_commute
160 lemmas scaleR_zero_left = real_vector.scale_zero_left
161 lemmas scaleR_minus_left = real_vector.scale_minus_left
162 lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib
163 lemmas scaleR_zero_right = real_vector.scale_zero_right
164 lemmas scaleR_minus_right = real_vector.scale_minus_right
165 lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib
166 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
167 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
168 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
169 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
170 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
172 lemma scaleR_minus1_left [simp]:
173   fixes x :: "'a::real_vector"
174   shows "scaleR (-1) x = - x"
175   using scaleR_minus_left [of 1 x] by simp
177 class real_algebra = real_vector + ring +
178   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
179   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
181 class real_algebra_1 = real_algebra + ring_1
183 class real_div_algebra = real_algebra_1 + division_ring
185 class real_field = real_div_algebra + field
187 instantiation real :: real_field
188 begin
190 definition
191   real_scaleR_def [simp]: "scaleR a x = a * x"
193 instance proof
194 qed (simp_all add: algebra_simps)
196 end
198 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
199 proof qed (rule scaleR_left_distrib)
201 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
202 proof qed (rule scaleR_right_distrib)
204 lemma nonzero_inverse_scaleR_distrib:
205   fixes x :: "'a::real_div_algebra" shows
206   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
207 by (rule inverse_unique, simp)
209 lemma inverse_scaleR_distrib:
210   fixes x :: "'a::{real_div_algebra,division_by_zero}"
211   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
212 apply (case_tac "a = 0", simp)
213 apply (case_tac "x = 0", simp)
214 apply (erule (1) nonzero_inverse_scaleR_distrib)
215 done
218 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
219 @{term of_real} *}
221 definition
222   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
223   "of_real r = scaleR r 1"
225 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
226 by (simp add: of_real_def)
228 lemma of_real_0 [simp]: "of_real 0 = 0"
229 by (simp add: of_real_def)
231 lemma of_real_1 [simp]: "of_real 1 = 1"
232 by (simp add: of_real_def)
234 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
235 by (simp add: of_real_def scaleR_left_distrib)
237 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
238 by (simp add: of_real_def)
240 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
241 by (simp add: of_real_def scaleR_left_diff_distrib)
243 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
244 by (simp add: of_real_def mult_commute)
246 lemma nonzero_of_real_inverse:
247   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
248    inverse (of_real x :: 'a::real_div_algebra)"
249 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
251 lemma of_real_inverse [simp]:
252   "of_real (inverse x) =
253    inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
254 by (simp add: of_real_def inverse_scaleR_distrib)
256 lemma nonzero_of_real_divide:
257   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
258    (of_real x / of_real y :: 'a::real_field)"
259 by (simp add: divide_inverse nonzero_of_real_inverse)
261 lemma of_real_divide [simp]:
262   "of_real (x / y) =
263    (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
264 by (simp add: divide_inverse)
266 lemma of_real_power [simp]:
267   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
268 by (induct n) simp_all
270 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
271 by (simp add: of_real_def)
273 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
275 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
276 proof
277   fix r
278   show "of_real r = id r"
279     by (simp add: of_real_def)
280 qed
282 text{*Collapse nested embeddings*}
283 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
284 by (induct n) auto
286 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
287 by (cases z rule: int_diff_cases, simp)
289 lemma of_real_number_of_eq:
290   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
291 by (simp add: number_of_eq)
293 text{*Every real algebra has characteristic zero*}
294 instance real_algebra_1 < ring_char_0
295 proof
296   fix m n :: nat
297   have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
298     by (simp only: of_real_eq_iff of_nat_eq_iff)
299   thus "(of_nat m = (of_nat n::'a)) = (m = n)"
300     by (simp only: of_real_of_nat_eq)
301 qed
303 instance real_field < field_char_0 ..
306 subsection {* The Set of Real Numbers *}
308 definition
309   Reals :: "'a::real_algebra_1 set" where
310   [code del]: "Reals = range of_real"
312 notation (xsymbols)
313   Reals  ("\<real>")
315 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
316 by (simp add: Reals_def)
318 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
319 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
321 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
322 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
324 lemma Reals_number_of [simp]:
325   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
326 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
328 lemma Reals_0 [simp]: "0 \<in> Reals"
329 apply (unfold Reals_def)
330 apply (rule range_eqI)
331 apply (rule of_real_0 [symmetric])
332 done
334 lemma Reals_1 [simp]: "1 \<in> Reals"
335 apply (unfold Reals_def)
336 apply (rule range_eqI)
337 apply (rule of_real_1 [symmetric])
338 done
340 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
341 apply (auto simp add: Reals_def)
342 apply (rule range_eqI)
343 apply (rule of_real_add [symmetric])
344 done
346 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
347 apply (auto simp add: Reals_def)
348 apply (rule range_eqI)
349 apply (rule of_real_minus [symmetric])
350 done
352 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
353 apply (auto simp add: Reals_def)
354 apply (rule range_eqI)
355 apply (rule of_real_diff [symmetric])
356 done
358 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
359 apply (auto simp add: Reals_def)
360 apply (rule range_eqI)
361 apply (rule of_real_mult [symmetric])
362 done
364 lemma nonzero_Reals_inverse:
365   fixes a :: "'a::real_div_algebra"
366   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
367 apply (auto simp add: Reals_def)
368 apply (rule range_eqI)
369 apply (erule nonzero_of_real_inverse [symmetric])
370 done
372 lemma Reals_inverse [simp]:
373   fixes a :: "'a::{real_div_algebra,division_by_zero}"
374   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
375 apply (auto simp add: Reals_def)
376 apply (rule range_eqI)
377 apply (rule of_real_inverse [symmetric])
378 done
380 lemma nonzero_Reals_divide:
381   fixes a b :: "'a::real_field"
382   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
383 apply (auto simp add: Reals_def)
384 apply (rule range_eqI)
385 apply (erule nonzero_of_real_divide [symmetric])
386 done
388 lemma Reals_divide [simp]:
389   fixes a b :: "'a::{real_field,division_by_zero}"
390   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
391 apply (auto simp add: Reals_def)
392 apply (rule range_eqI)
393 apply (rule of_real_divide [symmetric])
394 done
396 lemma Reals_power [simp]:
397   fixes a :: "'a::{real_algebra_1}"
398   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
399 apply (auto simp add: Reals_def)
400 apply (rule range_eqI)
401 apply (rule of_real_power [symmetric])
402 done
404 lemma Reals_cases [cases set: Reals]:
405   assumes "q \<in> \<real>"
406   obtains (of_real) r where "q = of_real r"
407   unfolding Reals_def
408 proof -
409   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
410   then obtain r where "q = of_real r" ..
411   then show thesis ..
412 qed
414 lemma Reals_induct [case_names of_real, induct set: Reals]:
415   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
416   by (rule Reals_cases) auto
419 subsection {* Topological spaces *}
421 class "open" =
422   fixes "open" :: "'a set \<Rightarrow> bool"
424 class topological_space = "open" +
425   assumes open_UNIV [simp, intro]: "open UNIV"
426   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
427   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
428 begin
430 definition
431   closed :: "'a set \<Rightarrow> bool" where
432   "closed S \<longleftrightarrow> open (- S)"
434 lemma open_empty [intro, simp]: "open {}"
435   using open_Union [of "{}"] by simp
437 lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
438   using open_Union [of "{S, T}"] by simp
440 lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
441   unfolding UN_eq by (rule open_Union) auto
443 lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
444   by (induct set: finite) auto
446 lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
447   unfolding Inter_def by (rule open_INT)
449 lemma closed_empty [intro, simp]:  "closed {}"
450   unfolding closed_def by simp
452 lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
453   unfolding closed_def by auto
455 lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
456   unfolding closed_def Inter_def by auto
458 lemma closed_UNIV [intro, simp]: "closed UNIV"
459   unfolding closed_def by simp
461 lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
462   unfolding closed_def by auto
464 lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
465   unfolding closed_def by auto
467 lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
468   by (induct set: finite) auto
470 lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
471   unfolding Union_def by (rule closed_UN)
473 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
474   unfolding closed_def by simp
476 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
477   unfolding closed_def by simp
479 lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
480   unfolding closed_open Diff_eq by (rule open_Int)
482 lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
483   unfolding open_closed Diff_eq by (rule closed_Int)
485 lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
486   unfolding closed_open .
488 lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
489   unfolding open_closed .
491 end
494 subsection {* Metric spaces *}
496 class dist =
497   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
499 class open_dist = "open" + dist +
500   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
502 class metric_space = open_dist +
503   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
504   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
505 begin
507 lemma dist_self [simp]: "dist x x = 0"
508 by simp
510 lemma zero_le_dist [simp]: "0 \<le> dist x y"
511 using dist_triangle2 [of x x y] by simp
513 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
514 by (simp add: less_le)
516 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
517 by (simp add: not_less)
519 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
520 by (simp add: le_less)
522 lemma dist_commute: "dist x y = dist y x"
523 proof (rule order_antisym)
524   show "dist x y \<le> dist y x"
525     using dist_triangle2 [of x y x] by simp
526   show "dist y x \<le> dist x y"
527     using dist_triangle2 [of y x y] by simp
528 qed
530 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
531 using dist_triangle2 [of x z y] by (simp add: dist_commute)
533 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
534 using dist_triangle2 [of x y a] by (simp add: dist_commute)
536 subclass topological_space
537 proof
538   have "\<exists>e::real. 0 < e"
539     by (fast intro: zero_less_one)
540   then show "open UNIV"
541     unfolding open_dist by simp
542 next
543   fix S T assume "open S" "open T"
544   then show "open (S \<inter> T)"
545     unfolding open_dist
546     apply clarify
547     apply (drule (1) bspec)+
548     apply (clarify, rename_tac r s)
549     apply (rule_tac x="min r s" in exI, simp)
550     done
551 next
552   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
553     unfolding open_dist by fast
554 qed
556 end
559 subsection {* Real normed vector spaces *}
561 class norm =
562   fixes norm :: "'a \<Rightarrow> real"
564 class sgn_div_norm = scaleR + norm + sgn +
565   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
567 class dist_norm = dist + norm + minus +
568   assumes dist_norm: "dist x y = norm (x - y)"
570 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
571   assumes norm_ge_zero [simp]: "0 \<le> norm x"
572   and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
573   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
574   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
576 class real_normed_algebra = real_algebra + real_normed_vector +
577   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
579 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
580   assumes norm_one [simp]: "norm 1 = 1"
582 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
583   assumes norm_mult: "norm (x * y) = norm x * norm y"
585 class real_normed_field = real_field + real_normed_div_algebra
587 instance real_normed_div_algebra < real_normed_algebra_1
588 proof
589   fix x y :: 'a
590   show "norm (x * y) \<le> norm x * norm y"
591     by (simp add: norm_mult)
592 next
593   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
594     by (rule norm_mult)
595   thus "norm (1::'a) = 1" by simp
596 qed
598 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
599 by simp
601 lemma zero_less_norm_iff [simp]:
602   fixes x :: "'a::real_normed_vector"
603   shows "(0 < norm x) = (x \<noteq> 0)"
604 by (simp add: order_less_le)
606 lemma norm_not_less_zero [simp]:
607   fixes x :: "'a::real_normed_vector"
608   shows "\<not> norm x < 0"
609 by (simp add: linorder_not_less)
611 lemma norm_le_zero_iff [simp]:
612   fixes x :: "'a::real_normed_vector"
613   shows "(norm x \<le> 0) = (x = 0)"
614 by (simp add: order_le_less)
616 lemma norm_minus_cancel [simp]:
617   fixes x :: "'a::real_normed_vector"
618   shows "norm (- x) = norm x"
619 proof -
620   have "norm (- x) = norm (scaleR (- 1) x)"
621     by (simp only: scaleR_minus_left scaleR_one)
622   also have "\<dots> = \<bar>- 1\<bar> * norm x"
623     by (rule norm_scaleR)
624   finally show ?thesis by simp
625 qed
627 lemma norm_minus_commute:
628   fixes a b :: "'a::real_normed_vector"
629   shows "norm (a - b) = norm (b - a)"
630 proof -
631   have "norm (- (b - a)) = norm (b - a)"
632     by (rule norm_minus_cancel)
633   thus ?thesis by simp
634 qed
636 lemma norm_triangle_ineq2:
637   fixes a b :: "'a::real_normed_vector"
638   shows "norm a - norm b \<le> norm (a - b)"
639 proof -
640   have "norm (a - b + b) \<le> norm (a - b) + norm b"
641     by (rule norm_triangle_ineq)
642   thus ?thesis by simp
643 qed
645 lemma norm_triangle_ineq3:
646   fixes a b :: "'a::real_normed_vector"
647   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
648 apply (subst abs_le_iff)
649 apply auto
650 apply (rule norm_triangle_ineq2)
651 apply (subst norm_minus_commute)
652 apply (rule norm_triangle_ineq2)
653 done
655 lemma norm_triangle_ineq4:
656   fixes a b :: "'a::real_normed_vector"
657   shows "norm (a - b) \<le> norm a + norm b"
658 proof -
659   have "norm (a + - b) \<le> norm a + norm (- b)"
660     by (rule norm_triangle_ineq)
661   thus ?thesis
662     by (simp only: diff_minus norm_minus_cancel)
663 qed
665 lemma norm_diff_ineq:
666   fixes a b :: "'a::real_normed_vector"
667   shows "norm a - norm b \<le> norm (a + b)"
668 proof -
669   have "norm a - norm (- b) \<le> norm (a - - b)"
670     by (rule norm_triangle_ineq2)
671   thus ?thesis by simp
672 qed
674 lemma norm_diff_triangle_ineq:
675   fixes a b c d :: "'a::real_normed_vector"
676   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
677 proof -
678   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
680   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
681     by (rule norm_triangle_ineq)
682   finally show ?thesis .
683 qed
685 lemma abs_norm_cancel [simp]:
686   fixes a :: "'a::real_normed_vector"
687   shows "\<bar>norm a\<bar> = norm a"
688 by (rule abs_of_nonneg [OF norm_ge_zero])
691   fixes x y :: "'a::real_normed_vector"
692   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
693 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
695 lemma norm_mult_less:
696   fixes x y :: "'a::real_normed_algebra"
697   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
698 apply (rule order_le_less_trans [OF norm_mult_ineq])
699 apply (simp add: mult_strict_mono')
700 done
702 lemma norm_of_real [simp]:
703   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
704 unfolding of_real_def by simp
706 lemma norm_number_of [simp]:
707   "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
708     = \<bar>number_of w\<bar>"
709 by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
711 lemma norm_of_int [simp]:
712   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
713 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
715 lemma norm_of_nat [simp]:
716   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
717 apply (subst of_real_of_nat_eq [symmetric])
718 apply (subst norm_of_real, simp)
719 done
721 lemma nonzero_norm_inverse:
722   fixes a :: "'a::real_normed_div_algebra"
723   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
724 apply (rule inverse_unique [symmetric])
725 apply (simp add: norm_mult [symmetric])
726 done
728 lemma norm_inverse:
729   fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
730   shows "norm (inverse a) = inverse (norm a)"
731 apply (case_tac "a = 0", simp)
732 apply (erule nonzero_norm_inverse)
733 done
735 lemma nonzero_norm_divide:
736   fixes a b :: "'a::real_normed_field"
737   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
738 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
740 lemma norm_divide:
741   fixes a b :: "'a::{real_normed_field,division_by_zero}"
742   shows "norm (a / b) = norm a / norm b"
743 by (simp add: divide_inverse norm_mult norm_inverse)
745 lemma norm_power_ineq:
746   fixes x :: "'a::{real_normed_algebra_1}"
747   shows "norm (x ^ n) \<le> norm x ^ n"
748 proof (induct n)
749   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
750 next
751   case (Suc n)
752   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
753     by (rule norm_mult_ineq)
754   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
755     using norm_ge_zero by (rule mult_left_mono)
756   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
757     by simp
758 qed
760 lemma norm_power:
761   fixes x :: "'a::{real_normed_div_algebra}"
762   shows "norm (x ^ n) = norm x ^ n"
763 by (induct n) (simp_all add: norm_mult)
765 text {* Every normed vector space is a metric space. *}
767 instance real_normed_vector < metric_space
768 proof
769   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
770     unfolding dist_norm by simp
771 next
772   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
773     unfolding dist_norm
774     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
775 qed
778 subsection {* Class instances for real numbers *}
780 instantiation real :: real_normed_field
781 begin
783 definition real_norm_def [simp]:
784   "norm r = \<bar>r\<bar>"
786 definition dist_real_def:
787   "dist x y = \<bar>x - y\<bar>"
789 definition open_real_def [code del]:
790   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
792 instance
793 apply (intro_classes, unfold real_norm_def real_scaleR_def)
794 apply (rule dist_real_def)
795 apply (rule open_real_def)
796 apply (simp add: real_sgn_def)
797 apply (rule abs_ge_zero)
798 apply (rule abs_eq_0)
799 apply (rule abs_triangle_ineq)
800 apply (rule abs_mult)
801 apply (rule abs_mult)
802 done
804 end
806 lemma open_real_lessThan [simp]:
807   fixes a :: real shows "open {..<a}"
808 unfolding open_real_def dist_real_def
809 proof (clarify)
810   fix x assume "x < a"
811   hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
812   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
813 qed
815 lemma open_real_greaterThan [simp]:
816   fixes a :: real shows "open {a<..}"
817 unfolding open_real_def dist_real_def
818 proof (clarify)
819   fix x assume "a < x"
820   hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
821   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
822 qed
824 lemma open_real_greaterThanLessThan [simp]:
825   fixes a b :: real shows "open {a<..<b}"
826 proof -
827   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
828   thus "open {a<..<b}" by (simp add: open_Int)
829 qed
831 lemma closed_real_atMost [simp]:
832   fixes a :: real shows "closed {..a}"
833 unfolding closed_open by simp
835 lemma closed_real_atLeast [simp]:
836   fixes a :: real shows "closed {a..}"
837 unfolding closed_open by simp
839 lemma closed_real_atLeastAtMost [simp]:
840   fixes a b :: real shows "closed {a..b}"
841 proof -
842   have "{a..b} = {a..} \<inter> {..b}" by auto
843   thus "closed {a..b}" by (simp add: closed_Int)
844 qed
847 subsection {* Extra type constraints *}
849 text {* Only allow @{term "open"} in class @{text topological_space}. *}
851 setup {* Sign.add_const_constraint
852   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
854 text {* Only allow @{term dist} in class @{text metric_space}. *}
856 setup {* Sign.add_const_constraint
857   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
859 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
861 setup {* Sign.add_const_constraint
862   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
865 subsection {* Sign function *}
867 lemma norm_sgn:
868   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
869 by (simp add: sgn_div_norm)
871 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
872 by (simp add: sgn_div_norm)
874 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
875 by (simp add: sgn_div_norm)
877 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
878 by (simp add: sgn_div_norm)
880 lemma sgn_scaleR:
881   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
882 by (simp add: sgn_div_norm mult_ac)
884 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
885 by (simp add: sgn_div_norm)
887 lemma sgn_of_real:
888   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
889 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
891 lemma sgn_mult:
892   fixes x y :: "'a::real_normed_div_algebra"
893   shows "sgn (x * y) = sgn x * sgn y"
894 by (simp add: sgn_div_norm norm_mult mult_commute)
896 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
897 by (simp add: sgn_div_norm divide_inverse)
899 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
900 unfolding real_sgn_eq by simp
902 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
903 unfolding real_sgn_eq by simp
906 subsection {* Bounded Linear and Bilinear Operators *}
908 locale bounded_linear = additive +
909   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
910   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
911   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
912 begin
914 lemma pos_bounded:
915   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
916 proof -
917   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
918     using bounded by fast
919   show ?thesis
920   proof (intro exI impI conjI allI)
921     show "0 < max 1 K"
922       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
923   next
924     fix x
925     have "norm (f x) \<le> norm x * K" using K .
926     also have "\<dots> \<le> norm x * max 1 K"
927       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
928     finally show "norm (f x) \<le> norm x * max 1 K" .
929   qed
930 qed
932 lemma nonneg_bounded:
933   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
934 proof -
935   from pos_bounded
936   show ?thesis by (auto intro: order_less_imp_le)
937 qed
939 end
941 locale bounded_bilinear =
942   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
943                  \<Rightarrow> 'c::real_normed_vector"
944     (infixl "**" 70)
945   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
946   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
947   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
948   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
949   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
950 begin
952 lemma pos_bounded:
953   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
954 apply (cut_tac bounded, erule exE)
955 apply (rule_tac x="max 1 K" in exI, safe)
956 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
957 apply (drule spec, drule spec, erule order_trans)
958 apply (rule mult_left_mono [OF le_maxI2])
959 apply (intro mult_nonneg_nonneg norm_ge_zero)
960 done
962 lemma nonneg_bounded:
963   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
964 proof -
965   from pos_bounded
966   show ?thesis by (auto intro: order_less_imp_le)
967 qed
969 lemma additive_right: "additive (\<lambda>b. prod a b)"
972 lemma additive_left: "additive (\<lambda>a. prod a b)"
975 lemma zero_left: "prod 0 b = 0"
978 lemma zero_right: "prod a 0 = 0"
981 lemma minus_left: "prod (- a) b = - prod a b"
984 lemma minus_right: "prod a (- b) = - prod a b"
987 lemma diff_left:
988   "prod (a - a') b = prod a b - prod a' b"
991 lemma diff_right:
992   "prod a (b - b') = prod a b - prod a b'"
995 lemma bounded_linear_left:
996   "bounded_linear (\<lambda>a. a ** b)"
997 apply (unfold_locales)
998 apply (rule add_left)
999 apply (rule scaleR_left)
1000 apply (cut_tac bounded, safe)
1001 apply (rule_tac x="norm b * K" in exI)
1002 apply (simp add: mult_ac)
1003 done
1005 lemma bounded_linear_right:
1006   "bounded_linear (\<lambda>b. a ** b)"
1007 apply (unfold_locales)
1008 apply (rule add_right)
1009 apply (rule scaleR_right)
1010 apply (cut_tac bounded, safe)
1011 apply (rule_tac x="norm a * K" in exI)
1012 apply (simp add: mult_ac)
1013 done
1015 lemma prod_diff_prod:
1016   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
1017 by (simp add: diff_left diff_right)
1019 end
1021 interpretation mult:
1022   bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
1023 apply (rule bounded_bilinear.intro)
1024 apply (rule left_distrib)
1025 apply (rule right_distrib)
1026 apply (rule mult_scaleR_left)
1027 apply (rule mult_scaleR_right)
1028 apply (rule_tac x="1" in exI)
1029 apply (simp add: norm_mult_ineq)
1030 done
1032 interpretation mult_left:
1033   bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
1034 by (rule mult.bounded_linear_left)
1036 interpretation mult_right:
1037   bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
1038 by (rule mult.bounded_linear_right)
1040 interpretation divide:
1041   bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
1042 unfolding divide_inverse by (rule mult.bounded_linear_left)
1044 interpretation scaleR: bounded_bilinear "scaleR"
1045 apply (rule bounded_bilinear.intro)
1046 apply (rule scaleR_left_distrib)
1047 apply (rule scaleR_right_distrib)
1048 apply simp
1049 apply (rule scaleR_left_commute)
1050 apply (rule_tac x="1" in exI, simp)
1051 done
1053 interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
1054 by (rule scaleR.bounded_linear_left)
1056 interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x"
1057 by (rule scaleR.bounded_linear_right)
1059 interpretation of_real: bounded_linear "\<lambda>r. of_real r"
1060 unfolding of_real_def by (rule scaleR.bounded_linear_left)
1062 end