src/HOL/Analysis/Finite_Cartesian_Product.thy
 author nipkow Thu Dec 07 15:48:50 2017 +0100 (10 months ago) changeset 67155 9e5b05d54f9d parent 66453 cc19f7ca2ed6 child 67399 eab6ce8368fa permissions -rw-r--r--
canonical name
1 (*  Title:      HOL/Analysis/Finite_Cartesian_Product.thy
2     Author:     Amine Chaieb, University of Cambridge
3 *)
5 section \<open>Definition of finite Cartesian product types.\<close>
7 theory Finite_Cartesian_Product
8 imports
9   Euclidean_Space
10   L2_Norm
11   "HOL-Library.Numeral_Type"
12   "HOL-Library.Countable_Set"
13   "HOL-Library.FuncSet"
14 begin
16 subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
18 typedef ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
19   morphisms vec_nth vec_lambda ..
21 notation
22   vec_nth (infixl "\$" 90) and
23   vec_lambda (binder "\<chi>" 10)
25 (*
26   Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
27   the finite type class write "vec 'b 'n"
28 *)
30 syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
32 parse_translation \<open>
33   let
34     fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
35     fun finite_vec_tr [t, u] =
36       (case Term_Position.strip_positions u of
37         v as Free (x, _) =>
38           if Lexicon.is_tid x then
39             vec t (Syntax.const @{syntax_const "_ofsort"} \$ v \$
40               Syntax.const @{class_syntax finite})
41           else vec t u
42       | _ => vec t u)
43   in
44     [(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
45   end
46 \<close>
48 lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x\$i = y\$i)"
49   by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
51 lemma vec_lambda_beta [simp]: "vec_lambda g \$ i = g i"
54 lemma vec_lambda_unique: "(\<forall>i. f\$i = g i) \<longleftrightarrow> vec_lambda g = f"
55   by (auto simp add: vec_eq_iff)
57 lemma vec_lambda_eta: "(\<chi> i. (g\$i)) = g"
60 subsection \<open>Cardinality of vectors\<close>
62 instance vec :: (finite, finite) finite
63 proof
64   show "finite (UNIV :: ('a, 'b) vec set)"
65   proof (subst bij_betw_finite)
66     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
67       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
68     have "finite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
69       by (intro finite_PiE) auto
70     also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
71       by auto
72     finally show "finite \<dots>" .
73   qed
74 qed
76 lemma countable_PiE:
77   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
78   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
80 instance vec :: (countable, finite) countable
81 proof
82   have "countable (UNIV :: ('a, 'b) vec set)"
83   proof (rule countableI_bij2)
84     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
85       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
86     have "countable (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
87       by (intro countable_PiE) auto
88     also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
89       by auto
90     finally show "countable \<dots>" .
91   qed
92   thus "\<exists>t::('a, 'b) vec \<Rightarrow> nat. inj t"
93     by (auto elim!: countableE)
94 qed
96 lemma infinite_UNIV_vec:
97   assumes "infinite (UNIV :: 'a set)"
98   shows   "infinite (UNIV :: ('a, 'b :: finite) vec set)"
99 proof (subst bij_betw_finite)
100   show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
101     by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
102   have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
103   proof
104     assume "finite ?A"
105     hence "finite ((\<lambda>f. f undefined) ` ?A)"
106       by (rule finite_imageI)
107     also have "(\<lambda>f. f undefined) ` ?A = UNIV"
108       by auto
109     finally show False
110       using \<open>infinite (UNIV :: 'a set)\<close> by contradiction
111   qed
112   also have "?A = Pi UNIV (\<lambda>_. UNIV)"
113     by auto
114   finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
115 qed
117 lemma CARD_vec [simp]:
118   "CARD(('a,'b::finite) vec) = CARD('a) ^ CARD('b)"
119 proof (cases "finite (UNIV :: 'a set)")
120   case True
121   show ?thesis
122   proof (subst bij_betw_same_card)
123     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
124       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
125     have "CARD('a) ^ CARD('b) = card (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
126       (is "_ = card ?A")
127       by (subst card_PiE) (auto simp: prod_constant)
129     also have "?A = Pi UNIV (\<lambda>_. UNIV)"
130       by auto
131     finally show "card \<dots> = CARD('a) ^ CARD('b)" ..
132   qed
135 subsection \<open>Group operations and class instances\<close>
137 instantiation vec :: (zero, finite) zero
138 begin
139   definition "0 \<equiv> (\<chi> i. 0)"
140   instance ..
141 end
143 instantiation vec :: (plus, finite) plus
144 begin
145   definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x\$i + y\$i))"
146   instance ..
147 end
149 instantiation vec :: (minus, finite) minus
150 begin
151   definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x\$i - y\$i))"
152   instance ..
153 end
155 instantiation vec :: (uminus, finite) uminus
156 begin
157   definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x\$i)))"
158   instance ..
159 end
161 lemma zero_index [simp]: "0 \$ i = 0"
162   unfolding zero_vec_def by simp
164 lemma vector_add_component [simp]: "(x + y)\$i = x\$i + y\$i"
165   unfolding plus_vec_def by simp
167 lemma vector_minus_component [simp]: "(x - y)\$i = x\$i - y\$i"
168   unfolding minus_vec_def by simp
170 lemma vector_uminus_component [simp]: "(- x)\$i = - (x\$i)"
171   unfolding uminus_vec_def by simp
180   by standard (simp_all add: vec_eq_iff)
183   by standard (simp add: vec_eq_iff)
186   by standard (simp_all add: vec_eq_iff)
189   by standard (simp_all add: vec_eq_iff diff_diff_eq)
194   by standard (simp_all add: vec_eq_iff)
197   by standard (simp_all add: vec_eq_iff)
200 subsection \<open>Real vector space\<close>
202 instantiation vec :: (real_vector, finite) real_vector
203 begin
205 definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x\$i)))"
207 lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
208   unfolding scaleR_vec_def by simp
210 instance
211   by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
213 end
216 subsection \<open>Topological space\<close>
218 instantiation vec :: (topological_space, finite) topological_space
219 begin
221 definition [code del]:
222   "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
223     (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x\$i \<in> A i) \<and>
224       (\<forall>y. (\<forall>i. y\$i \<in> A i) \<longrightarrow> y \<in> S))"
226 instance proof
227   show "open (UNIV :: ('a ^ 'b) set)"
228     unfolding open_vec_def by auto
229 next
230   fix S T :: "('a ^ 'b) set"
231   assume "open S" "open T" thus "open (S \<inter> T)"
232     unfolding open_vec_def
233     apply clarify
234     apply (drule (1) bspec)+
235     apply (clarify, rename_tac Sa Ta)
236     apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
238     done
239 next
240   fix K :: "('a ^ 'b) set set"
241   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
242     unfolding open_vec_def
243     apply clarify
244     apply (drule (1) bspec)
245     apply (drule (1) bspec)
246     apply clarify
247     apply (rule_tac x=A in exI)
248     apply fast
249     done
250 qed
252 end
254 lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x \$ i \<in> S i}"
255   unfolding open_vec_def by auto
257 lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x \$ i) -` S)"
258   unfolding open_vec_def
259   apply clarify
260   apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
261   done
263 lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x \$ i) -` S)"
264   unfolding closed_open vimage_Compl [symmetric]
265   by (rule open_vimage_vec_nth)
267 lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
268 proof -
269   have "{x. \<forall>i. x \$ i \<in> S i} = (\<Inter>i. (\<lambda>x. x \$ i) -` S i)" by auto
270   thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
271     by (simp add: closed_INT closed_vimage_vec_nth)
272 qed
274 lemma tendsto_vec_nth [tendsto_intros]:
275   assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
276   shows "((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
277 proof (rule topological_tendstoI)
278   fix S assume "open S" "a \$ i \<in> S"
279   then have "open ((\<lambda>y. y \$ i) -` S)" "a \<in> ((\<lambda>y. y \$ i) -` S)"
281   with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y \$ i) -` S) net"
282     by (rule topological_tendstoD)
283   then show "eventually (\<lambda>x. f x \$ i \<in> S) net"
284     by simp
285 qed
287 lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x \$ i) a"
288   unfolding isCont_def by (rule tendsto_vec_nth)
290 lemma vec_tendstoI:
291   assumes "\<And>i. ((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
292   shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
293 proof (rule topological_tendstoI)
294   fix S assume "open S" and "a \<in> S"
295   then obtain A where A: "\<And>i. open (A i)" "\<And>i. a \$ i \<in> A i"
296     and S: "\<And>y. \<forall>i. y \$ i \<in> A i \<Longrightarrow> y \<in> S"
297     unfolding open_vec_def by metis
298   have "\<And>i. eventually (\<lambda>x. f x \$ i \<in> A i) net"
299     using assms A by (rule topological_tendstoD)
300   hence "eventually (\<lambda>x. \<forall>i. f x \$ i \<in> A i) net"
301     by (rule eventually_all_finite)
302   thus "eventually (\<lambda>x. f x \<in> S) net"
303     by (rule eventually_mono, simp add: S)
304 qed
306 lemma tendsto_vec_lambda [tendsto_intros]:
307   assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
308   shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
309   using assms by (simp add: vec_tendstoI)
311 lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
312 proof (rule openI)
313   fix a assume "a \<in> (\<lambda>x. x \$ i) ` S"
314   then obtain z where "a = z \$ i" and "z \<in> S" ..
315   then obtain A where A: "\<forall>i. open (A i) \<and> z \$ i \<in> A i"
316     and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
317     using \<open>open S\<close> unfolding open_vec_def by auto
318   hence "A i \<subseteq> (\<lambda>x. x \$ i) ` S"
319     by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z \$ j" in image_eqI,
320       simp_all)
321   hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x \$ i) ` S"
322     using A \<open>a = z \$ i\<close> by simp
323   then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x \$ i) ` S" by - (rule exI)
324 qed
326 instance vec :: (perfect_space, finite) perfect_space
327 proof
328   fix x :: "'a ^ 'b" show "\<not> open {x}"
329   proof
330     assume "open {x}"
331     hence "\<forall>i. open ((\<lambda>x. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
332     hence "\<forall>i. open {x \$ i}" by simp
333     thus "False" by (simp add: not_open_singleton)
334   qed
335 qed
338 subsection \<open>Metric space\<close>
339 (* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
341 instantiation vec :: (metric_space, finite) dist
342 begin
344 definition
345   "dist x y = L2_set (\<lambda>i. dist (x\$i) (y\$i)) UNIV"
347 instance ..
348 end
350 instantiation vec :: (metric_space, finite) uniformity_dist
351 begin
353 definition [code del]:
354   "(uniformity :: (('a, 'b) vec \<times> ('a, 'b) vec) filter) =
355     (INF e:{0 <..}. principal {(x, y). dist x y < e})"
357 instance
358   by standard (rule uniformity_vec_def)
359 end
361 declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
363 instantiation vec :: (metric_space, finite) metric_space
364 begin
366 lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) \<le> dist x y"
367   unfolding dist_vec_def by (rule member_le_L2_set) simp_all
369 instance proof
370   fix x y :: "'a ^ 'b"
371   show "dist x y = 0 \<longleftrightarrow> x = y"
372     unfolding dist_vec_def
373     by (simp add: L2_set_eq_0_iff vec_eq_iff)
374 next
375   fix x y z :: "'a ^ 'b"
376   show "dist x y \<le> dist x z + dist y z"
377     unfolding dist_vec_def
378     apply (rule order_trans [OF _ L2_set_triangle_ineq])
379     apply (simp add: L2_set_mono dist_triangle2)
380     done
381 next
382   fix S :: "('a ^ 'b) set"
383   have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
384   proof
385     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
386     proof
387       fix x assume "x \<in> S"
388       obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x \$ i \<in> A i"
389         and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
390         using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
391       have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x \$ i) < r \<longrightarrow> y \<in> A i"
392         using A unfolding open_dist by simp
393       hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i)"
394         by (rule finite_set_choice [OF finite])
395       then obtain r where r1: "\<forall>i. 0 < r i"
396         and r2: "\<forall>i y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i" by fast
397       have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
398         by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
399       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
400     qed
401   next
402     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
403     proof (unfold open_vec_def, rule)
404       fix x assume "x \<in> S"
405       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
406         using * by fast
407       define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
408       from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
409         unfolding r_def by simp_all
410       from \<open>0 < e\<close> have e: "e = L2_set r UNIV"
411         unfolding r_def by (simp add: L2_set_constant)
412       define A where "A i = {y. dist (x \$ i) y < r i}" for i
413       have "\<forall>i. open (A i) \<and> x \$ i \<in> A i"
414         unfolding A_def by (simp add: open_ball r)
415       moreover have "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
416         by (simp add: A_def S dist_vec_def e L2_set_strict_mono dist_commute)
417       ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x \$ i \<in> A i) \<and>
418         (\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
419     qed
420   qed
421   show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
422     unfolding * eventually_uniformity_metric
423     by (simp del: split_paired_All add: dist_vec_def dist_commute)
424 qed
426 end
428 lemma Cauchy_vec_nth:
429   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n \$ i)"
430   unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
432 lemma vec_CauchyI:
433   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
434   assumes X: "\<And>i. Cauchy (\<lambda>n. X n \$ i)"
435   shows "Cauchy (\<lambda>n. X n)"
436 proof (rule metric_CauchyI)
437   fix r :: real assume "0 < r"
438   hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
439   define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s)" for i
440   define M where "M = Max (range N)"
441   have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s"
442     using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
443   hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m \$ i) (X n \$ i) < ?s"
444     unfolding N_def by (rule LeastI_ex)
445   hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m \$ i) (X n \$ i) < ?s"
446     unfolding M_def by simp
447   {
448     fix m n :: nat
449     assume "M \<le> m" "M \<le> n"
450     have "dist (X m) (X n) = L2_set (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
451       unfolding dist_vec_def ..
452     also have "\<dots> \<le> sum (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
453       by (rule L2_set_le_sum [OF zero_le_dist])
454     also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
455       by (rule sum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
456     also have "\<dots> = r"
457       by simp
458     finally have "dist (X m) (X n) < r" .
459   }
460   hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
461     by simp
462   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
463 qed
465 instance vec :: (complete_space, finite) complete_space
466 proof
467   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
468   have "\<And>i. (\<lambda>n. X n \$ i) \<longlonglongrightarrow> lim (\<lambda>n. X n \$ i)"
469     using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
470     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
471   hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n \$ i))"
473   then show "convergent X"
474     by (rule convergentI)
475 qed
478 subsection \<open>Normed vector space\<close>
480 instantiation vec :: (real_normed_vector, finite) real_normed_vector
481 begin
483 definition "norm x = L2_set (\<lambda>i. norm (x\$i)) UNIV"
485 definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
487 instance proof
488   fix a :: real and x y :: "'a ^ 'b"
489   show "norm x = 0 \<longleftrightarrow> x = 0"
490     unfolding norm_vec_def
491     by (simp add: L2_set_eq_0_iff vec_eq_iff)
492   show "norm (x + y) \<le> norm x + norm y"
493     unfolding norm_vec_def
494     apply (rule order_trans [OF _ L2_set_triangle_ineq])
495     apply (simp add: L2_set_mono norm_triangle_ineq)
496     done
497   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
498     unfolding norm_vec_def
500   show "sgn x = scaleR (inverse (norm x)) x"
501     by (rule sgn_vec_def)
502   show "dist x y = norm (x - y)"
503     unfolding dist_vec_def norm_vec_def
505 qed
507 end
509 lemma norm_nth_le: "norm (x \$ i) \<le> norm x"
510 unfolding norm_vec_def
511 by (rule member_le_L2_set) simp_all
513 lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x \$ i)"
514 apply standard
516 apply (rule vector_scaleR_component)
517 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
518 done
520 instance vec :: (banach, finite) banach ..
523 subsection \<open>Inner product space\<close>
525 instantiation vec :: (real_inner, finite) real_inner
526 begin
528 definition "inner x y = sum (\<lambda>i. inner (x\$i) (y\$i)) UNIV"
530 instance proof
531   fix r :: real and x y z :: "'a ^ 'b"
532   show "inner x y = inner y x"
533     unfolding inner_vec_def
535   show "inner (x + y) z = inner x z + inner y z"
536     unfolding inner_vec_def
538   show "inner (scaleR r x) y = r * inner x y"
539     unfolding inner_vec_def
541   show "0 \<le> inner x x"
542     unfolding inner_vec_def
544   show "inner x x = 0 \<longleftrightarrow> x = 0"
545     unfolding inner_vec_def
546     by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
547   show "norm x = sqrt (inner x x)"
548     unfolding inner_vec_def norm_vec_def L2_set_def
550 qed
552 end
555 subsection \<open>Euclidean space\<close>
557 text \<open>Vectors pointing along a single axis.\<close>
559 definition "axis k x = (\<chi> i. if i = k then x else 0)"
561 lemma axis_nth [simp]: "axis i x \$ i = x"
562   unfolding axis_def by simp
564 lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
565   unfolding axis_def vec_eq_iff by auto
567 lemma inner_axis_axis:
568   "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
569   unfolding inner_vec_def
570   apply (cases "i = j")
571   apply clarsimp
572   apply (subst sum.remove [of _ j], simp_all)
573   apply (rule sum.neutral, simp add: axis_def)
574   apply (rule sum.neutral, simp add: axis_def)
575   done
577 lemma sum_single:
578   assumes "finite A" and "k \<in> A" and "f k = y"
579   assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
580   shows "(\<Sum>i\<in>A. f i) = y"
581   apply (subst sum.remove [OF assms(1,2)])
582   apply (simp add: sum.neutral assms(3,4))
583   done
585 lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
586   unfolding inner_vec_def
587   apply (rule_tac k=i in sum_single)
588   apply simp_all
590   done
592 instantiation vec :: (euclidean_space, finite) euclidean_space
593 begin
595 definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
597 instance proof
598   show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
599     unfolding Basis_vec_def by simp
600 next
601   show "finite (Basis :: ('a ^ 'b) set)"
602     unfolding Basis_vec_def by simp
603 next
604   fix u v :: "'a ^ 'b"
605   assume "u \<in> Basis" and "v \<in> Basis"
606   thus "inner u v = (if u = v then 1 else 0)"
607     unfolding Basis_vec_def
608     by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
609 next
610   fix x :: "'a ^ 'b"
611   show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
612     unfolding Basis_vec_def
613     by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
614 qed
616 lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"