src/HOL/Library/Product_Vector.thy
 author haftmann Tue Feb 19 19:44:10 2013 +0100 (2013-02-19) changeset 51188 9b5bf1a9a710 parent 51002 496013a6eb38 child 51478 270b21f3ae0a permissions -rw-r--r--
dropped spurious left-over from 0a2371e7ced3
1 (*  Title:      HOL/Library/Product_Vector.thy
2     Author:     Brian Huffman
3 *)
5 header {* Cartesian Products as Vector Spaces *}
7 theory Product_Vector
8 imports Inner_Product Product_plus
9 begin
11 subsection {* Product is a real vector space *}
13 instantiation prod :: (real_vector, real_vector) real_vector
14 begin
16 definition scaleR_prod_def:
17   "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
19 lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
20   unfolding scaleR_prod_def by simp
22 lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
23   unfolding scaleR_prod_def by simp
25 lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
26   unfolding scaleR_prod_def by simp
28 instance proof
29   fix a b :: real and x y :: "'a \<times> 'b"
30   show "scaleR a (x + y) = scaleR a x + scaleR a y"
31     by (simp add: prod_eq_iff scaleR_right_distrib)
32   show "scaleR (a + b) x = scaleR a x + scaleR b x"
33     by (simp add: prod_eq_iff scaleR_left_distrib)
34   show "scaleR a (scaleR b x) = scaleR (a * b) x"
35     by (simp add: prod_eq_iff)
36   show "scaleR 1 x = x"
37     by (simp add: prod_eq_iff)
38 qed
40 end
42 subsection {* Product is a topological space *}
44 instantiation prod :: (topological_space, topological_space) topological_space
45 begin
47 definition open_prod_def:
48   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
49     (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
51 lemma open_prod_elim:
52   assumes "open S" and "x \<in> S"
53   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
54 using assms unfolding open_prod_def by fast
56 lemma open_prod_intro:
57   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
58   shows "open S"
59 using assms unfolding open_prod_def by fast
61 instance proof
62   show "open (UNIV :: ('a \<times> 'b) set)"
63     unfolding open_prod_def by auto
64 next
65   fix S T :: "('a \<times> 'b) set"
66   assume "open S" "open T"
67   show "open (S \<inter> T)"
68   proof (rule open_prod_intro)
69     fix x assume x: "x \<in> S \<inter> T"
70     from x have "x \<in> S" by simp
71     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
72       using `open S` and `x \<in> S` by (rule open_prod_elim)
73     from x have "x \<in> T" by simp
74     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
75       using `open T` and `x \<in> T` by (rule open_prod_elim)
76     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
77     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
78       using A B by (auto simp add: open_Int)
79     thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
80       by fast
81   qed
82 next
83   fix K :: "('a \<times> 'b) set set"
84   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
85     unfolding open_prod_def by fast
86 qed
88 end
90 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
91 unfolding open_prod_def by auto
93 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
94 by auto
96 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
97 by auto
99 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
100 by (simp add: fst_vimage_eq_Times open_Times)
102 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
103 by (simp add: snd_vimage_eq_Times open_Times)
105 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
106 unfolding closed_open vimage_Compl [symmetric]
107 by (rule open_vimage_fst)
109 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
110 unfolding closed_open vimage_Compl [symmetric]
111 by (rule open_vimage_snd)
113 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
114 proof -
115   have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
116   thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
117     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
118 qed
120 lemma openI: (* TODO: move *)
121   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
122   shows "open S"
123 proof -
124   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
125   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
126   ultimately show "open S" by simp
127 qed
129 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
130   unfolding image_def subset_eq by force
132 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
133   unfolding image_def subset_eq by force
135 lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
136 proof (rule openI)
137   fix x assume "x \<in> fst ` S"
138   then obtain y where "(x, y) \<in> S" by auto
139   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
140     using `open S` unfolding open_prod_def by auto
141   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
142   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
143   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
144 qed
146 lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
147 proof (rule openI)
148   fix y assume "y \<in> snd ` S"
149   then obtain x where "(x, y) \<in> S" by auto
150   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
151     using `open S` unfolding open_prod_def by auto
152   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
153   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
154   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
155 qed
157 subsubsection {* Continuity of operations *}
159 lemma tendsto_fst [tendsto_intros]:
160   assumes "(f ---> a) F"
161   shows "((\<lambda>x. fst (f x)) ---> fst a) F"
162 proof (rule topological_tendstoI)
163   fix S assume "open S" and "fst a \<in> S"
164   then have "open (fst -` S)" and "a \<in> fst -` S"
165     by (simp_all add: open_vimage_fst)
166   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
167     by (rule topological_tendstoD)
168   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
169     by simp
170 qed
172 lemma tendsto_snd [tendsto_intros]:
173   assumes "(f ---> a) F"
174   shows "((\<lambda>x. snd (f x)) ---> snd a) F"
175 proof (rule topological_tendstoI)
176   fix S assume "open S" and "snd a \<in> S"
177   then have "open (snd -` S)" and "a \<in> snd -` S"
178     by (simp_all add: open_vimage_snd)
179   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
180     by (rule topological_tendstoD)
181   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
182     by simp
183 qed
185 lemma tendsto_Pair [tendsto_intros]:
186   assumes "(f ---> a) F" and "(g ---> b) F"
187   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
188 proof (rule topological_tendstoI)
189   fix S assume "open S" and "(a, b) \<in> S"
190   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
191     unfolding open_prod_def by fast
192   have "eventually (\<lambda>x. f x \<in> A) F"
193     using `(f ---> a) F` `open A` `a \<in> A`
194     by (rule topological_tendstoD)
195   moreover
196   have "eventually (\<lambda>x. g x \<in> B) F"
197     using `(g ---> b) F` `open B` `b \<in> B`
198     by (rule topological_tendstoD)
199   ultimately
200   show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
201     by (rule eventually_elim2)
202        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
203 qed
205 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
206   unfolding isCont_def by (rule tendsto_fst)
208 lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
209   unfolding isCont_def by (rule tendsto_snd)
211 lemma isCont_Pair [simp]:
212   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
213   unfolding isCont_def by (rule tendsto_Pair)
215 subsubsection {* Separation axioms *}
217 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
218   by (induct x) simp (* TODO: move elsewhere *)
220 instance prod :: (t0_space, t0_space) t0_space
221 proof
222   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
223   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
224     by (simp add: prod_eq_iff)
225   thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
226     apply (rule disjE)
227     apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
228     apply (simp add: open_Times mem_Times_iff)
229     apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
230     apply (simp add: open_Times mem_Times_iff)
231     done
232 qed
234 instance prod :: (t1_space, t1_space) t1_space
235 proof
236   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
237   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
238     by (simp add: prod_eq_iff)
239   thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
240     apply (rule disjE)
241     apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
242     apply (simp add: open_Times mem_Times_iff)
243     apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
244     apply (simp add: open_Times mem_Times_iff)
245     done
246 qed
248 instance prod :: (t2_space, t2_space) t2_space
249 proof
250   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
251   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
252     by (simp add: prod_eq_iff)
253   thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
254     apply (rule disjE)
255     apply (drule hausdorff, clarify)
256     apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI)
257     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
258     apply (drule hausdorff, clarify)
259     apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI)
260     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
261     done
262 qed
264 subsection {* Product is a metric space *}
266 instantiation prod :: (metric_space, metric_space) metric_space
267 begin
269 definition dist_prod_def:
270   "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
272 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
273   unfolding dist_prod_def by simp
275 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
276 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
278 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
279 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
281 instance proof
282   fix x y :: "'a \<times> 'b"
283   show "dist x y = 0 \<longleftrightarrow> x = y"
284     unfolding dist_prod_def prod_eq_iff by simp
285 next
286   fix x y z :: "'a \<times> 'b"
287   show "dist x y \<le> dist x z + dist y z"
288     unfolding dist_prod_def
289     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
290         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
291 next
292   fix S :: "('a \<times> 'b) set"
293   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
294   proof
295     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
296     proof
297       fix x assume "x \<in> S"
298       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
299         using `open S` and `x \<in> S` by (rule open_prod_elim)
300       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
301         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
302       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
303         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
304       let ?e = "min r s"
305       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
306       proof (intro allI impI conjI)
307         show "0 < min r s" by (simp add: r(1) s(1))
308       next
309         fix y assume "dist y x < min r s"
310         hence "dist y x < r" and "dist y x < s"
311           by simp_all
312         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
313           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
314         hence "fst y \<in> A" and "snd y \<in> B"
315           by (simp_all add: r(2) s(2))
316         hence "y \<in> A \<times> B" by (induct y, simp)
317         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
318       qed
319       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
320     qed
321   next
322     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
323     proof (rule open_prod_intro)
324       fix x assume "x \<in> S"
325       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
326         using * by fast
327       def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
328       from `0 < e` have "0 < r" and "0 < s"
329         unfolding r_def s_def by (simp_all add: divide_pos_pos)
330       from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)"
331         unfolding r_def s_def by (simp add: power_divide)
332       def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
333       have "open A" and "open B"
334         unfolding A_def B_def by (simp_all add: open_ball)
335       moreover have "x \<in> A \<times> B"
336         unfolding A_def B_def mem_Times_iff
337         using `0 < r` and `0 < s` by simp
338       moreover have "A \<times> B \<subseteq> S"
339       proof (clarify)
340         fix a b assume "a \<in> A" and "b \<in> B"
341         hence "dist a (fst x) < r" and "dist b (snd x) < s"
342           unfolding A_def B_def by (simp_all add: dist_commute)
343         hence "dist (a, b) x < e"
344           unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)`
346         thus "(a, b) \<in> S"
347           by (simp add: S)
348       qed
349       ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
350     qed
351   qed
352 qed
354 end
356 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
357 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
359 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
360 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
362 lemma Cauchy_Pair:
363   assumes "Cauchy X" and "Cauchy Y"
364   shows "Cauchy (\<lambda>n. (X n, Y n))"
365 proof (rule metric_CauchyI)
366   fix r :: real assume "0 < r"
367   then have "0 < r / sqrt 2" (is "0 < ?s")
368     by (simp add: divide_pos_pos)
369   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
370     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
371   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
372     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
373   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
374     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
375   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
376 qed
378 subsection {* Product is a complete metric space *}
380 instance prod :: (complete_space, complete_space) complete_space
381 proof
382   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
383   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
384     using Cauchy_fst [OF `Cauchy X`]
385     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
386   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
387     using Cauchy_snd [OF `Cauchy X`]
388     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
389   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
390     using tendsto_Pair [OF 1 2] by simp
391   then show "convergent X"
392     by (rule convergentI)
393 qed
395 subsection {* Product is a normed vector space *}
397 instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
398 begin
400 definition norm_prod_def:
401   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
403 definition sgn_prod_def:
404   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
406 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
407   unfolding norm_prod_def by simp
409 instance proof
410   fix r :: real and x y :: "'a \<times> 'b"
411   show "norm x = 0 \<longleftrightarrow> x = 0"
412     unfolding norm_prod_def
413     by (simp add: prod_eq_iff)
414   show "norm (x + y) \<le> norm x + norm y"
415     unfolding norm_prod_def
416     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
417     apply (simp add: add_mono power_mono norm_triangle_ineq)
418     done
419   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
420     unfolding norm_prod_def
421     apply (simp add: power_mult_distrib)
422     apply (simp add: distrib_left [symmetric])
423     apply (simp add: real_sqrt_mult_distrib)
424     done
425   show "sgn x = scaleR (inverse (norm x)) x"
426     by (rule sgn_prod_def)
427   show "dist x y = norm (x - y)"
428     unfolding dist_prod_def norm_prod_def
429     by (simp add: dist_norm)
430 qed
432 end
434 instance prod :: (banach, banach) banach ..
436 subsubsection {* Pair operations are linear *}
438 lemma bounded_linear_fst: "bounded_linear fst"
439   using fst_add fst_scaleR
440   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
442 lemma bounded_linear_snd: "bounded_linear snd"
443   using snd_add snd_scaleR
444   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
446 text {* TODO: move to NthRoot *}
448   assumes x: "0 \<le> x" and y: "0 \<le> y"
449   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
450 apply (rule power2_le_imp_le)
451 apply (simp add: power2_sum x y)
452 apply (simp add: mult_nonneg_nonneg x y)
453 apply (simp add: x y)
454 done
456 lemma bounded_linear_Pair:
457   assumes f: "bounded_linear f"
458   assumes g: "bounded_linear g"
459   shows "bounded_linear (\<lambda>x. (f x, g x))"
460 proof
461   interpret f: bounded_linear f by fact
462   interpret g: bounded_linear g by fact
463   fix x y and r :: real
464   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
466   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
467     by (simp add: f.scaleR g.scaleR)
468   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
469     using f.pos_bounded by fast
470   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
471     using g.pos_bounded by fast
472   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
473     apply (rule allI)
474     apply (simp add: norm_Pair)
475     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
476     apply (simp add: distrib_left)
477     apply (rule add_mono [OF norm_f norm_g])
478     done
479   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
480 qed
482 subsubsection {* Frechet derivatives involving pairs *}
484 lemma FDERIV_Pair:
485   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
486   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
487 proof (rule FDERIV_I)
488   show "bounded_linear (\<lambda>h. (f' h, g' h))"
489     using f g by (intro bounded_linear_Pair FDERIV_bounded_linear)
490   let ?Rf = "\<lambda>h. f (x + h) - f x - f' h"
491   let ?Rg = "\<lambda>h. g (x + h) - g x - g' h"
492   let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))"
493   show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0"
494   proof (rule real_LIM_sandwich_zero)
495     show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0"
496       using f g by (intro tendsto_add_zero FDERIV_D)
497     fix h :: 'a assume "h \<noteq> 0"
498     thus "0 \<le> norm (?R h) / norm h"
499       by (simp add: divide_nonneg_pos)
500     show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h"
501       unfolding add_divide_distrib [symmetric]
502       by (simp add: norm_Pair divide_right_mono
504   qed
505 qed
507 subsection {* Product is an inner product space *}
509 instantiation prod :: (real_inner, real_inner) real_inner
510 begin
512 definition inner_prod_def:
513   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
515 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
516   unfolding inner_prod_def by simp
518 instance proof
519   fix r :: real
520   fix x y z :: "'a::real_inner \<times> 'b::real_inner"
521   show "inner x y = inner y x"
522     unfolding inner_prod_def
523     by (simp add: inner_commute)
524   show "inner (x + y) z = inner x z + inner y z"
525     unfolding inner_prod_def
527   show "inner (scaleR r x) y = r * inner x y"
528     unfolding inner_prod_def
529     by (simp add: distrib_left)
530   show "0 \<le> inner x x"
531     unfolding inner_prod_def
532     by (intro add_nonneg_nonneg inner_ge_zero)
533   show "inner x x = 0 \<longleftrightarrow> x = 0"
534     unfolding inner_prod_def prod_eq_iff