src/HOL/Library/Product_Vector.thy
 author huffman Tue May 04 10:06:05 2010 -0700 (2010-05-04) changeset 36660 1cc4ab4b7ff7 parent 36332 3ddb2bc07784 child 36661 0a5b7b818d65 permissions -rw-r--r--
make (X ----> L) an abbreviation for (X ---> L) sequentially
```     1 (*  Title:      HOL/Library/Product_Vector.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Cartesian Products as Vector Spaces *}
```
```     6
```
```     7 theory Product_Vector
```
```     8 imports Inner_Product Product_plus
```
```     9 begin
```
```    10
```
```    11 subsection {* Product is a real vector space *}
```
```    12
```
```    13 instantiation "*" :: (real_vector, real_vector) real_vector
```
```    14 begin
```
```    15
```
```    16 definition scaleR_prod_def:
```
```    17   "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
```
```    18
```
```    19 lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
```
```    20   unfolding scaleR_prod_def by simp
```
```    21
```
```    22 lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
```
```    23   unfolding scaleR_prod_def by simp
```
```    24
```
```    25 lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
```
```    26   unfolding scaleR_prod_def by simp
```
```    27
```
```    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: expand_prod_eq scaleR_right_distrib)
```
```    32   show "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```    33     by (simp add: expand_prod_eq scaleR_left_distrib)
```
```    34   show "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```    35     by (simp add: expand_prod_eq)
```
```    36   show "scaleR 1 x = x"
```
```    37     by (simp add: expand_prod_eq)
```
```    38 qed
```
```    39
```
```    40 end
```
```    41
```
```    42 subsection {* Product is a topological space *}
```
```    43
```
```    44 instantiation
```
```    45   "*" :: (topological_space, topological_space) topological_space
```
```    46 begin
```
```    47
```
```    48 definition open_prod_def:
```
```    49   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
```
```    50     (\<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
```
```    52 lemma open_prod_elim:
```
```    53   assumes "open S" and "x \<in> S"
```
```    54   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
```
```    55 using assms unfolding open_prod_def by fast
```
```    56
```
```    57 lemma open_prod_intro:
```
```    58   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"
```
```    59   shows "open S"
```
```    60 using assms unfolding open_prod_def by fast
```
```    61
```
```    62 instance proof
```
```    63   show "open (UNIV :: ('a \<times> 'b) set)"
```
```    64     unfolding open_prod_def by auto
```
```    65 next
```
```    66   fix S T :: "('a \<times> 'b) set"
```
```    67   assume "open S" "open T"
```
```    68   show "open (S \<inter> T)"
```
```    69   proof (rule open_prod_intro)
```
```    70     fix x assume x: "x \<in> S \<inter> T"
```
```    71     from x have "x \<in> S" by simp
```
```    72     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
```
```    73       using `open S` and `x \<in> S` by (rule open_prod_elim)
```
```    74     from x have "x \<in> T" by simp
```
```    75     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
```
```    76       using `open T` and `x \<in> T` by (rule open_prod_elim)
```
```    77     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
```
```    78     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
```
```    79       using A B by (auto simp add: open_Int)
```
```    80     thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
```
```    81       by fast
```
```    82   qed
```
```    83 next
```
```    84   fix K :: "('a \<times> 'b) set set"
```
```    85   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```    86     unfolding open_prod_def by fast
```
```    87 qed
```
```    88
```
```    89 end
```
```    90
```
```    91 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
```
```    92 unfolding open_prod_def by auto
```
```    93
```
```    94 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
```
```    95 by auto
```
```    96
```
```    97 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
```
```    98 by auto
```
```    99
```
```   100 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
```
```   101 by (simp add: fst_vimage_eq_Times open_Times)
```
```   102
```
```   103 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
```
```   104 by (simp add: snd_vimage_eq_Times open_Times)
```
```   105
```
```   106 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
```
```   107 unfolding closed_open vimage_Compl [symmetric]
```
```   108 by (rule open_vimage_fst)
```
```   109
```
```   110 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
```
```   111 unfolding closed_open vimage_Compl [symmetric]
```
```   112 by (rule open_vimage_snd)
```
```   113
```
```   114 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```   115 proof -
```
```   116   have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
```
```   117   thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```   118     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
```
```   119 qed
```
```   120
```
```   121 lemma openI: (* TODO: move *)
```
```   122   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
```
```   123   shows "open S"
```
```   124 proof -
```
```   125   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
```
```   126   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
```
```   127   ultimately show "open S" by simp
```
```   128 qed
```
```   129
```
```   130 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
```
```   131   unfolding image_def subset_eq by force
```
```   132
```
```   133 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
```
```   134   unfolding image_def subset_eq by force
```
```   135
```
```   136 lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
```
```   137 proof (rule openI)
```
```   138   fix x assume "x \<in> fst ` S"
```
```   139   then obtain y where "(x, y) \<in> S" by auto
```
```   140   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   141     using `open S` unfolding open_prod_def by auto
```
```   142   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
```
```   143   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
```
```   144   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
```
```   145 qed
```
```   146
```
```   147 lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
```
```   148 proof (rule openI)
```
```   149   fix y assume "y \<in> snd ` S"
```
```   150   then obtain x where "(x, y) \<in> S" by auto
```
```   151   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   152     using `open S` unfolding open_prod_def by auto
```
```   153   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
```
```   154   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
```
```   155   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
```
```   156 qed
```
```   157
```
```   158 subsection {* Product is a metric space *}
```
```   159
```
```   160 instantiation
```
```   161   "*" :: (metric_space, metric_space) metric_space
```
```   162 begin
```
```   163
```
```   164 definition dist_prod_def:
```
```   165   "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
```
```   166
```
```   167 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
```
```   168   unfolding dist_prod_def by simp
```
```   169
```
```   170 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
```
```   171 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
```
```   172
```
```   173 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
```
```   174 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
```
```   175
```
```   176 instance proof
```
```   177   fix x y :: "'a \<times> 'b"
```
```   178   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   179     unfolding dist_prod_def expand_prod_eq by simp
```
```   180 next
```
```   181   fix x y z :: "'a \<times> 'b"
```
```   182   show "dist x y \<le> dist x z + dist y z"
```
```   183     unfolding dist_prod_def
```
```   184     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
```
```   185         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
```
```   186 next
```
```   187   (* FIXME: long proof! *)
```
```   188   (* Maybe it would be easier to define topological spaces *)
```
```   189   (* in terms of neighborhoods instead of open sets? *)
```
```   190   fix S :: "('a \<times> 'b) set"
```
```   191   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   192   proof
```
```   193     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   194     proof
```
```   195       fix x assume "x \<in> S"
```
```   196       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
```
```   197         using `open S` and `x \<in> S` by (rule open_prod_elim)
```
```   198       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
```
```   199         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   200       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
```
```   201         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   202       let ?e = "min r s"
```
```   203       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
```
```   204       proof (intro allI impI conjI)
```
```   205         show "0 < min r s" by (simp add: r(1) s(1))
```
```   206       next
```
```   207         fix y assume "dist y x < min r s"
```
```   208         hence "dist y x < r" and "dist y x < s"
```
```   209           by simp_all
```
```   210         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
```
```   211           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
```
```   212         hence "fst y \<in> A" and "snd y \<in> B"
```
```   213           by (simp_all add: r(2) s(2))
```
```   214         hence "y \<in> A \<times> B" by (induct y, simp)
```
```   215         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
```
```   216       qed
```
```   217       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
```
```   218     qed
```
```   219   next
```
```   220     assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
```
```   221     unfolding open_prod_def open_dist
```
```   222     apply safe
```
```   223     apply (drule (1) bspec)
```
```   224     apply clarify
```
```   225     apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
```
```   226     apply clarify
```
```   227     apply (rule_tac x="{y. dist y a < r}" in exI)
```
```   228     apply (rule_tac x="{y. dist y b < s}" in exI)
```
```   229     apply (rule conjI)
```
```   230     apply clarify
```
```   231     apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
```
```   232     apply clarify
```
```   233     apply (simp add: less_diff_eq)
```
```   234     apply (erule le_less_trans [OF dist_triangle])
```
```   235     apply (rule conjI)
```
```   236     apply clarify
```
```   237     apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
```
```   238     apply clarify
```
```   239     apply (simp add: less_diff_eq)
```
```   240     apply (erule le_less_trans [OF dist_triangle])
```
```   241     apply (rule conjI)
```
```   242     apply simp
```
```   243     apply (clarify, rename_tac c d)
```
```   244     apply (drule spec, erule mp)
```
```   245     apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
```
```   246     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   247     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
```
```   248     apply (simp add: power_divide)
```
```   249     done
```
```   250   qed
```
```   251 qed
```
```   252
```
```   253 end
```
```   254
```
```   255 subsection {* Continuity of operations *}
```
```   256
```
```   257 lemma tendsto_fst [tendsto_intros]:
```
```   258   assumes "(f ---> a) net"
```
```   259   shows "((\<lambda>x. fst (f x)) ---> fst a) net"
```
```   260 proof (rule topological_tendstoI)
```
```   261   fix S assume "open S" "fst a \<in> S"
```
```   262   then have "open (fst -` S)" "a \<in> fst -` S"
```
```   263     unfolding open_prod_def
```
```   264     apply simp_all
```
```   265     apply clarify
```
```   266     apply (rule exI, erule conjI)
```
```   267     apply (rule exI, rule conjI [OF open_UNIV])
```
```   268     apply auto
```
```   269     done
```
```   270   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
```
```   271     by (rule topological_tendstoD)
```
```   272   then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
```
```   273     by simp
```
```   274 qed
```
```   275
```
```   276 lemma tendsto_snd [tendsto_intros]:
```
```   277   assumes "(f ---> a) net"
```
```   278   shows "((\<lambda>x. snd (f x)) ---> snd a) net"
```
```   279 proof (rule topological_tendstoI)
```
```   280   fix S assume "open S" "snd a \<in> S"
```
```   281   then have "open (snd -` S)" "a \<in> snd -` S"
```
```   282     unfolding open_prod_def
```
```   283     apply simp_all
```
```   284     apply clarify
```
```   285     apply (rule exI, rule conjI [OF open_UNIV])
```
```   286     apply (rule exI, erule conjI)
```
```   287     apply auto
```
```   288     done
```
```   289   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
```
```   290     by (rule topological_tendstoD)
```
```   291   then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
```
```   292     by simp
```
```   293 qed
```
```   294
```
```   295 lemma tendsto_Pair [tendsto_intros]:
```
```   296   assumes "(f ---> a) net" and "(g ---> b) net"
```
```   297   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
```
```   298 proof (rule topological_tendstoI)
```
```   299   fix S assume "open S" "(a, b) \<in> S"
```
```   300   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
```
```   301     unfolding open_prod_def by auto
```
```   302   have "eventually (\<lambda>x. f x \<in> A) net"
```
```   303     using `(f ---> a) net` `open A` `a \<in> A`
```
```   304     by (rule topological_tendstoD)
```
```   305   moreover
```
```   306   have "eventually (\<lambda>x. g x \<in> B) net"
```
```   307     using `(g ---> b) net` `open B` `b \<in> B`
```
```   308     by (rule topological_tendstoD)
```
```   309   ultimately
```
```   310   show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
```
```   311     by (rule eventually_elim2)
```
```   312        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
```
```   313 qed
```
```   314
```
```   315 lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
```
```   316 unfolding LIM_conv_tendsto by (rule tendsto_fst)
```
```   317
```
```   318 lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
```
```   319 unfolding LIM_conv_tendsto by (rule tendsto_snd)
```
```   320
```
```   321 lemma LIM_Pair:
```
```   322   assumes "f -- x --> a" and "g -- x --> b"
```
```   323   shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
```
```   324 using assms unfolding LIM_conv_tendsto
```
```   325 by (rule tendsto_Pair)
```
```   326
```
```   327 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
```
```   328 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
```
```   329
```
```   330 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
```
```   331 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
```
```   332
```
```   333 lemma Cauchy_Pair:
```
```   334   assumes "Cauchy X" and "Cauchy Y"
```
```   335   shows "Cauchy (\<lambda>n. (X n, Y n))"
```
```   336 proof (rule metric_CauchyI)
```
```   337   fix r :: real assume "0 < r"
```
```   338   then have "0 < r / sqrt 2" (is "0 < ?s")
```
```   339     by (simp add: divide_pos_pos)
```
```   340   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
```
```   341     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
```
```   342   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
```
```   343     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
```
```   344   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
```
```   345     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   346   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
```
```   347 qed
```
```   348
```
```   349 lemma isCont_Pair [simp]:
```
```   350   "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
```
```   351   unfolding isCont_def by (rule LIM_Pair)
```
```   352
```
```   353 subsection {* Product is a complete metric space *}
```
```   354
```
```   355 instance "*" :: (complete_space, complete_space) complete_space
```
```   356 proof
```
```   357   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
```
```   358   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
```
```   359     using Cauchy_fst [OF `Cauchy X`]
```
```   360     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   361   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
```
```   362     using Cauchy_snd [OF `Cauchy X`]
```
```   363     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   364   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
```
```   365     using tendsto_Pair [OF 1 2] by simp
```
```   366   then show "convergent X"
```
```   367     by (rule convergentI)
```
```   368 qed
```
```   369
```
```   370 subsection {* Product is a normed vector space *}
```
```   371
```
```   372 instantiation
```
```   373   "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   374 begin
```
```   375
```
```   376 definition norm_prod_def:
```
```   377   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
```
```   378
```
```   379 definition sgn_prod_def:
```
```   380   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
```
```   381
```
```   382 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
```
```   383   unfolding norm_prod_def by simp
```
```   384
```
```   385 instance proof
```
```   386   fix r :: real and x y :: "'a \<times> 'b"
```
```   387   show "0 \<le> norm x"
```
```   388     unfolding norm_prod_def by simp
```
```   389   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   390     unfolding norm_prod_def
```
```   391     by (simp add: expand_prod_eq)
```
```   392   show "norm (x + y) \<le> norm x + norm y"
```
```   393     unfolding norm_prod_def
```
```   394     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```   395     apply (simp add: add_mono power_mono norm_triangle_ineq)
```
```   396     done
```
```   397   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   398     unfolding norm_prod_def
```
```   399     apply (simp add: power_mult_distrib)
```
```   400     apply (simp add: right_distrib [symmetric])
```
```   401     apply (simp add: real_sqrt_mult_distrib)
```
```   402     done
```
```   403   show "sgn x = scaleR (inverse (norm x)) x"
```
```   404     by (rule sgn_prod_def)
```
```   405   show "dist x y = norm (x - y)"
```
```   406     unfolding dist_prod_def norm_prod_def
```
```   407     by (simp add: dist_norm)
```
```   408 qed
```
```   409
```
```   410 end
```
```   411
```
```   412 instance "*" :: (banach, banach) banach ..
```
```   413
```
```   414 subsection {* Product is an inner product space *}
```
```   415
```
```   416 instantiation "*" :: (real_inner, real_inner) real_inner
```
```   417 begin
```
```   418
```
```   419 definition inner_prod_def:
```
```   420   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
```
```   421
```
```   422 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
```
```   423   unfolding inner_prod_def by simp
```
```   424
```
```   425 instance proof
```
```   426   fix r :: real
```
```   427   fix x y z :: "'a::real_inner * 'b::real_inner"
```
```   428   show "inner x y = inner y x"
```
```   429     unfolding inner_prod_def
```
```   430     by (simp add: inner_commute)
```
```   431   show "inner (x + y) z = inner x z + inner y z"
```
```   432     unfolding inner_prod_def
```
```   433     by (simp add: inner_add_left)
```
```   434   show "inner (scaleR r x) y = r * inner x y"
```
```   435     unfolding inner_prod_def
```
```   436     by (simp add: right_distrib)
```
```   437   show "0 \<le> inner x x"
```
```   438     unfolding inner_prod_def
```
```   439     by (intro add_nonneg_nonneg inner_ge_zero)
```
```   440   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   441     unfolding inner_prod_def expand_prod_eq
```
```   442     by (simp add: add_nonneg_eq_0_iff)
```
```   443   show "norm x = sqrt (inner x x)"
```
```   444     unfolding norm_prod_def inner_prod_def
```
```   445     by (simp add: power2_norm_eq_inner)
```
```   446 qed
```
```   447
```
```   448 end
```
```   449
```
```   450 subsection {* Pair operations are linear *}
```
```   451
```
```   452 interpretation fst: bounded_linear fst
```
```   453   apply (unfold_locales)
```
```   454   apply (rule fst_add)
```
```   455   apply (rule fst_scaleR)
```
```   456   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   457   done
```
```   458
```
```   459 interpretation snd: bounded_linear snd
```
```   460   apply (unfold_locales)
```
```   461   apply (rule snd_add)
```
```   462   apply (rule snd_scaleR)
```
```   463   apply (rule_tac x="1" in exI, simp add: norm_Pair)
```
```   464   done
```
```   465
```
```   466 text {* TODO: move to NthRoot *}
```
```   467 lemma sqrt_add_le_add_sqrt:
```
```   468   assumes x: "0 \<le> x" and y: "0 \<le> y"
```
```   469   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
```
```   470 apply (rule power2_le_imp_le)
```
```   471 apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
```
```   472 apply (simp add: mult_nonneg_nonneg x y)
```
```   473 apply (simp add: add_nonneg_nonneg x y)
```
```   474 done
```
```   475
```
```   476 lemma bounded_linear_Pair:
```
```   477   assumes f: "bounded_linear f"
```
```   478   assumes g: "bounded_linear g"
```
```   479   shows "bounded_linear (\<lambda>x. (f x, g x))"
```
```   480 proof
```
```   481   interpret f: bounded_linear f by fact
```
```   482   interpret g: bounded_linear g by fact
```
```   483   fix x y and r :: real
```
```   484   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
```
```   485     by (simp add: f.add g.add)
```
```   486   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
```
```   487     by (simp add: f.scaleR g.scaleR)
```
```   488   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
```
```   489     using f.pos_bounded by fast
```
```   490   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
```
```   491     using g.pos_bounded by fast
```
```   492   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
```
```   493     apply (rule allI)
```
```   494     apply (simp add: norm_Pair)
```
```   495     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
```
```   496     apply (simp add: right_distrib)
```
```   497     apply (rule add_mono [OF norm_f norm_g])
```
```   498     done
```
```   499   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
```
```   500 qed
```
```   501
```
```   502 subsection {* Frechet derivatives involving pairs *}
```
```   503
```
```   504 lemma FDERIV_Pair:
```
```   505   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
```
```   506   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
```
```   507 apply (rule FDERIV_I)
```
```   508 apply (rule bounded_linear_Pair)
```
```   509 apply (rule FDERIV_bounded_linear [OF f])
```
```   510 apply (rule FDERIV_bounded_linear [OF g])
```
```   511 apply (simp add: norm_Pair)
```
```   512 apply (rule real_LIM_sandwich_zero)
```
```   513 apply (rule LIM_add_zero)
```
```   514 apply (rule FDERIV_D [OF f])
```
```   515 apply (rule FDERIV_D [OF g])
```
```   516 apply (rename_tac h)
```
```   517 apply (simp add: divide_nonneg_pos)
```
```   518 apply (rename_tac h)
```
```   519 apply (subst add_divide_distrib [symmetric])
```
```   520 apply (rule divide_right_mono [OF _ norm_ge_zero])
```
```   521 apply (rule order_trans [OF sqrt_add_le_add_sqrt])
```
```   522 apply simp
```
```   523 apply simp
```
```   524 apply simp
```
```   525 done
```
```   526
```
```   527 end
```