src/HOL/Library/Product_Vector.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 56541 0e3abadbef39 child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     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 prod :: (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: 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
```
```    39
```
```    40 end
```
```    41
```
```    42 subsection {* Product is a topological space *}
```
```    43
```
```    44 instantiation prod :: (topological_space, topological_space) topological_space
```
```    45 begin
```
```    46
```
```    47 definition open_prod_def[code del]:
```
```    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)"
```
```    50
```
```    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
```
```    55
```
```    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
```
```    60
```
```    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
```
```    87
```
```    88 end
```
```    89
```
```    90 declare [[code abort: "open::('a::topological_space*'b::topological_space) set \<Rightarrow> bool"]]
```
```    91
```
```    92 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
```
```    93 unfolding open_prod_def by auto
```
```    94
```
```    95 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
```
```    96 by auto
```
```    97
```
```    98 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
```
```    99 by auto
```
```   100
```
```   101 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
```
```   102 by (simp add: fst_vimage_eq_Times open_Times)
```
```   103
```
```   104 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
```
```   105 by (simp add: snd_vimage_eq_Times open_Times)
```
```   106
```
```   107 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
```
```   108 unfolding closed_open vimage_Compl [symmetric]
```
```   109 by (rule open_vimage_fst)
```
```   110
```
```   111 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
```
```   112 unfolding closed_open vimage_Compl [symmetric]
```
```   113 by (rule open_vimage_snd)
```
```   114
```
```   115 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```   116 proof -
```
```   117   have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
```
```   118   thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```   119     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
```
```   120 qed
```
```   121
```
```   122 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
```
```   123   unfolding image_def subset_eq by force
```
```   124
```
```   125 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
```
```   126   unfolding image_def subset_eq by force
```
```   127
```
```   128 lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
```
```   129 proof (rule openI)
```
```   130   fix x assume "x \<in> fst ` S"
```
```   131   then obtain y where "(x, y) \<in> S" by auto
```
```   132   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   133     using `open S` unfolding open_prod_def by auto
```
```   134   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
```
```   135   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
```
```   136   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
```
```   137 qed
```
```   138
```
```   139 lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
```
```   140 proof (rule openI)
```
```   141   fix y assume "y \<in> snd ` S"
```
```   142   then obtain x where "(x, y) \<in> S" by auto
```
```   143   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   144     using `open S` unfolding open_prod_def by auto
```
```   145   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
```
```   146   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
```
```   147   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
```
```   148 qed
```
```   149
```
```   150 subsubsection {* Continuity of operations *}
```
```   151
```
```   152 lemma tendsto_fst [tendsto_intros]:
```
```   153   assumes "(f ---> a) F"
```
```   154   shows "((\<lambda>x. fst (f x)) ---> fst a) F"
```
```   155 proof (rule topological_tendstoI)
```
```   156   fix S assume "open S" and "fst a \<in> S"
```
```   157   then have "open (fst -` S)" and "a \<in> fst -` S"
```
```   158     by (simp_all add: open_vimage_fst)
```
```   159   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
```
```   160     by (rule topological_tendstoD)
```
```   161   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
```
```   162     by simp
```
```   163 qed
```
```   164
```
```   165 lemma tendsto_snd [tendsto_intros]:
```
```   166   assumes "(f ---> a) F"
```
```   167   shows "((\<lambda>x. snd (f x)) ---> snd a) F"
```
```   168 proof (rule topological_tendstoI)
```
```   169   fix S assume "open S" and "snd a \<in> S"
```
```   170   then have "open (snd -` S)" and "a \<in> snd -` S"
```
```   171     by (simp_all add: open_vimage_snd)
```
```   172   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
```
```   173     by (rule topological_tendstoD)
```
```   174   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
```
```   175     by simp
```
```   176 qed
```
```   177
```
```   178 lemma tendsto_Pair [tendsto_intros]:
```
```   179   assumes "(f ---> a) F" and "(g ---> b) F"
```
```   180   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
```
```   181 proof (rule topological_tendstoI)
```
```   182   fix S assume "open S" and "(a, b) \<in> S"
```
```   183   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
```
```   184     unfolding open_prod_def by fast
```
```   185   have "eventually (\<lambda>x. f x \<in> A) F"
```
```   186     using `(f ---> a) F` `open A` `a \<in> A`
```
```   187     by (rule topological_tendstoD)
```
```   188   moreover
```
```   189   have "eventually (\<lambda>x. g x \<in> B) F"
```
```   190     using `(g ---> b) F` `open B` `b \<in> B`
```
```   191     by (rule topological_tendstoD)
```
```   192   ultimately
```
```   193   show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
```
```   194     by (rule eventually_elim2)
```
```   195        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
```
```   196 qed
```
```   197
```
```   198 lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
```
```   199   unfolding continuous_def by (rule tendsto_fst)
```
```   200
```
```   201 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
```
```   202   unfolding continuous_def by (rule tendsto_snd)
```
```   203
```
```   204 lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
```
```   205   unfolding continuous_def by (rule tendsto_Pair)
```
```   206
```
```   207 lemma continuous_on_fst[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))"
```
```   208   unfolding continuous_on_def by (auto intro: tendsto_fst)
```
```   209
```
```   210 lemma continuous_on_snd[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))"
```
```   211   unfolding continuous_on_def by (auto intro: tendsto_snd)
```
```   212
```
```   213 lemma continuous_on_Pair[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))"
```
```   214   unfolding continuous_on_def by (auto intro: tendsto_Pair)
```
```   215
```
```   216 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
```
```   217   by (fact continuous_fst)
```
```   218
```
```   219 lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
```
```   220   by (fact continuous_snd)
```
```   221
```
```   222 lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
```
```   223   by (fact continuous_Pair)
```
```   224
```
```   225 subsubsection {* Separation axioms *}
```
```   226
```
```   227 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
```
```   228   by (induct x) simp (* TODO: move elsewhere *)
```
```   229
```
```   230 instance prod :: (t0_space, t0_space) t0_space
```
```   231 proof
```
```   232   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
```
```   233   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```   234     by (simp add: prod_eq_iff)
```
```   235   thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
```
```   236     by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
```
```   237 qed
```
```   238
```
```   239 instance prod :: (t1_space, t1_space) t1_space
```
```   240 proof
```
```   241   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
```
```   242   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```   243     by (simp add: prod_eq_iff)
```
```   244   thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```   245     by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
```
```   246 qed
```
```   247
```
```   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     by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
```
```   255 qed
```
```   256
```
```   257 subsection {* Product is a metric space *}
```
```   258
```
```   259 instantiation prod :: (metric_space, metric_space) metric_space
```
```   260 begin
```
```   261
```
```   262 definition dist_prod_def[code del]:
```
```   263   "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
```
```   264
```
```   265 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
```
```   266   unfolding dist_prod_def by simp
```
```   267
```
```   268 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
```
```   269   unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
```
```   270
```
```   271 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
```
```   272   unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
```
```   273
```
```   274 instance proof
```
```   275   fix x y :: "'a \<times> 'b"
```
```   276   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   277     unfolding dist_prod_def prod_eq_iff by simp
```
```   278 next
```
```   279   fix x y z :: "'a \<times> 'b"
```
```   280   show "dist x y \<le> dist x z + dist y z"
```
```   281     unfolding dist_prod_def
```
```   282     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
```
```   283         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
```
```   284 next
```
```   285   fix S :: "('a \<times> 'b) set"
```
```   286   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   287   proof
```
```   288     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   289     proof
```
```   290       fix x assume "x \<in> S"
```
```   291       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
```
```   292         using `open S` and `x \<in> S` by (rule open_prod_elim)
```
```   293       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
```
```   294         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   295       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
```
```   296         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   297       let ?e = "min r s"
```
```   298       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
```
```   299       proof (intro allI impI conjI)
```
```   300         show "0 < min r s" by (simp add: r(1) s(1))
```
```   301       next
```
```   302         fix y assume "dist y x < min r s"
```
```   303         hence "dist y x < r" and "dist y x < s"
```
```   304           by simp_all
```
```   305         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
```
```   306           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
```
```   307         hence "fst y \<in> A" and "snd y \<in> B"
```
```   308           by (simp_all add: r(2) s(2))
```
```   309         hence "y \<in> A \<times> B" by (induct y, simp)
```
```   310         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
```
```   311       qed
```
```   312       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
```
```   313     qed
```
```   314   next
```
```   315     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
```
```   316     proof (rule open_prod_intro)
```
```   317       fix x assume "x \<in> S"
```
```   318       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   319         using * by fast
```
```   320       def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
```
```   321       from `0 < e` have "0 < r" and "0 < s"
```
```   322         unfolding r_def s_def by simp_all
```
```   323       from `0 < e` have "e = sqrt (r\<^sup>2 + s\<^sup>2)"
```
```   324         unfolding r_def s_def by (simp add: power_divide)
```
```   325       def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
```
```   326       have "open A" and "open B"
```
```   327         unfolding A_def B_def by (simp_all add: open_ball)
```
```   328       moreover have "x \<in> A \<times> B"
```
```   329         unfolding A_def B_def mem_Times_iff
```
```   330         using `0 < r` and `0 < s` by simp
```
```   331       moreover have "A \<times> B \<subseteq> S"
```
```   332       proof (clarify)
```
```   333         fix a b assume "a \<in> A" and "b \<in> B"
```
```   334         hence "dist a (fst x) < r" and "dist b (snd x) < s"
```
```   335           unfolding A_def B_def by (simp_all add: dist_commute)
```
```   336         hence "dist (a, b) x < e"
```
```   337           unfolding dist_prod_def `e = sqrt (r\<^sup>2 + s\<^sup>2)`
```
```   338           by (simp add: add_strict_mono power_strict_mono)
```
```   339         thus "(a, b) \<in> S"
```
```   340           by (simp add: S)
```
```   341       qed
```
```   342       ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
```
```   343     qed
```
```   344   qed
```
```   345 qed
```
```   346
```
```   347 end
```
```   348
```
```   349 declare [[code abort: "dist::('a::metric_space*'b::metric_space)\<Rightarrow>('a*'b) \<Rightarrow> real"]]
```
```   350
```
```   351 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
```
```   352   unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
```
```   353
```
```   354 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
```
```   355   unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
```
```   356
```
```   357 lemma Cauchy_Pair:
```
```   358   assumes "Cauchy X" and "Cauchy Y"
```
```   359   shows "Cauchy (\<lambda>n. (X n, Y n))"
```
```   360 proof (rule metric_CauchyI)
```
```   361   fix r :: real assume "0 < r"
```
```   362   hence "0 < r / sqrt 2" (is "0 < ?s") by simp
```
```   363   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
```
```   364     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
```
```   365   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
```
```   366     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
```
```   367   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
```
```   368     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   369   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
```
```   370 qed
```
```   371
```
```   372 subsection {* Product is a complete metric space *}
```
```   373
```
```   374 instance prod :: (complete_space, complete_space) complete_space
```
```   375 proof
```
```   376   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
```
```   377   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
```
```   378     using Cauchy_fst [OF `Cauchy X`]
```
```   379     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   380   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
```
```   381     using Cauchy_snd [OF `Cauchy X`]
```
```   382     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   383   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
```
```   384     using tendsto_Pair [OF 1 2] by simp
```
```   385   then show "convergent X"
```
```   386     by (rule convergentI)
```
```   387 qed
```
```   388
```
```   389 subsection {* Product is a normed vector space *}
```
```   390
```
```   391 instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   392 begin
```
```   393
```
```   394 definition norm_prod_def[code del]:
```
```   395   "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
```
```   396
```
```   397 definition sgn_prod_def:
```
```   398   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
```
```   399
```
```   400 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
```
```   401   unfolding norm_prod_def by simp
```
```   402
```
```   403 instance proof
```
```   404   fix r :: real and x y :: "'a \<times> 'b"
```
```   405   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   406     unfolding norm_prod_def
```
```   407     by (simp add: prod_eq_iff)
```
```   408   show "norm (x + y) \<le> norm x + norm y"
```
```   409     unfolding norm_prod_def
```
```   410     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```   411     apply (simp add: add_mono power_mono norm_triangle_ineq)
```
```   412     done
```
```   413   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   414     unfolding norm_prod_def
```
```   415     apply (simp add: power_mult_distrib)
```
```   416     apply (simp add: distrib_left [symmetric])
```
```   417     apply (simp add: real_sqrt_mult_distrib)
```
```   418     done
```
```   419   show "sgn x = scaleR (inverse (norm x)) x"
```
```   420     by (rule sgn_prod_def)
```
```   421   show "dist x y = norm (x - y)"
```
```   422     unfolding dist_prod_def norm_prod_def
```
```   423     by (simp add: dist_norm)
```
```   424 qed
```
```   425
```
```   426 end
```
```   427
```
```   428 declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) \<Rightarrow> real"]]
```
```   429
```
```   430 instance prod :: (banach, banach) banach ..
```
```   431
```
```   432 subsubsection {* Pair operations are linear *}
```
```   433
```
```   434 lemma bounded_linear_fst: "bounded_linear fst"
```
```   435   using fst_add fst_scaleR
```
```   436   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
```
```   437
```
```   438 lemma bounded_linear_snd: "bounded_linear snd"
```
```   439   using snd_add snd_scaleR
```
```   440   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
```
```   441
```
```   442 text {* TODO: move to NthRoot *}
```
```   443 lemma sqrt_add_le_add_sqrt:
```
```   444   assumes x: "0 \<le> x" and y: "0 \<le> y"
```
```   445   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
```
```   446 apply (rule power2_le_imp_le)
```
```   447 apply (simp add: power2_sum x y)
```
```   448 apply (simp add: x y)
```
```   449 done
```
```   450
```
```   451 lemma bounded_linear_Pair:
```
```   452   assumes f: "bounded_linear f"
```
```   453   assumes g: "bounded_linear g"
```
```   454   shows "bounded_linear (\<lambda>x. (f x, g x))"
```
```   455 proof
```
```   456   interpret f: bounded_linear f by fact
```
```   457   interpret g: bounded_linear g by fact
```
```   458   fix x y and r :: real
```
```   459   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
```
```   460     by (simp add: f.add g.add)
```
```   461   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
```
```   462     by (simp add: f.scaleR g.scaleR)
```
```   463   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
```
```   464     using f.pos_bounded by fast
```
```   465   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
```
```   466     using g.pos_bounded by fast
```
```   467   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
```
```   468     apply (rule allI)
```
```   469     apply (simp add: norm_Pair)
```
```   470     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
```
```   471     apply (simp add: distrib_left)
```
```   472     apply (rule add_mono [OF norm_f norm_g])
```
```   473     done
```
```   474   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
```
```   475 qed
```
```   476
```
```   477 subsubsection {* Frechet derivatives involving pairs *}
```
```   478
```
```   479 lemma has_derivative_Pair [derivative_intros]:
```
```   480   assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
```
```   481   shows "((\<lambda>x. (f x, g x)) has_derivative (\<lambda>h. (f' h, g' h))) (at x within s)"
```
```   482 proof (rule has_derivativeI_sandwich[of 1])
```
```   483   show "bounded_linear (\<lambda>h. (f' h, g' h))"
```
```   484     using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
```
```   485   let ?Rf = "\<lambda>y. f y - f x - f' (y - x)"
```
```   486   let ?Rg = "\<lambda>y. g y - g x - g' (y - x)"
```
```   487   let ?R = "\<lambda>y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
```
```   488
```
```   489   show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) ---> 0) (at x within s)"
```
```   490     using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
```
```   491
```
```   492   fix y :: 'a assume "y \<noteq> x"
```
```   493   show "norm (?R y) / norm (y - x) \<le> norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)"
```
```   494     unfolding add_divide_distrib [symmetric]
```
```   495     by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt])
```
```   496 qed simp
```
```   497
```
```   498 lemmas has_derivative_fst [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_fst]
```
```   499 lemmas has_derivative_snd [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_snd]
```
```   500
```
```   501 lemma has_derivative_split [derivative_intros]:
```
```   502   "((\<lambda>p. f (fst p) (snd p)) has_derivative f') F \<Longrightarrow> ((\<lambda>(a, b). f a b) has_derivative f') F"
```
```   503   unfolding split_beta' .
```
```   504
```
```   505 subsection {* Product is an inner product space *}
```
```   506
```
```   507 instantiation prod :: (real_inner, real_inner) real_inner
```
```   508 begin
```
```   509
```
```   510 definition inner_prod_def:
```
```   511   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
```
```   512
```
```   513 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
```
```   514   unfolding inner_prod_def by simp
```
```   515
```
```   516 instance proof
```
```   517   fix r :: real
```
```   518   fix x y z :: "'a::real_inner \<times> 'b::real_inner"
```
```   519   show "inner x y = inner y x"
```
```   520     unfolding inner_prod_def
```
```   521     by (simp add: inner_commute)
```
```   522   show "inner (x + y) z = inner x z + inner y z"
```
```   523     unfolding inner_prod_def
```
```   524     by (simp add: inner_add_left)
```
```   525   show "inner (scaleR r x) y = r * inner x y"
```
```   526     unfolding inner_prod_def
```
```   527     by (simp add: distrib_left)
```
```   528   show "0 \<le> inner x x"
```
```   529     unfolding inner_prod_def
```
```   530     by (intro add_nonneg_nonneg inner_ge_zero)
```
```   531   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   532     unfolding inner_prod_def prod_eq_iff
```
```   533     by (simp add: add_nonneg_eq_0_iff)
```
```   534   show "norm x = sqrt (inner x x)"
```
```   535     unfolding norm_prod_def inner_prod_def
```
```   536     by (simp add: power2_norm_eq_inner)
```
```   537 qed
```
```   538
```
```   539 end
```
```   540
```
```   541 end
```