src/HOL/Library/Product_Vector.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51478 270b21f3ae0a parent 51002 496013a6eb38 child 51642 400ec5ae7f8f permissions -rw-r--r--
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
```     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:
```
```    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 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
```
```    91 unfolding open_prod_def by auto
```
```    92
```
```    93 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
```
```    94 by auto
```
```    95
```
```    96 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
```
```    97 by auto
```
```    98
```
```    99 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
```
```   100 by (simp add: fst_vimage_eq_Times open_Times)
```
```   101
```
```   102 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
```
```   103 by (simp add: snd_vimage_eq_Times open_Times)
```
```   104
```
```   105 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
```
```   106 unfolding closed_open vimage_Compl [symmetric]
```
```   107 by (rule open_vimage_fst)
```
```   108
```
```   109 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
```
```   110 unfolding closed_open vimage_Compl [symmetric]
```
```   111 by (rule open_vimage_snd)
```
```   112
```
```   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
```
```   119
```
```   120 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
```
```   121   unfolding image_def subset_eq by force
```
```   122
```
```   123 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
```
```   124   unfolding image_def subset_eq by force
```
```   125
```
```   126 lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
```
```   127 proof (rule openI)
```
```   128   fix x assume "x \<in> fst ` S"
```
```   129   then obtain y where "(x, y) \<in> S" by auto
```
```   130   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   131     using `open S` unfolding open_prod_def by auto
```
```   132   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
```
```   133   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
```
```   134   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
```
```   135 qed
```
```   136
```
```   137 lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
```
```   138 proof (rule openI)
```
```   139   fix y assume "y \<in> snd ` S"
```
```   140   then obtain x where "(x, y) \<in> S" by auto
```
```   141   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```   142     using `open S` unfolding open_prod_def by auto
```
```   143   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
```
```   144   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
```
```   145   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
```
```   146 qed
```
```   147
```
```   148 subsubsection {* Continuity of operations *}
```
```   149
```
```   150 lemma tendsto_fst [tendsto_intros]:
```
```   151   assumes "(f ---> a) F"
```
```   152   shows "((\<lambda>x. fst (f x)) ---> fst a) F"
```
```   153 proof (rule topological_tendstoI)
```
```   154   fix S assume "open S" and "fst a \<in> S"
```
```   155   then have "open (fst -` S)" and "a \<in> fst -` S"
```
```   156     by (simp_all add: open_vimage_fst)
```
```   157   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
```
```   158     by (rule topological_tendstoD)
```
```   159   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
```
```   160     by simp
```
```   161 qed
```
```   162
```
```   163 lemma tendsto_snd [tendsto_intros]:
```
```   164   assumes "(f ---> a) F"
```
```   165   shows "((\<lambda>x. snd (f x)) ---> snd a) F"
```
```   166 proof (rule topological_tendstoI)
```
```   167   fix S assume "open S" and "snd a \<in> S"
```
```   168   then have "open (snd -` S)" and "a \<in> snd -` S"
```
```   169     by (simp_all add: open_vimage_snd)
```
```   170   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
```
```   171     by (rule topological_tendstoD)
```
```   172   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
```
```   173     by simp
```
```   174 qed
```
```   175
```
```   176 lemma tendsto_Pair [tendsto_intros]:
```
```   177   assumes "(f ---> a) F" and "(g ---> b) F"
```
```   178   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
```
```   179 proof (rule topological_tendstoI)
```
```   180   fix S assume "open S" and "(a, b) \<in> S"
```
```   181   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
```
```   182     unfolding open_prod_def by fast
```
```   183   have "eventually (\<lambda>x. f x \<in> A) F"
```
```   184     using `(f ---> a) F` `open A` `a \<in> A`
```
```   185     by (rule topological_tendstoD)
```
```   186   moreover
```
```   187   have "eventually (\<lambda>x. g x \<in> B) F"
```
```   188     using `(g ---> b) F` `open B` `b \<in> B`
```
```   189     by (rule topological_tendstoD)
```
```   190   ultimately
```
```   191   show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
```
```   192     by (rule eventually_elim2)
```
```   193        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
```
```   194 qed
```
```   195
```
```   196 lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
```
```   197   unfolding continuous_def by (rule tendsto_fst)
```
```   198
```
```   199 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
```
```   200   unfolding continuous_def by (rule tendsto_snd)
```
```   201
```
```   202 lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
```
```   203   unfolding continuous_def by (rule tendsto_Pair)
```
```   204
```
```   205 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
```
```   206   by (fact continuous_fst)
```
```   207
```
```   208 lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
```
```   209   by (fact continuous_snd)
```
```   210
```
```   211 lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
```
```   212   by (fact continuous_Pair)
```
```   213
```
```   214 subsubsection {* Separation axioms *}
```
```   215
```
```   216 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
```
```   217   by (induct x) simp (* TODO: move elsewhere *)
```
```   218
```
```   219 instance prod :: (t0_space, t0_space) t0_space
```
```   220 proof
```
```   221   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
```
```   222   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```   223     by (simp add: prod_eq_iff)
```
```   224   thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
```
```   225     apply (rule disjE)
```
```   226     apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
```
```   227     apply (simp add: open_Times mem_Times_iff)
```
```   228     apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
```
```   229     apply (simp add: open_Times mem_Times_iff)
```
```   230     done
```
```   231 qed
```
```   232
```
```   233 instance prod :: (t1_space, t1_space) t1_space
```
```   234 proof
```
```   235   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
```
```   236   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```   237     by (simp add: prod_eq_iff)
```
```   238   thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```   239     apply (rule disjE)
```
```   240     apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
```
```   241     apply (simp add: open_Times mem_Times_iff)
```
```   242     apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
```
```   243     apply (simp add: open_Times mem_Times_iff)
```
```   244     done
```
```   245 qed
```
```   246
```
```   247 instance prod :: (t2_space, t2_space) t2_space
```
```   248 proof
```
```   249   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
```
```   250   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```   251     by (simp add: prod_eq_iff)
```
```   252   thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```   253     apply (rule disjE)
```
```   254     apply (drule hausdorff, clarify)
```
```   255     apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI)
```
```   256     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
```
```   257     apply (drule hausdorff, clarify)
```
```   258     apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI)
```
```   259     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
```
```   260     done
```
```   261 qed
```
```   262
```
```   263 subsection {* Product is a metric space *}
```
```   264
```
```   265 instantiation prod :: (metric_space, metric_space) metric_space
```
```   266 begin
```
```   267
```
```   268 definition dist_prod_def:
```
```   269   "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
```
```   270
```
```   271 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
```
```   272   unfolding dist_prod_def by simp
```
```   273
```
```   274 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
```
```   275 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
```
```   276
```
```   277 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
```
```   278 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
```
```   279
```
```   280 instance proof
```
```   281   fix x y :: "'a \<times> 'b"
```
```   282   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   283     unfolding dist_prod_def prod_eq_iff by simp
```
```   284 next
```
```   285   fix x y z :: "'a \<times> 'b"
```
```   286   show "dist x y \<le> dist x z + dist y z"
```
```   287     unfolding dist_prod_def
```
```   288     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
```
```   289         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
```
```   290 next
```
```   291   fix S :: "('a \<times> 'b) set"
```
```   292   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   293   proof
```
```   294     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   295     proof
```
```   296       fix x assume "x \<in> S"
```
```   297       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
```
```   298         using `open S` and `x \<in> S` by (rule open_prod_elim)
```
```   299       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
```
```   300         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   301       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
```
```   302         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
```
```   303       let ?e = "min r s"
```
```   304       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
```
```   305       proof (intro allI impI conjI)
```
```   306         show "0 < min r s" by (simp add: r(1) s(1))
```
```   307       next
```
```   308         fix y assume "dist y x < min r s"
```
```   309         hence "dist y x < r" and "dist y x < s"
```
```   310           by simp_all
```
```   311         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
```
```   312           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
```
```   313         hence "fst y \<in> A" and "snd y \<in> B"
```
```   314           by (simp_all add: r(2) s(2))
```
```   315         hence "y \<in> A \<times> B" by (induct y, simp)
```
```   316         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
```
```   317       qed
```
```   318       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
```
```   319     qed
```
```   320   next
```
```   321     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
```
```   322     proof (rule open_prod_intro)
```
```   323       fix x assume "x \<in> S"
```
```   324       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   325         using * by fast
```
```   326       def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
```
```   327       from `0 < e` have "0 < r" and "0 < s"
```
```   328         unfolding r_def s_def by (simp_all add: divide_pos_pos)
```
```   329       from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)"
```
```   330         unfolding r_def s_def by (simp add: power_divide)
```
```   331       def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
```
```   332       have "open A" and "open B"
```
```   333         unfolding A_def B_def by (simp_all add: open_ball)
```
```   334       moreover have "x \<in> A \<times> B"
```
```   335         unfolding A_def B_def mem_Times_iff
```
```   336         using `0 < r` and `0 < s` by simp
```
```   337       moreover have "A \<times> B \<subseteq> S"
```
```   338       proof (clarify)
```
```   339         fix a b assume "a \<in> A" and "b \<in> B"
```
```   340         hence "dist a (fst x) < r" and "dist b (snd x) < s"
```
```   341           unfolding A_def B_def by (simp_all add: dist_commute)
```
```   342         hence "dist (a, b) x < e"
```
```   343           unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)`
```
```   344           by (simp add: add_strict_mono power_strict_mono)
```
```   345         thus "(a, b) \<in> S"
```
```   346           by (simp add: S)
```
```   347       qed
```
```   348       ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
```
```   349     qed
```
```   350   qed
```
```   351 qed
```
```   352
```
```   353 end
```
```   354
```
```   355 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
```
```   356 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
```
```   357
```
```   358 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
```
```   359 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
```
```   360
```
```   361 lemma Cauchy_Pair:
```
```   362   assumes "Cauchy X" and "Cauchy Y"
```
```   363   shows "Cauchy (\<lambda>n. (X n, Y n))"
```
```   364 proof (rule metric_CauchyI)
```
```   365   fix r :: real assume "0 < r"
```
```   366   then have "0 < r / sqrt 2" (is "0 < ?s")
```
```   367     by (simp add: divide_pos_pos)
```
```   368   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
```
```   369     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
```
```   370   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
```
```   371     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
```
```   372   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
```
```   373     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
```
```   374   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
```
```   375 qed
```
```   376
```
```   377 subsection {* Product is a complete metric space *}
```
```   378
```
```   379 instance prod :: (complete_space, complete_space) complete_space
```
```   380 proof
```
```   381   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
```
```   382   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
```
```   383     using Cauchy_fst [OF `Cauchy X`]
```
```   384     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   385   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
```
```   386     using Cauchy_snd [OF `Cauchy X`]
```
```   387     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   388   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
```
```   389     using tendsto_Pair [OF 1 2] by simp
```
```   390   then show "convergent X"
```
```   391     by (rule convergentI)
```
```   392 qed
```
```   393
```
```   394 subsection {* Product is a normed vector space *}
```
```   395
```
```   396 instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   397 begin
```
```   398
```
```   399 definition norm_prod_def:
```
```   400   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
```
```   401
```
```   402 definition sgn_prod_def:
```
```   403   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
```
```   404
```
```   405 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
```
```   406   unfolding norm_prod_def by simp
```
```   407
```
```   408 instance proof
```
```   409   fix r :: real and x y :: "'a \<times> 'b"
```
```   410   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   411     unfolding norm_prod_def
```
```   412     by (simp add: prod_eq_iff)
```
```   413   show "norm (x + y) \<le> norm x + norm y"
```
```   414     unfolding norm_prod_def
```
```   415     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
```
```   416     apply (simp add: add_mono power_mono norm_triangle_ineq)
```
```   417     done
```
```   418   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   419     unfolding norm_prod_def
```
```   420     apply (simp add: power_mult_distrib)
```
```   421     apply (simp add: distrib_left [symmetric])
```
```   422     apply (simp add: real_sqrt_mult_distrib)
```
```   423     done
```
```   424   show "sgn x = scaleR (inverse (norm x)) x"
```
```   425     by (rule sgn_prod_def)
```
```   426   show "dist x y = norm (x - y)"
```
```   427     unfolding dist_prod_def norm_prod_def
```
```   428     by (simp add: dist_norm)
```
```   429 qed
```
```   430
```
```   431 end
```
```   432
```
```   433 instance prod :: (banach, banach) banach ..
```
```   434
```
```   435 subsubsection {* Pair operations are linear *}
```
```   436
```
```   437 lemma bounded_linear_fst: "bounded_linear fst"
```
```   438   using fst_add fst_scaleR
```
```   439   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
```
```   440
```
```   441 lemma bounded_linear_snd: "bounded_linear snd"
```
```   442   using snd_add snd_scaleR
```
```   443   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
```
```   444
```
```   445 text {* TODO: move to NthRoot *}
```
```   446 lemma sqrt_add_le_add_sqrt:
```
```   447   assumes x: "0 \<le> x" and y: "0 \<le> y"
```
```   448   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
```
```   449 apply (rule power2_le_imp_le)
```
```   450 apply (simp add: power2_sum x y)
```
```   451 apply (simp add: mult_nonneg_nonneg x y)
```
```   452 apply (simp add: x y)
```
```   453 done
```
```   454
```
```   455 lemma bounded_linear_Pair:
```
```   456   assumes f: "bounded_linear f"
```
```   457   assumes g: "bounded_linear g"
```
```   458   shows "bounded_linear (\<lambda>x. (f x, g x))"
```
```   459 proof
```
```   460   interpret f: bounded_linear f by fact
```
```   461   interpret g: bounded_linear g by fact
```
```   462   fix x y and r :: real
```
```   463   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
```
```   464     by (simp add: f.add g.add)
```
```   465   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
```
```   466     by (simp add: f.scaleR g.scaleR)
```
```   467   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
```
```   468     using f.pos_bounded by fast
```
```   469   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
```
```   470     using g.pos_bounded by fast
```
```   471   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
```
```   472     apply (rule allI)
```
```   473     apply (simp add: norm_Pair)
```
```   474     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
```
```   475     apply (simp add: distrib_left)
```
```   476     apply (rule add_mono [OF norm_f norm_g])
```
```   477     done
```
```   478   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
```
```   479 qed
```
```   480
```
```   481 subsubsection {* Frechet derivatives involving pairs *}
```
```   482
```
```   483 lemma FDERIV_Pair:
```
```   484   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
```
```   485   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
```
```   486 proof (rule FDERIV_I)
```
```   487   show "bounded_linear (\<lambda>h. (f' h, g' h))"
```
```   488     using f g by (intro bounded_linear_Pair FDERIV_bounded_linear)
```
```   489   let ?Rf = "\<lambda>h. f (x + h) - f x - f' h"
```
```   490   let ?Rg = "\<lambda>h. g (x + h) - g x - g' h"
```
```   491   let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))"
```
```   492   show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0"
```
```   493   proof (rule real_LIM_sandwich_zero)
```
```   494     show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0"
```
```   495       using f g by (intro tendsto_add_zero FDERIV_D)
```
```   496     fix h :: 'a assume "h \<noteq> 0"
```
```   497     thus "0 \<le> norm (?R h) / norm h"
```
```   498       by (simp add: divide_nonneg_pos)
```
```   499     show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h"
```
```   500       unfolding add_divide_distrib [symmetric]
```
```   501       by (simp add: norm_Pair divide_right_mono
```
```   502         order_trans [OF sqrt_add_le_add_sqrt])
```
```   503   qed
```
```   504 qed
```
```   505
```
```   506 subsection {* Product is an inner product space *}
```
```   507
```
```   508 instantiation prod :: (real_inner, real_inner) real_inner
```
```   509 begin
```
```   510
```
```   511 definition inner_prod_def:
```
```   512   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
```
```   513
```
```   514 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
```
```   515   unfolding inner_prod_def by simp
```
```   516
```
```   517 instance proof
```
```   518   fix r :: real
```
```   519   fix x y z :: "'a::real_inner \<times> 'b::real_inner"
```
```   520   show "inner x y = inner y x"
```
```   521     unfolding inner_prod_def
```
```   522     by (simp add: inner_commute)
```
```   523   show "inner (x + y) z = inner x z + inner y z"
```
```   524     unfolding inner_prod_def
```
```   525     by (simp add: inner_add_left)
```
```   526   show "inner (scaleR r x) y = r * inner x y"
```
```   527     unfolding inner_prod_def
```
```   528     by (simp add: distrib_left)
```
```   529   show "0 \<le> inner x x"
```
```   530     unfolding inner_prod_def
```
```   531     by (intro add_nonneg_nonneg inner_ge_zero)
```
```   532   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   533     unfolding inner_prod_def prod_eq_iff
```
```   534     by (simp add: add_nonneg_eq_0_iff)
```
```   535   show "norm x = sqrt (inner x x)"
```
```   536     unfolding norm_prod_def inner_prod_def
```
```   537     by (simp add: power2_norm_eq_inner)
```
```   538 qed
```
```   539
```
```   540 end
```
```   541
```
```   542 end
```