src/HOL/Analysis/Finite_Cartesian_Product.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63627 6ddb43c6b711 child 63918 6bf55e6e0b75 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Analysis/Finite_Cartesian_Product.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section \<open>Definition of finite Cartesian product types.\<close>
```
```     6
```
```     7 theory Finite_Cartesian_Product
```
```     8 imports
```
```     9   Euclidean_Space
```
```    10   L2_Norm
```
```    11   "~~/src/HOL/Library/Numeral_Type"
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
```
```    15
```
```    16 typedef ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
```
```    17   morphisms vec_nth vec_lambda ..
```
```    18
```
```    19 notation
```
```    20   vec_nth (infixl "\$" 90) and
```
```    21   vec_lambda (binder "\<chi>" 10)
```
```    22
```
```    23 (*
```
```    24   Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
```
```    25   the finite type class write "vec 'b 'n"
```
```    26 *)
```
```    27
```
```    28 syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
```
```    29
```
```    30 parse_translation \<open>
```
```    31   let
```
```    32     fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
```
```    33     fun finite_vec_tr [t, u] =
```
```    34       (case Term_Position.strip_positions u of
```
```    35         v as Free (x, _) =>
```
```    36           if Lexicon.is_tid x then
```
```    37             vec t (Syntax.const @{syntax_const "_ofsort"} \$ v \$
```
```    38               Syntax.const @{class_syntax finite})
```
```    39           else vec t u
```
```    40       | _ => vec t u)
```
```    41   in
```
```    42     [(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
```
```    43   end
```
```    44 \<close>
```
```    45
```
```    46 lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x\$i = y\$i)"
```
```    47   by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
```
```    48
```
```    49 lemma vec_lambda_beta [simp]: "vec_lambda g \$ i = g i"
```
```    50   by (simp add: vec_lambda_inverse)
```
```    51
```
```    52 lemma vec_lambda_unique: "(\<forall>i. f\$i = g i) \<longleftrightarrow> vec_lambda g = f"
```
```    53   by (auto simp add: vec_eq_iff)
```
```    54
```
```    55 lemma vec_lambda_eta: "(\<chi> i. (g\$i)) = g"
```
```    56   by (simp add: vec_eq_iff)
```
```    57
```
```    58
```
```    59 subsection \<open>Group operations and class instances\<close>
```
```    60
```
```    61 instantiation vec :: (zero, finite) zero
```
```    62 begin
```
```    63   definition "0 \<equiv> (\<chi> i. 0)"
```
```    64   instance ..
```
```    65 end
```
```    66
```
```    67 instantiation vec :: (plus, finite) plus
```
```    68 begin
```
```    69   definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x\$i + y\$i))"
```
```    70   instance ..
```
```    71 end
```
```    72
```
```    73 instantiation vec :: (minus, finite) minus
```
```    74 begin
```
```    75   definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x\$i - y\$i))"
```
```    76   instance ..
```
```    77 end
```
```    78
```
```    79 instantiation vec :: (uminus, finite) uminus
```
```    80 begin
```
```    81   definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x\$i)))"
```
```    82   instance ..
```
```    83 end
```
```    84
```
```    85 lemma zero_index [simp]: "0 \$ i = 0"
```
```    86   unfolding zero_vec_def by simp
```
```    87
```
```    88 lemma vector_add_component [simp]: "(x + y)\$i = x\$i + y\$i"
```
```    89   unfolding plus_vec_def by simp
```
```    90
```
```    91 lemma vector_minus_component [simp]: "(x - y)\$i = x\$i - y\$i"
```
```    92   unfolding minus_vec_def by simp
```
```    93
```
```    94 lemma vector_uminus_component [simp]: "(- x)\$i = - (x\$i)"
```
```    95   unfolding uminus_vec_def by simp
```
```    96
```
```    97 instance vec :: (semigroup_add, finite) semigroup_add
```
```    98   by standard (simp add: vec_eq_iff add.assoc)
```
```    99
```
```   100 instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
```
```   101   by standard (simp add: vec_eq_iff add.commute)
```
```   102
```
```   103 instance vec :: (monoid_add, finite) monoid_add
```
```   104   by standard (simp_all add: vec_eq_iff)
```
```   105
```
```   106 instance vec :: (comm_monoid_add, finite) comm_monoid_add
```
```   107   by standard (simp add: vec_eq_iff)
```
```   108
```
```   109 instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
```
```   110   by standard (simp_all add: vec_eq_iff)
```
```   111
```
```   112 instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
```
```   113   by standard (simp_all add: vec_eq_iff diff_diff_eq)
```
```   114
```
```   115 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
```
```   116
```
```   117 instance vec :: (group_add, finite) group_add
```
```   118   by standard (simp_all add: vec_eq_iff)
```
```   119
```
```   120 instance vec :: (ab_group_add, finite) ab_group_add
```
```   121   by standard (simp_all add: vec_eq_iff)
```
```   122
```
```   123
```
```   124 subsection \<open>Real vector space\<close>
```
```   125
```
```   126 instantiation vec :: (real_vector, finite) real_vector
```
```   127 begin
```
```   128
```
```   129 definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x\$i)))"
```
```   130
```
```   131 lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
```
```   132   unfolding scaleR_vec_def by simp
```
```   133
```
```   134 instance
```
```   135   by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
```
```   136
```
```   137 end
```
```   138
```
```   139
```
```   140 subsection \<open>Topological space\<close>
```
```   141
```
```   142 instantiation vec :: (topological_space, finite) topological_space
```
```   143 begin
```
```   144
```
```   145 definition [code del]:
```
```   146   "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
```
```   147     (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x\$i \<in> A i) \<and>
```
```   148       (\<forall>y. (\<forall>i. y\$i \<in> A i) \<longrightarrow> y \<in> S))"
```
```   149
```
```   150 instance proof
```
```   151   show "open (UNIV :: ('a ^ 'b) set)"
```
```   152     unfolding open_vec_def by auto
```
```   153 next
```
```   154   fix S T :: "('a ^ 'b) set"
```
```   155   assume "open S" "open T" thus "open (S \<inter> T)"
```
```   156     unfolding open_vec_def
```
```   157     apply clarify
```
```   158     apply (drule (1) bspec)+
```
```   159     apply (clarify, rename_tac Sa Ta)
```
```   160     apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
```
```   161     apply (simp add: open_Int)
```
```   162     done
```
```   163 next
```
```   164   fix K :: "('a ^ 'b) set set"
```
```   165   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   166     unfolding open_vec_def
```
```   167     apply clarify
```
```   168     apply (drule (1) bspec)
```
```   169     apply (drule (1) bspec)
```
```   170     apply clarify
```
```   171     apply (rule_tac x=A in exI)
```
```   172     apply fast
```
```   173     done
```
```   174 qed
```
```   175
```
```   176 end
```
```   177
```
```   178 lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x \$ i \<in> S i}"
```
```   179   unfolding open_vec_def by auto
```
```   180
```
```   181 lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x \$ i) -` S)"
```
```   182   unfolding open_vec_def
```
```   183   apply clarify
```
```   184   apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
```
```   185   done
```
```   186
```
```   187 lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x \$ i) -` S)"
```
```   188   unfolding closed_open vimage_Compl [symmetric]
```
```   189   by (rule open_vimage_vec_nth)
```
```   190
```
```   191 lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
```
```   192 proof -
```
```   193   have "{x. \<forall>i. x \$ i \<in> S i} = (\<Inter>i. (\<lambda>x. x \$ i) -` S i)" by auto
```
```   194   thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
```
```   195     by (simp add: closed_INT closed_vimage_vec_nth)
```
```   196 qed
```
```   197
```
```   198 lemma tendsto_vec_nth [tendsto_intros]:
```
```   199   assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
```
```   200   shows "((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
```
```   201 proof (rule topological_tendstoI)
```
```   202   fix S assume "open S" "a \$ i \<in> S"
```
```   203   then have "open ((\<lambda>y. y \$ i) -` S)" "a \<in> ((\<lambda>y. y \$ i) -` S)"
```
```   204     by (simp_all add: open_vimage_vec_nth)
```
```   205   with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y \$ i) -` S) net"
```
```   206     by (rule topological_tendstoD)
```
```   207   then show "eventually (\<lambda>x. f x \$ i \<in> S) net"
```
```   208     by simp
```
```   209 qed
```
```   210
```
```   211 lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x \$ i) a"
```
```   212   unfolding isCont_def by (rule tendsto_vec_nth)
```
```   213
```
```   214 lemma vec_tendstoI:
```
```   215   assumes "\<And>i. ((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
```
```   216   shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
```
```   217 proof (rule topological_tendstoI)
```
```   218   fix S assume "open S" and "a \<in> S"
```
```   219   then obtain A where A: "\<And>i. open (A i)" "\<And>i. a \$ i \<in> A i"
```
```   220     and S: "\<And>y. \<forall>i. y \$ i \<in> A i \<Longrightarrow> y \<in> S"
```
```   221     unfolding open_vec_def by metis
```
```   222   have "\<And>i. eventually (\<lambda>x. f x \$ i \<in> A i) net"
```
```   223     using assms A by (rule topological_tendstoD)
```
```   224   hence "eventually (\<lambda>x. \<forall>i. f x \$ i \<in> A i) net"
```
```   225     by (rule eventually_all_finite)
```
```   226   thus "eventually (\<lambda>x. f x \<in> S) net"
```
```   227     by (rule eventually_mono, simp add: S)
```
```   228 qed
```
```   229
```
```   230 lemma tendsto_vec_lambda [tendsto_intros]:
```
```   231   assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
```
```   232   shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
```
```   233   using assms by (simp add: vec_tendstoI)
```
```   234
```
```   235 lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
```
```   236 proof (rule openI)
```
```   237   fix a assume "a \<in> (\<lambda>x. x \$ i) ` S"
```
```   238   then obtain z where "a = z \$ i" and "z \<in> S" ..
```
```   239   then obtain A where A: "\<forall>i. open (A i) \<and> z \$ i \<in> A i"
```
```   240     and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
```
```   241     using \<open>open S\<close> unfolding open_vec_def by auto
```
```   242   hence "A i \<subseteq> (\<lambda>x. x \$ i) ` S"
```
```   243     by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z \$ j" in image_eqI,
```
```   244       simp_all)
```
```   245   hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x \$ i) ` S"
```
```   246     using A \<open>a = z \$ i\<close> by simp
```
```   247   then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x \$ i) ` S" by - (rule exI)
```
```   248 qed
```
```   249
```
```   250 instance vec :: (perfect_space, finite) perfect_space
```
```   251 proof
```
```   252   fix x :: "'a ^ 'b" show "\<not> open {x}"
```
```   253   proof
```
```   254     assume "open {x}"
```
```   255     hence "\<forall>i. open ((\<lambda>x. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
```
```   256     hence "\<forall>i. open {x \$ i}" by simp
```
```   257     thus "False" by (simp add: not_open_singleton)
```
```   258   qed
```
```   259 qed
```
```   260
```
```   261
```
```   262 subsection \<open>Metric space\<close>
```
```   263 (* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
```
```   264
```
```   265 instantiation vec :: (metric_space, finite) dist
```
```   266 begin
```
```   267
```
```   268 definition
```
```   269   "dist x y = setL2 (\<lambda>i. dist (x\$i) (y\$i)) UNIV"
```
```   270
```
```   271 instance ..
```
```   272 end
```
```   273
```
```   274 instantiation vec :: (metric_space, finite) uniformity_dist
```
```   275 begin
```
```   276
```
```   277 definition [code del]:
```
```   278   "(uniformity :: (('a, 'b) vec \<times> ('a, 'b) vec) filter) =
```
```   279     (INF e:{0 <..}. principal {(x, y). dist x y < e})"
```
```   280
```
```   281 instance
```
```   282   by standard (rule uniformity_vec_def)
```
```   283 end
```
```   284
```
```   285 declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
```
```   286
```
```   287 instantiation vec :: (metric_space, finite) metric_space
```
```   288 begin
```
```   289
```
```   290 lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) \<le> dist x y"
```
```   291   unfolding dist_vec_def by (rule member_le_setL2) simp_all
```
```   292
```
```   293 instance proof
```
```   294   fix x y :: "'a ^ 'b"
```
```   295   show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   296     unfolding dist_vec_def
```
```   297     by (simp add: setL2_eq_0_iff vec_eq_iff)
```
```   298 next
```
```   299   fix x y z :: "'a ^ 'b"
```
```   300   show "dist x y \<le> dist x z + dist y z"
```
```   301     unfolding dist_vec_def
```
```   302     apply (rule order_trans [OF _ setL2_triangle_ineq])
```
```   303     apply (simp add: setL2_mono dist_triangle2)
```
```   304     done
```
```   305 next
```
```   306   fix S :: "('a ^ 'b) set"
```
```   307   have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   308   proof
```
```   309     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   310     proof
```
```   311       fix x assume "x \<in> S"
```
```   312       obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x \$ i \<in> A i"
```
```   313         and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
```
```   314         using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
```
```   315       have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x \$ i) < r \<longrightarrow> y \<in> A i"
```
```   316         using A unfolding open_dist by simp
```
```   317       hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i)"
```
```   318         by (rule finite_set_choice [OF finite])
```
```   319       then obtain r where r1: "\<forall>i. 0 < r i"
```
```   320         and r2: "\<forall>i y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i" by fast
```
```   321       have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
```
```   322         by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
```
```   323       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
```
```   324     qed
```
```   325   next
```
```   326     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
```
```   327     proof (unfold open_vec_def, rule)
```
```   328       fix x assume "x \<in> S"
```
```   329       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
```
```   330         using * by fast
```
```   331       define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
```
```   332       from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
```
```   333         unfolding r_def by simp_all
```
```   334       from \<open>0 < e\<close> have e: "e = setL2 r UNIV"
```
```   335         unfolding r_def by (simp add: setL2_constant)
```
```   336       define A where "A i = {y. dist (x \$ i) y < r i}" for i
```
```   337       have "\<forall>i. open (A i) \<and> x \$ i \<in> A i"
```
```   338         unfolding A_def by (simp add: open_ball r)
```
```   339       moreover have "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
```
```   340         by (simp add: A_def S dist_vec_def e setL2_strict_mono dist_commute)
```
```   341       ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x \$ i \<in> A i) \<and>
```
```   342         (\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
```
```   343     qed
```
```   344   qed
```
```   345   show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
```
```   346     unfolding * eventually_uniformity_metric
```
```   347     by (simp del: split_paired_All add: dist_vec_def dist_commute)
```
```   348 qed
```
```   349
```
```   350 end
```
```   351
```
```   352 lemma Cauchy_vec_nth:
```
```   353   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n \$ i)"
```
```   354   unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
```
```   355
```
```   356 lemma vec_CauchyI:
```
```   357   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
```
```   358   assumes X: "\<And>i. Cauchy (\<lambda>n. X n \$ i)"
```
```   359   shows "Cauchy (\<lambda>n. X n)"
```
```   360 proof (rule metric_CauchyI)
```
```   361   fix r :: real assume "0 < r"
```
```   362   hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
```
```   363   define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s)" for i
```
```   364   define M where "M = Max (range N)"
```
```   365   have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s"
```
```   366     using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
```
```   367   hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m \$ i) (X n \$ i) < ?s"
```
```   368     unfolding N_def by (rule LeastI_ex)
```
```   369   hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m \$ i) (X n \$ i) < ?s"
```
```   370     unfolding M_def by simp
```
```   371   {
```
```   372     fix m n :: nat
```
```   373     assume "M \<le> m" "M \<le> n"
```
```   374     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
```
```   375       unfolding dist_vec_def ..
```
```   376     also have "\<dots> \<le> setsum (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
```
```   377       by (rule setL2_le_setsum [OF zero_le_dist])
```
```   378     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
```
```   379       by (rule setsum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
```
```   380     also have "\<dots> = r"
```
```   381       by simp
```
```   382     finally have "dist (X m) (X n) < r" .
```
```   383   }
```
```   384   hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
```
```   385     by simp
```
```   386   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
```
```   387 qed
```
```   388
```
```   389 instance vec :: (complete_space, finite) complete_space
```
```   390 proof
```
```   391   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
```
```   392   have "\<And>i. (\<lambda>n. X n \$ i) \<longlonglongrightarrow> lim (\<lambda>n. X n \$ i)"
```
```   393     using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
```
```   394     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   395   hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n \$ i))"
```
```   396     by (simp add: vec_tendstoI)
```
```   397   then show "convergent X"
```
```   398     by (rule convergentI)
```
```   399 qed
```
```   400
```
```   401
```
```   402 subsection \<open>Normed vector space\<close>
```
```   403
```
```   404 instantiation vec :: (real_normed_vector, finite) real_normed_vector
```
```   405 begin
```
```   406
```
```   407 definition "norm x = setL2 (\<lambda>i. norm (x\$i)) UNIV"
```
```   408
```
```   409 definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
```
```   410
```
```   411 instance proof
```
```   412   fix a :: real and x y :: "'a ^ 'b"
```
```   413   show "norm x = 0 \<longleftrightarrow> x = 0"
```
```   414     unfolding norm_vec_def
```
```   415     by (simp add: setL2_eq_0_iff vec_eq_iff)
```
```   416   show "norm (x + y) \<le> norm x + norm y"
```
```   417     unfolding norm_vec_def
```
```   418     apply (rule order_trans [OF _ setL2_triangle_ineq])
```
```   419     apply (simp add: setL2_mono norm_triangle_ineq)
```
```   420     done
```
```   421   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   422     unfolding norm_vec_def
```
```   423     by (simp add: setL2_right_distrib)
```
```   424   show "sgn x = scaleR (inverse (norm x)) x"
```
```   425     by (rule sgn_vec_def)
```
```   426   show "dist x y = norm (x - y)"
```
```   427     unfolding dist_vec_def norm_vec_def
```
```   428     by (simp add: dist_norm)
```
```   429 qed
```
```   430
```
```   431 end
```
```   432
```
```   433 lemma norm_nth_le: "norm (x \$ i) \<le> norm x"
```
```   434 unfolding norm_vec_def
```
```   435 by (rule member_le_setL2) simp_all
```
```   436
```
```   437 lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x \$ i)"
```
```   438 apply standard
```
```   439 apply (rule vector_add_component)
```
```   440 apply (rule vector_scaleR_component)
```
```   441 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
```
```   442 done
```
```   443
```
```   444 instance vec :: (banach, finite) banach ..
```
```   445
```
```   446
```
```   447 subsection \<open>Inner product space\<close>
```
```   448
```
```   449 instantiation vec :: (real_inner, finite) real_inner
```
```   450 begin
```
```   451
```
```   452 definition "inner x y = setsum (\<lambda>i. inner (x\$i) (y\$i)) UNIV"
```
```   453
```
```   454 instance proof
```
```   455   fix r :: real and x y z :: "'a ^ 'b"
```
```   456   show "inner x y = inner y x"
```
```   457     unfolding inner_vec_def
```
```   458     by (simp add: inner_commute)
```
```   459   show "inner (x + y) z = inner x z + inner y z"
```
```   460     unfolding inner_vec_def
```
```   461     by (simp add: inner_add_left setsum.distrib)
```
```   462   show "inner (scaleR r x) y = r * inner x y"
```
```   463     unfolding inner_vec_def
```
```   464     by (simp add: setsum_right_distrib)
```
```   465   show "0 \<le> inner x x"
```
```   466     unfolding inner_vec_def
```
```   467     by (simp add: setsum_nonneg)
```
```   468   show "inner x x = 0 \<longleftrightarrow> x = 0"
```
```   469     unfolding inner_vec_def
```
```   470     by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
```
```   471   show "norm x = sqrt (inner x x)"
```
```   472     unfolding inner_vec_def norm_vec_def setL2_def
```
```   473     by (simp add: power2_norm_eq_inner)
```
```   474 qed
```
```   475
```
```   476 end
```
```   477
```
```   478
```
```   479 subsection \<open>Euclidean space\<close>
```
```   480
```
```   481 text \<open>Vectors pointing along a single axis.\<close>
```
```   482
```
```   483 definition "axis k x = (\<chi> i. if i = k then x else 0)"
```
```   484
```
```   485 lemma axis_nth [simp]: "axis i x \$ i = x"
```
```   486   unfolding axis_def by simp
```
```   487
```
```   488 lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
```
```   489   unfolding axis_def vec_eq_iff by auto
```
```   490
```
```   491 lemma inner_axis_axis:
```
```   492   "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
```
```   493   unfolding inner_vec_def
```
```   494   apply (cases "i = j")
```
```   495   apply clarsimp
```
```   496   apply (subst setsum.remove [of _ j], simp_all)
```
```   497   apply (rule setsum.neutral, simp add: axis_def)
```
```   498   apply (rule setsum.neutral, simp add: axis_def)
```
```   499   done
```
```   500
```
```   501 lemma setsum_single:
```
```   502   assumes "finite A" and "k \<in> A" and "f k = y"
```
```   503   assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
```
```   504   shows "(\<Sum>i\<in>A. f i) = y"
```
```   505   apply (subst setsum.remove [OF assms(1,2)])
```
```   506   apply (simp add: setsum.neutral assms(3,4))
```
```   507   done
```
```   508
```
```   509 lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
```
```   510   unfolding inner_vec_def
```
```   511   apply (rule_tac k=i in setsum_single)
```
```   512   apply simp_all
```
```   513   apply (simp add: axis_def)
```
```   514   done
```
```   515
```
```   516 instantiation vec :: (euclidean_space, finite) euclidean_space
```
```   517 begin
```
```   518
```
```   519 definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
```
```   520
```
```   521 instance proof
```
```   522   show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
```
```   523     unfolding Basis_vec_def by simp
```
```   524 next
```
```   525   show "finite (Basis :: ('a ^ 'b) set)"
```
```   526     unfolding Basis_vec_def by simp
```
```   527 next
```
```   528   fix u v :: "'a ^ 'b"
```
```   529   assume "u \<in> Basis" and "v \<in> Basis"
```
```   530   thus "inner u v = (if u = v then 1 else 0)"
```
```   531     unfolding Basis_vec_def
```
```   532     by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
```
```   533 next
```
```   534   fix x :: "'a ^ 'b"
```
```   535   show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
```
```   536     unfolding Basis_vec_def
```
```   537     by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
```
```   538 qed
```
```   539
```
```   540 lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
```
```   541   apply (simp add: Basis_vec_def)
```
```   542   apply (subst card_UN_disjoint)
```
```   543      apply simp
```
```   544     apply simp
```
```   545    apply (auto simp: axis_eq_axis) [1]
```
```   546   apply (subst card_UN_disjoint)
```
```   547      apply (auto simp: axis_eq_axis)
```
```   548   done
```
```   549
```
```   550 end
```
```   551
```
```   552 lemma cart_eq_inner_axis: "a \$ i = inner a (axis i 1)"
```
```   553   by (simp add: inner_axis)
```
```   554
```
```   555 lemma axis_in_Basis: "a \<in> Basis \<Longrightarrow> axis i a \<in> Basis"
```
```   556   by (auto simp add: Basis_vec_def axis_eq_axis)
```
```   557
```
```   558 end
```