src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
 changeset 63627 6ddb43c6b711 parent 63626 44ce6b524ff3 child 63631 2edc8da89edc child 63633 2accfb71e33b
```     1.1 --- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Aug 05 18:34:57 2016 +0200
1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,558 +0,0 @@
1.4 -(*  Title:      HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
1.5 -    Author:     Amine Chaieb, University of Cambridge
1.6 -*)
1.7 -
1.8 -section \<open>Definition of finite Cartesian product types.\<close>
1.9 -
1.10 -theory Finite_Cartesian_Product
1.11 -imports
1.12 -  Euclidean_Space
1.13 -  L2_Norm
1.14 -  "~~/src/HOL/Library/Numeral_Type"
1.15 -begin
1.16 -
1.17 -subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
1.18 -
1.19 -typedef ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
1.20 -  morphisms vec_nth vec_lambda ..
1.21 -
1.22 -notation
1.23 -  vec_nth (infixl "\$" 90) and
1.24 -  vec_lambda (binder "\<chi>" 10)
1.25 -
1.26 -(*
1.27 -  Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
1.28 -  the finite type class write "vec 'b 'n"
1.29 -*)
1.30 -
1.31 -syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
1.32 -
1.33 -parse_translation \<open>
1.34 -  let
1.35 -    fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
1.36 -    fun finite_vec_tr [t, u] =
1.37 -      (case Term_Position.strip_positions u of
1.38 -        v as Free (x, _) =>
1.39 -          if Lexicon.is_tid x then
1.40 -            vec t (Syntax.const @{syntax_const "_ofsort"} \$ v \$
1.41 -              Syntax.const @{class_syntax finite})
1.42 -          else vec t u
1.43 -      | _ => vec t u)
1.44 -  in
1.45 -    [(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
1.46 -  end
1.47 -\<close>
1.48 -
1.49 -lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x\$i = y\$i)"
1.50 -  by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
1.51 -
1.52 -lemma vec_lambda_beta [simp]: "vec_lambda g \$ i = g i"
1.53 -  by (simp add: vec_lambda_inverse)
1.54 -
1.55 -lemma vec_lambda_unique: "(\<forall>i. f\$i = g i) \<longleftrightarrow> vec_lambda g = f"
1.56 -  by (auto simp add: vec_eq_iff)
1.57 -
1.58 -lemma vec_lambda_eta: "(\<chi> i. (g\$i)) = g"
1.59 -  by (simp add: vec_eq_iff)
1.60 -
1.61 -
1.62 -subsection \<open>Group operations and class instances\<close>
1.63 -
1.64 -instantiation vec :: (zero, finite) zero
1.65 -begin
1.66 -  definition "0 \<equiv> (\<chi> i. 0)"
1.67 -  instance ..
1.68 -end
1.69 -
1.70 -instantiation vec :: (plus, finite) plus
1.71 -begin
1.72 -  definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x\$i + y\$i))"
1.73 -  instance ..
1.74 -end
1.75 -
1.76 -instantiation vec :: (minus, finite) minus
1.77 -begin
1.78 -  definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x\$i - y\$i))"
1.79 -  instance ..
1.80 -end
1.81 -
1.82 -instantiation vec :: (uminus, finite) uminus
1.83 -begin
1.84 -  definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x\$i)))"
1.85 -  instance ..
1.86 -end
1.87 -
1.88 -lemma zero_index [simp]: "0 \$ i = 0"
1.89 -  unfolding zero_vec_def by simp
1.90 -
1.91 -lemma vector_add_component [simp]: "(x + y)\$i = x\$i + y\$i"
1.92 -  unfolding plus_vec_def by simp
1.93 -
1.94 -lemma vector_minus_component [simp]: "(x - y)\$i = x\$i - y\$i"
1.95 -  unfolding minus_vec_def by simp
1.96 -
1.97 -lemma vector_uminus_component [simp]: "(- x)\$i = - (x\$i)"
1.98 -  unfolding uminus_vec_def by simp
1.99 -
1.102 -
1.105 -
1.107 -  by standard (simp_all add: vec_eq_iff)
1.108 -
1.110 -  by standard (simp add: vec_eq_iff)
1.111 -
1.113 -  by standard (simp_all add: vec_eq_iff)
1.114 -
1.116 -  by standard (simp_all add: vec_eq_iff diff_diff_eq)
1.117 -
1.119 -
1.121 -  by standard (simp_all add: vec_eq_iff)
1.122 -
1.124 -  by standard (simp_all add: vec_eq_iff)
1.125 -
1.126 -
1.127 -subsection \<open>Real vector space\<close>
1.128 -
1.129 -instantiation vec :: (real_vector, finite) real_vector
1.130 -begin
1.131 -
1.132 -definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x\$i)))"
1.133 -
1.134 -lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
1.135 -  unfolding scaleR_vec_def by simp
1.136 -
1.137 -instance
1.138 -  by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
1.139 -
1.140 -end
1.141 -
1.142 -
1.143 -subsection \<open>Topological space\<close>
1.144 -
1.145 -instantiation vec :: (topological_space, finite) topological_space
1.146 -begin
1.147 -
1.148 -definition [code del]:
1.149 -  "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
1.150 -    (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x\$i \<in> A i) \<and>
1.151 -      (\<forall>y. (\<forall>i. y\$i \<in> A i) \<longrightarrow> y \<in> S))"
1.152 -
1.153 -instance proof
1.154 -  show "open (UNIV :: ('a ^ 'b) set)"
1.155 -    unfolding open_vec_def by auto
1.156 -next
1.157 -  fix S T :: "('a ^ 'b) set"
1.158 -  assume "open S" "open T" thus "open (S \<inter> T)"
1.159 -    unfolding open_vec_def
1.160 -    apply clarify
1.161 -    apply (drule (1) bspec)+
1.162 -    apply (clarify, rename_tac Sa Ta)
1.163 -    apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
1.164 -    apply (simp add: open_Int)
1.165 -    done
1.166 -next
1.167 -  fix K :: "('a ^ 'b) set set"
1.168 -  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
1.169 -    unfolding open_vec_def
1.170 -    apply clarify
1.171 -    apply (drule (1) bspec)
1.172 -    apply (drule (1) bspec)
1.173 -    apply clarify
1.174 -    apply (rule_tac x=A in exI)
1.175 -    apply fast
1.176 -    done
1.177 -qed
1.178 -
1.179 -end
1.180 -
1.181 -lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x \$ i \<in> S i}"
1.182 -  unfolding open_vec_def by auto
1.183 -
1.184 -lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x \$ i) -` S)"
1.185 -  unfolding open_vec_def
1.186 -  apply clarify
1.187 -  apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
1.188 -  done
1.189 -
1.190 -lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x \$ i) -` S)"
1.191 -  unfolding closed_open vimage_Compl [symmetric]
1.192 -  by (rule open_vimage_vec_nth)
1.193 -
1.194 -lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
1.195 -proof -
1.196 -  have "{x. \<forall>i. x \$ i \<in> S i} = (\<Inter>i. (\<lambda>x. x \$ i) -` S i)" by auto
1.197 -  thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
1.198 -    by (simp add: closed_INT closed_vimage_vec_nth)
1.199 -qed
1.200 -
1.201 -lemma tendsto_vec_nth [tendsto_intros]:
1.202 -  assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
1.203 -  shows "((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
1.204 -proof (rule topological_tendstoI)
1.205 -  fix S assume "open S" "a \$ i \<in> S"
1.206 -  then have "open ((\<lambda>y. y \$ i) -` S)" "a \<in> ((\<lambda>y. y \$ i) -` S)"
1.207 -    by (simp_all add: open_vimage_vec_nth)
1.208 -  with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y \$ i) -` S) net"
1.209 -    by (rule topological_tendstoD)
1.210 -  then show "eventually (\<lambda>x. f x \$ i \<in> S) net"
1.211 -    by simp
1.212 -qed
1.213 -
1.214 -lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x \$ i) a"
1.215 -  unfolding isCont_def by (rule tendsto_vec_nth)
1.216 -
1.217 -lemma vec_tendstoI:
1.218 -  assumes "\<And>i. ((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
1.219 -  shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
1.220 -proof (rule topological_tendstoI)
1.221 -  fix S assume "open S" and "a \<in> S"
1.222 -  then obtain A where A: "\<And>i. open (A i)" "\<And>i. a \$ i \<in> A i"
1.223 -    and S: "\<And>y. \<forall>i. y \$ i \<in> A i \<Longrightarrow> y \<in> S"
1.224 -    unfolding open_vec_def by metis
1.225 -  have "\<And>i. eventually (\<lambda>x. f x \$ i \<in> A i) net"
1.226 -    using assms A by (rule topological_tendstoD)
1.227 -  hence "eventually (\<lambda>x. \<forall>i. f x \$ i \<in> A i) net"
1.228 -    by (rule eventually_all_finite)
1.229 -  thus "eventually (\<lambda>x. f x \<in> S) net"
1.230 -    by (rule eventually_mono, simp add: S)
1.231 -qed
1.232 -
1.233 -lemma tendsto_vec_lambda [tendsto_intros]:
1.234 -  assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
1.235 -  shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
1.236 -  using assms by (simp add: vec_tendstoI)
1.237 -
1.238 -lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
1.239 -proof (rule openI)
1.240 -  fix a assume "a \<in> (\<lambda>x. x \$ i) ` S"
1.241 -  then obtain z where "a = z \$ i" and "z \<in> S" ..
1.242 -  then obtain A where A: "\<forall>i. open (A i) \<and> z \$ i \<in> A i"
1.243 -    and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
1.244 -    using \<open>open S\<close> unfolding open_vec_def by auto
1.245 -  hence "A i \<subseteq> (\<lambda>x. x \$ i) ` S"
1.246 -    by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z \$ j" in image_eqI,
1.247 -      simp_all)
1.248 -  hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x \$ i) ` S"
1.249 -    using A \<open>a = z \$ i\<close> by simp
1.250 -  then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x \$ i) ` S" by - (rule exI)
1.251 -qed
1.252 -
1.253 -instance vec :: (perfect_space, finite) perfect_space
1.254 -proof
1.255 -  fix x :: "'a ^ 'b" show "\<not> open {x}"
1.256 -  proof
1.257 -    assume "open {x}"
1.258 -    hence "\<forall>i. open ((\<lambda>x. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
1.259 -    hence "\<forall>i. open {x \$ i}" by simp
1.260 -    thus "False" by (simp add: not_open_singleton)
1.261 -  qed
1.262 -qed
1.263 -
1.264 -
1.265 -subsection \<open>Metric space\<close>
1.266 -(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
1.267 -
1.268 -instantiation vec :: (metric_space, finite) dist
1.269 -begin
1.270 -
1.271 -definition
1.272 -  "dist x y = setL2 (\<lambda>i. dist (x\$i) (y\$i)) UNIV"
1.273 -
1.274 -instance ..
1.275 -end
1.276 -
1.277 -instantiation vec :: (metric_space, finite) uniformity_dist
1.278 -begin
1.279 -
1.280 -definition [code del]:
1.281 -  "(uniformity :: (('a, 'b) vec \<times> ('a, 'b) vec) filter) =
1.282 -    (INF e:{0 <..}. principal {(x, y). dist x y < e})"
1.283 -
1.284 -instance
1.285 -  by standard (rule uniformity_vec_def)
1.286 -end
1.287 -
1.288 -declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
1.289 -
1.290 -instantiation vec :: (metric_space, finite) metric_space
1.291 -begin
1.292 -
1.293 -lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) \<le> dist x y"
1.294 -  unfolding dist_vec_def by (rule member_le_setL2) simp_all
1.295 -
1.296 -instance proof
1.297 -  fix x y :: "'a ^ 'b"
1.298 -  show "dist x y = 0 \<longleftrightarrow> x = y"
1.299 -    unfolding dist_vec_def
1.300 -    by (simp add: setL2_eq_0_iff vec_eq_iff)
1.301 -next
1.302 -  fix x y z :: "'a ^ 'b"
1.303 -  show "dist x y \<le> dist x z + dist y z"
1.304 -    unfolding dist_vec_def
1.305 -    apply (rule order_trans [OF _ setL2_triangle_ineq])
1.306 -    apply (simp add: setL2_mono dist_triangle2)
1.307 -    done
1.308 -next
1.309 -  fix S :: "('a ^ 'b) set"
1.310 -  have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
1.311 -  proof
1.312 -    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
1.313 -    proof
1.314 -      fix x assume "x \<in> S"
1.315 -      obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x \$ i \<in> A i"
1.316 -        and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
1.317 -        using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
1.318 -      have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x \$ i) < r \<longrightarrow> y \<in> A i"
1.319 -        using A unfolding open_dist by simp
1.320 -      hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i)"
1.321 -        by (rule finite_set_choice [OF finite])
1.322 -      then obtain r where r1: "\<forall>i. 0 < r i"
1.323 -        and r2: "\<forall>i y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i" by fast
1.324 -      have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
1.325 -        by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
1.326 -      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
1.327 -    qed
1.328 -  next
1.329 -    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
1.330 -    proof (unfold open_vec_def, rule)
1.331 -      fix x assume "x \<in> S"
1.332 -      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
1.333 -        using * by fast
1.334 -      define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
1.335 -      from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
1.336 -        unfolding r_def by simp_all
1.337 -      from \<open>0 < e\<close> have e: "e = setL2 r UNIV"
1.338 -        unfolding r_def by (simp add: setL2_constant)
1.339 -      define A where "A i = {y. dist (x \$ i) y < r i}" for i
1.340 -      have "\<forall>i. open (A i) \<and> x \$ i \<in> A i"
1.341 -        unfolding A_def by (simp add: open_ball r)
1.342 -      moreover have "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
1.343 -        by (simp add: A_def S dist_vec_def e setL2_strict_mono dist_commute)
1.344 -      ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x \$ i \<in> A i) \<and>
1.345 -        (\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
1.346 -    qed
1.347 -  qed
1.348 -  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
1.349 -    unfolding * eventually_uniformity_metric
1.350 -    by (simp del: split_paired_All add: dist_vec_def dist_commute)
1.351 -qed
1.352 -
1.353 -end
1.354 -
1.355 -lemma Cauchy_vec_nth:
1.356 -  "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n \$ i)"
1.357 -  unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
1.358 -
1.359 -lemma vec_CauchyI:
1.360 -  fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
1.361 -  assumes X: "\<And>i. Cauchy (\<lambda>n. X n \$ i)"
1.362 -  shows "Cauchy (\<lambda>n. X n)"
1.363 -proof (rule metric_CauchyI)
1.364 -  fix r :: real assume "0 < r"
1.365 -  hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
1.366 -  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
1.367 -  define M where "M = Max (range N)"
1.368 -  have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s"
1.369 -    using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
1.370 -  hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m \$ i) (X n \$ i) < ?s"
1.371 -    unfolding N_def by (rule LeastI_ex)
1.372 -  hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m \$ i) (X n \$ i) < ?s"
1.373 -    unfolding M_def by simp
1.374 -  {
1.375 -    fix m n :: nat
1.376 -    assume "M \<le> m" "M \<le> n"
1.377 -    have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
1.378 -      unfolding dist_vec_def ..
1.379 -    also have "\<dots> \<le> setsum (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
1.380 -      by (rule setL2_le_setsum [OF zero_le_dist])
1.381 -    also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
1.382 -      by (rule setsum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
1.383 -    also have "\<dots> = r"
1.384 -      by simp
1.385 -    finally have "dist (X m) (X n) < r" .
1.386 -  }
1.387 -  hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
1.388 -    by simp
1.389 -  then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
1.390 -qed
1.391 -
1.392 -instance vec :: (complete_space, finite) complete_space
1.393 -proof
1.394 -  fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
1.395 -  have "\<And>i. (\<lambda>n. X n \$ i) \<longlonglongrightarrow> lim (\<lambda>n. X n \$ i)"
1.396 -    using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
1.397 -    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1.398 -  hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n \$ i))"
1.399 -    by (simp add: vec_tendstoI)
1.400 -  then show "convergent X"
1.401 -    by (rule convergentI)
1.402 -qed
1.403 -
1.404 -
1.405 -subsection \<open>Normed vector space\<close>
1.406 -
1.407 -instantiation vec :: (real_normed_vector, finite) real_normed_vector
1.408 -begin
1.409 -
1.410 -definition "norm x = setL2 (\<lambda>i. norm (x\$i)) UNIV"
1.411 -
1.412 -definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
1.413 -
1.414 -instance proof
1.415 -  fix a :: real and x y :: "'a ^ 'b"
1.416 -  show "norm x = 0 \<longleftrightarrow> x = 0"
1.417 -    unfolding norm_vec_def
1.418 -    by (simp add: setL2_eq_0_iff vec_eq_iff)
1.419 -  show "norm (x + y) \<le> norm x + norm y"
1.420 -    unfolding norm_vec_def
1.421 -    apply (rule order_trans [OF _ setL2_triangle_ineq])
1.422 -    apply (simp add: setL2_mono norm_triangle_ineq)
1.423 -    done
1.424 -  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
1.425 -    unfolding norm_vec_def
1.426 -    by (simp add: setL2_right_distrib)
1.427 -  show "sgn x = scaleR (inverse (norm x)) x"
1.428 -    by (rule sgn_vec_def)
1.429 -  show "dist x y = norm (x - y)"
1.430 -    unfolding dist_vec_def norm_vec_def
1.431 -    by (simp add: dist_norm)
1.432 -qed
1.433 -
1.434 -end
1.435 -
1.436 -lemma norm_nth_le: "norm (x \$ i) \<le> norm x"
1.437 -unfolding norm_vec_def
1.438 -by (rule member_le_setL2) simp_all
1.439 -
1.440 -lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x \$ i)"
1.441 -apply standard
1.443 -apply (rule vector_scaleR_component)
1.444 -apply (rule_tac x="1" in exI, simp add: norm_nth_le)
1.445 -done
1.446 -
1.447 -instance vec :: (banach, finite) banach ..
1.448 -
1.449 -
1.450 -subsection \<open>Inner product space\<close>
1.451 -
1.452 -instantiation vec :: (real_inner, finite) real_inner
1.453 -begin
1.454 -
1.455 -definition "inner x y = setsum (\<lambda>i. inner (x\$i) (y\$i)) UNIV"
1.456 -
1.457 -instance proof
1.458 -  fix r :: real and x y z :: "'a ^ 'b"
1.459 -  show "inner x y = inner y x"
1.460 -    unfolding inner_vec_def
1.461 -    by (simp add: inner_commute)
1.462 -  show "inner (x + y) z = inner x z + inner y z"
1.463 -    unfolding inner_vec_def
1.465 -  show "inner (scaleR r x) y = r * inner x y"
1.466 -    unfolding inner_vec_def
1.467 -    by (simp add: setsum_right_distrib)
1.468 -  show "0 \<le> inner x x"
1.469 -    unfolding inner_vec_def
1.470 -    by (simp add: setsum_nonneg)
1.471 -  show "inner x x = 0 \<longleftrightarrow> x = 0"
1.472 -    unfolding inner_vec_def
1.473 -    by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
1.474 -  show "norm x = sqrt (inner x x)"
1.475 -    unfolding inner_vec_def norm_vec_def setL2_def
1.476 -    by (simp add: power2_norm_eq_inner)
1.477 -qed
1.478 -
1.479 -end
1.480 -
1.481 -
1.482 -subsection \<open>Euclidean space\<close>
1.483 -
1.484 -text \<open>Vectors pointing along a single axis.\<close>
1.485 -
1.486 -definition "axis k x = (\<chi> i. if i = k then x else 0)"
1.487 -
1.488 -lemma axis_nth [simp]: "axis i x \$ i = x"
1.489 -  unfolding axis_def by simp
1.490 -
1.491 -lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
1.492 -  unfolding axis_def vec_eq_iff by auto
1.493 -
1.494 -lemma inner_axis_axis:
1.495 -  "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
1.496 -  unfolding inner_vec_def
1.497 -  apply (cases "i = j")
1.498 -  apply clarsimp
1.499 -  apply (subst setsum.remove [of _ j], simp_all)
1.500 -  apply (rule setsum.neutral, simp add: axis_def)
1.501 -  apply (rule setsum.neutral, simp add: axis_def)
1.502 -  done
1.503 -
1.504 -lemma setsum_single:
1.505 -  assumes "finite A" and "k \<in> A" and "f k = y"
1.506 -  assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
1.507 -  shows "(\<Sum>i\<in>A. f i) = y"
1.508 -  apply (subst setsum.remove [OF assms(1,2)])
1.509 -  apply (simp add: setsum.neutral assms(3,4))
1.510 -  done
1.511 -
1.512 -lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
1.513 -  unfolding inner_vec_def
1.514 -  apply (rule_tac k=i in setsum_single)
1.515 -  apply simp_all
1.516 -  apply (simp add: axis_def)
1.517 -  done
1.518 -
1.519 -instantiation vec :: (euclidean_space, finite) euclidean_space
1.520 -begin
1.521 -
1.522 -definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
1.523 -
1.524 -instance proof
1.525 -  show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
1.526 -    unfolding Basis_vec_def by simp
1.527 -next
1.528 -  show "finite (Basis :: ('a ^ 'b) set)"
1.529 -    unfolding Basis_vec_def by simp
1.530 -next
1.531 -  fix u v :: "'a ^ 'b"
1.532 -  assume "u \<in> Basis" and "v \<in> Basis"
1.533 -  thus "inner u v = (if u = v then 1 else 0)"
1.534 -    unfolding Basis_vec_def
1.535 -    by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
1.536 -next
1.537 -  fix x :: "'a ^ 'b"
1.538 -  show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
1.539 -    unfolding Basis_vec_def
1.540 -    by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
1.541 -qed
1.542 -
1.543 -lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
1.544 -  apply (simp add: Basis_vec_def)
1.545 -  apply (subst card_UN_disjoint)
1.546 -     apply simp
1.547 -    apply simp
1.548 -   apply (auto simp: axis_eq_axis) [1]
1.549 -  apply (subst card_UN_disjoint)
1.550 -     apply (auto simp: axis_eq_axis)
1.551 -  done
1.552 -
1.553 -end
1.554 -
1.555 -lemma cart_eq_inner_axis: "a \$ i = inner a (axis i 1)"
1.556 -  by (simp add: inner_axis)
1.557 -
1.558 -lemma axis_in_Basis: "a \<in> Basis \<Longrightarrow> axis i a \<in> Basis"
1.559 -  by (auto simp add: Basis_vec_def axis_eq_axis)
1.560 -
1.561 -end
```