src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
 changeset 63627 6ddb43c6b711 parent 63626 44ce6b524ff3 child 63631 2edc8da89edc child 63633 2accfb71e33b
```--- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Aug 05 18:34:57 2016 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,558 +0,0 @@
-(*  Title:      HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
-    Author:     Amine Chaieb, University of Cambridge
-*)
-
-section \<open>Definition of finite Cartesian product types.\<close>
-
-theory Finite_Cartesian_Product
-imports
-  Euclidean_Space
-  L2_Norm
-  "~~/src/HOL/Library/Numeral_Type"
-begin
-
-subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
-
-typedef ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
-  morphisms vec_nth vec_lambda ..
-
-notation
-  vec_nth (infixl "\$" 90) and
-  vec_lambda (binder "\<chi>" 10)
-
-(*
-  Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
-  the finite type class write "vec 'b 'n"
-*)
-
-syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
-
-parse_translation \<open>
-  let
-    fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
-    fun finite_vec_tr [t, u] =
-      (case Term_Position.strip_positions u of
-        v as Free (x, _) =>
-          if Lexicon.is_tid x then
-            vec t (Syntax.const @{syntax_const "_ofsort"} \$ v \$
-              Syntax.const @{class_syntax finite})
-          else vec t u
-      | _ => vec t u)
-  in
-    [(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
-  end
-\<close>
-
-lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x\$i = y\$i)"
-  by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
-
-lemma vec_lambda_beta [simp]: "vec_lambda g \$ i = g i"
-
-lemma vec_lambda_unique: "(\<forall>i. f\$i = g i) \<longleftrightarrow> vec_lambda g = f"
-  by (auto simp add: vec_eq_iff)
-
-lemma vec_lambda_eta: "(\<chi> i. (g\$i)) = g"
-
-
-subsection \<open>Group operations and class instances\<close>
-
-instantiation vec :: (zero, finite) zero
-begin
-  definition "0 \<equiv> (\<chi> i. 0)"
-  instance ..
-end
-
-instantiation vec :: (plus, finite) plus
-begin
-  definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x\$i + y\$i))"
-  instance ..
-end
-
-instantiation vec :: (minus, finite) minus
-begin
-  definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x\$i - y\$i))"
-  instance ..
-end
-
-instantiation vec :: (uminus, finite) uminus
-begin
-  definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x\$i)))"
-  instance ..
-end
-
-lemma zero_index [simp]: "0 \$ i = 0"
-  unfolding zero_vec_def by simp
-
-lemma vector_add_component [simp]: "(x + y)\$i = x\$i + y\$i"
-  unfolding plus_vec_def by simp
-
-lemma vector_minus_component [simp]: "(x - y)\$i = x\$i - y\$i"
-  unfolding minus_vec_def by simp
-
-lemma vector_uminus_component [simp]: "(- x)\$i = - (x\$i)"
-  unfolding uminus_vec_def by simp
-
-
-
-  by standard (simp_all add: vec_eq_iff)
-
-  by standard (simp add: vec_eq_iff)
-
-  by standard (simp_all add: vec_eq_iff)
-
-  by standard (simp_all add: vec_eq_iff diff_diff_eq)
-
-
-  by standard (simp_all add: vec_eq_iff)
-
-  by standard (simp_all add: vec_eq_iff)
-
-
-subsection \<open>Real vector space\<close>
-
-instantiation vec :: (real_vector, finite) real_vector
-begin
-
-definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x\$i)))"
-
-lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
-  unfolding scaleR_vec_def by simp
-
-instance
-  by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
-
-end
-
-
-subsection \<open>Topological space\<close>
-
-instantiation vec :: (topological_space, finite) topological_space
-begin
-
-definition [code del]:
-  "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
-    (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x\$i \<in> A i) \<and>
-      (\<forall>y. (\<forall>i. y\$i \<in> A i) \<longrightarrow> y \<in> S))"
-
-instance proof
-  show "open (UNIV :: ('a ^ 'b) set)"
-    unfolding open_vec_def by auto
-next
-  fix S T :: "('a ^ 'b) set"
-  assume "open S" "open T" thus "open (S \<inter> T)"
-    unfolding open_vec_def
-    apply clarify
-    apply (drule (1) bspec)+
-    apply (clarify, rename_tac Sa Ta)
-    apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
-    done
-next
-  fix K :: "('a ^ 'b) set set"
-  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
-    unfolding open_vec_def
-    apply clarify
-    apply (drule (1) bspec)
-    apply (drule (1) bspec)
-    apply clarify
-    apply (rule_tac x=A in exI)
-    apply fast
-    done
-qed
-
-end
-
-lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x \$ i \<in> S i}"
-  unfolding open_vec_def by auto
-
-lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x \$ i) -` S)"
-  unfolding open_vec_def
-  apply clarify
-  apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
-  done
-
-lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x \$ i) -` S)"
-  unfolding closed_open vimage_Compl [symmetric]
-  by (rule open_vimage_vec_nth)
-
-lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
-proof -
-  have "{x. \<forall>i. x \$ i \<in> S i} = (\<Inter>i. (\<lambda>x. x \$ i) -` S i)" by auto
-  thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x \$ i \<in> S i}"
-    by (simp add: closed_INT closed_vimage_vec_nth)
-qed
-
-lemma tendsto_vec_nth [tendsto_intros]:
-  assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
-  shows "((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
-proof (rule topological_tendstoI)
-  fix S assume "open S" "a \$ i \<in> S"
-  then have "open ((\<lambda>y. y \$ i) -` S)" "a \<in> ((\<lambda>y. y \$ i) -` S)"
-  with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y \$ i) -` S) net"
-    by (rule topological_tendstoD)
-  then show "eventually (\<lambda>x. f x \$ i \<in> S) net"
-    by simp
-qed
-
-lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x \$ i) a"
-  unfolding isCont_def by (rule tendsto_vec_nth)
-
-lemma vec_tendstoI:
-  assumes "\<And>i. ((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
-  shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
-proof (rule topological_tendstoI)
-  fix S assume "open S" and "a \<in> S"
-  then obtain A where A: "\<And>i. open (A i)" "\<And>i. a \$ i \<in> A i"
-    and S: "\<And>y. \<forall>i. y \$ i \<in> A i \<Longrightarrow> y \<in> S"
-    unfolding open_vec_def by metis
-  have "\<And>i. eventually (\<lambda>x. f x \$ i \<in> A i) net"
-    using assms A by (rule topological_tendstoD)
-  hence "eventually (\<lambda>x. \<forall>i. f x \$ i \<in> A i) net"
-    by (rule eventually_all_finite)
-  thus "eventually (\<lambda>x. f x \<in> S) net"
-    by (rule eventually_mono, simp add: S)
-qed
-
-lemma tendsto_vec_lambda [tendsto_intros]:
-  assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
-  shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
-  using assms by (simp add: vec_tendstoI)
-
-lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
-proof (rule openI)
-  fix a assume "a \<in> (\<lambda>x. x \$ i) ` S"
-  then obtain z where "a = z \$ i" and "z \<in> S" ..
-  then obtain A where A: "\<forall>i. open (A i) \<and> z \$ i \<in> A i"
-    and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
-    using \<open>open S\<close> unfolding open_vec_def by auto
-  hence "A i \<subseteq> (\<lambda>x. x \$ i) ` S"
-    by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z \$ j" in image_eqI,
-      simp_all)
-  hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x \$ i) ` S"
-    using A \<open>a = z \$ i\<close> by simp
-  then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x \$ i) ` S" by - (rule exI)
-qed
-
-instance vec :: (perfect_space, finite) perfect_space
-proof
-  fix x :: "'a ^ 'b" show "\<not> open {x}"
-  proof
-    assume "open {x}"
-    hence "\<forall>i. open ((\<lambda>x. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
-    hence "\<forall>i. open {x \$ i}" by simp
-    thus "False" by (simp add: not_open_singleton)
-  qed
-qed
-
-
-subsection \<open>Metric space\<close>
-(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
-
-instantiation vec :: (metric_space, finite) dist
-begin
-
-definition
-  "dist x y = setL2 (\<lambda>i. dist (x\$i) (y\$i)) UNIV"
-
-instance ..
-end
-
-instantiation vec :: (metric_space, finite) uniformity_dist
-begin
-
-definition [code del]:
-  "(uniformity :: (('a, 'b) vec \<times> ('a, 'b) vec) filter) =
-    (INF e:{0 <..}. principal {(x, y). dist x y < e})"
-
-instance
-  by standard (rule uniformity_vec_def)
-end
-
-declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
-
-instantiation vec :: (metric_space, finite) metric_space
-begin
-
-lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) \<le> dist x y"
-  unfolding dist_vec_def by (rule member_le_setL2) simp_all
-
-instance proof
-  fix x y :: "'a ^ 'b"
-  show "dist x y = 0 \<longleftrightarrow> x = y"
-    unfolding dist_vec_def
-    by (simp add: setL2_eq_0_iff vec_eq_iff)
-next
-  fix x y z :: "'a ^ 'b"
-  show "dist x y \<le> dist x z + dist y z"
-    unfolding dist_vec_def
-    apply (rule order_trans [OF _ setL2_triangle_ineq])
-    apply (simp add: setL2_mono dist_triangle2)
-    done
-next
-  fix S :: "('a ^ 'b) set"
-  have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-  proof
-    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
-    proof
-      fix x assume "x \<in> S"
-      obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x \$ i \<in> A i"
-        and S: "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
-        using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
-      have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x \$ i) < r \<longrightarrow> y \<in> A i"
-        using A unfolding open_dist by simp
-      hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i)"
-        by (rule finite_set_choice [OF finite])
-      then obtain r where r1: "\<forall>i. 0 < r i"
-        and r2: "\<forall>i y. dist y (x \$ i) < r i \<longrightarrow> y \<in> A i" by fast
-      have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
-        by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
-      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
-    qed
-  next
-    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
-    proof (unfold open_vec_def, rule)
-      fix x assume "x \<in> S"
-      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
-        using * by fast
-      define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
-      from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
-        unfolding r_def by simp_all
-      from \<open>0 < e\<close> have e: "e = setL2 r UNIV"
-        unfolding r_def by (simp add: setL2_constant)
-      define A where "A i = {y. dist (x \$ i) y < r i}" for i
-      have "\<forall>i. open (A i) \<and> x \$ i \<in> A i"
-        unfolding A_def by (simp add: open_ball r)
-      moreover have "\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S"
-        by (simp add: A_def S dist_vec_def e setL2_strict_mono dist_commute)
-      ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x \$ i \<in> A i) \<and>
-        (\<forall>y. (\<forall>i. y \$ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
-    qed
-  qed
-  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
-    unfolding * eventually_uniformity_metric
-    by (simp del: split_paired_All add: dist_vec_def dist_commute)
-qed
-
-end
-
-lemma Cauchy_vec_nth:
-  "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n \$ i)"
-  unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
-
-lemma vec_CauchyI:
-  fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
-  assumes X: "\<And>i. Cauchy (\<lambda>n. X n \$ i)"
-  shows "Cauchy (\<lambda>n. X n)"
-proof (rule metric_CauchyI)
-  fix r :: real assume "0 < r"
-  hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
-  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
-  define M where "M = Max (range N)"
-  have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s"
-    using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
-  hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m \$ i) (X n \$ i) < ?s"
-    unfolding N_def by (rule LeastI_ex)
-  hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m \$ i) (X n \$ i) < ?s"
-    unfolding M_def by simp
-  {
-    fix m n :: nat
-    assume "M \<le> m" "M \<le> n"
-    have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
-      unfolding dist_vec_def ..
-    also have "\<dots> \<le> setsum (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
-      by (rule setL2_le_setsum [OF zero_le_dist])
-    also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
-      by (rule setsum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
-    also have "\<dots> = r"
-      by simp
-    finally have "dist (X m) (X n) < r" .
-  }
-  hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
-    by simp
-  then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
-qed
-
-instance vec :: (complete_space, finite) complete_space
-proof
-  fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
-  have "\<And>i. (\<lambda>n. X n \$ i) \<longlonglongrightarrow> lim (\<lambda>n. X n \$ i)"
-    using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
-    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
-  hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n \$ i))"
-  then show "convergent X"
-    by (rule convergentI)
-qed
-
-
-subsection \<open>Normed vector space\<close>
-
-instantiation vec :: (real_normed_vector, finite) real_normed_vector
-begin
-
-definition "norm x = setL2 (\<lambda>i. norm (x\$i)) UNIV"
-
-definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
-
-instance proof
-  fix a :: real and x y :: "'a ^ 'b"
-  show "norm x = 0 \<longleftrightarrow> x = 0"
-    unfolding norm_vec_def
-    by (simp add: setL2_eq_0_iff vec_eq_iff)
-  show "norm (x + y) \<le> norm x + norm y"
-    unfolding norm_vec_def
-    apply (rule order_trans [OF _ setL2_triangle_ineq])
-    apply (simp add: setL2_mono norm_triangle_ineq)
-    done
-  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
-    unfolding norm_vec_def
-  show "sgn x = scaleR (inverse (norm x)) x"
-    by (rule sgn_vec_def)
-  show "dist x y = norm (x - y)"
-    unfolding dist_vec_def norm_vec_def
-qed
-
-end
-
-lemma norm_nth_le: "norm (x \$ i) \<le> norm x"
-unfolding norm_vec_def
-by (rule member_le_setL2) simp_all
-
-lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x \$ i)"
-apply standard
-apply (rule vector_scaleR_component)
-apply (rule_tac x="1" in exI, simp add: norm_nth_le)
-done
-
-instance vec :: (banach, finite) banach ..
-
-
-subsection \<open>Inner product space\<close>
-
-instantiation vec :: (real_inner, finite) real_inner
-begin
-
-definition "inner x y = setsum (\<lambda>i. inner (x\$i) (y\$i)) UNIV"
-
-instance proof
-  fix r :: real and x y z :: "'a ^ 'b"
-  show "inner x y = inner y x"
-    unfolding inner_vec_def
-  show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_vec_def
-  show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_vec_def
-  show "0 \<le> inner x x"
-    unfolding inner_vec_def
-  show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_vec_def
-    by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
-  show "norm x = sqrt (inner x x)"
-    unfolding inner_vec_def norm_vec_def setL2_def
-qed
-
-end
-
-
-subsection \<open>Euclidean space\<close>
-
-text \<open>Vectors pointing along a single axis.\<close>
-
-definition "axis k x = (\<chi> i. if i = k then x else 0)"
-
-lemma axis_nth [simp]: "axis i x \$ i = x"
-  unfolding axis_def by simp
-
-lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
-  unfolding axis_def vec_eq_iff by auto
-
-lemma inner_axis_axis:
-  "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
-  unfolding inner_vec_def
-  apply (cases "i = j")
-  apply clarsimp
-  apply (subst setsum.remove [of _ j], simp_all)
-  apply (rule setsum.neutral, simp add: axis_def)
-  apply (rule setsum.neutral, simp add: axis_def)
-  done
-
-lemma setsum_single:
-  assumes "finite A" and "k \<in> A" and "f k = y"
-  assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
-  shows "(\<Sum>i\<in>A. f i) = y"
-  apply (subst setsum.remove [OF assms(1,2)])
-  apply (simp add: setsum.neutral assms(3,4))
-  done
-
-lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
-  unfolding inner_vec_def
-  apply (rule_tac k=i in setsum_single)
-  apply simp_all
-  done
-
-instantiation vec :: (euclidean_space, finite) euclidean_space
-begin
-
-definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
-
-instance proof
-  show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
-    unfolding Basis_vec_def by simp
-next
-  show "finite (Basis :: ('a ^ 'b) set)"
-    unfolding Basis_vec_def by simp
-next
-  fix u v :: "'a ^ 'b"
-  assume "u \<in> Basis" and "v \<in> Basis"
-  thus "inner u v = (if u = v then 1 else 0)"
-    unfolding Basis_vec_def
-    by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
-next
-  fix x :: "'a ^ 'b"
-  show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
-    unfolding Basis_vec_def
-    by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
-qed
-
-lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-  apply (subst card_UN_disjoint)
-     apply simp
-    apply simp
-   apply (auto simp: axis_eq_axis) [1]
-  apply (subst card_UN_disjoint)
-     apply (auto simp: axis_eq_axis)
-  done
-
-end
-
-lemma cart_eq_inner_axis: "a \$ i = inner a (axis i 1)"