src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy

   "~~/src/HOL/Library/Numeral_Type"
 begin
 
 subsection {* Finite Cartesian products, with indexing and lambdas. *}
 
-typedef (open Cart)
-  ('a, 'b) cart = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
-  morphisms Cart_nth Cart_lambda ..
+typedef (open)
+  ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
+  morphisms vec_nth vec_lambda ..
 
 notation
-  Cart_nth (infixl "$" 90) and
-  Cart_lambda (binder "\<chi>" 10)
+  vec_nth (infixl "$" 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 "cart 'b 'n"
+  the finite type class write "vec 'b 'n"
 *)
 
-syntax "_finite_cart" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
+syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
 
 parse_translation {*
 let
-  fun cart t u = Syntax.const @{type_syntax cart} $ t $ u;
-  fun finite_cart_tr [t, u as Free (x, _)] =
+  fun vec t u = Syntax.const @{type_syntax vec} $ t $ u;
+  fun finite_vec_tr [t, u as Free (x, _)] =
         if Lexicon.is_tid x then
-          cart t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
-        else cart t u
-    | finite_cart_tr [t, u] = cart t u
+          vec t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
+        else vec t u
+    | finite_vec_tr [t, u] = vec t u
 in
-  [(@{syntax_const "_finite_cart"}, finite_cart_tr)]
+  [(@{syntax_const "_finite_vec"}, finite_vec_tr)]
 end
 *}
 
 lemma stupid_ext: "(\<forall>x. f x = g x) \<longleftrightarrow> (f = g)"
   by (auto intro: ext)
 
-lemma Cart_eq: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
-  by (simp add: Cart_nth_inject [symmetric] fun_eq_iff)
-
-lemma Cart_lambda_beta [simp]: "Cart_lambda g $ i = g i"
-  by (simp add: Cart_lambda_inverse)
-
-lemma Cart_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
-  by (auto simp add: Cart_eq)
-
-lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g"
-  by (simp add: Cart_eq)
+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"
+  by (simp add: vec_lambda_inverse)
+
+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"
+  by (simp add: vec_eq_iff)
 
 
 subsection {* Group operations and class instances *}
 
-instantiation cart :: (zero,finite) zero
-begin
-  definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
+instantiation vec :: (zero, finite) zero
+begin
+  definition "0 \<equiv> (\<chi> i. 0)"
   instance ..
 end
 
-instantiation cart :: (plus,finite) plus
-begin
-  definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))"
+instantiation vec :: (plus, finite) plus
+begin
+  definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
   instance ..
 end
 
-instantiation cart :: (minus,finite) minus
-begin
-  definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
+instantiation vec :: (minus, finite) minus
+begin
+  definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
   instance ..
 end
 
-instantiation cart :: (uminus,finite) uminus
-begin
-  definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
+instantiation vec :: (uminus, finite) uminus
+begin
+  definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
   instance ..
 end
 
 lemma zero_index [simp]: "0 $ i = 0"
-  unfolding vector_zero_def by simp
+  unfolding zero_vec_def by simp
 
 lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
-  unfolding vector_add_def by simp
+  unfolding plus_vec_def by simp
 
 lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
-  unfolding vector_minus_def by simp
+  unfolding minus_vec_def by simp
 
 lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
-  unfolding vector_uminus_def by simp
-
-instance cart :: (semigroup_add, finite) semigroup_add
-  by default (simp add: Cart_eq add_assoc)
-
-instance cart :: (ab_semigroup_add, finite) ab_semigroup_add
-  by default (simp add: Cart_eq add_commute)
-
-instance cart :: (monoid_add, finite) monoid_add
-  by default (simp_all add: Cart_eq)
-
-instance cart :: (comm_monoid_add, finite) comm_monoid_add
-  by default (simp add: Cart_eq)
-
-instance cart :: (cancel_semigroup_add, finite) cancel_semigroup_add
-  by default (simp_all add: Cart_eq)
-
-instance cart :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
-  by default (simp add: Cart_eq)
-
-instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
-
-instance cart :: (group_add, finite) group_add
-  by default (simp_all add: Cart_eq diff_minus)
-
-instance cart :: (ab_group_add, finite) ab_group_add
-  by default (simp_all add: Cart_eq)
+  unfolding uminus_vec_def by simp
+
+instance vec :: (semigroup_add, finite) semigroup_add
+  by default (simp add: vec_eq_iff add_assoc)
+
+instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
+  by default (simp add: vec_eq_iff add_commute)
+
+instance vec :: (monoid_add, finite) monoid_add
+  by default (simp_all add: vec_eq_iff)
+
+instance vec :: (comm_monoid_add, finite) comm_monoid_add
+  by default (simp add: vec_eq_iff)
+
+instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
+  by default (simp_all add: vec_eq_iff)
+
+instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
+  by default (simp add: vec_eq_iff)
+
+instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
+
+instance vec :: (group_add, finite) group_add
+  by default (simp_all add: vec_eq_iff diff_minus)
+
+instance vec :: (ab_group_add, finite) ab_group_add
+  by default (simp_all add: vec_eq_iff)
 
 
 subsection {* Real vector space *}
 
-instantiation cart :: (real_vector, finite) real_vector
-begin
-
-definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
+instantiation vec :: (real_vector, finite) real_vector
+begin
+
+definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
 
 lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
-  unfolding vector_scaleR_def by simp
+  unfolding scaleR_vec_def by simp
 
 instance
-  by default (simp_all add: Cart_eq scaleR_left_distrib scaleR_right_distrib)
+  by default (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
 
 end
 
 
 subsection {* Topological space *}
 
-instantiation cart :: (topological_space, finite) topological_space
-begin
-
-definition open_vector_def:
+instantiation vec :: (topological_space, finite) topological_space
+begin
+
+definition
   "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_vector_def by auto
+    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_vector_def
+    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)
     apply (simp add: open_Int)
     done
 next
   fix K :: "('a ^ 'b) set set"
   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
-    unfolding open_vector_def
+    unfolding open_vec_def
     apply clarify
     apply (drule (1) bspec)
     apply (drule (1) bspec)
     apply clarify
     apply (rule_tac x=A in exI)

 qed
 
 end
 
 lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
-unfolding open_vector_def by auto
-
-lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
-unfolding open_vector_def
-apply clarify
-apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
-done
-
-lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
-unfolding closed_open vimage_Compl [symmetric]
-by (rule open_vimage_Cart_nth)
+  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_Cart_nth)
-qed
-
-lemma tendsto_Cart_nth [tendsto_intros]:
+    by (simp add: closed_INT closed_vimage_vec_nth)
+qed
+
+lemma tendsto_vec_nth [tendsto_intros]:
   assumes "((\<lambda>x. f x) ---> a) net"
   shows "((\<lambda>x. f x $ i) ---> 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)"
-    by (simp_all add: open_vimage_Cart_nth)
+    by (simp_all add: open_vimage_vec_nth)
   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

   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
 using eventually_Ball_finite [of UNIV P] assms by simp
 
-lemma tendsto_vector:
+lemma vec_tendstoI:
   assumes "\<And>i. ((\<lambda>x. f x $ i) ---> a $ i) net"
   shows "((\<lambda>x. f x) ---> 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_vector_def by metis
+    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_elim1, simp add: S)
 qed
 
-lemma tendsto_Cart_lambda [tendsto_intros]:
+lemma tendsto_vec_lambda [tendsto_intros]:
   assumes "\<And>i. ((\<lambda>x. f x i) ---> a i) net"
   shows "((\<lambda>x. \<chi> i. f x i) ---> (\<chi> i. a i)) net"
-using assms by (simp add: tendsto_vector)
+  using assms by (simp add: vec_tendstoI)
 
 
 subsection {* Metric *}
 
 (* TODO: move somewhere else *)

 apply (induct set: finite, simp_all)
 apply (clarify, rename_tac y)
 apply (rule_tac x="f(x:=y)" in exI, simp)
 done
 
-instantiation cart :: (metric_space, finite) metric_space
-begin
-
-definition dist_vector_def:
+instantiation vec :: (metric_space, finite) metric_space
+begin
+
+definition
   "dist x y = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
 
-lemma dist_nth_le_cart: "dist (x $ i) (y $ i) \<le> dist x y"
-unfolding dist_vector_def
-by (rule member_le_setL2) simp_all
+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_vector_def
-    by (simp add: setL2_eq_0_iff Cart_eq)
+    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_vector_def
+    unfolding dist_vec_def
     apply (rule order_trans [OF _ setL2_triangle_ineq])
     apply (simp add: setL2_mono dist_triangle2)
     done
 next
   (* FIXME: long proof! *)
   fix S :: "('a ^ 'b) set"
   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-    unfolding open_vector_def open_dist
+    unfolding open_vec_def open_dist
     apply safe
      apply (drule (1) bspec)
      apply clarify
      apply (subgoal_tac "\<exists>e>0. \<forall>i y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
       apply clarify
       apply (rule_tac x=e in exI, clarify)
       apply (drule spec, erule mp, clarify)
       apply (drule spec, drule spec, erule mp)
-      apply (erule le_less_trans [OF dist_nth_le_cart])
+      apply (erule le_less_trans [OF dist_vec_nth_le])
      apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>e>0. \<forall>y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
       apply (drule finite_choice [OF finite], clarify)
       apply (rule_tac x="Min (range f)" in exI, simp)
      apply clarify
      apply (drule_tac x=i in spec, clarify)

        apply (simp only: less_diff_eq)
        apply (erule le_less_trans [OF dist_triangle])
       apply simp
      apply clarify
      apply (drule spec, erule mp)
-     apply (simp add: dist_vector_def setL2_strict_mono)
+     apply (simp add: dist_vec_def setL2_strict_mono)
     apply (rule_tac x="\<lambda>i. e / sqrt (of_nat CARD('b))" in exI)
     apply (simp add: divide_pos_pos setL2_constant)
     done
 qed
 
 end
 
-lemma Cauchy_Cart_nth:
+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_nth_le_cart])
-
-lemma Cauchy_vector:
+  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"

     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_vector_def ..
+      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 `M \<le> m` `M \<le> n`)
     also have "\<dots> = 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 cart :: (complete_space, finite) complete_space
+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) ----> lim (\<lambda>n. X n $ i)"
-    using Cauchy_Cart_nth [OF `Cauchy X`]
+    using Cauchy_vec_nth [OF `Cauchy X`]
     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
-  hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
-    by (simp add: tendsto_vector)
+  hence "X ----> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
+    by (simp add: vec_tendstoI)
   then show "convergent X"
     by (rule convergentI)
 qed
 
 
 subsection {* Normed vector space *}
 
-instantiation cart :: (real_normed_vector, finite) real_normed_vector
-begin
-
-definition norm_vector_def:
-  "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
+instantiation vec :: (real_normed_vector, finite) real_normed_vector
+begin
+
+definition "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
 
 definition vector_sgn_def:
   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
 instance proof
   fix a :: real and x y :: "'a ^ 'b"
   show "0 \<le> norm x"
-    unfolding norm_vector_def
+    unfolding norm_vec_def
     by (rule setL2_nonneg)
   show "norm x = 0 \<longleftrightarrow> x = 0"
-    unfolding norm_vector_def
-    by (simp add: setL2_eq_0_iff Cart_eq)
+    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_vector_def
+    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_vector_def
+    unfolding norm_vec_def
     by (simp add: setL2_right_distrib)
   show "sgn x = scaleR (inverse (norm x)) x"
     by (rule vector_sgn_def)
   show "dist x y = norm (x - y)"
-    unfolding dist_vector_def norm_vector_def
+    unfolding dist_vec_def norm_vec_def
     by (simp add: dist_norm)
 qed
 
 end
 
 lemma norm_nth_le: "norm (x $ i) \<le> norm x"
-unfolding norm_vector_def
+unfolding norm_vec_def
 by (rule member_le_setL2) simp_all
 
-interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
+interpretation vec_nth: bounded_linear "\<lambda>x. x $ i"
 apply default
 apply (rule vector_add_component)
 apply (rule vector_scaleR_component)
 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
 done
 
-instance cart :: (banach, finite) banach ..
+instance vec :: (banach, finite) banach ..
 
 
 subsection {* Inner product space *}
 
-instantiation cart :: (real_inner, finite) real_inner
-begin
-
-definition inner_vector_def:
-  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
+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_vector_def
+    unfolding inner_vec_def
     by (simp add: inner_commute)
   show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: inner_add_left setsum_addf)
   show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: setsum_right_distrib)
   show "0 \<le> inner x x"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: setsum_nonneg)
   show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_vector_def
-    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
+    unfolding inner_vec_def
+    by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
   show "norm x = sqrt (inner x x)"
-    unfolding inner_vector_def norm_vector_def setL2_def
+    unfolding inner_vec_def norm_vec_def setL2_def
     by (simp add: power2_norm_eq_inner)
 qed
 
 end
 
 subsection {* Euclidean space *}
 
 text {* A bijection between @{text "'n::finite"} and @{text "{..<CARD('n)}"} *}
 
-definition cart_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
-  "cart_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
-
-abbreviation "\<pi> \<equiv> cart_bij_nat"
+definition vec_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
+  "vec_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
+
+abbreviation "\<pi> \<equiv> vec_bij_nat"
 definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"
 
 lemma bij_betw_pi:
   "bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
   using ex_bij_betw_nat_finite[of "UNIV::'n set"]
-  by (auto simp: cart_bij_nat_def atLeast0LessThan
+  by (auto simp: vec_bij_nat_def atLeast0LessThan
     intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])
 
 lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
   using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto
 

   by auto
 
 lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
   using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)
 
-instantiation cart :: (euclidean_space, finite) euclidean_space
+instantiation vec :: (euclidean_space, finite) euclidean_space
 begin
 
 definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
 
 definition "(basis i::'a^'b) =

   shows "basis (j + i * DIM('a)) = (\<chi> k. if k = \<pi> i then basis j else 0)"
 proof -
   have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
   also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
   finally show ?thesis
-    unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps)
+    unfolding basis_vec_def using assms by (auto simp: vec_eq_iff not_less field_simps)
 qed
 
 lemma basis_eq_pi':
   assumes "j < DIM('a)"
   shows "basis (j + \<pi>' i * DIM('a)) $ k = (if k = i then basis j else 0)"

   also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp
   finally show ?thesis by simp
 qed
 
 lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-  by (rule dimension_cart_def)
+  by (rule dimension_vec_def)
 
 lemma all_less_DIM_cart:
   fixes m n :: nat
   shows "(\<forall>i<DIM('a^'b). P i) \<longleftrightarrow> (\<forall>x::'b. \<forall>i<DIM('a). P (i + \<pi>' x * DIM('a)))"
 unfolding DIM_cart

   "inner x (if a then y else z) = (if a then inner x y else inner x z)"
   by simp_all
 
 instance proof
   show "0 < DIM('a ^ 'b)"
-    unfolding dimension_cart_def
+    unfolding dimension_vec_def
     by (intro mult_pos_pos zero_less_card_finite DIM_positive)
 next
   fix i :: nat
   assume "DIM('a ^ 'b) \<le> i" thus "basis i = (0::'a^'b)"
-    unfolding dimension_cart_def basis_cart_def
+    unfolding dimension_vec_def basis_vec_def
     by simp
 next
   show "\<forall>i<DIM('a ^ 'b). \<forall>j<DIM('a ^ 'b).
     inner (basis i :: 'a ^ 'b) (basis j) = (if i = j then 1 else 0)"
-    apply (simp add: inner_vector_def)
+    apply (simp add: inner_vec_def)
     apply safe
     apply (erule split_CARD_DIM, simp add: basis_eq_pi')
     apply (simp add: inner_if setsum_delta cong: if_cong)
     apply (simp add: basis_orthonormal)
     apply (elim split_CARD_DIM, simp add: basis_eq_pi')

     done
 next
   fix x :: "'a ^ 'b"
   show "(\<forall>i<DIM('a ^ 'b). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
     unfolding all_less_DIM_cart
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     apply (simp add: basis_eq_pi')
     apply (simp add: inner_if setsum_delta cong: if_cong)
     apply (simp add: euclidean_all_zero)
-    apply (simp add: Cart_eq)
+    apply (simp add: vec_eq_iff)
     done
 qed
 
 end