diff --git a/src/HOL/Analysis/Finite_Cartesian_Product.thy b/src/HOL/Analysis/Finite_Cartesian_Product.thy
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy
@@ -1,1284 +1,1264 @@
 (*  Title:      HOL/Analysis/Finite_Cartesian_Product.thy
     Author:     Amine Chaieb, University of Cambridge
 *)
 
 section \<open>Definition of Finite Cartesian Product Type\<close>
 
 theory Finite_Cartesian_Product
 imports
   Euclidean_Space
   L2_Norm
   "HOL-Library.Numeral_Type"
   "HOL-Library.Countable_Set"
   "HOL-Library.FuncSet"
 begin
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Finite Cartesian products, with indexing and lambdas\<close>
 
 typedef ('a, 'b) vec = "UNIV :: ('b::finite \<Rightarrow> 'a) set"
   morphisms vec_nth vec_lambda ..
 
 declare vec_lambda_inject [simplified, simp]
 
 bundle vec_syntax begin
 notation
   vec_nth (infixl "$" 90) and
   vec_lambda (binder "\<chi>" 10)
 end
 
 bundle no_vec_syntax begin
 no_notation
   vec_nth (infixl "$" 90) and
   vec_lambda (binder "\<chi>" 10)
 end
 
 unbundle vec_syntax
 
 text \<open>
   Concrete syntax for \<open>('a, 'b) vec\<close>:
     \<^item> \<open>'a^'b\<close> becomes \<open>('a, 'b::finite) vec\<close>
     \<^item> \<open>'a^'b::_\<close> becomes \<open>('a, 'b) vec\<close> without extra sort-constraint
 \<close>
 syntax "_vec_type" :: "type \<Rightarrow> type \<Rightarrow> type" (infixl "^" 15)
 parse_translation \<open>
   let
     fun vec t u = Syntax.const \<^type_syntax>\<open>vec\<close> $ 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>\<open>_ofsort\<close> $ v $
               Syntax.const \<^class_syntax>\<open>finite\<close>)
           else vec t u
       | _ => vec t u)
   in
     [(\<^syntax_const>\<open>_vec_type\<close>, 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"
   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 [simp]: "(\<chi> i. (g$i)) = g"
   by (simp add: vec_eq_iff)
 
 subsection \<open>Cardinality of vectors\<close>
 
 instance vec :: (finite, finite) finite
 proof
   show "finite (UNIV :: ('a, 'b) vec set)"
   proof (subst bij_betw_finite)
     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
     have "finite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       by (intro finite_PiE) auto
     also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
       by auto
     finally show "finite \<dots>" .
   qed
 qed
 
 lemma countable_PiE:
   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
 
 instance vec :: (countable, finite) countable
 proof
   have "countable (UNIV :: ('a, 'b) vec set)"
   proof (rule countableI_bij2)
     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
     have "countable (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       by (intro countable_PiE) auto
     also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
       by auto
     finally show "countable \<dots>" .
   qed
   thus "\<exists>t::('a, 'b) vec \<Rightarrow> nat. inj t"
     by (auto elim!: countableE)
 qed
 
 lemma infinite_UNIV_vec:
   assumes "infinite (UNIV :: 'a set)"
   shows   "infinite (UNIV :: ('a^'b) set)"
 proof (subst bij_betw_finite)
   show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
     by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
   have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
   proof
     assume "finite ?A"
     hence "finite ((\<lambda>f. f undefined) ` ?A)"
       by (rule finite_imageI)
     also have "(\<lambda>f. f undefined) ` ?A = UNIV"
       by auto
     finally show False 
       using \<open>infinite (UNIV :: 'a set)\<close> by contradiction
   qed
   also have "?A = Pi UNIV (\<lambda>_. UNIV)" 
     by auto
   finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
 qed
 
 proposition CARD_vec [simp]:
   "CARD('a^'b) = CARD('a) ^ CARD('b)"
 proof (cases "finite (UNIV :: 'a set)")
   case True
   show ?thesis
   proof (subst bij_betw_same_card)
     show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
     have "CARD('a) ^ CARD('b) = card (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
       (is "_ = card ?A")
       by (subst card_PiE) (auto)
     also have "?A = Pi UNIV (\<lambda>_. UNIV)" 
       by auto
     finally show "card \<dots> = CARD('a) ^ CARD('b)" ..
   qed
 qed (simp_all add: infinite_UNIV_vec)
 
 lemma countable_vector:
   fixes B:: "'n::finite \<Rightarrow> 'a set"
   assumes "\<And>i. countable (B i)"
   shows "countable {V. \<forall>i::'n::finite. V $ i \<in> B i}"
 proof -
   have "f \<in> ($) ` {V. \<forall>i. V $ i \<in> B i}" if "f \<in> Pi\<^sub>E UNIV B" for f
   proof -
     have "\<exists>W. (\<forall>i. W $ i \<in> B i) \<and> ($) W = f"
       by (metis that PiE_iff UNIV_I vec_lambda_inverse)
     then show "f \<in> ($) ` {v. \<forall>i. v $ i \<in> B i}"
       by blast
   qed
   then have "Pi\<^sub>E UNIV B = vec_nth ` {V. \<forall>i::'n. V $ i \<in> B i}"
     by blast
   then have "countable (vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
     by (metis finite_class.finite_UNIV countable_PiE assms)
   then have "countable (vec_lambda ` vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
     by auto
   then show ?thesis
     by (simp add: image_comp o_def vec_nth_inverse)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<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 "(+) \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
   instance ..
 end
 
 instantiation vec :: (minus, finite) minus
 begin
   definition "(-) \<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
 
 instance vec :: (semigroup_add, finite) semigroup_add
   by standard (simp add: vec_eq_iff add.assoc)
 
 instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
   by standard (simp add: vec_eq_iff add.commute)
 
 instance vec :: (monoid_add, finite) monoid_add
   by standard (simp_all add: vec_eq_iff)
 
 instance vec :: (comm_monoid_add, finite) comm_monoid_add
   by standard (simp add: vec_eq_iff)
 
 instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
   by standard (simp_all add: vec_eq_iff)
 
 instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
   by standard (simp_all add: vec_eq_iff diff_diff_eq)
 
 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
 
 instance vec :: (group_add, finite) group_add
   by standard (simp_all add: vec_eq_iff)
 
 instance vec :: (ab_group_add, finite) ab_group_add
   by standard (simp_all add: vec_eq_iff)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic componentwise operations on vectors\<close>
 
 instantiation vec :: (times, finite) times
 begin
 
 definition "(*) \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
 instance ..
 
 end
 
 instantiation vec :: (one, finite) one
 begin
 
 definition "1 \<equiv> (\<chi> i. 1)"
 instance ..
 
 end
 
 instantiation vec :: (ord, finite) ord
 begin
 
 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
 instance ..
 
 end
 
 text\<open>The ordering on one-dimensional vectors is linear.\<close>
 
 instance vec:: (order, finite) order
   by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
       intro: order.trans order.antisym order.strict_implies_order)
 
 instance vec :: (linorder, CARD_1) linorder
 proof
   obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
   proof -
     have "CARD ('b) = 1" by (rule CARD_1)
     then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
     then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
     then show thesis by (auto intro: that)
   qed
   fix x y :: "'a^'b::CARD_1"
   note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
   show "x \<le> y \<or> y \<le> x" by auto
 qed
 
 text\<open>Constant Vectors\<close>
 
 definition "vec x = (\<chi> i. x)"
 
 text\<open>Also the scalar-vector multiplication.\<close>
 
 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
   where "c *s x = (\<chi> i. c * (x$i))"
 
 text \<open>scalar product\<close>
 
 definition scalar_product :: "'a :: semiring_1 ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" where
   "scalar_product v w = (\<Sum> i \<in> UNIV. v $ i * w $ i)"
 
 
 subsection \<open>Real vector space\<close>
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (real_vector, finite) real_vector
 begin
 
 definition\<^marker>\<open>tag important\<close> "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\<^marker>\<open>tag unimportant\<close>
   by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
 
 end
 
 
 subsection \<open>Topological space\<close>
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (topological_space, finite) topological_space
 begin
 
 definition\<^marker>\<open>tag important\<close> [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\<^marker>\<open>tag unimportant\<close> 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)
     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_vec_def
-    apply clarify
-    apply (drule (1) bspec)
-    apply (drule (1) bspec)
-    apply clarify
-    apply (rule_tac x=A in exI)
-    apply fast
-    done
+    by (metis Union_iff)
 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)"
     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
 
 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\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> vec :: (metric_space, finite) dist
 begin
 
 definition\<^marker>\<open>tag important\<close>
   "dist x y = L2_set (\<lambda>i. dist (x$i) (y$i)) UNIV"
 
 instance ..
 end
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (metric_space, finite) uniformity_dist
 begin
 
 definition\<^marker>\<open>tag important\<close> [code del]:
   "(uniformity :: (('a^'b::_) \<times> ('a^'b::_)) filter) =
     (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
 
 instance\<^marker>\<open>tag unimportant\<close>
   by standard (rule uniformity_vec_def)
 end
 
 declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (metric_space, finite) metric_space
 begin
 
 proposition dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
   unfolding dist_vec_def by (rule member_le_L2_set) simp_all
 
 instance proof
   fix x y :: "'a ^ 'b"
   show "dist x y = 0 \<longleftrightarrow> x = y"
     unfolding dist_vec_def
     by (simp add: L2_set_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 _ L2_set_triangle_ineq])
     apply (simp add: L2_set_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 = L2_set r UNIV"
         unfolding r_def by (simp add: L2_set_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 L2_set_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) = L2_set (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
       unfolding dist_vec_def ..
     also have "\<dots> \<le> sum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
       by (rule L2_set_le_sum [OF zero_le_dist])
     also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
       by (rule sum_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\<^marker>\<open>tag unimportant\<close> 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))"
     by (simp add: vec_tendstoI)
   then show "convergent X"
     by (rule convergentI)
 qed
 
 
 subsection \<open>Normed vector space\<close>
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (real_normed_vector, finite) real_normed_vector
 begin
 
 definition\<^marker>\<open>tag important\<close> "norm x = L2_set (\<lambda>i. norm (x$i)) UNIV"
 
 definition\<^marker>\<open>tag important\<close> "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
 instance\<^marker>\<open>tag unimportant\<close> proof
   fix a :: real and x y :: "'a ^ 'b"
   show "norm x = 0 \<longleftrightarrow> x = 0"
     unfolding norm_vec_def
     by (simp add: L2_set_eq_0_iff vec_eq_iff)
   show "norm (x + y) \<le> norm x + norm y"
     unfolding norm_vec_def
     apply (rule order_trans [OF _ L2_set_triangle_ineq])
     apply (simp add: L2_set_mono norm_triangle_ineq)
     done
   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
     unfolding norm_vec_def
     by (simp add: L2_set_right_distrib)
   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
     by (simp add: dist_norm)
 qed
 
 end
 
 lemma norm_nth_le: "norm (x $ i) \<le> norm x"
 unfolding norm_vec_def
 by (rule member_le_L2_set) simp_all
 
 lemma norm_le_componentwise_cart:
   fixes x :: "'a::real_normed_vector^'n"
   assumes "\<And>i. norm(x$i) \<le> norm(y$i)"
   shows "norm x \<le> norm y"
   unfolding norm_vec_def
   by (rule L2_set_mono) (auto simp: assms)
 
 lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
-  apply (simp add: norm_vec_def)
-  apply (rule member_le_L2_set, simp_all)
-  done
+  by (metis norm_nth_le real_norm_def)
 
 lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
   by (metis component_le_norm_cart order_trans)
 
 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
   by (metis component_le_norm_cart le_less_trans)
 
 lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   by (simp add: norm_vec_def L2_set_le_sum)
 
 lemma bounded_linear_vec_nth[intro]: "bounded_linear (\<lambda>x. x $ i)"
-apply standard
-apply (rule vector_add_component)
-apply (rule vector_scaleR_component)
-apply (rule_tac x="1" in exI, simp add: norm_nth_le)
-done
+proof
+  show "\<exists>K. \<forall>x. norm (x $ i) \<le> norm x * K"
+    by (metis mult.commute mult.left_neutral norm_nth_le)
+qed auto
 
 instance vec :: (banach, finite) banach ..
 
 
 subsection \<open>Inner product space\<close>
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (real_inner, finite) real_inner
 begin
 
 definition\<^marker>\<open>tag important\<close> "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
 instance\<^marker>\<open>tag unimportant\<close> proof
   fix r :: real and x y z :: "'a ^ 'b"
   show "inner x y = inner y x"
     unfolding inner_vec_def
     by (simp add: inner_commute)
   show "inner (x + y) z = inner x z + inner y z"
     unfolding inner_vec_def
     by (simp add: inner_add_left sum.distrib)
   show "inner (scaleR r x) y = r * inner x y"
     unfolding inner_vec_def
     by (simp add: sum_distrib_left)
   show "0 \<le> inner x x"
     unfolding inner_vec_def
     by (simp add: sum_nonneg)
   show "inner x x = 0 \<longleftrightarrow> x = 0"
     unfolding inner_vec_def
     by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
   show "norm x = sqrt (inner x x)"
     unfolding inner_vec_def norm_vec_def L2_set_def
     by (simp add: power2_norm_eq_inner)
 qed
 
 end
 
 
 subsection \<open>Euclidean space\<close>
 
 text \<open>Vectors pointing along a single axis.\<close>
 
 definition\<^marker>\<open>tag important\<close> "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 sum.remove [of _ j], simp_all)
-  apply (rule sum.neutral, simp add: axis_def)
-  apply (rule sum.neutral, simp add: axis_def)
-  done
+  by (simp add: inner_vec_def axis_def sum.neutral sum.remove [of _ j])
 
 lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
   by (simp add: inner_vec_def axis_def sum.remove [where x=i])
 
 lemma inner_axis': "inner(axis i y) x = inner y (x $ i)"
   by (simp add: inner_axis inner_commute)
 
 instantiation\<^marker>\<open>tag unimportant\<close> vec :: (euclidean_space, finite) euclidean_space
 begin
 
 definition\<^marker>\<open>tag important\<close> "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
 
 instance\<^marker>\<open>tag unimportant\<close> 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
 
 proposition DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
 proof -
   have "card (\<Union>i::'b. \<Union>u::'a\<in>Basis. {axis i u}) = (\<Sum>i::'b\<in>UNIV. card (\<Union>u::'a\<in>Basis. {axis i u}))"
     by (rule card_UN_disjoint) (auto simp: axis_eq_axis) 
   also have "... = CARD('b) * DIM('a)"
     by (subst card_UN_disjoint) (auto simp: axis_eq_axis)
   finally show ?thesis
     by (simp add: Basis_vec_def)
 qed
 
 end
 
 lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
   by (simp add: inner_axis' norm_eq_1)
 
 lemma sum_norm_allsubsets_bound_cart:
   fixes f:: "'a \<Rightarrow> real ^'n"
   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
   shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
   using sum_norm_allsubsets_bound[OF assms]
   by simp
 
 lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
   by (simp add: inner_axis)
 
 lemma axis_eq_0_iff [simp]:
   shows "axis m x = 0 \<longleftrightarrow> x = 0"
   by (simp add: axis_def vec_eq_iff)
 
 lemma axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
   by (auto simp: Basis_vec_def axis_eq_axis)
 
 text\<open>Mapping each basis element to the corresponding finite index\<close>
 definition axis_index :: "('a::comm_ring_1)^'n \<Rightarrow> 'n" where "axis_index v \<equiv> SOME i. v = axis i 1"
 
 lemma axis_inverse:
   fixes v :: "real^'n"
   assumes "v \<in> Basis"
   shows "\<exists>i. v = axis i 1"
 proof -
   have "v \<in> (\<Union>n. \<Union>r\<in>Basis. {axis n r})"
     using assms Basis_vec_def by blast
   then show ?thesis
     by (force simp add: vec_eq_iff)
 qed
 
 lemma axis_index:
   fixes v :: "real^'n"
   assumes "v \<in> Basis"
   shows "v = axis (axis_index v) 1"
   by (metis (mono_tags) assms axis_inverse axis_index_def someI_ex)
 
 lemma axis_index_axis [simp]:
   fixes UU :: "real^'n"
   shows "(axis_index (axis u 1 :: real^'n)) = (u::'n)"
   by (simp add: axis_eq_axis axis_index_def)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
 
 lemma sum_cong_aux:
   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
   by (auto intro: sum.cong)
 
 hide_fact (open) sum_cong_aux
 
 method_setup vector = \<open>
 let
   val ss1 =
     simpset_of (put_simpset HOL_basic_ss \<^context>
       addsimps [@{thm sum.distrib} RS sym,
       @{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
       @{thm sum_distrib_right}, @{thm sum_negf} RS sym])
   val ss2 =
     simpset_of (\<^context> addsimps
              [@{thm plus_vec_def}, @{thm times_vec_def},
               @{thm minus_vec_def}, @{thm uminus_vec_def},
               @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
               @{thm scaleR_vec_def}, @{thm vector_scalar_mult_def}])
   fun vector_arith_tac ctxt ths =
     simp_tac (put_simpset ss1 ctxt)
     THEN' (fn i => resolve_tac ctxt @{thms Finite_Cartesian_Product.sum_cong_aux} i
          ORELSE resolve_tac ctxt @{thms sum.neutral} i
          ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
     (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
     THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
 in
   Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
 end
 \<close> "lift trivial vector statements to real arith statements"
 
 lemma vec_0[simp]: "vec 0 = 0" by vector
 lemma vec_1[simp]: "vec 1 = 1" by vector
 
 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
 
 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
 
 lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
 lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
 lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
 lemma vec_neg: "vec(- x) = - vec x " by vector
 
 lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
   by vector
 
 lemma vec_sum:
   assumes "finite S"
   shows "vec(sum f S) = sum (vec \<circ> f) S"
   using assms
 proof induct
   case empty
   then show ?case by simp
 next
   case insert
   then show ?case by (auto simp add: vec_add)
 qed
 
 text\<open>Obvious "component-pushing".\<close>
 
 lemma vec_component [simp]: "vec x $ i = x"
   by vector
 
 lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
   by vector
 
 lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
   by vector
 
 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
 
 lemmas\<^marker>\<open>tag unimportant\<close> vector_component =
   vec_component vector_add_component vector_mult_component
   vector_smult_component vector_minus_component vector_uminus_component
   vector_scaleR_component cond_component
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some frequently useful arithmetic lemmas over vectors\<close>
 
 instance vec :: (semigroup_mult, finite) semigroup_mult
   by standard (vector mult.assoc)
 
 instance vec :: (monoid_mult, finite) monoid_mult
   by standard vector+
 
 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
   by standard (vector mult.commute)
 
 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
   by standard vector
 
 instance vec :: (semiring, finite) semiring
   by standard (vector field_simps)+
 
 instance vec :: (semiring_0, finite) semiring_0
   by standard (vector field_simps)+
 instance vec :: (semiring_1, finite) semiring_1
   by standard vector
 instance vec :: (comm_semiring, finite) comm_semiring
   by standard (vector field_simps)+
 
 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
 instance vec :: (ring, finite) ring ..
 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
 
 instance vec :: (ring_1, finite) ring_1 ..
 
 instance vec :: (real_algebra, finite) real_algebra
   by standard (simp_all add: vec_eq_iff)
 
 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
 
 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
 proof (induct n)
   case 0
   then show ?case by vector
 next
   case Suc
   then show ?case by vector
 qed
 
 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
   by vector
 
 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
   by vector
 
 instance vec :: (semiring_char_0, finite) semiring_char_0
 proof
   fix m n :: nat
   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
     by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
 qed
 
 instance vec :: (numeral, finite) numeral ..
 instance vec :: (semiring_numeral, finite) semiring_numeral ..
 
 lemma numeral_index [simp]: "numeral w $ i = numeral w"
   by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
 
 lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
   by (simp only: vector_uminus_component numeral_index)
 
 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
 instance vec :: (ring_char_0, finite) ring_char_0 ..
 
 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
   by (vector mult.assoc)
 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
   by (vector field_simps)
 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
   by (vector field_simps)
 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
   by (vector field_simps)
 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
   by (vector field_simps)
 
 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
   by (simp add: vec_eq_iff)
 
 lemma Vector_Spaces_linear_vec [simp]: "Vector_Spaces.linear (*) vector_scalar_mult vec"
   by unfold_locales (vector algebra_simps)+
 
 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   by vector
 
 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::'a::field) \<or> x = y"
   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
 
 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::'a::field) = b \<or> x = 0"
   by (subst eq_iff_diff_eq_0, subst vector_sub_rdistrib [symmetric]) simp
 
 lemma scalar_mult_eq_scaleR [abs_def]: "c *s x = c *\<^sub>R x"
   unfolding scaleR_vec_def vector_scalar_mult_def by simp
 
 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
   unfolding dist_norm scalar_mult_eq_scaleR
   unfolding scaleR_right_diff_distrib[symmetric] by simp
 
 lemma sum_component [simp]:
   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
   shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
 proof (cases "finite S")
   case True
   then show ?thesis by induct simp_all
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
   by (simp add: vec_eq_iff)
 
 lemma sum_cmul:
   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
   shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
   by (simp add: vec_eq_iff sum_distrib_left)
 
 lemma linear_vec [simp]: "linear vec"
   using Vector_Spaces_linear_vec
-  apply (auto )
   by unfold_locales (vector algebra_simps)+
 
 subsection \<open>Matrix operations\<close>
 
 text\<open>Matrix notation. NB: an MxN matrix is of type \<^typ>\<open>'a^'n^'m\<close>, not \<^typ>\<open>'a^'m^'n\<close>\<close>
 
 definition\<^marker>\<open>tag important\<close> map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
   "map_matrix f x = (\<chi> i j. f (x $ i $ j))"
 
 lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
   by (simp add: map_matrix_def)
 
 definition\<^marker>\<open>tag important\<close> matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
     (infixl "**" 70)
   where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
 
 definition\<^marker>\<open>tag important\<close> matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
     (infixl "*v" 70)
   where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
 
 definition\<^marker>\<open>tag important\<close> vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
     (infixl "v*" 70)
   where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
 
 definition\<^marker>\<open>tag unimportant\<close> "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
 definition\<^marker>\<open>tag unimportant\<close> transpose where
   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
 definition\<^marker>\<open>tag unimportant\<close> "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
 definition\<^marker>\<open>tag unimportant\<close> "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
 definition\<^marker>\<open>tag unimportant\<close> "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
 definition\<^marker>\<open>tag unimportant\<close> "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
 
 lemma times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0" 
   by (simp add: matrix_matrix_mult_def zero_vec_def)
 
 lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0" 
   by (simp add: matrix_matrix_mult_def zero_vec_def)
 
 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
   by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
 
 lemma matrix_mul_lid [simp]:
   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   shows "mat 1 ** A = A"
-  apply (simp add: matrix_matrix_mult_def mat_def)
-  apply vector
-  apply (auto simp only: if_distrib if_distribR sum.delta'[OF finite]
-    mult_1_left mult_zero_left if_True UNIV_I)
-  done
+  unfolding matrix_matrix_mult_def mat_def
+  by (auto simp: if_distrib if_distribR sum.delta'[OF finite] cong: if_cong)
 
 lemma matrix_mul_rid [simp]:
   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   shows "A ** mat 1 = A"
-  apply (simp add: matrix_matrix_mult_def mat_def)
-  apply vector
-  apply (auto simp only: if_distrib if_distribR sum.delta[OF finite]
-    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
-  done
+  unfolding matrix_matrix_mult_def mat_def
+  by (auto simp: if_distrib if_distribR sum.delta'[OF finite] cong: if_cong)
 
 proposition matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
   apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
   apply (subst sum.swap)
   apply simp
   done
 
 proposition matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
     sum_distrib_left sum_distrib_right mult.assoc)
   apply (subst sum.swap)
   apply simp
   done
 
 proposition scalar_matrix_assoc:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
   by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
 
 proposition matrix_scalar_ac:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
   by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
 
 lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
   apply (vector matrix_vector_mult_def mat_def)
   apply (simp add: if_distrib if_distribR cong del: if_weak_cong)
   done
 
 lemma matrix_transpose_mul:
     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
 
 lemma matrix_mult_transpose_dot_column:
   shows "transpose A ** A = (\<chi> i j. inner (column i A) (column j A))"
   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
 
 lemma matrix_mult_transpose_dot_row:
   shows "A ** transpose A = (\<chi> i j. inner (row i A) (row j A))"
   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
 
 lemma matrix_eq:
   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
-  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
-  apply auto
-  apply (subst vec_eq_iff)
-  apply clarify
-  apply (clarsimp simp add: matrix_vector_mult_def if_distrib if_distribR vec_eq_iff cong del: if_weak_cong)
-  apply (erule_tac x="axis ia 1" in allE)
-  apply (erule_tac x="i" in allE)
-  apply (auto simp add: if_distrib if_distribR axis_def
-    sum.delta[OF finite] cong del: if_weak_cong)
-  done
+  shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?rhs
+  then show ?lhs
+    apply (subst vec_eq_iff)
+    apply (clarsimp simp add: matrix_vector_mult_def if_distrib if_distribR vec_eq_iff cong: if_cong)
+    apply (erule_tac x="axis ia 1" in allE)
+    apply (erule_tac x="i" in allE)
+    apply (auto simp add: if_distrib if_distribR axis_def
+        sum.delta[OF finite] cong del: if_weak_cong)
+    done
+qed auto
 
 lemma matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
 lemma dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
   apply (subst sum.swap)
   apply simp
   done
 
 lemma transpose_mat [simp]: "transpose (mat n) = mat n"
   by (vector transpose_def mat_def)
 
 lemma transpose_transpose [simp]: "transpose(transpose A) = A"
   by (vector transpose_def)
 
 lemma row_transpose [simp]: "row i (transpose A) = column i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
 lemma column_transpose [simp]: "column i (transpose A) = row i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
 lemma rows_transpose [simp]: "rows(transpose A) = columns A"
   by (auto simp add: rows_def columns_def intro: set_eqI)
 
 lemma columns_transpose [simp]: "columns(transpose A) = rows A"
   by (metis transpose_transpose rows_transpose)
 
 lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
   unfolding transpose_def
   by (simp add: vec_eq_iff)
 
 lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
   by (metis transpose_transpose)
 
 lemma matrix_mult_sum:
   "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
 
 lemma vector_componentwise:
   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
   by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
 
 lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
   by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
 
 
 text\<open>Correspondence between matrices and linear operators.\<close>
 
 definition\<^marker>\<open>tag important\<close> matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
   where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
 
 lemma matrix_id_mat_1: "matrix id = mat 1"
   by (simp add: mat_def matrix_def axis_def)
 
 lemma matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
   by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
 
 lemma matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum_distrib_left
       sum.distrib scaleR_right.sum)
 
 lemma vector_matrix_left_distrib [algebra_simps]:
   shows "(x + y) v* A = x v* A + y v* A"
   unfolding vector_matrix_mult_def
   by (simp add: algebra_simps sum.distrib vec_eq_iff)
 
 lemma matrix_vector_right_distrib [algebra_simps]:
   "A *v (x + y) = A *v x + A *v y"
   by (vector matrix_vector_mult_def sum.distrib distrib_left)
 
 lemma matrix_vector_mult_diff_distrib [algebra_simps]:
   fixes A :: "'a::ring_1^'n^'m"
   shows "A *v (x - y) = A *v x - A *v y"
   by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
 
 lemma matrix_vector_mult_scaleR[algebra_simps]:
   fixes A :: "real^'n^'m"
   shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
   using linear_iff matrix_vector_mul_linear by blast
 
 lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
 lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
 lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
   "(A + B) *v x = (A *v x) + (B *v x)"
   by (vector matrix_vector_mult_def sum.distrib distrib_right)
 
 lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
   fixes A :: "'a :: ring_1^'n^'m"
   shows "(A - B) *v x = (A *v x) - (B *v x)"
   by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
 
 lemma matrix_vector_column:
   "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
 
 subsection\<open>Inverse matrices  (not necessarily square)\<close>
 
 definition\<^marker>\<open>tag important\<close>
   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
 
 definition\<^marker>\<open>tag important\<close>
   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
 
 lemma inj_matrix_vector_mult:
   fixes A::"'a::field^'n^'m"
   assumes "invertible A"
   shows "inj ((*v) A)"
   by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
 
 lemma scalar_invertible:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A)"
 proof -
   obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
     using assms unfolding invertible_def by auto
   with \<open>k \<noteq> 0\<close>
   have "(k *\<^sub>R A) ** ((1/k) *\<^sub>R A') = mat 1" "((1/k) *\<^sub>R A') ** (k *\<^sub>R A) = mat 1"
     by (simp_all add: assms matrix_scalar_ac)
   thus "invertible (k *\<^sub>R A)"
     unfolding invertible_def by auto
 qed
 
 proposition scalar_invertible_iff:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
   by (simp add: assms scalar_invertible)
 
 lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
 lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
 lemma vector_scalar_commute:
   fixes A :: "'a::{field}^'m^'n"
   shows "A *v (c *s x) = c *s (A *v x)"
   by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
 
 lemma scalar_vector_matrix_assoc:
   fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
   shows "(k *s x) v* A = k *s (x v* A)"
   by (metis transpose_matrix_vector vector_scalar_commute)
  
 lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
 lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
 lemma vector_matrix_mul_rid [simp]:
   fixes v :: "('a::semiring_1)^'n"
   shows "v v* mat 1 = v"
   by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
 
 lemma scaleR_vector_matrix_assoc:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
   by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
 
 proposition vector_scaleR_matrix_ac:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
 proof -
   have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
     unfolding vector_matrix_mult_def
     by (simp add: algebra_simps)
   with scaleR_vector_matrix_assoc
   show "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
     by auto
 qed
 
 end