Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
authorhoelzl
Fri, 14 Dec 2012 15:46:01 +0100
changeset 50526 899c9c4e4a4c
parent 50525 46be26e02456
child 50527 2f9b5b0e388d
child 50534 75e02cd16533
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
NEWS
src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Derivative.thy
src/HOL/Multivariate_Analysis/Determinants.thy
src/HOL/Multivariate_Analysis/Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Fashoda.thy
src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Linear_Algebra.thy
src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy
src/HOL/Multivariate_Analysis/Operator_Norm.thy
src/HOL/Multivariate_Analysis/Path_Connected.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
src/HOL/Probability/Borel_Space.thy
src/HOL/Probability/Lebesgue_Measure.thy
src/HOL/Probability/Sigma_Algebra.thy
--- a/NEWS	Fri Dec 14 14:46:01 2012 +0100
+++ b/NEWS	Fri Dec 14 15:46:01 2012 +0100
@@ -227,6 +227,36 @@
 * HOL/Cardinals: Theories of ordinals and cardinals
 (supersedes the AFP entry "Ordinals_and_Cardinals").
 
+* HOL/Multivariate_Analysis:
+  Replaced "basis :: 'a::euclidean_space => nat => real" and
+  "\<Chi>\<Chi> :: (nat => real) => 'a::euclidean_space" on euclidean spaces by
+  using the inner product "_ \<bullet> _" with vectors from the Basis set.
+  "\<Chi>\<Chi> i. f i" is replaced by "SUM i : Basis. f i *r i".
+
+  With this change the following constants are also chanegd or removed:
+
+    DIM('a) :: nat   ~>   card (Basis :: 'a set)   (is an abbreviation)
+    a $$ i    ~>    inner a i  (where i : Basis)
+    cart_base i     removed
+    \<pi>, \<pi>'   removed
+
+  Theorems about these constants where removed.
+
+  Renamed lemmas:
+
+    component_le_norm   ~>   Basis_le_norm
+    euclidean_eq   ~>   euclidean_eq_iff
+    differential_zero_maxmin_component   ~>   differential_zero_maxmin_cart
+    euclidean_simps   ~>   inner_simps
+    independent_basis   ~>   independent_Basis
+    span_basis   ~>   span_Basis
+    in_span_basis   ~>   in_span_Basis
+    norm_bound_component_le   ~>   norm_boound_Basis_le
+    norm_bound_component_lt   ~>   norm_boound_Basis_lt
+    component_le_infnorm   ~>   Basis_le_infnorm
+
+  INCOMPATIBILITY.
+
 * HOL/Probability:
   - Add simproc "measurable" to automatically prove measurability
 
--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -22,6 +22,18 @@
   imports Convex_Euclidean_Space
 begin
 
+(** move this **)
+lemma divide_nonneg_nonneg:assumes "a \<ge> 0" "b \<ge> 0" shows "0 \<le> a / (b::real)"
+  apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto
+
+lemma continuous_setsum:
+  fixes f :: "'i \<Rightarrow> 'a::t2_space \<Rightarrow> 'b::real_normed_vector"
+  assumes f: "\<And>i. i \<in> I \<Longrightarrow> continuous F (f i)" shows "continuous F (\<lambda>x. \<Sum>i\<in>I. f i x)"
+proof cases
+  assume "finite I" from this f show ?thesis
+    by (induct I) (auto intro!: continuous_intros)
+qed (auto intro!: continuous_intros)
+
 lemma brouwer_compactness_lemma:
   assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::_::euclidean_space)))"
   obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)"
@@ -39,39 +51,39 @@
 qed
 
 lemma kuhn_labelling_lemma:
-  fixes type::"'a::euclidean_space"
-  assumes "(\<forall>x::'a. P x \<longrightarrow> P (f x))"
-    and "\<forall>x. P x \<longrightarrow> (\<forall>i<DIM('a). Q i \<longrightarrow> 0 \<le> x$$i \<and> x$$i \<le> 1)"
-  shows "\<exists>l. (\<forall>x.\<forall> i<DIM('a). l x i \<le> (1::nat)) \<and>
-             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (x$$i = 0) \<longrightarrow> (l x i = 0)) \<and>
-             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (x$$i = 1) \<longrightarrow> (l x i = 1)) \<and>
-             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$$i \<le> f(x)$$i) \<and>
-             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$$i \<le> x$$i)"
+  fixes P Q :: "'a::euclidean_space \<Rightarrow> bool"
+  assumes "(\<forall>x. P x \<longrightarrow> P (f x))"
+    and "\<forall>x. P x \<longrightarrow> (\<forall>i\<in>Basis. Q i \<longrightarrow> 0 \<le> x\<bullet>i \<and> x\<bullet>i \<le> 1)"
+  shows "\<exists>l. (\<forall>x.\<forall>i\<in>Basis. l x i \<le> (1::nat)) \<and>
+             (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 0) \<longrightarrow> (l x i = 0)) \<and>
+             (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 1) \<longrightarrow> (l x i = 1)) \<and>
+             (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x\<bullet>i \<le> f(x)\<bullet>i) \<and>
+             (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)\<bullet>i \<le> x\<bullet>i)"
 proof -
   have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)"
     by auto
   have *: "\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)"
     by auto
   show ?thesis
-    unfolding and_forall_thm
+    unfolding and_forall_thm Ball_def
     apply(subst choice_iff[THEN sym])+
     apply rule
     apply rule
   proof -
-    case goal1
-    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $$ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
-        (P x \<and> Q xa \<and> x $$ xa = 1 \<longrightarrow> y = 1) \<and>
-        (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $$ xa \<le> f x $$ xa) \<and>
-        (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $$ xa \<le> x $$ xa)"
+    case (goal1 x)
+    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x \<bullet> xa = 0 \<longrightarrow> y = (0::nat)) \<and>
+        (P x \<and> Q xa \<and> x \<bullet> xa = 1 \<longrightarrow> y = 1) \<and>
+        (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x \<bullet> xa \<le> f x \<bullet> xa) \<and>
+        (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x \<bullet> xa \<le> x \<bullet> xa)"
     {
-      assume "P x" "Q xa" "xa<DIM('a)"
-      then have "0 \<le> f x $$ xa \<and> f x $$ xa \<le> 1"
+      assume "P x" "Q xa" "xa\<in>Basis"
+      then have "0 \<le> f x \<bullet> xa \<and> f x \<bullet> xa \<le> 1"
         using assms(2)[rule_format,of "f x" xa]
         apply (drule_tac assms(1)[rule_format])
         apply auto
         done
     }
-    then have "xa<DIM('a) \<Longrightarrow> ?R 0 \<or> ?R 1" by auto
+    then have "xa\<in>Basis \<Longrightarrow> ?R 0 \<or> ?R 1" by auto
     then show ?case by auto
   qed
 qed
@@ -1363,50 +1375,56 @@
         apply(drule_tac assms(1)[rule_format]) by auto }
     hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 
 
-lemma brouwer_cube: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
-  assumes "continuous_on {0..\<chi>\<chi> i. 1} f" "f ` {0..\<chi>\<chi> i. 1} \<subseteq> {0..\<chi>\<chi> i. 1}"
-  shows "\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x = x" apply(rule ccontr) proof-
-  def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by(auto simp add:Suc_le_eq)
-  assume "\<not> (\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x = x)" hence *:"\<not> (\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x - x = 0)" by auto
+lemma brouwer_cube:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
+  assumes "continuous_on {0..(\<Sum>Basis)} f" "f ` {0..(\<Sum>Basis)} \<subseteq> {0..(\<Sum>Basis)}"
+  shows "\<exists>x\<in>{0..(\<Sum>Basis)}. f x = x"
+  proof (rule ccontr)
+  def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by(auto simp add: Suc_le_eq DIM_positive)
+  assume "\<not> (\<exists>x\<in>{0..\<Sum>Basis}. f x = x)" hence *:"\<not> (\<exists>x\<in>{0..\<Sum>Basis}. f x - x = 0)" by auto
   guess d apply(rule brouwer_compactness_lemma[OF compact_interval _ *]) 
     apply(rule continuous_on_intros assms)+ . note d=this[rule_format]
-  have *:"\<forall>x. x \<in> {0..\<chi>\<chi> i. 1} \<longrightarrow> f x \<in> {0..\<chi>\<chi> i. 1}"  "\<forall>x. x \<in> {0..(\<chi>\<chi> i. 1)::'a} \<longrightarrow>
-    (\<forall>i<DIM('a). True \<longrightarrow> 0 \<le> x $$ i \<and> x $$ i \<le> 1)"
+  have *:"\<forall>x. x \<in> {0..\<Sum>Basis} \<longrightarrow> f x \<in> {0..\<Sum>Basis}"  "\<forall>x. x \<in> {0..(\<Sum>Basis)::'a} \<longrightarrow>
+    (\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
     using assms(2)[unfolded image_subset_iff Ball_def] unfolding mem_interval by auto
   guess label using kuhn_labelling_lemma[OF *] apply-apply(erule exE,(erule conjE)+) . note label = this[rule_format]
-  have lem1:"\<forall>x\<in>{0..\<chi>\<chi> i. 1}.\<forall>y\<in>{0..\<chi>\<chi> i. 1}.\<forall>i<DIM('a). label x i \<noteq> label y i
-            \<longrightarrow> abs(f x $$ i - x $$ i) \<le> norm(f y - f x) + norm(y - x)" proof safe
-    fix x y::'a assume xy:"x\<in>{0..\<chi>\<chi> i. 1}" "y\<in>{0..\<chi>\<chi> i. 1}" fix i assume i:"label x i \<noteq> label y i" "i<DIM('a)"
+  have lem1:"\<forall>x\<in>{0..\<Sum>Basis}.\<forall>y\<in>{0..\<Sum>Basis}.\<forall>i\<in>Basis. label x i \<noteq> label y i
+            \<longrightarrow> abs(f x \<bullet> i - x \<bullet> i) \<le> norm(f y - f x) + norm(y - x)" proof safe
+    fix x y::'a assume xy:"x\<in>{0..\<Sum>Basis}" "y\<in>{0..\<Sum>Basis}"
+    fix i assume i:"label x i \<noteq> label y i" "i\<in>Basis"
     have *:"\<And>x y fx fy::real. (x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy)
              \<Longrightarrow> abs(fx - x) \<le> abs(fy - fx) + abs(y - x)" by auto
-    have "\<bar>(f x - x) $$ i\<bar> \<le> abs((f y - f x)$$i) + abs((y - x)$$i)" unfolding euclidean_simps
+    have "\<bar>(f x - x) \<bullet> i\<bar> \<le> abs((f y - f x)\<bullet>i) + abs((y - x)\<bullet>i)"
+      unfolding inner_simps
       apply(rule *) apply(cases "label x i = 0") apply(rule disjI1,rule) prefer 3 proof(rule disjI2,rule)
       assume lx:"label x i = 0" hence ly:"label y i = 1" using i label(1)[of i y] by auto
-      show "x $$ i \<le> f x $$ i" apply(rule label(4)[rule_format]) using xy lx i(2) by auto
-      show "f y $$ i \<le> y $$ i" apply(rule label(5)[rule_format]) using xy ly i(2) by auto next
+      show "x \<bullet> i \<le> f x \<bullet> i" apply(rule label(4)[rule_format]) using xy lx i(2) by auto
+      show "f y \<bullet> i \<le> y \<bullet> i" apply(rule label(5)[rule_format]) using xy ly i(2) by auto next
       assume "label x i \<noteq> 0" hence l:"label x i = 1" "label y i = 0"
         using i label(1)[of i x] label(1)[of i y] by auto
-      show "f x $$ i \<le> x $$ i" apply(rule label(5)[rule_format]) using xy l i(2) by auto
-      show "y $$ i \<le> f y $$ i" apply(rule label(4)[rule_format]) using xy l i(2) by auto qed 
-    also have "\<dots> \<le> norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule component_le_norm)+
-    finally show "\<bar>f x $$ i - x $$ i\<bar> \<le> norm (f y - f x) + norm (y - x)" unfolding euclidean_simps . qed
-  have "\<exists>e>0. \<forall>x\<in>{0..\<chi>\<chi> i. 1}. \<forall>y\<in>{0..\<chi>\<chi> i. 1}. \<forall>z\<in>{0..\<chi>\<chi> i. 1}. \<forall>i<DIM('a).
-    norm(x - z) < e \<and> norm(y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow> abs((f(z) - z)$$i) < d / (real n)" proof-
-    have d':"d / real n / 8 > 0" apply(rule divide_pos_pos)+ using d(1) unfolding n_def by auto
-    have *:"uniformly_continuous_on {0..\<chi>\<chi> i. 1} f" by(rule compact_uniformly_continuous[OF assms(1) compact_interval])
+      show "f x \<bullet> i \<le> x \<bullet> i" apply(rule label(5)[rule_format]) using xy l i(2) by auto
+      show "y \<bullet> i \<le> f y \<bullet> i" apply(rule label(4)[rule_format]) using xy l i(2) by auto qed 
+    also have "\<dots> \<le> norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule Basis_le_norm[OF i(2)])+
+    finally show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)" unfolding inner_simps .
+  qed
+  have "\<exists>e>0. \<forall>x\<in>{0..\<Sum>Basis}. \<forall>y\<in>{0..\<Sum>Basis}. \<forall>z\<in>{0..\<Sum>Basis}. \<forall>i\<in>Basis.
+    norm(x - z) < e \<and> norm(y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow> abs((f(z) - z)\<bullet>i) < d / (real n)" proof-
+    have d':"d / real n / 8 > 0" apply(rule divide_pos_pos)+ using d(1) unfolding n_def by (auto simp:  DIM_positive)
+    have *:"uniformly_continuous_on {0..\<Sum>Basis} f" by(rule compact_uniformly_continuous[OF assms(1) compact_interval])
     guess e using *[unfolded uniformly_continuous_on_def,rule_format,OF d'] apply-apply(erule exE,(erule conjE)+) .
     note e=this[rule_format,unfolded dist_norm]
     show ?thesis apply(rule_tac x="min (e/2) (d/real n/8)" in exI)
-    proof safe show "0 < min (e / 2) (d / real n / 8)" using d' e by auto
-      fix x y z i assume as:"x \<in> {0..\<chi>\<chi> i. 1}" "y \<in> {0..\<chi>\<chi> i. 1}" "z \<in> {0..\<chi>\<chi> i. 1}"
+    proof safe
+      show "0 < min (e / 2) (d / real n / 8)" using d' e by auto
+      fix x y z i assume as:"x \<in> {0..\<Sum>Basis}" "y \<in> {0..\<Sum>Basis}" "z \<in> {0..\<Sum>Basis}"
         "norm (x - z) < min (e / 2) (d / real n / 8)"
-        "norm (y - z) < min (e / 2) (d / real n / 8)" "label x i \<noteq> label y i" and i:"i<DIM('a)"
+        "norm (y - z) < min (e / 2) (d / real n / 8)" "label x i \<noteq> label y i" and i:"i\<in>Basis"
       have *:"\<And>z fz x fx n1 n2 n3 n4 d4 d::real. abs(fx - x) \<le> n1 + n2 \<Longrightarrow> abs(fx - fz) \<le> n3 \<Longrightarrow> abs(x - z) \<le> n4 \<Longrightarrow>
         n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow> (8 * d4 = d) \<Longrightarrow> abs(fz - z) < d" by auto
-      show "\<bar>(f z - z) $$ i\<bar> < d / real n" unfolding euclidean_simps proof(rule *)
-        show "\<bar>f x $$ i - x $$ i\<bar> \<le> norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as i  by auto
-        show "\<bar>f x $$ i - f z $$ i\<bar> \<le> norm (f x - f z)" "\<bar>x $$ i - z $$ i\<bar> \<le> norm (x - z)"
-          unfolding euclidean_component_diff[THEN sym] by(rule component_le_norm)+
+      show "\<bar>(f z - z) \<bullet> i\<bar> < d / real n" unfolding inner_simps proof(rule *)
+        show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as i  by auto
+        show "\<bar>f x \<bullet> i - f z \<bullet> i\<bar> \<le> norm (f x - f z)" "\<bar>x \<bullet> i - z \<bullet> i\<bar> \<le> norm (x - z)"
+          unfolding inner_diff_left[THEN sym] by(rule Basis_le_norm[OF i])+
         have tria:"norm (y - x) \<le> norm (y - z) + norm (x - z)" using dist_triangle[of y x z,unfolded dist_norm]
           unfolding norm_minus_commute by auto
         also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_mono) using as(4,5) by auto
@@ -1418,95 +1436,99 @@
   guess p using real_arch_simple[of "1 + real n / e"] .. note p=this
   have "1 + real n / e > 0" apply(rule add_pos_pos) defer apply(rule divide_pos_pos) using e(1) n by auto
   hence "p > 0" using p by auto
-  def b \<equiv> "\<lambda>i. i - 1::nat" have b:"bij_betw b {1..n} {..<DIM('a)}"
-    unfolding bij_betw_def inj_on_def b_def n_def apply rule defer
-    apply safe defer unfolding image_iff apply(rule_tac x="Suc x" in bexI) by auto
+
+  obtain b :: "nat \<Rightarrow> 'a" where b: "bij_betw b {1..n} Basis"
+    by atomize_elim (auto simp: n_def intro!: finite_same_card_bij)
   def b' \<equiv> "inv_into {1..n} b"
-  have b':"bij_betw b' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF b] unfolding b'_def n_def by auto
-  have bb'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> b (b' i) = i" unfolding b'_def apply(rule f_inv_into_f) using b  
-    unfolding bij_betw_def by auto
-  have b'b[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> b' (b i) = i" unfolding b'_def apply(rule inv_into_f_eq)
-    using b unfolding n_def bij_betw_def by auto
+  then have b': "bij_betw b' Basis {1..n}"
+    using bij_betw_inv_into[OF b] by auto
+  then have b'_Basis: "\<And>i. i \<in> Basis \<Longrightarrow> b' i \<in> {Suc 0 .. n}"
+    unfolding bij_betw_def by (auto simp: set_eq_iff)
+  have bb'[simp]:"\<And>i. i \<in> Basis \<Longrightarrow> b (b' i) = i"
+    unfolding b'_def using b by (auto simp: f_inv_into_f bij_betw_def)
+  have b'b[simp]:"\<And>i. i \<in> {1..n} \<Longrightarrow> b' (b i) = i"
+    unfolding b'_def using b by (auto simp: inv_into_f_eq bij_betw_def)
   have *:"\<And>x::nat. x=0 \<or> x=1 \<longleftrightarrow> x\<le>1" by auto
-  have b'':"\<And>j. j\<in>{1..n} \<Longrightarrow> b j <DIM('a)" using b unfolding bij_betw_def by auto
+  have b'':"\<And>j. j\<in>{Suc 0..n} \<Longrightarrow> b j \<in>Basis" using b unfolding bij_betw_def by auto
   have q1:"0 < p" "0 < n"  "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow>
-    (\<forall>i\<in>{1..n}. (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0 \<or> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
+    (\<forall>i\<in>{1..n}. (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0 \<or>
+                (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
     unfolding * using `p>0` `n>0` using label(1)[OF b'']  by auto
-  have q2:"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = 0 \<longrightarrow> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0)"
-    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = p \<longrightarrow> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
-    apply(rule,rule,rule,rule) defer proof(rule,rule,rule,rule) fix x i 
-    assume as:"\<forall>i\<in>{1..n}. x i \<le> p" "i \<in> {1..n}"
-    { assume "x i = p \<or> x i = 0" 
-      have "(\<chi>\<chi> i. real (x (b' i)) / real p) \<in> {0::'a..\<chi>\<chi> i. 1}" unfolding mem_interval 
-        apply safe unfolding euclidean_lambda_beta euclidean_component_zero
-      proof (simp_all only: if_P) fix j assume j':"j<DIM('a)"
-        hence j:"b' j \<in> {1..n}" using b' unfolding n_def bij_betw_def by auto
-        show "0 \<le> real (x (b' j)) / real p"
-          apply(rule divide_nonneg_pos) using `p>0` using as(1)[rule_format,OF j] by auto
-        show "real (x (b' j)) / real p \<le> 1" unfolding divide_le_eq_1
-          using as(1)[rule_format,OF j] by auto qed } note cube=this
-    { assume "x i = p" thus "(label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1" unfolding o_def
-        apply- apply(rule label(3)) apply(rule b'') using cube using as `p>0`
-      proof safe assume i:"i\<in>{1..n}"
-        show "((\<chi>\<chi> ia. real (x (b' ia)) / real (x i))::'a) $$ b i = 1"
-          unfolding euclidean_lambda_beta apply(subst if_P) apply(rule b''[OF i]) unfolding b'b[OF i] 
-          unfolding  `x i = p` using q1(1) by auto
-      qed auto }
-    { assume "x i = 0" thus "(label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0" unfolding o_def
-        apply-apply(rule label(2)[OF b'']) using cube using as `p>0`
-      proof safe assume i:"i\<in>{1..n}"
-        show "((\<chi>\<chi> ia. real (x (b' ia)) / real p)::'a) $$ b i = 0"
-          unfolding euclidean_lambda_beta apply (subst if_P) apply(rule b''[OF i]) unfolding b'b[OF i] 
-          unfolding `x i = 0` using q1(1) by auto 
-      qed auto }
+  have q2:"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = 0 \<longrightarrow> 
+      (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
+    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = p \<longrightarrow>
+      (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
+    apply(rule,rule,rule,rule)
+    defer
+  proof(rule,rule,rule,rule)
+    fix x i assume as:"\<forall>i\<in>{1..n}. x i \<le> p" "i \<in> {1..n}"
+    { assume "x i = p \<or> x i = 0"
+      have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> {0::'a..\<Sum>Basis}"
+        unfolding mem_interval using as b'_Basis
+        by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) }
+    note cube=this
+    { assume "x i = p" thus "(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1"
+        unfolding o_def using cube as `p>0`
+        by (intro label(3)) (auto simp add: b'') }
+    { assume "x i = 0" thus "(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0"
+        unfolding o_def using cube as `p>0`
+        by (intro label(2)) (auto simp add: b'') }
   qed
   guess q apply(rule kuhn_lemma[OF q1 q2]) . note q=this
-  def z \<equiv> "(\<chi>\<chi> i. real (q (b' i)) / real p)::'a"
-  have "\<exists>i<DIM('a). d / real n \<le> abs((f z - z)$$i)" proof(rule ccontr)
-    have "\<forall>i<DIM('a). q (b' i) \<in> {0..<p}" using q(1) b'[unfolded bij_betw_def] by auto 
-    hence "\<forall>i<DIM('a). q (b' i) \<in> {0..p}" apply-apply(rule,erule_tac x=i in allE) by auto
-    hence "z\<in>{0..\<chi>\<chi> i.1}" unfolding z_def mem_interval apply safe unfolding euclidean_lambda_beta
-      unfolding euclidean_component_zero apply (simp_all only: if_P)
-      apply(rule divide_nonneg_pos) using `p>0` unfolding divide_le_eq_1 by auto
-    hence d_fz_z:"d \<le> norm (f z - z)" apply(drule_tac d) .
-    case goal1 hence as:"\<forall>i<DIM('a). \<bar>f z $$ i - z $$ i\<bar> < d / real n" using `n>0` by(auto simp add:not_le)
-    have "norm (f z - z) \<le> (\<Sum>i<DIM('a). \<bar>f z $$ i - z $$ i\<bar>)" unfolding euclidean_component_diff[THEN sym] by(rule norm_le_l1)
-    also have "\<dots> < (\<Sum>i<DIM('a). d / real n)" apply(rule setsum_strict_mono) using as by auto
-    also have "\<dots> = d" unfolding real_eq_of_nat n_def using n using DIM_positive[where 'a='a] by auto
+  def z \<equiv> "(\<Sum>i\<in>Basis. (real (q (b' i)) / real p) *\<^sub>R i)::'a"
+  have "\<exists>i\<in>Basis. d / real n \<le> abs((f z - z)\<bullet>i)"
+  proof(rule ccontr)
+    have "\<forall>i\<in>Basis. q (b' i) \<in> {0..p}"
+      using q(1) b' by (auto intro: less_imp_le simp: bij_betw_def)
+    hence "z\<in>{0..\<Sum>Basis}"
+      unfolding z_def mem_interval using b'_Basis
+      by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1)
+    hence d_fz_z:"d \<le> norm (f z - z)" by (rule d)
+    case goal1
+    hence as:"\<forall>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar> < d / real n"
+      using `n>0` by(auto simp add: not_le inner_simps)
+    have "norm (f z - z) \<le> (\<Sum>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar>)"
+      unfolding inner_diff_left[symmetric] by(rule norm_le_l1)
+    also have "\<dots> < (\<Sum>(i::'a)\<in>Basis. d / real n)" apply(rule setsum_strict_mono) using as by auto
+    also have "\<dots> = d" using DIM_positive[where 'a='a] by (auto simp: real_eq_of_nat n_def)
     finally show False using d_fz_z by auto qed then guess i .. note i=this
   have *:"b' i \<in> {1..n}" using i using b'[unfolded bij_betw_def] by auto
   guess r using q(2)[rule_format,OF *] .. then guess s apply-apply(erule exE,(erule conjE)+) . note rs=this[rule_format]
-  have b'_im:"\<And>i. i<DIM('a) \<Longrightarrow>  b' i \<in> {1..n}" using b' unfolding bij_betw_def by auto
-  def r' \<equiv> "(\<chi>\<chi> i. real (r (b' i)) / real p)::'a"
-  have "\<And>i. i<DIM('a) \<Longrightarrow> r (b' i) \<le> p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2])
+  have b'_im:"\<And>i. i\<in>Basis \<Longrightarrow>  b' i \<in> {1..n}" using b' unfolding bij_betw_def by auto
+  def r' \<equiv> "(\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i)::'a"
+  have "\<And>i. i\<in>Basis \<Longrightarrow> r (b' i) \<le> p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2])
     using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
-  hence "r' \<in> {0..\<chi>\<chi> i. 1}"  unfolding r'_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
-    apply (simp only: if_P)
-    apply(rule divide_nonneg_pos) using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
-  def s' \<equiv> "(\<chi>\<chi> i. real (s (b' i)) / real p)::'a"
-  have "\<And>i. i<DIM('a) \<Longrightarrow> s (b' i) \<le> p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2])
+  hence "r' \<in> {0..\<Sum>Basis}"
+    unfolding r'_def mem_interval using b'_Basis
+    by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1)
+  def s' \<equiv> "(\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)::'a"
+  have "\<And>i. i\<in>Basis \<Longrightarrow> s (b' i) \<le> p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2])
     using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
-  hence "s' \<in> {0..\<chi>\<chi> i.1}" unfolding s'_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
-    apply (simp_all only: if_P) apply(rule divide_nonneg_pos) using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
-  have "z\<in>{0..\<chi>\<chi> i.1}" unfolding z_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
-    apply (simp_all only: if_P) apply(rule divide_nonneg_pos) using q(1)[rule_format,OF b'_im] `p>0` by(auto intro:less_imp_le)
+  hence "s' \<in> {0..\<Sum>Basis}"
+    unfolding s'_def mem_interval using b'_Basis
+    by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1)
+  have "z\<in>{0..\<Sum>Basis}"
+    unfolding z_def mem_interval using b'_Basis q(1)[rule_format,OF b'_im] `p>0`
+    by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1 less_imp_le)
   have *:"\<And>x. 1 + real x = real (Suc x)" by auto
-  { have "(\<Sum>i<DIM('a). \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>i<DIM('a). 1)" 
+  { have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)" 
       apply(rule setsum_mono) using rs(1)[OF b'_im] by(auto simp add:* field_simps)
-    also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
-      by(auto simp add:field_simps)
-    finally have "(\<Sum>i<DIM('a). \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" . } moreover
-  { have "(\<Sum>i<DIM('a). \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>i<DIM('a). 1)" 
+    also have "\<dots> < e * real p" using p `e>0` `p>0`
+      by(auto simp add: field_simps n_def real_of_nat_def)
+    finally have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" . } moreover
+  { have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)" 
       apply(rule setsum_mono) using rs(2)[OF b'_im] by(auto simp add:* field_simps)
-    also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
-      by(auto simp add:field_simps)
-    finally have "(\<Sum>i<DIM('a). \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" . } ultimately
-  have "norm (r' - z) < e" "norm (s' - z) < e" unfolding r'_def s'_def z_def apply-
-    apply(rule_tac[!] le_less_trans[OF norm_le_l1]) using `p>0`
-    by(auto simp add:field_simps setsum_divide_distrib[THEN sym])
-  hence "\<bar>(f z - z) $$ i\<bar> < d / real n" apply-apply(rule e(2)[OF `r'\<in>{0..\<chi>\<chi> i.1}` `s'\<in>{0..\<chi>\<chi> i.1}` `z\<in>{0..\<chi>\<chi> i.1}`])
-    using rs(3) unfolding r'_def[symmetric] s'_def[symmetric] o_def bb' using i by auto
-  thus False using i by auto qed
+    also have "\<dots> < e * real p" using p `e>0` `p>0`
+      by(auto simp add: field_simps n_def real_of_nat_def)
+    finally have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" . } ultimately
+  have "norm (r' - z) < e" "norm (s' - z) < e" unfolding r'_def s'_def z_def using `p>0`
+    by (rule_tac[!] le_less_trans[OF norm_le_l1])
+       (auto simp add: field_simps setsum_divide_distrib[symmetric] inner_diff_left)
+  hence "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
+    using rs(3) i unfolding r'_def[symmetric] s'_def[symmetric] o_def bb'
+    by (intro e(2)[OF `r'\<in>{0..\<Sum>Basis}` `s'\<in>{0..\<Sum>Basis}` `z\<in>{0..\<Sum>Basis}`]) auto
+  thus False using i by auto
+qed
 
 subsection {* Retractions. *}
 
@@ -1551,12 +1573,14 @@
 
 subsection {*So the Brouwer theorem for any set with nonempty interior. *}
 
-lemma brouwer_weak: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
+lemma brouwer_weak:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
   assumes "compact s" "convex s" "interior s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
   obtains x where "x \<in> s" "f x = x" proof-
-  have *:"interior {0::'a..\<chi>\<chi> i.1} \<noteq> {}" unfolding interior_closed_interval interval_eq_empty by auto
-  have *:"{0::'a..\<chi>\<chi> i.1} homeomorphic s" using homeomorphic_convex_compact[OF convex_interval(1) compact_interval * assms(2,1,3)] .
-  have "\<forall>f. continuous_on {0::'a..\<chi>\<chi> i.1} f \<and> f ` {0::'a..\<chi>\<chi> i.1} \<subseteq> {0::'a..\<chi>\<chi> i.1} \<longrightarrow> (\<exists>x\<in>{0::'a..\<chi>\<chi> i.1}. f x = x)"
+  have *:"interior {0::'a..\<Sum>Basis} \<noteq> {}" unfolding interior_closed_interval interval_eq_empty by auto
+  have *:"{0::'a..\<Sum>Basis} homeomorphic s" using homeomorphic_convex_compact[OF convex_interval(1) compact_interval * assms(2,1,3)] .
+  have "\<forall>f. continuous_on {0::'a..\<Sum>Basis} f \<and> f ` {0::'a..\<Sum>Basis} \<subseteq> {0::'a..\<Sum>Basis} \<longrightarrow> 
+    (\<exists>x\<in>{0::'a..\<Sum>Basis}. f x = x)"
     using brouwer_cube by auto
   thus ?thesis unfolding homeomorphic_fixpoint_property[OF *] apply(erule_tac x=f in allE)
     apply(erule impE) defer apply(erule bexE) apply(rule_tac x=y in that) using assms by auto qed
@@ -1609,49 +1633,45 @@
 
 subsection {*Bijections between intervals. *}
 
-definition "interval_bij = (\<lambda> (a::'a,b::'a) (u::'a,v::'a) (x::'a::ordered_euclidean_space).
-    (\<chi>\<chi> i. u$$i + (x$$i - a$$i) / (b$$i - a$$i) * (v$$i - u$$i))::'a)"
+definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::ordered_euclidean_space" where
+  "interval_bij \<equiv> \<lambda>(a, b) (u, v) x. (\<Sum>i\<in>Basis. (u\<bullet>i + (x\<bullet>i - a\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (v\<bullet>i - u\<bullet>i)) *\<^sub>R i)"
 
 lemma interval_bij_affine:
- "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi>\<chi> i. (v$$i - u$$i) / (b$$i - a$$i) * x$$i) +
-            (\<chi>\<chi> i. u$$i - (v$$i - u$$i) / (b$$i - a$$i) * a$$i))"
-  apply rule apply(subst euclidean_eq,safe) unfolding euclidean_simps interval_bij_def euclidean_lambda_beta
-  by(auto simp add: field_simps add_divide_distrib[THEN sym])
+  "interval_bij (a,b) (u,v) = (\<lambda>x. (\<Sum>i\<in>Basis. ((v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (x\<bullet>i)) *\<^sub>R i) +
+    (\<Sum>i\<in>Basis. (u\<bullet>i - (v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (a\<bullet>i)) *\<^sub>R i))"
+  by (auto simp: setsum_addf[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff
+                 field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum_cong)
 
 lemma continuous_interval_bij:
   "continuous (at x) (interval_bij (a,b::'a::ordered_euclidean_space) (u,v::'a))" 
-  unfolding interval_bij_affine apply(rule continuous_intros)
-    apply(rule linear_continuous_at) unfolding linear_conv_bounded_linear[THEN sym]
-    unfolding linear_def euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta prefer 3
-    apply(rule continuous_intros) by(auto simp add:field_simps add_divide_distrib[THEN sym])
+  by (auto simp add: divide_inverse interval_bij_def intro!: continuous_setsum continuous_intros)
 
 lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a,b) (u,v))"
   apply(rule continuous_at_imp_continuous_on) by(rule, rule continuous_interval_bij)
 
-(** move this **)
-lemma divide_nonneg_nonneg:assumes "a \<ge> 0" "b \<ge> 0" shows "0 \<le> a / (b::real)"
-  apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto
-
-lemma in_interval_interval_bij: assumes "x \<in> {a..b}" "{u..v} \<noteq> {}"
-  shows "interval_bij (a,b) (u,v) x \<in> {u..v::'a::ordered_euclidean_space}" 
-  unfolding interval_bij_def split_conv mem_interval apply safe unfolding euclidean_lambda_beta
-proof (simp_all only: if_P)
-  fix i assume i:"i<DIM('a)" have "{a..b} \<noteq> {}" using assms by auto
-  hence *:"a$$i \<le> b$$i" "u$$i \<le> v$$i" using assms(2) unfolding interval_eq_empty not_ex apply-
-    apply(erule_tac[!] x=i in allE)+ by auto
-  have x:"a$$i\<le>x$$i" "x$$i\<le>b$$i" using assms(1)[unfolded mem_interval] using i by auto
-  have "0 \<le> (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i)"
-    apply(rule mult_nonneg_nonneg) apply(rule divide_nonneg_nonneg)
-    using * x by(auto simp add:field_simps)
-  thus "u $$ i \<le> u $$ i + (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i)" using * by auto
-  have "((x $$ i - a $$ i) / (b $$ i - a $$ i)) * (v $$ i - u $$ i) \<le> 1 * (v $$ i - u $$ i)"
+lemma in_interval_interval_bij:
+  fixes a b u v x :: "'a::ordered_euclidean_space"
+  assumes "x \<in> {a..b}" "{u..v} \<noteq> {}" shows "interval_bij (a,b) (u,v) x \<in> {u..v}"
+  apply (simp only: interval_bij_def split_conv mem_interval inner_setsum_left_Basis cong: ball_cong)
+proof safe
+  fix i :: 'a assume i:"i\<in>Basis"
+  have "{a..b} \<noteq> {}" using assms by auto
+  with i have *: "a\<bullet>i \<le> b\<bullet>i" "u\<bullet>i \<le> v\<bullet>i"
+    using assms(2) by (auto simp add: interval_eq_empty not_less)
+  have x: "a\<bullet>i\<le>x\<bullet>i" "x\<bullet>i\<le>b\<bullet>i"
+    using assms(1)[unfolded mem_interval] using i by auto
+  have "0 \<le> (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)"
+    using * x by (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
+  thus "u \<bullet> i \<le> u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)"
+    using * by auto
+  have "((x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (v \<bullet> i - u \<bullet> i) \<le> 1 * (v \<bullet> i - u \<bullet> i)"
     apply(rule mult_right_mono) unfolding divide_le_eq_1 using * x by auto
-  thus "u $$ i + (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i) \<le> v $$ i" using * by auto
+  thus "u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i) \<le> v \<bullet> i" using * by auto
 qed
 
-lemma interval_bij_bij: fixes x::"'a::ordered_euclidean_space" assumes "\<forall>i. a$$i < b$$i \<and> u$$i < v$$i" 
-  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
-  unfolding interval_bij_def split_conv euclidean_eq[where 'a='a]
-  apply(rule,insert assms,erule_tac x=i in allE) by auto
+lemma interval_bij_bij: 
+  "\<forall>(i::'a::ordered_euclidean_space)\<in>Basis. a\<bullet>i < b\<bullet>i \<and> u\<bullet>i < v\<bullet>i \<Longrightarrow>
+    interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
+  by (auto simp: interval_bij_def euclidean_eq_iff[where 'a='a])
 
 end
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -328,283 +328,18 @@
   fixes f:: "'a \<Rightarrow> real ^'n"
   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
-proof -
-  let ?d = "real CARD('n)"
-  let ?nf = "\<lambda>x. norm (f x)"
-  let ?U = "UNIV :: 'n set"
-  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
-    by (rule setsum_commute)
-  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
-  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
-    apply (rule setsum_mono)
-    apply (rule norm_le_l1_cart)
-    done
-  also have "\<dots> \<le> 2 * ?d * e"
-    unfolding th0 th1
-  proof(rule setsum_bounded)
-    fix i assume i: "i \<in> ?U"
-    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
-    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
-    have thp: "P = ?Pp \<union> ?Pn" by auto
-    have thp0: "?Pp \<inter> ?Pn ={}" by auto
-    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
-    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
-      using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
-      by (auto intro: abs_le_D1)
-    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
-      using component_le_norm_cart[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
-      by (auto simp add: setsum_negf intro: abs_le_D1)
-    have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
-      apply (subst thp)
-      apply (rule setsum_Un_zero)
-      using fP thp0 apply auto
-      done
-    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
-    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
-  qed
-  finally show ?thesis .
-qed
-
-lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
-
-lemma split_dimensions'[consumes 1]:
-  assumes "k < DIM('a::euclidean_space^'b)"
-  obtains i j where "i < CARD('b::finite)"
-    and "j < DIM('a::euclidean_space)"
-    and "k = j + i * DIM('a::euclidean_space)"
-  using split_times_into_modulo[OF assms[simplified]] .
-
-lemma cart_euclidean_bound[intro]:
-  assumes j:"j < DIM('a::euclidean_space)"
-  shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::euclidean_space)"
-  using linear_less_than_times[OF pi'_range j, of i] .
-
-lemma (in euclidean_space) forall_CARD_DIM:
-  "(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"
-   (is "?l \<longleftrightarrow> ?r")
-proof (safe elim!: split_times_into_modulo)
-  fix i :: 'b and j
-  assume "j < DIM('a)"
-  note linear_less_than_times[OF pi'_range[of i] this]
-  moreover assume "?l"
-  ultimately show "P (j + \<pi>' i * DIM('a))" by auto
-next
-  fix i j
-  assume "i < CARD('b)" "j < DIM('a)" and "?r"
-  from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`
-  show "P (j + i * DIM('a))" by simp
-qed
-
-lemma (in euclidean_space) exists_CARD_DIM:
-  "(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"
-  using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast
-
-lemma forall_CARD:
-  "(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))"
-  using forall_CARD_DIM[where 'a=real, of P] by simp
-
-lemma exists_CARD:
-  "(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))"
-  using exists_CARD_DIM[where 'a=real, of P] by simp
-
-lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
-
-lemma cart_euclidean_nth[simp]:
-  fixes x :: "('a::euclidean_space, 'b::finite) vec"
-  assumes j:"j < DIM('a)"
-  shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
-  unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta
-  by (simp add: setsum_cases)
-
-lemma real_euclidean_nth:
-  fixes x :: "real^'n"
-  shows "x $$ \<pi>' i = (x $ i :: real)"
-  using cart_euclidean_nth[where 'a=real, of 0 x i] by simp
-
-lemmas nth_conv_component = real_euclidean_nth[symmetric]
-
-lemma mult_split_eq:
-  fixes A :: nat assumes "x < A" "y < A"
-  shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j"
-proof
-  assume *: "x + i * A = y + j * A"
-  { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A"
-    hence "x + i * A < Suc i * A" using `x < A` by simp
-    also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp
-    also have "\<dots> \<le> y + j * A" by simp
-    finally have "i = j" using * by simp }
-  note eq = this
-
-  have "i = j"
-  proof (cases rule: linorder_cases)
-    assume "i < j"
-    from eq[OF this `x < A` *] show "i = j" by simp
-  next
-    assume "j < i"
-    from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
-  qed simp
-  thus "x = y \<and> i = j" using * by simp
-qed simp
+  using setsum_norm_allsubsets_bound[OF assms]
+  by (simp add: DIM_cart Basis_real_def)
 
 instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
 proof
   fix x y::"'a^'b"
-  show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)"
-    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps)
-  show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)"
-    unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
-qed
-
-
-subsection{* Basis vectors in coordinate directions. *}
-
-definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)"
-
-lemma basis_component [simp]: "cart_basis k $ i = (if k=i then 1 else 0)"
-  unfolding cart_basis_def by simp
-
-lemma norm_basis[simp]:
-  shows "norm (cart_basis k :: real ^'n) = 1"
-  apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def
-  apply (vector delta_mult_idempotent)
-  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] apply auto
-  done
-
-lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1"
-  by (rule norm_basis)
-
-lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
-  by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp
-
-lemma vector_choose_dist:
-  assumes e: "0 <= e"
-  shows "\<exists>(y::real^'n). dist x y = e"
-proof -
-  from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
-    by blast
-  then have "dist x (x - c) = e" by (simp add: dist_norm)
-  then show ?thesis by blast
+  show "(x \<le> y) = (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
+    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: Basis_vec_def inner_axis)
+  show"(x < y) = (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"
+    unfolding less_vec_def apply(subst eucl_less) by (simp add: Basis_vec_def inner_axis)
 qed
 
-lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
-  by (simp add: inj_on_def vec_eq_iff)
-
-lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)"
-  (is "?lhs = ?rhs" is "setsum ?f ?S = _")
-  by (auto simp add: vec_eq_iff
-      if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
-
-lemma smult_conv_scaleR: "c *s x = scaleR c x"
-  unfolding vector_scalar_mult_def scaleR_vec_def by simp
-
-lemma basis_expansion': "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
-  by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])
-
-lemma basis_expansion_unique:
-  "setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
-  by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
-
-lemma dot_basis: "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
-  by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
-           cong del: if_weak_cong)
-
-lemma inner_basis:
-  fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
-  shows "inner (cart_basis i) x = inner 1 (x $ i)"
-    and "inner x (cart_basis i) = inner (x $ i) 1"
-  unfolding inner_vec_def cart_basis_def
-  by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
-
-lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
-  by (auto simp add: vec_eq_iff)
-
-lemma basis_nonzero: "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
-  by (simp add: basis_eq_0)
-
-text {* some lemmas to map between Eucl and Cart *}
-lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
-  unfolding basis_vec_def using pi'_range[where 'n='a]
-  by (auto simp: vec_eq_iff axis_def)
-
-subsection {* Orthogonality on cartesian products *}
-
-lemma orthogonal_basis: "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
-  by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib
-                     cond_application_beta setsum_delta cong del: if_weak_cong)
-
-lemma orthogonal_basis_basis: "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
-  unfolding orthogonal_basis[of i] basis_component[of j] by simp
-
-subsection {* Linearity on cartesian products *}
-
-lemma linear_vmul_component:
-  assumes "linear f"
-  shows "linear (\<lambda>x. f x $ k *\<^sub>R v)"
-  using assms by (auto simp add: linear_def algebra_simps)
-
-
-subsection {* Adjoints on cartesian products *}
-
-text {* TODO: The following lemmas about adjoints should hold for any
-Hilbert space (i.e. complete inner product space).
-(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})
-*}
-
-lemma adjoint_works_lemma:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
-  assumes lf: "linear f"
-  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
-proof -
-  let ?N = "UNIV :: 'n set"
-  let ?M = "UNIV :: 'm set"
-  have fN: "finite ?N" by simp
-  have fM: "finite ?M" by simp
-  { fix y:: "real ^ 'm"
-    let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> y)) :: real ^ 'n"
-    { fix x
-      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"
-        by (simp only: basis_expansion')
-      also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"
-        unfolding linear_setsum[OF lf fN]
-        by (simp add: linear_cmul[OF lf])
-      finally have "f x \<bullet> y = x \<bullet> ?w"
-        by (simp add: inner_vec_def setsum_left_distrib
-            setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
-    }
-  }
-  then show ?thesis
-    unfolding adjoint_def some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
-    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
-    by metis
-qed
-
-lemma adjoint_works:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
-  assumes lf: "linear f"
-  shows "x \<bullet> adjoint f y = f x \<bullet> y"
-  using adjoint_works_lemma[OF lf] by metis
-
-lemma adjoint_linear:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
-  assumes lf: "linear f"
-  shows "linear (adjoint f)"
-  unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe
-  unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto
-
-lemma adjoint_clauses:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
-  assumes lf: "linear f"
-  shows "x \<bullet> adjoint f y = f x \<bullet> y"
-    and "adjoint f y \<bullet> x = y \<bullet> f x"
-  by (simp_all add: adjoint_works[OF lf] inner_commute)
-
-lemma adjoint_adjoint:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
-  assumes lf: "linear f"
-  shows "adjoint (adjoint f) = f"
-  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
-
-
 subsection {* Matrix operations *}
 
 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
@@ -680,10 +415,10 @@
   apply auto
   apply (subst vec_eq_iff)
   apply clarify
-  apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
-  apply (erule_tac x="cart_basis ia" in allE)
+  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta 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: cart_basis_def if_distrib cond_application_beta
+  apply (auto simp add: if_distrib cond_application_beta axis_def
     setsum_delta[OF finite] cong del: if_weak_cong)
   done
 
@@ -728,25 +463,27 @@
   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
 
 lemma vector_componentwise:
-  "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
-  apply (subst basis_expansion[symmetric])
-  apply (vector vec_eq_iff setsum_component)
-  done
+  "(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 setsum_cases vec_eq_iff)
+
+lemma basis_expansion: "setsum (\<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 setsum_cases cong del: if_weak_cong)
 
 lemma linear_componentwise:
   fixes f:: "real ^'m \<Rightarrow> real ^ _"
   assumes lf: "linear f"
-  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
+  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
 proof -
   let ?M = "(UNIV :: 'm set)"
   let ?N = "(UNIV :: 'n set)"
   have fM: "finite ?M" by simp
-  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j"
-    unfolding vector_smult_component[symmetric] smult_conv_scaleR
-    unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]
-    ..
+  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
+    unfolding setsum_component by simp
   then show ?thesis
-    unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
+    unfolding linear_setsum_mul[OF lf fM, symmetric]
+    unfolding scalar_mult_eq_scaleR[symmetric]
+    unfolding basis_expansion
+    by simp
 qed
 
 text{* Inverse matrices  (not necessarily square) *}
@@ -761,7 +498,7 @@
 text{* Correspondence between matrices and linear operators. *}
 
 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
-  where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
+  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
 
 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
   by (simp add: linear_def matrix_vector_mult_def vec_eq_iff
@@ -831,103 +568,8 @@
   ultimately show ?thesis by metis
 qed
 
-
-subsection {* Standard bases are a spanning set, and obviously finite. *}
-
-lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
-  apply (rule set_eqI)
-  apply auto
-  apply (subst basis_expansion'[symmetric])
-  apply (rule span_setsum)
-  apply simp
-  apply auto
-  apply (rule span_mul)
-  apply (rule span_superset)
-  apply auto
-  done
-
-lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
-proof -
-  have "?S = cart_basis ` UNIV" by blast
-  then show ?thesis by auto
-qed
-
-lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
-proof -
-  have "?S = cart_basis ` UNIV" by blast
-  then show ?thesis using card_image[OF basis_inj] by simp
-qed
-
-lemma independent_stdbasis_lemma:
-  assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"
-    and iS: "i \<notin> S"
-  shows "(x$i) = 0"
-proof -
-  let ?U = "UNIV :: 'n set"
-  let ?B = "cart_basis ` S"
-  let ?P = "{(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0}"
-  { fix x::"real^_" assume xS: "x\<in> ?B"
-    from xS have "x \<in> ?P" by auto }
-  moreover
-  have "subspace ?P"
-    by (auto simp add: subspace_def)
-  ultimately show ?thesis
-    using x span_induct[of x ?B ?P] iS by blast
-qed
-
-lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
-proof -
-  let ?I = "UNIV :: 'n set"
-  let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"
-  let ?B = "?b ` ?I"
-  have eq: "{?b i|i. i \<in> ?I} = ?B" by auto
-  { assume d: "dependent ?B"
-    then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
-      unfolding dependent_def by auto
-    have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
-    have eq2: "?B - {?b k} = ?b ` (?I - {k})"
-      unfolding eq1
-      apply (rule inj_on_image_set_diff[symmetric])
-      apply (rule basis_inj) using k(1)
-      apply auto
-      done
-    from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
-    from independent_stdbasis_lemma[OF th0, of k, simplified]
-    have False by simp }
-  then show ?thesis unfolding eq dependent_def ..
-qed
-
 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
-  unfolding inner_simps smult_conv_scaleR by auto
-
-lemma linear_eq_stdbasis_cart:
-  assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g"
-    and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
-  shows "f = g"
-proof -
-  let ?U = "UNIV :: 'm set"
-  let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"
-  { fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
-    from equalityD2[OF span_stdbasis]
-    have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast
-    from linear_eq[OF lf lg IU] fg x
-    have "f x = g x" unfolding Ball_def mem_Collect_eq by metis
-  }
-  then show ?thesis by auto
-qed
-
-lemma bilinear_eq_stdbasis_cart:
-  assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"
-    and bg: "bilinear g"
-    and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
-  shows "f = g"
-proof -
-  from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}.
-      \<forall>y\<in>  {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y"
-    by blast
-  from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th]
-  show ?thesis by blast
-qed
+  unfolding inner_simps scalar_mult_eq_scaleR by auto
 
 lemma left_invertible_transpose:
   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
@@ -1043,7 +685,7 @@
         unfolding y[symmetric]
         apply (rule span_setsum[OF fU])
         apply clarify
-        unfolding smult_conv_scaleR
+        unfolding scalar_mult_eq_scaleR
         apply (rule span_mul)
         apply (rule span_superset)
         unfolding columns_def
@@ -1056,7 +698,7 @@
     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
     { fix y
       have "?P y"
-      proof (rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
+      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
           by (rule exI[where x=0], simp)
       next
@@ -1159,25 +801,12 @@
     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
 
 
-lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
-  unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe
-  apply(rule_tac x="\<pi> i" in exI) defer
-  apply(rule_tac x="\<pi>' i" in exI)
-  unfolding nth_conv_component
-  using pi'_range apply auto
-  done
-
-lemma infnorm_set_image_cart: "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
-  (\<lambda>i. abs(x$i)) ` (UNIV)" by blast
-
-lemma infnorm_set_lemma_cart:
-  "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
-  "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
-  unfolding infnorm_set_image_cart by auto
+lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
+  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
 
 lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
-  unfolding nth_conv_component
-  using component_le_infnorm[of x] .
+  using Basis_le_infnorm[of "axis i 1" x]
+  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
 
 lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
   unfolding continuous_def by (rule tendsto_vec_nth)
@@ -1371,7 +1000,7 @@
     and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
     and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
     and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
-  using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)
+  using subset_interval[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
 
 lemma disjoint_interval_cart:
   fixes a::"real^'n"
@@ -1379,7 +1008,7 @@
     and "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
     and "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
     and "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
-  using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)
+  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
 
 lemma inter_interval_cart:
   fixes a :: "'a::linorder^'n"
@@ -1400,7 +1029,7 @@
 lemma is_interval_cart:
   "is_interval (s::(real^'n) set) \<longleftrightarrow>
     (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
-  by (simp add: is_interval_def Ball_def cart_simps real_euclidean_nth)
+  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
 
 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
   by (simp add: closed_Collect_le)
@@ -1416,27 +1045,15 @@
 
 lemma Lim_component_le_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
-  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
+  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
   shows "l$i \<le> b"
-proof -
-  { fix x
-    have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b"
-      unfolding inner_basis by auto }
-  then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l]
-    using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto
-qed
+  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
 
 lemma Lim_component_ge_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
   shows "b \<le> l$i"
-proof -
-  { fix x
-    have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b"
-      unfolding inner_basis by auto }
-  then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l]
-    using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto
-qed
+  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
 
 lemma Lim_component_eq_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
@@ -1449,8 +1066,8 @@
 lemma connected_ivt_component_cart:
   fixes x :: "real^'n"
   shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
-  using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a]
-  by (auto simp add: inner_basis)
+  using connected_ivt_hyperplane[of s x y "axis k 1" a]
+  by (auto simp add: inner_axis inner_commute)
 
 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
   unfolding subspace_def by auto
@@ -1468,20 +1085,14 @@
 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
   (is "dim ?A = _")
 proof -
-  have *: "{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} =
-      {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
-    apply safe
-    apply (erule_tac x="\<pi>' i" in allE) defer
-    apply (erule_tac x="\<pi> i" in allE)
-    unfolding image_iff real_euclidean_nth[symmetric]
-    apply (auto simp: pi'_inj[THEN inj_eq])
-    done
-  have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}"
-    using pi'_range[where 'n='n] by auto
+  let ?a = "\<lambda>x. axis x 1 :: real^'n"
+  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
+    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
+  have "?a ` d \<subseteq> Basis"
+    by (auto simp: Basis_vec_def)
   thus ?thesis
-    using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 
-    unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def
-    by auto
+    using dim_substandard[of "?a ` d"] card_image[of ?a d]
+    by (auto simp: axis_eq_axis inj_on_def *)
 qed
 
 lemma affinity_inverses:
@@ -1513,32 +1124,24 @@
   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
   by metis
 
-lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"
-  apply(subst euclidean_eq)
-proof safe
-  case goal1
-  hence *: "(basis i::real^'n) = cart_basis (\<pi> i)"
-    unfolding basis_real_n[symmetric] by auto
-  have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *
-    unfolding dot_basis by auto
-  thus ?case using goal1 by auto
-qed
-
+lemma vector_cart:
+  fixes f :: "real^'n \<Rightarrow> real"
+  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
+  unfolding euclidean_eq_iff[where 'a="real^'n"]
+  by simp (simp add: Basis_vec_def inner_axis)
+  
+lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
+  by (rule vector_cart)
 
 subsection "Convex Euclidean Space"
 
-lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"
-  apply(subst euclidean_eq)
-proof safe
-  case goal1
-  thus ?case
-    using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
-qed
+lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
+  using const_vector_cart[of 1] by (simp add: one_vec_def)
 
 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
 
-lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component
+lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
 
 lemma convex_box_cart:
   assumes "\<And>i. convex {x. P i x}"
@@ -1551,95 +1154,20 @@
 lemma unit_interval_convex_hull_cart:
   "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
-  apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_eqI) unfolding mem_Collect_eq
-  apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer
-  apply(erule_tac x="\<pi> i" in allE)
-  apply auto
-  done
+  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
 
 lemma cube_convex_hull_cart:
   assumes "0 < d"
   obtains s::"(real^'n) set"
     where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
 proof -
-  obtain s where s: "finite s" "{x - (\<chi>\<chi> i. d)..x + (\<chi>\<chi> i. d)} = convex hull s"
-    by (rule cube_convex_hull [OF assms])
-  show thesis
-    apply(rule that[OF s(1)]) unfolding s(2)[symmetric] const_vector_cart ..
+  from cube_convex_hull [OF assms, of x] guess s .
+  with that[of s] show thesis by (simp add: const_vector_cart)
 qed
 
-lemma std_simplex_cart:
-  "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
-    (insert 0 { basis i | i. i<DIM((real,'n) vec)})"
-  apply (rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
-  unfolding basis_real_n[symmetric]
-  apply safe
-  apply (rule_tac x="\<pi>' i" in exI) defer
-  apply (rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n]
-  apply auto
-  done
-
-
-subsection "Brouwer Fixpoint"
-
-lemma kuhn_labelling_lemma_cart:
-  assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
-  shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
-             (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
-             (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
-             (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
-             (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)"
-proof -
-  have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)"
-    by auto
-  have *: "\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)"
-    by auto
-  show ?thesis
-    unfolding and_forall_thm apply(subst choice_iff[symmetric])+
-  proof (rule, rule)
-    case goal1
-    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
-        (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and>
-        (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and>
-        (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
-    { assume "P x" "Q xa"
-      hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1"
-        using assms(2)[rule_format,of "f x" xa]
-        apply (drule_tac assms(1)[rule_format])
-        apply auto
-        done
-    }
-    hence "?R 0 \<or> ?R 1" by auto
-    thus ?case by auto
-  qed
-qed 
-
-lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
-    (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
-  unfolding interval_bij_def apply(rule ext)+ apply safe
-  unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
-  apply rule
-  apply (subst euclidean_lambda_beta)
-  using pi'_range apply auto
-  done
-
-lemma interval_bij_affine_cart:
- "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
-            (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
-  apply rule
-  unfolding vec_eq_iff interval_bij_cart vector_component_simps
-  apply (auto simp add: field_simps add_divide_distrib[symmetric]) 
-  done
-
 
 subsection "Derivative"
 
-lemma has_derivative_vmul_component_cart:
-  fixes c :: "real^'a \<Rightarrow> real^'b" and v :: "real^'c"
-  assumes "(c has_derivative c') net"
-  shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net"
-  unfolding nth_conv_component by (intro has_derivative_intros assms)
-
 lemma differentiable_at_imp_differentiable_on:
   "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
@@ -1662,56 +1190,14 @@
 subsection {* Component of the differential must be zero if it exists at a local
   maximum or minimum for that corresponding component. *}
 
-lemma differential_zero_maxmin_component:
+lemma differential_zero_maxmin_cart:
   fixes f::"real^'a \<Rightarrow> real^'b"
   assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
-    "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
-(* FIXME: reuse proof of generic differential_zero_maxmin_component*)
-proof (rule ccontr)
-  def D \<equiv> "jacobian f (at x)"
-  assume "jacobian f (at x) $ k \<noteq> 0"
-  then obtain j where j:"D$k$j \<noteq> 0" unfolding vec_eq_iff D_def by auto
-  hence *: "abs (jacobian f (at x) $ k $ j) / 2 > 0"
-    unfolding D_def by auto
-  note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
-  guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
-  guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
-  { fix c
-    assume "abs c \<le> d" 
-    hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'"
-      using norm_basis[of j] d by auto
-    have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
-        norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" 
-      by (rule component_le_norm_cart)
-    also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
-      using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j]
-      unfolding D_def[symmetric] by auto
-    finally
-    have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
-      \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
-    hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le>
-      \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
-      unfolding vector_component_simps matrix_vector_mul_component
-      unfolding smult_conv_scaleR[symmetric] 
-      unfolding inner_simps dot_basis smult_conv_scaleR by simp
-  } note * = this
-  have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"
-    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
-  hence **: "((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
-      ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)"
-    using assms(2) by auto
-  have ***: "\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow>
-    d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
-  show False
-    apply (rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"])
-    using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
-    unfolding mult_minus_left
-    unfolding abs_mult diff_minus_eq_add scaleR_minus_left
-    unfolding algebra_simps
-    apply (auto intro: mult_pos_pos)
-    done
-qed
-
+    "f differentiable (at x)"
+  shows "jacobian f (at x) $ k = 0"
+  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
+    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
+  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
 
 subsection {* Lemmas for working on @{typ "real^1"} *}
 
@@ -1775,25 +1261,6 @@
 
 end
 
-(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
-
-abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"
-
-abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a" where "dest_vec1 x \<equiv> (x$1)"
-
-lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x"
-  by (simp add: vec_eq_iff)
-
-lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
-  by (metis vec1_dest_vec1(1))
-
-lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
-  by (metis vec1_dest_vec1(1))
-
-lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
-  by (metis vec1_dest_vec1(1))
-
-
 subsection{* The collapse of the general concepts to dimension one. *}
 
 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
@@ -1863,438 +1330,10 @@
   apply (simp add: forall_3)
   done
 
-lemma range_vec1[simp]:"range vec1 = UNIV"
-  apply (rule set_eqI,rule) unfolding image_iff defer
-  apply (rule_tac x="dest_vec1 x" in bexI)
-  apply auto
-  done
-
-lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
-  by simp
-
-lemma dest_vec1_vec: "dest_vec1(vec x) = x"
-  by simp
-
-lemma dest_vec1_sum: assumes fS: "finite S"
-  shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
-  apply (induct rule: finite_induct[OF fS])
-  apply simp
-  apply auto
-  done
-
-lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)"
-  by (simp add: vec_def norm_real)
-
-lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
-  by (simp only: dist_real vec_component)
-lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
-  by (metis vec1_dest_vec1(1) norm_vec1)
-
-lemmas vec1_dest_vec1_simps =
-  forall_vec1 vec_add[symmetric] dist_vec1 vec_sub[symmetric] vec1_dest_vec1 norm_vec1 vector_smult_component
-  vec_inj[where 'b=1] vec_cmul[symmetric] smult_conv_scaleR[symmetric] o_def dist_real_def real_norm_def
-
-lemma bounded_linear_vec1: "bounded_linear (vec1::real\<Rightarrow>real^1)"
-  unfolding bounded_linear_def additive_def bounded_linear_axioms_def 
-  unfolding smult_conv_scaleR[symmetric]
-  unfolding vec1_dest_vec1_simps
-  apply (rule conjI) defer  
-  apply (rule conjI) defer
-  apply (rule_tac x=1 in exI)
-  apply auto
-  done
-
-lemma linear_vmul_dest_vec1:
-  fixes f:: "real^_ \<Rightarrow> real^1"
-  shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
-  unfolding smult_conv_scaleR
-  by (rule linear_vmul_component)
-
-lemma linear_from_scalars:
-  assumes lf: "linear (f::real^1 \<Rightarrow> real^_)"
-  shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
-  unfolding smult_conv_scaleR
-  apply (rule ext)
-  apply (subst matrix_works[OF lf, symmetric])
-  apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute)
-  done
-
-lemma linear_to_scalars:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real^1)"
-  shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
-  apply (rule ext)
-  apply (subst matrix_works[OF lf, symmetric])
-  apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute)
-  done
-
-lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
-  by (simp add: dest_vec1_eq[symmetric])
-
-lemma setsum_scalars:
-  assumes fS: "finite S"
-  shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
-  unfolding vec_setsum[OF fS] by simp
-
-lemma dest_vec1_wlog_le:
-  "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)
-    \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
-  apply (cases "dest_vec1 x \<le> dest_vec1 y")
-  apply simp
-  apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
-  apply auto
-  done
-
-text{* Lifting and dropping *}
-
-lemma continuous_on_o_dest_vec1:
-  fixes f::"real \<Rightarrow> 'a::real_normed_vector"
-  assumes "continuous_on {a..b::real} f"
-  shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
-  using assms unfolding continuous_on_iff apply safe
-  apply (erule_tac x="x$1" in ballE,erule_tac x=e in allE)
-  apply safe
-  apply (rule_tac x=d in exI)
-  apply safe
-  unfolding o_def dist_real_def dist_real
-  apply (erule_tac x="dest_vec1 x'" in ballE)
-  apply (auto simp add:less_eq_vec_def)
-  done
-
-lemma continuous_on_o_vec1:
-  fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
-  assumes "continuous_on {a..b} f"
-  shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
-  using assms unfolding continuous_on_iff
-  apply safe
-  apply (erule_tac x="vec x" in ballE,erule_tac x=e in allE)
-  apply safe
-  apply (rule_tac x=d in exI)
-  apply safe
-  unfolding o_def dist_real_def dist_real
-  apply (erule_tac x="vec1 x'" in ballE)
-  apply (auto simp add:less_eq_vec_def)
-  done
-
-lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
-  by (rule linear_continuous_on[OF bounded_linear_vec1])
-
-lemma mem_interval_1:
-  fixes x :: "real^1"
-  shows "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
-    and "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-  by (simp_all add: vec_eq_iff less_vec_def less_eq_vec_def)
-
-lemma vec1_interval:
-  fixes a::"real"
-  shows "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
-    and "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
-  apply (rule_tac[!] set_eqI)
-  unfolding image_iff less_vec_def
-  unfolding mem_interval_cart
-  unfolding forall_1 vec1_dest_vec1_simps
-  apply rule defer
-  apply (rule_tac x="dest_vec1 x" in bexI) prefer 3
-  apply rule defer
-  apply (rule_tac x="dest_vec1 x" in bexI)
-  apply auto
-  done
-
-(* Some special cases for intervals in R^1.                                  *)
-
-lemma interval_cases_1:
-  fixes x :: "real^1"
-  shows "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
-  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart
-  by (auto simp del:dest_vec1_eq)
-
-lemma in_interval_1:
-  fixes x :: "real^1"
-  shows "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
-    (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart
-  by (auto simp del:dest_vec1_eq)
-
-lemma interval_eq_empty_1:
-  fixes a :: "real^1"
-  shows "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
-    and "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
-  unfolding interval_eq_empty_cart and ex_1 by auto
-
-lemma subset_interval_1:
-  fixes a :: "real^1"
-  shows "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
-    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
-   "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
-    dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
-   "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>
-    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
-   "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
-    dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
-  unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto
-
-lemma eq_interval_1:
-  fixes a :: "real^1"
-  shows "{a .. b} = {c .. d} \<longleftrightarrow>
-          dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
-          dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
-  unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
-  unfolding subset_interval_1(1)[of a b c d]
-  unfolding subset_interval_1(1)[of c d a b]
-  by auto
-
-lemma disjoint_interval_1:
-  fixes a :: "real^1"
-  shows
-    "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow>
-      dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
-    "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow>
-      dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
-    "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow>
-      dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
-    "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow>
-      dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
-  unfolding disjoint_interval_cart and ex_1 by auto
-
-lemma open_closed_interval_1:
-  fixes a :: "real^1"
-  shows "{a<..<b} = {a .. b} - {a, b}"
-  unfolding set_eq_iff apply simp
-  unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
-  apply (auto simp del:dest_vec1_eq)
-  done
-
-lemma closed_open_interval_1:
-  "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
-  unfolding set_eq_iff
-  apply simp
-  unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
-  apply (auto simp del:dest_vec1_eq)
-  done
-
-lemma Lim_drop_le:
-  fixes f :: "'a \<Rightarrow> real^1"
-  shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
-    eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
-  using Lim_component_le_cart[of f l net 1 b] by auto
-
-lemma Lim_drop_ge:
-  fixes f :: "'a \<Rightarrow> real^1"
-  shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
-    eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
-  using Lim_component_ge_cart[of f l net b 1] by auto
-
-
-text{* Also more convenient formulations of monotone convergence.                *}
-
-lemma bounded_increasing_convergent:
-  fixes s :: "nat \<Rightarrow> real^1"
-  assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
-  shows "\<exists>l. (s ---> l) sequentially"
-proof -
-  obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le>  a"
-    using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
-  { fix m::nat
-    have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
-      apply (induct_tac n)
-      apply simp
-      using assms(2) apply (erule_tac x="na" in allE)
-      apply (auto simp add: not_less_eq_eq)
-      done
-  }
-  hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
-    by auto
-  then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e"
-    using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
-  thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="vec1 l" in exI)
-    unfolding dist_norm unfolding abs_dest_vec1 by auto
-qed
-
-lemma dest_vec1_simps[simp]:
-  fixes a :: "real^1"
-  shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
-    "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
-  by (auto simp add: less_eq_vec_def vec_eq_iff)
-
-lemma dest_vec1_inverval:
-  "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
-  "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
-  "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
-  "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
-  apply(rule_tac [!] equalityI)
-  unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
-  apply(rule_tac [!] allI)apply(rule_tac [!] impI)
-  apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
-  apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
-  apply (auto simp add: less_vec_def less_eq_vec_def)
-  done
-
-lemma dest_vec1_setsum:
-  assumes "finite S"
-  shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
-  using dest_vec1_sum[OF assms] by auto
-
-lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
-  unfolding open_vec_def forall_1 by auto
-
-lemma tendsto_dest_vec1 [tendsto_intros]:
-  "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
-  by (rule tendsto_vec_nth)
-
-lemma continuous_dest_vec1:
-  "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
-  unfolding continuous_def by (rule tendsto_dest_vec1)
-
-lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))" 
-  apply safe defer
-  apply (erule_tac x="vec1 x" in allE)
-  apply auto
-  done
-
-lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
-  apply rule
-  apply rule
-  apply (erule_tac x="vec1 \<circ> x" in allE)
-  unfolding o_def vec1_dest_vec1
-  apply auto
-  done
-
-lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
-  apply rule
-  apply rule
-  apply (erule_tac x="(vec1 x)" in allE) defer
-  apply rule 
-  apply (erule_tac x="dest_vec1 v" in allE)
-  unfolding o_def vec1_dest_vec1
-  apply auto
-  done
-
-lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x"
-  unfolding dist_norm by auto
-
-lemma bounded_linear_vec1_dest_vec1:
-  fixes f :: "real \<Rightarrow> real"
-  shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r")
-proof -
-  { assume ?l
-    then have "\<exists>K. \<forall>x. norm ((vec1 \<circ> f \<circ> dest_vec1) x) \<le> K * norm x" by (rule linear_bounded)
-    then guess K ..
-    hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K"
-      apply(rule_tac x=K in exI)
-      unfolding vec1_dest_vec1_simps by (auto simp add:field_simps)
-  }
-  thus ?thesis
-    unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
-    unfolding vec1_dest_vec1_simps by auto
-qed
-
-lemma vec1_le[simp]: fixes a :: real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
-  unfolding less_eq_vec_def by auto
-lemma vec1_less[simp]: fixes a :: real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
-  unfolding less_vec_def by auto
-
-
-subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
-
-lemma has_derivative_within_vec1_dest_vec1:
-  fixes f :: "real \<Rightarrow> real"
-  shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
-    = (f has_derivative f') (at x within s)"
-  unfolding has_derivative_within
-  unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
-  unfolding o_def Lim_within Ball_def unfolding forall_vec1 
-  unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff
-  by auto
-
-lemma has_derivative_at_vec1_dest_vec1:
-  fixes f :: "real \<Rightarrow> real"
-  shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
-  using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV]
-  by auto
-
-lemma bounded_linear_vec1':
-  fixes f :: "'a::real_normed_vector\<Rightarrow>real"
-  shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
-  unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
-  unfolding vec1_dest_vec1_simps by auto
-
-lemma bounded_linear_dest_vec1:
-  fixes f :: "real\<Rightarrow>'a::real_normed_vector"
-  shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
-  unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
-  unfolding vec1_dest_vec1_simps
-  by auto
-
-lemma has_derivative_at_vec1:
-  fixes f :: "'a::real_normed_vector\<Rightarrow>real"
-  shows "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
-  unfolding has_derivative_at
-  unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]
-  unfolding o_def Lim_at
-  unfolding vec1_dest_vec1_simps dist_vec1_0
-  by auto
-
-lemma has_derivative_within_dest_vec1:
-  fixes f :: "real\<Rightarrow>'a::real_normed_vector"
-  shows "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) =
-    (f has_derivative f') (at x within s)"
-  unfolding has_derivative_within bounded_linear_dest_vec1
-  unfolding o_def Lim_within Ball_def
-  unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff
-  by auto
-
-lemma has_derivative_at_dest_vec1:
-  fixes f :: "real\<Rightarrow>'a::real_normed_vector"
-  shows "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) =
-    (f has_derivative f') (at x)"
-  using has_derivative_within_dest_vec1[where s=UNIV] by simp
-
-
-subsection {* In particular if we have a mapping into @{typ "real^1"}. *}
-
-lemma onorm_vec1:
-  fixes f::"real \<Rightarrow> real"
-  shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f"
-proof -
-  have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}"
-    unfolding forall_vec1 by (auto simp add: vec_eq_iff)
-  hence 1: "{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto
-  have 2: "{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} =
-      (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}"
-    by auto
-  have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto
-  hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
-  have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
-  show ?thesis
-    unfolding onorm_def 1 2 3 4 by (simp add:Sup_finite_Max)
-qed
-
-lemma convex_vec1:"convex (vec1 ` s) = convex (s::real set)"
-  unfolding convex_def Ball_def forall_vec1
-  unfolding vec1_dest_vec1_simps image_iff
-  by auto
-
 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
   apply (rule bounded_linearI[where K=1])
   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
 
-lemma bounded_vec1[intro]: "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"
-  unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI)
-  apply (auto simp add: dist_real dist_real_def)
-  done
-
-(*lemma content_closed_interval_cases_cart:
-  "content {a..b::real^'n} =
-  (if {a..b} = {} then 0 else setprod (\<lambda>i. b$i - a$i) UNIV)" 
-proof(cases "{a..b} = {}")
-  case True thus ?thesis unfolding content_def by auto
-next case Falsethus ?thesis unfolding content_def unfolding if_not_P[OF False]
-  proof(cases "\<forall>i. a $ i \<le> b $ i")
-    case False thus ?thesis unfolding content_def using t by auto
-  next case True note interval_eq_empty
-   apply auto 
-  
-  sorry*)
-
 lemma integral_component_eq_cart[simp]:
   fixes f :: "'n::ordered_euclidean_space \<Rightarrow> real^'m"
   assumes "f integrable_on s"
@@ -2309,39 +1348,8 @@
   unfolding vec_lambda_beta
   by auto
 
-(*lemma content_split_cart:
-  "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
-proof- note simps = interval_split_cart content_closed_interval_cases_cart vec_lambda_beta less_eq_vec_def
-  { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
-  have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
-  have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
-    "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
-    apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
-  assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
-    \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
-    by  (auto simp add:field_simps)
-  moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
-    unfolding not_le using as[unfolded less_eq_vec_def,rule_format,of k] by auto
-  ultimately show ?thesis 
-    unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
-qed*)
-
-lemma has_integral_vec1:
-  assumes "(f has_integral k) {a..b}"
-  shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
-proof -
-  have *: "\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k =
-      vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
-    unfolding vec_sub vec_eq_iff by (auto simp add: split_beta)
-  show ?thesis
-    using assms unfolding has_integral
-    apply safe
-    apply(erule_tac x=e in allE,safe)
-    apply(rule_tac x=d in exI,safe)
-    apply(erule_tac x=p in allE,safe)
-    unfolding * norm_vector_1
-    apply auto
-    done
-qed
+lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
+  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
+  using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
 
 end
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -101,56 +101,22 @@
 lemma span_eq[simp]: "(span s = s) <-> subspace s"
   unfolding span_def by (rule hull_eq, rule subspace_Inter)
 
-lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
-  by (auto simp add: inj_on_def euclidean_eq[where 'a='n])
-
-lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
-proof -
-  have eq: "?S = basis ` d" by blast
-  show ?thesis
-    unfolding eq
-    apply (rule finite_subset[OF _ range_basis_finite])
-    apply auto
-    done
-qed
-
-lemma card_substdbasis:
-  assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
-  shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
-proof -
-  have eq: "?S = basis ` d" by blast
-  show ?thesis
-    unfolding eq
-    using card_image[OF basis_inj_on[of d]] assms by auto
-qed
-
 lemma substdbasis_expansion_unique:
-  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
-  shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
-      <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
+  assumes d: "d \<subseteq> Basis"
+  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space)
+      \<longleftrightarrow> (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
 proof -
   have *: "\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
-  have **: "finite d" apply(rule finite_subset[OF assms]) by fastforce
-  have ***: "\<And>i. (setsum (%i. f i *\<^sub>R ((basis i)::'a)) d) $$ i = (\<Sum>x\<in>d. if x = i then f x else 0)"
-    unfolding euclidean_component_setsum euclidean_component_scaleR basis_component *
-    apply (rule setsum_cong2)
-    using assms apply auto
-    done
+  have **: "finite d" by (auto intro: finite_subset[OF assms])
+  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
+    using d
+    by (auto intro!: setsum_cong simp: inner_Basis inner_setsum_left)
   show ?thesis
-    unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
-qed
-
-lemma independent_substdbasis:
-  assumes "d \<subseteq> {..<DIM('a::euclidean_space)}"
-  shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}"
-  (is "independent ?A")
-proof -
-  have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
-  show ?thesis
-    apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
-    using independent_basis[where 'a='a] assms apply (auto simp: *)
-    done
-qed
+    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum_delta[OF **] ***)
+qed
+
+lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
+  by (rule independent_mono[OF independent_Basis])
 
 lemma dim_cball:
   assumes "0<e"
@@ -321,8 +287,8 @@
 
 lemma vector_choose_size:
   "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
-  apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"])
-  using DIM_positive[where 'a='a] apply auto
+  apply (rule exI[where x="c *\<^sub>R (SOME i. i \<in> Basis)"])
+  apply (auto simp: SOME_Basis)
   done
 
 lemma setsum_delta_notmem:
@@ -1291,11 +1257,12 @@
 text {* Balls, being convex, are connected. *}
 
 lemma convex_box: fixes a::"'a::euclidean_space"
-  assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
-  shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
-  using assms unfolding convex_def by auto
-
-lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
+  assumes "\<And>i. i\<in>Basis \<Longrightarrow> convex {x. P i x}"
+  shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
+  using assms unfolding convex_def
+  by (auto simp: inner_add_left)
+
+lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
   by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
 
 lemma convex_local_global_minimum:
@@ -2073,40 +2040,39 @@
 from this show ?thesis using assms `span B=S` by auto
 qed
 
-lemma span_substd_basis:  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
-  shows "(span {basis i | i. i : d}) = {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
-  (is "span ?A = ?B")
+lemma span_substd_basis:
+  assumes d: "d \<subseteq> Basis"
+  shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}" (is "_ = ?B")
 proof-
-have "?A <= ?B" by auto
+have "d <= ?B" using d by (auto simp: inner_Basis)
 moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
-ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
-moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"]
-   independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
-moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
-ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"]
-  subspace_span[of "?A"] by auto
+ultimately have "span d <= ?B" using span_mono[of d "?B"] span_eq[of "?B"] by blast
+moreover have "card d <= dim (span d)" using independent_card_le_dim[of d "span d"]
+   independent_substdbasis[OF assms] span_inc[of d] by auto
+moreover hence "dim ?B <= dim (span d)" using dim_substandard[OF assms] by auto
+ultimately show ?thesis using s subspace_dim_equal[of "span d" "?B"]
+  subspace_span[of d] by auto
 qed
 
 lemma basis_to_substdbasis_subspace_isomorphism:
 fixes B :: "('a::euclidean_space) set"
 assumes "independent B"
-shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} &
-       f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}"
+shows "EX f (d::'a set). card d = card B \<and> linear f \<and> f ` B = d \<and>
+       f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
 proof-
   have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
-  def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
-  have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
-  hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
-  let ?t = "{x::'a::euclidean_space. !i<DIM('a). i ~: d --> x$$i = 0}"
-  have "EX f. linear f & f ` B = {basis i |i. i : d} &
-    f ` span B = ?t & inj_on f (span B)"
-    apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "{basis i |i. i : d}"])
+  have "dim B \<le> card (Basis :: 'a set)" using dim_subset_UNIV[of B] by simp
+  from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B" by auto
+  let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0}"
+  have "EX f. linear f & f ` B = d & f ` span B = ?t & inj_on f (span B)"
+    apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "d"])
     apply(rule subspace_span) apply(rule subspace_substandard) defer
     apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
-    unfolding span_substd_basis[OF d,symmetric] card_substdbasis[OF d] apply(rule span_inc)
+    unfolding span_substd_basis[OF d, symmetric] 
+    apply(rule span_inc)
     apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
     unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d] by auto
-  from this t `card B=dim B` show ?thesis using d by auto
+  with t `card B = dim B` d show ?thesis by auto
 qed
 
 lemma aff_dim_empty:
@@ -2492,7 +2458,7 @@
         using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"] by (simp add: algebra_simps)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm unfolding norm_scaleR[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`
-      by(auto simp add:euclidean_eq[where 'a='a] field_simps)
+      by(auto simp add:euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     also have "... = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "... < d" using as[unfolded dist_norm] and `e>0`
       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
@@ -2770,9 +2736,9 @@
 
 subsection{* Some Properties of subset of standard basis *}
 
-lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
-  shows "affine hull (insert 0 {basis i | i. i : d}) =
-  {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
+lemma affine_hull_substd_basis: assumes "d\<subseteq>Basis"
+  shows "affine hull (insert 0 d) =
+  {x::'a::euclidean_space. (\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)}"
  (is "affine hull (insert 0 ?A) = ?B")
 proof- have *:"\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A" by auto
   show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
@@ -3230,10 +3196,10 @@
   assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
   proof(cases "s={}")
-  case True have "norm ((basis 0)::'a) = 1" by auto
-  hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
-    apply(subst norm_le_zero_iff[symmetric]) by auto
-  thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
+  case True
+  have "norm ((SOME i. i\<in>Basis)::'a) = 1" "(SOME i. i\<in>Basis) \<noteq> (0::'a)" defer
+    apply(subst norm_le_zero_iff[symmetric]) by (auto simp: SOME_Basis)
+  thus ?thesis apply(rule_tac x="SOME i. i\<in>Basis" in exI, rule_tac x=1 in exI)
     using True using DIM_positive[where 'a='a] by auto
 next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
     apply - apply(erule exE)+ unfolding inner_zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
@@ -3703,10 +3669,10 @@
       case False thus ?thesis apply (intro continuous_intros)
         using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
     next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
-      hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer
-        apply(erule_tac x="basis 0" in ballE)
+      hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="SOME i. i\<in>Basis" in ballE) defer
+        apply(erule_tac x="SOME i. i\<in>Basis" in ballE)
         unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
-        by auto
+        by (auto simp: SOME_Basis)
       case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
         apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
         unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
@@ -3849,7 +3815,8 @@
     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
-    using as(3-) DIM_positive[where 'a='a] by auto qed
+    using as(3-) DIM_positive[where 'a='a] by (auto simp: inner_simps)
+qed
 
 lemma is_interval_connected:
   fixes s :: "('a::euclidean_space) set"
@@ -3892,8 +3859,8 @@
 subsection {* Another intermediate value theorem formulation *}
 
 lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
-  assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
-  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
+  assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
+  shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
 proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI)
     using assms(1) by auto
   thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
@@ -3902,20 +3869,20 @@
 
 lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
-   \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
+   \<Longrightarrow> f a\<bullet>k \<le> y \<Longrightarrow> y \<le> f b\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
 by(rule ivt_increasing_component_on_1)
   (auto simp add: continuous_at_imp_continuous_on)
 
 lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
-  assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
-  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
+  assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)\<bullet>k \<le> y" "y \<le> (f a)\<bullet>k"
+  shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   apply(subst neg_equal_iff_equal[symmetric])
   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
   using assms using continuous_on_minus by auto
 
 lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
-    \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
+    \<Longrightarrow> f b\<bullet>k \<le> y \<Longrightarrow> y \<le> f a\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
 by(rule ivt_decreasing_component_on_1)
   (auto simp: continuous_at_imp_continuous_on)
 
@@ -3933,104 +3900,127 @@
   thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
     unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
 
+lemma inner_setsum_Basis[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
+  by (simp add: One_def inner_setsum_left setsum_cases inner_Basis)
+
 lemma unit_interval_convex_hull:
-  "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
+  defines "One \<equiv> (\<Sum>Basis)"
+  shows "{0::'a::ordered_euclidean_space .. One} =
+    convex hull {x. \<forall>i\<in>Basis. (x\<bullet>i = 0) \<or> (x\<bullet>i = 1)}"
   (is "?int = convex hull ?points")
-proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
-  { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n"
+proof -
+  have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
+    by (simp add: One_def inner_setsum_left setsum_cases inner_Basis)
+  have 01:"{0,One} \<subseteq> convex hull ?points" 
+    apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
+  { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. One}" "n \<le> DIM('a)" "card {i. i\<in>Basis \<and> x\<bullet>i \<noteq> 0} \<le> n"
   hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
-    case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
+    case 0 hence "x = 0" apply(subst euclidean_eq_iff) apply rule by auto
     thus "x\<in>convex hull ?points" using 01 by auto
   next
-    case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
-      case True hence "x = 0" apply(subst euclidean_eq) by auto
+    case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i\<in>Basis \<and> x\<bullet>i \<noteq> 0} = {}")
+      case True hence "x = 0" apply(subst euclidean_eq_iff) by auto
       thus "x\<in>convex hull ?points" using 01 by auto
     next
-      case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
-      have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
-      then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
-      have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
+      case False def xi \<equiv> "Min ((\<lambda>i. x\<bullet>i) ` {i. i\<in>Basis \<and> x\<bullet>i \<noteq> 0})"
+      have "xi \<in> (\<lambda>i. x\<bullet>i) ` {i. i\<in>Basis \<and> x\<bullet>i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
+      then obtain i where i':"x\<bullet>i = xi" "x\<bullet>i \<noteq> 0" "i\<in>Basis" by auto
+      have i:"\<And>j. j\<in>Basis \<Longrightarrow> x\<bullet>j > 0 \<Longrightarrow> x\<bullet>i \<le> x\<bullet>j"
         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
         defer apply(rule_tac x=j in bexI) using i' by auto
-      have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
-        using i'(2-) `x$$i \<noteq> 0` by auto
-      show ?thesis proof(cases "x$$i=1")
-        case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
-        proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
-          hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
-          hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto
-          hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
+      have i01:"x\<bullet>i \<le> 1" "x\<bullet>i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
+        using i'(2-) `x\<bullet>i \<noteq> 0` by auto
+      show ?thesis proof(cases "x\<bullet>i=1")
+        case True have "\<forall>j\<in>{i. i\<in>Basis \<and> x\<bullet>i \<noteq> 0}. x\<bullet>j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
+        proof(erule conjE) fix j assume as:"x \<bullet> j \<noteq> 0" "x \<bullet> j \<noteq> 1" "j\<in>Basis"
+          hence j:"x\<bullet>j \<in> {0<..<1}" using Suc(2)
+            by (auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
+          hence "x\<bullet>j \<in> op \<bullet> x ` {i. i\<in>Basis \<and> x \<bullet> i \<noteq> 0}" using as(3) by auto
+          hence "x\<bullet>j \<ge> x\<bullet>i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
           thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
           by auto
-      next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
-        case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
-          apply(subst euclidean_eq) by(auto simp add: field_simps)
-        { fix j assume j:"j<DIM('a)"
-          have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
+      next
+        let ?y = "\<Sum>j\<in>Basis. (if x\<bullet>j = 0 then 0 else (x\<bullet>j - x\<bullet>i) / (1 - x\<bullet>i)) *\<^sub>R j"
+        case False
+        then have *: "x = (x\<bullet>i) *\<^sub>R (\<Sum>j\<in>Basis. (if x\<bullet>j = 0 then 0 else 1) *\<^sub>R j) + (1 - x\<bullet>i) *\<^sub>R ?y"
+          by (subst euclidean_eq_iff) (simp add: inner_simps)
+        { fix j :: 'a assume j:"j\<in>Basis"
+          have "x\<bullet>j \<noteq> 0 \<Longrightarrow> 0 \<le> (x \<bullet> j - x \<bullet> i) / (1 - x \<bullet> i)" "(x \<bullet> j - x \<bullet> i) / (1 - x \<bullet> i) \<le> 1"
             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
             using Suc(2)[unfolded mem_interval, rule_format, of j] using j
-            by(auto simp add:field_simps)
-          hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
-        moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
+            by(auto simp add: field_simps)
+          with j have "0 \<le> ?y \<bullet> j \<and> ?y \<bullet> j \<le> 1" by (auto simp: inner_simps) }
+        moreover have "i\<in>{j. j\<in>Basis \<and> x\<bullet>j \<noteq> 0} - {j. j\<in>Basis \<and> ?y \<bullet> j \<noteq> 0}"
           using i01 using i'(3) by auto
-        hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
-        hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule
+        hence "{j. j\<in>Basis \<and> x\<bullet>j \<noteq> 0} \<noteq> {j. j\<in>Basis \<and> ?y \<bullet> j \<noteq> 0}" using i'(3) by blast
+        hence **:"{j. j\<in>Basis \<and> ?y \<bullet> j \<noteq> 0} \<subset> {j. j\<in>Basis \<and> x\<bullet>j \<noteq> 0}"
           by auto
-        have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
+        have "card {j. j\<in>Basis \<and> ?y \<bullet> j \<noteq> 0} \<le> n"
           using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
-        ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
-          apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
-          unfolding mem_interval using i01 Suc(3) by auto
-      qed qed qed } note * = this
-  have **:"DIM('a) = card {..<DIM('a)}" by auto
-  show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
-    apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **)
+        ultimately show ?thesis
+          apply(subst *)
+          apply(rule convex_convex_hull[unfolded convex_def, rule_format])
+          apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) 
+          defer 
+          apply(rule Suc(1))
+          unfolding mem_interval 
+          using i01 Suc(3)
+          by auto
+      qed
+    qed
+  qed } note * = this
+  show ?thesis 
+    apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
+    apply(rule_tac n2="DIM('a)" in *) prefer 3
     apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
-    unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
-    by auto qed
+    unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in ballE)
+    by auto
+qed
 
 text {* And this is a finite set of vertices. *}
 
-lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
-  apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
-  apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
+lemma unit_cube_convex_hull:
+  obtains s :: "'a::ordered_euclidean_space set" where "finite s" "{0 .. \<Sum>Basis} = convex hull s"
+  apply(rule that[of "{x::'a. \<forall>i\<in>Basis. x\<bullet>i=0 \<or> x\<bullet>i=1}"])
+  apply(rule finite_subset[of _ "(\<lambda>s. (\<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i)::'a) ` Pow Basis"])
   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
-  fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
-  show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
-    apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
-    using as apply(subst euclidean_eq) by auto qed auto
+  fix x::"'a" assume as:"\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
+  show "x \<in> (\<lambda>s. \<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i) ` Pow Basis"
+    apply(rule image_eqI[where x="{i. i\<in>Basis \<and> x\<bullet>i = 1}"])
+    using as apply(subst euclidean_eq_iff) by (auto simp: inner_setsum_left_Basis)
+qed auto
 
 text {* Hence any cube (could do any nonempty interval). *}
 
 lemma cube_convex_hull:
   assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
-  "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
-  let ?d = "(\<chi>\<chi> i. d)::'a"
-  have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_eqI, rule)
+  "finite s" "{x - (\<Sum>i\<in>Basis. d*\<^sub>Ri) .. x + (\<Sum>i\<in>Basis. d*\<^sub>Ri)} = convex hull s" proof-
+  let ?d = "(\<Sum>i\<in>Basis. d*\<^sub>Ri)::'a"
+  have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<Sum>Basis}" apply(rule set_eqI, rule)
     unfolding image_iff defer apply(erule bexE) proof-
     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
-    { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
-        using as[unfolded mem_interval, THEN spec[where x=i]] i
-        by auto
-      hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
+    { fix i :: 'a assume i:"i\<in>Basis" have "x \<bullet> i \<le> d + y \<bullet> i" "y \<bullet> i \<le> d + x \<bullet> i"
+        using as[unfolded mem_interval, THEN bspec[where x=i]] i
+        by (auto simp: inner_simps)
+      hence "1 \<ge> inverse d * (x \<bullet> i - y \<bullet> i)" "1 \<ge> inverse d * (y \<bullet> i - x \<bullet> i)"
         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[symmetric]
         using assms by(auto simp add: field_simps)
-      hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
-            "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
-    hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
-      by(auto simp add: field_simps)
-    thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
+      hence "inverse d * (x \<bullet> i * 2) \<le> 2 + inverse d * (y \<bullet> i * 2)"
+            "inverse d * (y \<bullet> i * 2) \<le> 2 + inverse d * (x \<bullet> i * 2)" by(auto simp add:field_simps) }
+    hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<Sum>Basis}" unfolding mem_interval using assms
+      by(auto simp add: field_simps inner_simps)
+    thus "\<exists>z\<in>{0..\<Sum>Basis}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
       using assms by auto
   next
-    fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z"
-    have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
-      using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
+    fix y z assume as:"z\<in>{0..\<Sum>Basis}" "y = x - ?d + (2*d) *\<^sub>R z"
+    have "\<And>i. i\<in>Basis \<Longrightarrow> 0 \<le> d * (z \<bullet> i) \<and> d * (z \<bullet> i) \<le> d"
+      using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in ballE)
       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
       using assms by auto
     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
-      apply(erule_tac x=i in allE) using assms by auto qed
-  obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
+      apply(erule_tac x=i in ballE) using assms by (auto simp: inner_simps) qed
+  obtain s where "finite s" "{0::'a..\<Sum>Basis} = convex hull s" using unit_cube_convex_hull by auto
   thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
 
 subsection {* Bounded convex function on open set is continuous *}
@@ -4103,6 +4093,9 @@
 
 subsubsection {* Hence a convex function on an open set is continuous *}
 
+lemma real_of_nat_ge_one_iff: "1 \<le> real (n::nat) \<longleftrightarrow> 1 \<le> n"
+  by auto
+
 lemma convex_on_continuous:
   assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f"
   (* FIXME: generalize to euclidean_space *)
@@ -4113,29 +4106,40 @@
   then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
   def d \<equiv> "e / real DIM('a)"
   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)
-  let ?d = "(\<chi>\<chi> i. d)::'a"
+  let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
-  have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
+  have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by (auto simp: inner_setsum_left_Basis inner_simps)
   hence "c\<noteq>{}" using c by auto
   def k \<equiv> "Max (f ` c)"
-  have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
-    apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
+  have "convex_on {x - ?d..x + ?d} f"
+    apply(rule convex_on_subset[OF assms(2)])
+    apply(rule subset_trans[OF _ e(1)])
+    unfolding subset_eq mem_cball
+  proof
     fix z assume z:"z\<in>{x - ?d..x + ?d}"
-    have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
+    have e:"e = setsum (\<lambda>i::'a. d) Basis" unfolding setsum_constant d_def using dimge1
       unfolding real_eq_of_nat by auto
     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
-      using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
+      using z[unfolded mem_interval] apply(erule_tac x=b in ballE) by (auto simp: inner_simps)
+  qed
   hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
-  have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 by auto
+  have "d \<le> e"
+    unfolding d_def
+    apply(rule mult_imp_div_pos_le)
+    using `e>0`
+    unfolding mult_le_cancel_left1
+    apply (auto simp: real_of_nat_ge_one_iff Suc_le_eq DIM_positive)
+    done
   hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
   have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
   hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
     fix y assume y:"y\<in>cball x d"
-    { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d"
-        using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto  }
+    { fix i :: 'a assume "i\<in>Basis" hence "x \<bullet> i - d \<le> y \<bullet> i"  "y \<bullet> i \<le> x \<bullet> i + d"
+        using order_trans[OF Basis_le_norm y[unfolded mem_cball dist_norm], of i] by (auto simp: inner_diff_left)  }
     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
-      by auto qed
+      by (auto simp: inner_simps)
+  qed
   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
     apply force
@@ -4266,25 +4270,26 @@
       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
         unfolding as[unfolded dist_norm] norm_ge_zero by auto
       thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
-        unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
-      proof(rule,rule) fix i assume i:"i<DIM('a)"
-          have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
-            ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
-            using Fal by(auto simp add: field_simps)
-          also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
+        unfolding dist_norm apply(subst euclidean_eq_iff) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
+      proof(rule) fix i :: 'a assume i:"i\<in>Basis"
+          have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i =
+            ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
+            using Fal by(auto simp add: field_simps inner_simps)
+          also have "\<dots> = x\<bullet>i" apply(rule divide_eq_imp[OF Fal])
             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
-            apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps)
-          finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i"
+            apply(subst (asm) euclidean_eq_iff) using i apply(erule_tac x=i in ballE) by(auto simp add:field_simps inner_simps)
+          finally show "x \<bullet> i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
             by auto
-        qed(insert Fal2, auto) qed qed
+        qed(insert Fal2, auto) qed
+qed
 
 lemma between_midpoint: fixes a::"'a::euclidean_space" shows
   "between (a,b) (midpoint a b)" (is ?t1)
   "between (b,a) (midpoint a b)" (is ?t2)
 proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
-    unfolding euclidean_eq[where 'a='a]
-    by(auto simp add:field_simps) qed
+    unfolding euclidean_eq_iff[where 'a='a]
+    by(auto simp add:field_simps inner_simps) qed
 
 lemma between_mem_convex_hull:
   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
@@ -4303,7 +4308,7 @@
     have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm unfolding norm_scaleR[symmetric] apply(rule arg_cong[where f=norm]) using `e>0`
-      by(auto simp add: euclidean_eq[where 'a='a] field_simps)
+      by(auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
       by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
@@ -4347,236 +4352,237 @@
   apply(rule_tac x=u in exI) defer apply(rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
   unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
 
-lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
-  shows "convex hull (insert 0 { basis i | i. i : d}) =
-        {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
-  (!i<DIM('a). i ~: d --> x$$i = 0)}"
+lemma substd_simplex:
+  assumes d: "d \<subseteq> Basis"
+  shows "convex hull (insert 0 d) = {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d --> x\<bullet>i = 0)}"
   (is "convex hull (insert 0 ?p) = ?s")
-(* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
-proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
+proof- let ?D = d
   have "0 ~: ?p" using assms by (auto simp: image_def)
-  have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
-  note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
-  show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`]
+  from d have "finite d" by (blast intro: finite_subset finite_Basis)
+  show ?thesis unfolding simplex[OF `finite d` `0 ~: ?p`]
     apply(rule set_eqI) unfolding mem_Collect_eq apply rule
     apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
-    fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
-      "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
-    have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3)
-      unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
-    hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas
+    fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>?D. 0 \<le> u x"
+      "setsum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+    have *:"\<forall>i\<in>Basis. i:d --> u i = x\<bullet>i" and "(\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)" using as(3)
+      unfolding substdbasis_expansion_unique[OF assms] by auto
+    hence **:"setsum u ?D = setsum (op \<bullet> x) ?D"
       apply-apply(rule setsum_cong2) using assms by auto
-    have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1"
-      apply - proof(rule,rule,rule)
-      fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,symmetric]
+    have " (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1"
+      apply - proof(rule,rule)
+      fix i :: 'a assume i:"i\<in>Basis" have "i : d ==> 0 \<le> x\<bullet>i" unfolding *[rule_format,OF i,symmetric]
          apply(rule_tac as(1)[rule_format]) by auto
-      moreover have "i ~: d ==> 0 \<le> x$$i"
-        using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
-      ultimately show "0 \<le> x$$i" by auto
+      moreover have "i ~: d ==> 0 \<le> x\<bullet>i"
+        using `(\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)`[rule_format, OF i] by auto
+      ultimately show "0 \<le> x\<bullet>i" by auto
     qed(insert as(2)[unfolded **], auto)
-    from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)"
-      using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
-  next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
-      "(!i<DIM('a). i ~: d --> x $$ i = 0)"
-    show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
-      setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
-      apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE)
-      using as(1) apply(erule_tac x=i in allE) unfolding sumbas apply safe unfolding not_less basis_zero
-      unfolding substdbasis_expansion_unique[OF assms] euclidean_component_def[symmetric]
-      using as(2,3) by(auto simp add:dot_basis not_less)
-  qed qed
+    from this show " (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1 & (\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)"
+      using `(\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)` by auto
+  next fix x::"'a::euclidean_space" assume as:"\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "setsum (op \<bullet> x) ?D \<le> 1"
+      "(\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)"
+    show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> setsum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+      using as d unfolding substdbasis_expansion_unique[OF assms]
+      by (rule_tac x="inner x" in exI) auto
+  qed
+qed
 
 lemma std_simplex:
-  "convex hull (insert 0 { basis i | i. i<DIM('a)}) =
-        {x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
-  using substd_simplex[of "{..<DIM('a)}"] by auto
+  "convex hull (insert 0 Basis) =
+        {x::'a::euclidean_space . (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (\<lambda>i. x\<bullet>i) Basis \<le> 1 }"
+  using substd_simplex[of Basis] by auto
 
 lemma interior_std_simplex:
-  "interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
-  {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
+  "interior (convex hull (insert 0 Basis)) =
+  {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> setsum (\<lambda>i. x\<bullet>i) Basis < 1 }"
   apply(rule set_eqI) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
   unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
-  fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
-  show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
-    fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
-      unfolding dist_norm by (auto elim!:allE[where x=i])
-  next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using  `e>0`
-      unfolding dist_norm by(auto intro!: mult_strict_left_mono)
-    have "\<And>i. i<DIM('a) \<Longrightarrow> (x + (e / 2) *\<^sub>R basis 0) $$ i = x$$i + (if i = 0 then e/2 else 0)"
-      by auto
-    hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)} = setsum (\<lambda>i. x$$i + (if 0 = i then e/2 else 0)) {..<DIM('a)}"
+  fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> setsum (op \<bullet> xa) Basis \<le> 1"
+  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> setsum (op \<bullet> x) Basis < 1" apply(safe) proof-
+    fix i :: 'a assume i:"i\<in>Basis" thus "0 < x \<bullet> i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R i"]] and `e>0`
+      unfolding dist_norm
+      by (auto elim!:ballE[where x=i] simp: inner_simps)
+  next have **:"dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using  `e>0`
+      unfolding dist_norm by(auto intro!: mult_strict_left_mono simp: SOME_Basis)
+    have "\<And>i. i\<in>Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i = x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
+      by (auto simp: SOME_Basis inner_Basis inner_simps)
+    hence *:"setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis = setsum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
       apply(rule_tac setsum_cong) by auto
-    have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
-      using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
+    have "setsum (op \<bullet> x) Basis < setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis" unfolding * setsum_addf
+      using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by (auto simp: SOME_Basis)
     also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
-    finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
-next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
+    finally show "setsum (op \<bullet> x) Basis < 1" by auto qed
+next fix x::"'a" assume as:"\<forall>i\<in>Basis. 0 < x \<bullet> i" "setsum (op \<bullet> x) Basis < 1"
   guess a using UNIV_witness[where 'a='b] ..
-  let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
-  have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
-  moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
-  ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
-    apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
-    fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
-    have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
-      fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
-        using component_le_norm[of "y - x" i]
-        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
-      thus "y $$ i \<le> x $$ i + ?d" by auto qed
+  let ?d = "(1 - setsum (op \<bullet> x) Basis) / real (DIM('a))"
+  have "Min ((op \<bullet> x) ` Basis) > 0" apply(rule Min_grI) using as(1) by auto
+  moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by (auto simp add: Suc_le_eq DIM_positive)
+  ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
+    apply(rule_tac x="min (Min ((op \<bullet> x) ` Basis)) ?D" in exI) apply rule defer apply(rule,rule) proof-
+    fix y assume y:"dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
+    have "setsum (op \<bullet> y) Basis \<le> setsum (\<lambda>i. x\<bullet>i + ?d) Basis" proof(rule setsum_mono)
+      fix i :: 'a assume i: "i\<in>Basis" hence "abs (y\<bullet>i - x\<bullet>i) < ?d" apply-apply(rule le_less_trans)
+        using Basis_le_norm[OF i, of "y - x"]
+        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute inner_diff_left)
+      thus "y \<bullet> i \<le> x \<bullet> i + ?d" by auto qed
     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat by(auto simp add: Suc_le_eq)
-    finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
-    proof safe fix i assume i:"i<DIM('a)"
-      have "norm (x - y) < x$$i" apply(rule less_le_trans)
+    finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
+    proof safe fix i :: 'a assume i:"i\<in>Basis"
+      have "norm (x - y) < x\<bullet>i" apply(rule less_le_trans)
         apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
-      thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
+      thus "0 \<le> y\<bullet>i" using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
+        by (auto simp: inner_simps)
     qed qed auto qed
 
 lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
-  "a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
-  let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
-  have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
-  { fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
-      unfolding euclidean_component_setsum * and setsum_reindex[OF basis_inj] and o_def
-      apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
-      defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
+  "a \<in> interior(convex hull (insert 0 Basis))" proof-
+  let ?D = "Basis :: 'a set" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
+  { fix i :: 'a assume i:"i\<in>Basis" have "?a \<bullet> i = inverse (2 * real DIM('a))"
+      by (rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
+         (simp_all add: setsum_cases i) }
   note ** = this
   show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
-    fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
-  next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
+    fix i :: 'a assume i:"i\<in>Basis" show "0 < ?a \<bullet> i" unfolding **[OF i] by(auto simp add: Suc_le_eq DIM_positive)
+  next have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
     also have "\<dots> < 1" unfolding setsum_constant real_eq_of_nat divide_inverse[symmetric] by (auto simp add:field_simps)
-    finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
-
-lemma rel_interior_substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
-  shows "rel_interior (convex hull (insert 0 { basis i| i. i : d})) =
-  {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x$$i) & setsum (%i. x$$i) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)}"
+    finally show "setsum (op \<bullet> ?a) ?D < 1" by auto qed qed
+
+lemma rel_interior_substd_simplex: assumes d: "d\<subseteq>Basis"
+  shows "rel_interior (convex hull (insert 0 d)) =
+  {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)}"
   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)
 proof-
 have "finite d" apply(rule finite_subset) using assms by auto
-{ assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
+{ assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto }
 moreover
 { assume "d~={}"
-have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
+have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)}"
    using affine_hull_convex_hull affine_hull_substd_basis assms by auto
-have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
+have aux: "!!x::'a. \<forall>i\<in>Basis. ((\<forall>i\<in>d. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)) \<longrightarrow> 0 \<le> x\<bullet>i" 
+  by auto
 { fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
   from this obtain e where e0: "e>0" and
-       "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)"
+       "ball x e Int {xa. (\<forall>i\<in>Basis. i ~: d --> xa\<bullet>i = 0)} <= convex hull (insert 0 ?p)"
        using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
-  hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
-    (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
+  hence as: "ALL xa. (dist x xa < e & (\<forall>i\<in>Basis. i ~: d --> xa\<bullet>i = 0)) -->
+    (!i : d. 0 <= xa \<bullet> i) & setsum (op \<bullet> xa) d <= 1"
     unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
-  have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)"
+  have x0: "(\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)"
     using x_def rel_interior_subset  substd_simplex[OF assms] by auto
-  have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule)
+  have "(\<forall>i\<in>d. 0 < x \<bullet> i) & setsum (op \<bullet> x) d < 1 & (\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)" apply(rule,rule)
   proof-
-    fix i::nat assume "i:d"
-    hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
-      unfolding dist_norm using assms `e>0` x0 by auto
-    thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
+    fix i::'a assume "i\<in>d"
+    hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> ia" apply-apply(rule as[rule_format,THEN conjunct1])
+      unfolding dist_norm using d `e>0` x0 by (auto simp: inner_simps inner_Basis)
+    thus "0 < x \<bullet> i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` d
+    by (auto simp: inner_simps inner_Basis)
   next obtain a where a:"a:d" using `d ~= {}` by auto
-    have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
-      using  `e>0` and Euclidean_Space.norm_basis[of a]
+    then have **:"dist x (x + (e / 2) *\<^sub>R a) < e"
+      using  `e>0` norm_Basis[of a] d
       unfolding dist_norm by auto
-    have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $$ i = x$$i + (if i = a then e/2 else 0)"
-      unfolding euclidean_simps using a assms by auto
-    hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d =
-      setsum (\<lambda>i. x$$i + (if a = i then e/2 else 0)) d" by(rule_tac setsum_cong, auto)
-    have h1: "(ALL i<DIM('a). i ~: d --> (x + (e / 2) *\<^sub>R basis a) $$ i = 0)"
-      using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
-      by(auto elim:allE[where x=a])
-    have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
+    have "\<And>i. i\<in>Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
+      using a d by (auto simp: inner_simps inner_Basis)
+    hence *:"setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
+      setsum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d" using d by (intro setsum_cong) auto
+    have "a \<in> Basis" using `a \<in> d` d by auto
+    then have h1: "(\<forall>i\<in>Basis. i ~: d --> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
+      using x0 d `a\<in>d` by (auto simp add: inner_add_left inner_Basis)
+    have "setsum (op \<bullet> x) d < setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d" unfolding * setsum_addf
       using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
-    also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
-    finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto
+    also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"] by auto
+    finally show "setsum (op \<bullet> x) d < 1 & (\<forall>i\<in>Basis. i ~: d --> x\<bullet>i = 0)" using x0 by auto
   qed
 }
 moreover
 {
   fix x::"'a::euclidean_space" assume as: "x : ?s"
-  have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
+  have "!i. ((0<x\<bullet>i) | (0=x\<bullet>i) --> 0<=x\<bullet>i)" by auto
   moreover have "!i. (i:d) | (i ~: d)" by auto
   ultimately
-  have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
+  have "!i. ( (ALL i:d. 0 < x\<bullet>i) & (ALL i. i ~: d --> x\<bullet>i = 0) ) --> 0 <= x\<bullet>i" by metis
   hence h2: "x : convex hull (insert 0 ?p)" using as assms
     unfolding substd_simplex[OF assms] by fastforce
   obtain a where a:"a:d" using `d ~= {}` by auto
-  let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
+  let ?d = "(1 - setsum (op \<bullet> x) d) / real (card d)"
   have "0 < card d" using `d ~={}` `finite d` by (simp add: card_gt_0_iff)
-  have "Min ((op $$ x) ` d) > 0" using as `d \<noteq> {}` `finite d` by (simp add: Min_grI)
+  have "Min ((op \<bullet> x) ` d) > 0" using as `d \<noteq> {}` `finite d` by (simp add: Min_grI)
   moreover have "?d > 0" apply(rule divide_pos_pos) using as using `0 < card d` by auto
-  ultimately have h3: "min (Min ((op $$ x) ` d)) ?d > 0" by auto
+  ultimately have h3: "min (Min ((op \<bullet> x) ` d)) ?d > 0" by auto
 
   have "x : rel_interior (convex hull (insert 0 ?p))"
     unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)
     unfolding substd_simplex[OF assms]
-    apply(rule_tac x="min (Min ((op $$ x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
-  proof- fix y::'a assume y:"dist x y < min (Min (op $$ x ` d)) ?d" and y2:"(!i<DIM('a). i ~: d --> y$$i = 0)"
-    have "setsum (op $$ y) d \<le> setsum (\<lambda>i. x$$i + ?d) d" proof(rule setsum_mono)
-      fix i assume i:"i\<in>d"
-      have "abs (y$$i - x$$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
+    apply(rule_tac x="min (Min ((op \<bullet> x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
+  proof-
+    fix y::'a assume y:"dist x y < min (Min (op \<bullet> x ` d)) ?d" and y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
+    have "setsum (op \<bullet> y) d \<le> setsum (\<lambda>i. x\<bullet>i + ?d) d"
+    proof(rule setsum_mono)
+      fix i assume "i \<in> d"
+      with d have i: "i \<in> Basis" by auto
+      have "abs (y\<bullet>i - x\<bullet>i) < ?d" apply(rule le_less_trans) using Basis_le_norm[OF i, of "y - x"]
         using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
-        by(auto simp add: norm_minus_commute)
-      thus "y $$ i \<le> x $$ i + ?d" by auto qed
+        by (auto simp add: norm_minus_commute inner_simps)
+      thus "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
+    qed
     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant real_eq_of_nat
       using `0 < card d` by auto
-    finally show "setsum (op $$ y) d \<le> 1" .
-
-    fix i assume "i<DIM('a)" thus "0 \<le> y$$i"
+    finally show "setsum (op \<bullet> y) d \<le> 1" .
+
+    fix i :: 'a assume i: "i \<in> Basis" thus "0 \<le> y\<bullet>i"
     proof(cases "i\<in>d") case True
-      have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
-        using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `0 < card d` `i:d`
+      have "norm (x - y) < x\<bullet>i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
+        using Min_gr_iff[of "op \<bullet> x ` d" "norm (x - y)"] `0 < card d` `i:d`
         by (simp add: card_gt_0_iff)
-      thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
+      thus "0 \<le> y\<bullet>i" using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
+        by (auto simp: inner_simps)
     qed(insert y2, auto)
   qed
 } ultimately have
-    "!!x :: 'a::euclidean_space. (x : rel_interior (convex hull insert 0 {basis i |i. i : d})) =
-    (x : {x. (ALL i:d. 0 < x $$ i) &
-    setsum (op $$ x) d < 1 & (ALL i<DIM('a). i ~: d --> x $$ i = 0)})" by blast
+    "\<And>x. (x : rel_interior (convex hull insert 0 d)) = (x \<in> {x. (ALL i:d. 0 < x \<bullet> i) &
+    setsum (op \<bullet> x) d < 1 & (\<forall>i\<in>Basis. i ~: d --> x \<bullet> i = 0)})" by blast
 from this have ?thesis by (rule set_eqI)
 } ultimately show ?thesis by blast
 qed
 
-lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>{..<DIM('a::euclidean_space)}"
+lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>Basis"
   obtains a::"'a::euclidean_space" where
-  "a : rel_interior(convex hull (insert 0 {basis i | i . i : d}))" proof-
+  "a : rel_interior(convex hull (insert 0 d))" proof-
 (* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)
-  let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
-  have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
+  let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
   have "finite d" apply(rule finite_subset) using assms(2) by auto
   hence d1: "0 < real(card d)" using `d ~={}` by auto
-  { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
-      unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
+  { fix i assume "i:d"
+    have "?a \<bullet> i = inverse (2 * real (card d))"
       apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
-      unfolding euclidean_component_setsum
+      unfolding inner_setsum_left
       apply(rule setsum_cong2)
       using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
-      by (auto simp add: Euclidean_Space.basis_component[of i])}
+      by (auto simp: inner_simps inner_Basis set_rev_mp[OF _ assms(2)]) }
   note ** = this
   show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
   proof safe fix i assume "i:d"
     have "0 < inverse (2 * real (card d))" using d1 by auto
-    also have "...=?a $$ i" using **[of i] `i:d` by auto
-    finally show "0 < ?a $$ i" by auto
-  next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
+    also have "...=?a \<bullet> i" using **[of i] `i:d` by auto
+    finally show "0 < ?a \<bullet> i" by auto
+  next have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
       by(rule setsum_cong2, rule **)
     also have "\<dots> < 1" unfolding setsum_constant real_eq_of_nat divide_real_def[symmetric]
       by (auto simp add:field_simps)
-    finally show "setsum (op $$ ?a) ?D < 1" by auto
-  next fix i assume "i<DIM('a)" and "i~:d"
-    have "?a : (span {basis i | i. i : d})"
-      apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"])
-      using finite_substdbasis[of d] apply blast
+    finally show "setsum (op \<bullet> ?a) ?D < 1" by auto
+  next fix i assume "i\<in>Basis" and "i~:d"
+    have "?a : (span d)"
+      apply (rule span_setsum[of d "(%b. b /\<^sub>R (2 * real (card d)))" d])
+      using finite_subset[OF assms(2) finite_Basis]
+      apply blast
     proof-
-      { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
-        hence "x : span {basis i |i. i : d}"
-          using span_superset[of _ "{basis i |i. i : d}"] by auto
-        hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
-          using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
-      } thus "\<forall>x\<in>{basis i |i. i \<in> d}. x /\<^sub>R (2 * real (card d)) \<in> span {basis i ::'a |i. i \<in> d}" by auto
+      { fix x assume "(x :: 'a::euclidean_space): d"
+        hence "x : span d"
+          using span_superset[of _ "d"] by auto
+        hence "(x /\<^sub>R (2 * real (card d))) : (span d)"
+          using span_mul[of x "d" "(inverse (real (card d)) / 2)"] by auto
+      } thus "\<forall>x\<in>d. x /\<^sub>R (2 * real (card d)) \<in> span d" by auto
     qed
-    thus "?a $$ i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i<DIM('a)` by auto
+    thus "?a \<bullet> i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i\<in>Basis` by auto
   qed
 qed
 
@@ -4608,14 +4614,14 @@
 ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
     assms  hull_subset[of S] by auto
-obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} &
-       f ` span B = {x. ALL i<DIM('n). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B)" and d:"d\<subseteq>{..<DIM('n)}"
+obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = d &
+       f ` span B = {x. \<forall>i\<in>Basis. i ~: d --> x \<bullet> i = (0::real)} &  inj_on f (span B)" and d:"d\<subseteq>Basis"
     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
 hence "bounded_linear f" using linear_conv_bounded_linear by auto
 have "d ~={}" using fd B_def `B ~={}` by auto
-have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
-hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
-   using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"]
+have "(insert 0 d) = f ` (insert 0 B)" using fd linear_0 by auto
+hence "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
+   using convex_hull_linear_image[of f "(insert 0 d)"]
    convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
 moreover have "rel_interior (f ` (convex hull insert 0 B)) =
    f ` rel_interior (convex hull insert 0 B)"
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -158,9 +158,6 @@
 lemmas mult_left_has_derivative =
   bounded_linear.has_derivative [OF bounded_linear_mult_left]
 
-lemmas euclidean_component_has_derivative =
-  bounded_linear.has_derivative [OF bounded_linear_euclidean_component]
-
 lemma has_derivative_neg:
   assumes "(f has_derivative f') net"
   shows "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
@@ -191,20 +188,12 @@
   using assms by (induct, simp_all add: has_derivative_const has_derivative_add)
 text {* Somewhat different results for derivative of scalar multiplier. *}
 
-(** move **)
-lemma linear_vmul_component: (* TODO: delete *)
-  assumes lf: "linear f"
-  shows "linear (\<lambda>x. f x $$ k *\<^sub>R v)"
-  using lf
-  by (auto simp add: linear_def algebra_simps)
-
 lemmas has_derivative_intros =
   has_derivative_id has_derivative_const
   has_derivative_add has_derivative_sub has_derivative_neg
   has_derivative_add_const
   scaleR_left_has_derivative scaleR_right_has_derivative
   inner_left_has_derivative inner_right_has_derivative
-  euclidean_component_has_derivative
 
 subsubsection {* Limit transformation for derivatives *}
 
@@ -531,7 +520,7 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_derivative f') (at x within s)"
   assumes "(f has_derivative f'') (at x within s)"
-  assumes "(\<forall>i<DIM('a). \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)"
+  assumes "(\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R i) \<in> s)"
   shows "f' = f''"
 proof-
   note as = assms(1,2)[unfolded has_derivative_def]
@@ -540,32 +529,32 @@
   have "x islimpt s" unfolding islimpt_approachable
   proof(rule,rule)
     fix e::real assume "0<e" guess d
-      using assms(3)[rule_format,OF DIM_positive `e>0`] ..
+      using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
     thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
-      apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
-      unfolding dist_norm by auto
+      apply(rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
+      unfolding dist_norm by (auto simp: SOME_Basis nonzero_Basis)
   qed
   hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
     unfolding trivial_limit_within by simp
   show ?thesis  apply(rule linear_eq_stdbasis)
     unfolding linear_conv_bounded_linear
     apply(rule as(1,2)[THEN conjunct1])+
-  proof(rule,rule,rule ccontr)
-    fix i assume i:"i<DIM('a)" def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
-    assume "f' (basis i) \<noteq> f'' (basis i)"
+  proof(rule,rule ccontr)
+    fix i :: 'a assume i:"i \<in> Basis" def e \<equiv> "norm (f' i - f'' i)"
+    assume "f' i \<noteq> f'' i"
     hence "e>0" unfolding e_def by auto
     guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
     guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
-    have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
+    have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
       unfolding scaleR_right_distrib by auto
-    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"  
+    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"  
       unfolding f'.scaleR f''.scaleR
       unfolding scaleR_right_distrib scaleR_minus_right by auto
     also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
-      using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"]
+      using norm_minus_cancel[of "f' i - f'' i"]
       by (auto simp add: add.commute ab_diff_minus)
     finally show False using c
-      using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"]
+      using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R i"]
       unfolding dist_norm
       unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
         scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
@@ -584,37 +573,38 @@
 
 lemma frechet_derivative_unique_within_closed_interval:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I")
+  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "x \<in> {a..b}" (is "x\<in>?I")
   assumes "(f has_derivative f' ) (at x within {a..b})"
   assumes "(f has_derivative f'') (at x within {a..b})"
   shows "f' = f''"
   apply(rule frechet_derivative_unique_within)
   apply(rule assms(3,4))+
-proof(rule,rule,rule,rule)
-  fix e::real and i assume "e>0" and i:"i<DIM('a)"
-  thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}"
-  proof(cases "x$$i=a$$i")
+proof(rule,rule,rule)
+  fix e::real and i :: 'a assume "e>0" and i:"i\<in>Basis"
+  thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
+  proof(cases "x\<bullet>i=a\<bullet>i")
     case True thus ?thesis
-      apply(rule_tac x="(min (b$$i - a$$i)  e) / 2" in exI)
-      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
-      unfolding mem_interval euclidean_simps
-      using i by (auto simp add: field_simps)
-  next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
-    case False moreover have "a $$ i < x $$ i" using False * by auto
+      apply(rule_tac x="(min (b\<bullet>i - a\<bullet>i)  e) / 2" in exI)
+      using assms(1)[THEN bspec[where x=i]] and `e>0` and assms(2)
+      unfolding mem_interval
+      using i by (auto simp add: field_simps inner_simps inner_Basis)
+  next 
+    note * = assms(2)[unfolded mem_interval, THEN bspec, OF i]
+    case False moreover have "a \<bullet> i < x \<bullet> i" using False * by auto
     moreover {
-      have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i"
+      have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
         by auto
-      also have "\<dots> = a$$i + x$$i" by auto
-      also have "\<dots> \<le> 2 * x$$i" using * by auto 
-      finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto
+      also have "\<dots> = a\<bullet>i + x\<bullet>i" by auto
+      also have "\<dots> \<le> 2 * (x\<bullet>i)" using * by auto
+      finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2" by auto
     }
-    moreover have "min (x $$ i - a $$ i) e \<ge> 0" using * and `e>0` by auto
-    hence "x $$ i * 2 \<le> b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto
+    moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0" using * and `e>0` by auto
+    hence "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e" using * by auto
     ultimately show ?thesis
-      apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
-      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
-      unfolding mem_interval euclidean_simps
-      using i by (auto simp add: field_simps)
+      apply(rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
+      using assms(1)[THEN bspec, OF i] and `e>0` and assms(2)
+      unfolding mem_interval
+      using i by (auto simp add: field_simps inner_simps inner_Basis)
   qed
 qed
 
@@ -638,7 +628,7 @@
 
 lemma frechet_derivative_within_closed_interval:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<forall>i<DIM('a). a$$i < b$$i" and "x \<in> {a..b}"
+  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and "x \<in> {a..b}"
   assumes "(f has_derivative f') (at x within {a.. b})"
   shows "frechet_derivative f (at x within {a.. b}) = f'"
   apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) 
@@ -650,14 +640,14 @@
 lemma linear_componentwise:
   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes lf: "linear f"
-  shows "(f x) $$ j = (\<Sum>i<DIM('a). (x$$i) * (f (basis i)$$j))" (is "?lhs = ?rhs")
+  shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
 proof -
-  have fA: "finite {..<DIM('a)}" by simp
-  have "?rhs = (\<Sum>i<DIM('a). x$$i *\<^sub>R f (basis i))$$j"
-    by simp
+  have fA: "finite Basis" by simp
+  have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
+    by (simp add: inner_setsum_left)
   then show ?thesis
     unfolding linear_setsum_mul[OF lf fA, symmetric]
-    unfolding euclidean_representation[symmetric] ..
+    unfolding euclidean_representation ..
 qed
 
 text {* We do not introduce @{text jacobian}, which is defined on matrices, instead we use
@@ -665,52 +655,54 @@
 
 lemma jacobian_works:
   "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
-   (f has_derivative (\<lambda>h. \<chi>\<chi> i.
-      \<Sum>j<DIM('a). frechet_derivative f net (basis j) $$ i * h $$ j)) net"
-  (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net")
+   (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
+      (\<Sum>j\<in>Basis. frechet_derivative f net (j) \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
+  (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
 proof
   assume *: ?differentiable
   { fix h i
-    have "?SUM h i = frechet_derivative f net h $$ i" using *
+    have "?SUM h i = frechet_derivative f net h \<bullet> i" using *
       by (auto intro!: setsum_cong
                simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
-  thus "(f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net"
-    using * by (simp add: frechet_derivative_works)
+  with * show "(f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net"
+    by (simp add: frechet_derivative_works euclidean_representation)
 qed (auto intro!: differentiableI)
 
 lemma differential_zero_maxmin_component:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes k: "k < DIM('b)"
-    and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)$$k \<le> (f x)$$k) \<or> (\<forall>y\<in>ball x e. (f x)$$k \<le> (f y)$$k))"
+  assumes k: "k \<in> Basis"
+    and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k))"
     and diff: "f differentiable (at x)"
-  shows "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j) $$ k) = (0::'a)" (is "?D k = 0")
+  shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
 proof (rule ccontr)
   assume "?D k \<noteq> 0"
-  then obtain j where j: "?D k $$ j \<noteq> 0" "j < DIM('a)"
-    unfolding euclidean_lambda_beta euclidean_eq[of _ "0::'a"] by auto
-  hence *: "\<bar>?D k $$ j\<bar> / 2 > 0" by auto
+  then obtain j where j: "?D k \<bullet> j \<noteq> 0" "j \<in> Basis"
+    unfolding euclidean_eq_iff[of _ "0::'a"] by auto
+  hence *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0" by auto
   note as = diff[unfolded jacobian_works has_derivative_at_alt]
   guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
   guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
   { fix c assume "abs c \<le> d"
-    hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
-    let ?v = "(\<chi>\<chi> i. \<Sum>l<DIM('a). ?D i $$ l * (c *\<^sub>R basis j :: 'a) $$ l)"
+    hence *:"norm (x + c *\<^sub>R j - x) < e'" using norm_Basis[OF j(2)] d by auto
+    let ?v = "(\<Sum>i\<in>Basis. (\<Sum>l\<in>Basis. ?D i \<bullet> l * ((c *\<^sub>R j :: 'a) \<bullet> l)) *\<^sub>R i)"
     have if_dist: "\<And> P a b c. a * (if P then b else c) = (if P then a * b else a * c)" by auto
-    have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le>
-        norm (f (x + c *\<^sub>R basis j) - f x - ?v)" by (rule component_le_norm)
-    also have "\<dots> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
-      using e'[THEN conjunct2, rule_format, OF *] and norm_basis[of j] by fastforce
-    finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>" by simp
-    hence "\<bar>f (x + c *\<^sub>R basis j) $$ k - f x $$ k - c * ?D k $$ j\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
-      unfolding euclidean_simps euclidean_lambda_beta using j k
-      by (simp add: if_dist setsum_cases field_simps) } note * = this
-  have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
-    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
-  hence **:"((f (x - d *\<^sub>R basis j))$$k \<le> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<le> (f x)$$k) \<or>
-         ((f (x - d *\<^sub>R basis j))$$k \<ge> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<ge> (f x)$$k)" using ball by auto
+    have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le>
+        norm (f (x + c *\<^sub>R j) - f x - ?v)" by (rule Basis_le_norm[OF k])
+    also have "\<dots> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
+      using e'[THEN conjunct2, rule_format, OF *] and norm_Basis[OF j(2)] j
+      by simp
+    finally have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>" by simp
+    hence "\<bar>f (x + c *\<^sub>R j) \<bullet> k - f x \<bullet> k - c * (?D k \<bullet> j)\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
+      using j k
+      by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist) }
+  note * = this
+  have "x + d *\<^sub>R j \<in> ball x e" "x - d *\<^sub>R j \<in> ball x e"
+    unfolding mem_ball dist_norm using norm_Basis[OF j(2)] d by auto
+  hence **:"((f (x - d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k) \<or>
+         ((f (x - d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k)" using ball by auto
   have ***: "\<And>y y1 y2 d dx::real.
     (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
-  show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\<bar>?D k $$ j\<bar> / 2 * \<bar>d\<bar>"])
+  show False apply(rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
     unfolding mult_minus_left
     unfolding abs_mult diff_minus_eq_add scaleR_minus_left
@@ -728,13 +720,13 @@
 proof -
   obtain e where e:"e>0" "ball x e \<subseteq> s"
     using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
-  with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified]
-  have "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j)) = (0::'a)"
-    unfolding differentiable_def using mono deriv by auto
+  with differential_zero_maxmin_component[where 'b=real, of 1 e x f] mono deriv
+  have "(\<Sum>j\<in>Basis. frechet_derivative f (at x) j *\<^sub>R j) = (0::'a)"
+    by (auto simp: Basis_real_def differentiable_def)
   with frechet_derivative_at[OF deriv, symmetric]
-  have "\<forall>i<DIM('a). f' (basis i) = 0"
-    by (simp add: euclidean_eq[of _ "0::'a"])
-  with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 0]
+  have "\<forall>i\<in>Basis. f' i = 0"
+    by (simp add: euclidean_eq_iff[of _ "0::'a"] inner_setsum_left_Basis)
+  with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 1]
   show ?thesis by (simp add: fun_eq_iff)
 qed
 
@@ -1281,8 +1273,8 @@
 proof-
   interpret bounded_linear g' using assms by auto
   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
-  have "g' (f' a (\<chi>\<chi> i.1)) = (\<chi>\<chi> i.1)" "(\<chi>\<chi> i.1) \<noteq> (0::'n)" defer 
-    apply(subst euclidean_eq) using f'g' by auto
+  have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)" defer 
+    apply(subst euclidean_eq_iff) using f'g' by auto
   hence *:"0 < onorm g'"
     unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastforce
   def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
@@ -1726,7 +1718,7 @@
   have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
     apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
     using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
-    by auto
+    by (auto simp: Basis_real_def)
   show ?thesis
   proof(rule ccontr)
     assume "f' \<noteq> f''"
--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -452,7 +452,7 @@
 
   ultimately show ?thesis
     apply -
-    apply (rule span_induct_alt[of ?P ?S, OF P0, folded smult_conv_scaleR])
+    apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR])
     apply blast
     apply (rule x)
     done
@@ -746,7 +746,7 @@
       apply (rule span_setsum)
       apply simp
       apply (rule ballI)
-      apply (rule span_mul [where 'a="real^'n", folded smult_conv_scaleR])+
+      apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
       apply (rule span_superset)
       apply auto
       done
@@ -782,7 +782,7 @@
     apply (rule det_row_span)
     apply (rule span_setsum[OF fUk])
     apply (rule ballI)
-    apply (rule span_mul [where 'a="real^'n", folded smult_conv_scaleR])+
+    apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
     apply (rule span_superset)
     apply auto
     done
@@ -879,9 +879,10 @@
       have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
         "\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
         by simp_all
-      from fd[rule_format, of "cart_basis i" "cart_basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
+      from fd[rule_format, of "axis i 1" "axis j 1", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
       have "?A$i$j = ?m1 $ i $ j"
-        by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def cart_basis_def th0 setsum_delta[OF fU] mat_def)}
+        by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
+            th0 setsum_delta[OF fU] mat_def axis_def) }
     hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
     with lf have ?rhs by blast}
   moreover
@@ -931,7 +932,9 @@
       unfolding dot_norm_neg dist_norm[symmetric]
       unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
   note fc = this
-  show ?thesis unfolding linear_def vector_eq[where 'a="real^'n"] smult_conv_scaleR by (simp add: inner_add fc field_simps)
+  show ?thesis
+    unfolding linear_def vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR 
+    by (simp add: inner_add fc field_simps)
 qed
 
 lemma isometry_linear:
@@ -981,7 +984,7 @@
       apply (subst H(4))
       using H(5-9)
       apply (simp add: norm_eq norm_eq_1)
-      apply (simp add: inner_diff smult_conv_scaleR) unfolding *
+      apply (simp add: inner_diff scalar_mult_eq_scaleR) unfolding *
       by (simp add: field_simps) }
   note th0 = this
   let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -23,24 +23,24 @@
   assumes euclidean_all_zero_iff:
     "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
 
-  -- "FIXME: make this a separate definition"
-  fixes dimension :: "'a itself \<Rightarrow> nat"
-  assumes dimension_def: "dimension TYPE('a) = card Basis"
-
-  -- "FIXME: eventually basis function can be removed"
-  fixes basis :: "nat \<Rightarrow> 'a"
-  assumes image_basis: "basis ` {..<dimension TYPE('a)} = Basis"
-  assumes basis_finite: "basis ` {dimension TYPE('a)..} = {0}"
+abbreviation dimension :: "('a::euclidean_space) itself \<Rightarrow> nat" where
+  "dimension TYPE('a) \<equiv> card (Basis :: 'a set)"
 
 syntax "_type_dimension" :: "type => nat" ("(1DIM/(1'(_')))")
 
 translations "DIM('t)" == "CONST dimension (TYPE('t))"
 
-lemma (in euclidean_space) norm_Basis: "u \<in> Basis \<Longrightarrow> norm u = 1"
+lemma (in euclidean_space) norm_Basis[simp]: "u \<in> Basis \<Longrightarrow> norm u = 1"
   unfolding norm_eq_sqrt_inner by (simp add: inner_Basis)
 
+lemma (in euclidean_space) inner_same_Basis[simp]: "u \<in> Basis \<Longrightarrow> inner u u = 1"
+  by (simp add: inner_Basis)
+
+lemma (in euclidean_space) inner_not_same_Basis: "u \<in> Basis \<Longrightarrow> v \<in> Basis \<Longrightarrow> u \<noteq> v \<Longrightarrow> inner u v = 0"
+  by (simp add: inner_Basis)
+
 lemma (in euclidean_space) sgn_Basis: "u \<in> Basis \<Longrightarrow> sgn u = u"
-  unfolding sgn_div_norm by (simp add: norm_Basis scaleR_one)
+  unfolding sgn_div_norm by (simp add: scaleR_one)
 
 lemma (in euclidean_space) Basis_zero [simp]: "0 \<notin> Basis"
 proof
@@ -51,184 +51,45 @@
 lemma (in euclidean_space) nonzero_Basis: "u \<in> Basis \<Longrightarrow> u \<noteq> 0"
   by clarsimp
 
-text {* Lemmas related to @{text basis} function. *}
-
-lemma (in euclidean_space) euclidean_all_zero:
-  "(\<forall>i<DIM('a). inner (basis i) x = 0) \<longleftrightarrow> (x = 0)"
-  using euclidean_all_zero_iff [of x, folded image_basis]
-  unfolding ball_simps by (simp add: Ball_def inner_commute)
-
-lemma (in euclidean_space) basis_zero [simp]:
-  "DIM('a) \<le> i \<Longrightarrow> basis i = 0"
-  using basis_finite by auto
+lemma (in euclidean_space) SOME_Basis: "(SOME i. i \<in> Basis) \<in> Basis"
+  by (metis ex_in_conv nonempty_Basis someI_ex)
 
-lemma (in euclidean_space) DIM_positive [intro]: "0 < DIM('a)"
-  unfolding dimension_def by (simp add: card_gt_0_iff)
-
-lemma (in euclidean_space) basis_inj [simp, intro]: "inj_on basis {..<DIM('a)}"
-  by (simp add: inj_on_iff_eq_card image_basis dimension_def [symmetric])
-
-lemma (in euclidean_space) basis_in_Basis [simp]:
-  "basis i \<in> Basis \<longleftrightarrow> i < DIM('a)"
-  by (cases "i < DIM('a)", simp add: image_basis [symmetric], simp)
-
-lemma (in euclidean_space) Basis_elim:
-  assumes "u \<in> Basis" obtains i where "i < DIM('a)" and "u = basis i"
-  using assms unfolding image_basis [symmetric] by fast
+lemma (in euclidean_space) inner_setsum_left_Basis[simp]:
+    "b \<in> Basis \<Longrightarrow> inner (\<Sum>i\<in>Basis. f i *\<^sub>R i) b = f b"
+  by (simp add: inner_setsum_left inner_Basis if_distrib setsum_cases)
 
-lemma (in euclidean_space) basis_orthonormal:
-    "\<forall>i<DIM('a). \<forall>j<DIM('a).
-      inner (basis i) (basis j) = (if i = j then 1 else 0)"
-  apply clarify
-  apply (simp add: inner_Basis)
-  apply (simp add: basis_inj [unfolded inj_on_def])
-  done
-
-lemma (in euclidean_space) dot_basis:
-  "inner (basis i) (basis j) = (if i = j \<and> i < DIM('a) then 1 else 0)"
-proof (cases "(i < DIM('a) \<and> j < DIM('a))")
-  case False
-  hence "inner (basis i) (basis j) = 0" by auto
-  thus ?thesis using False by auto
-next
-  case True thus ?thesis using basis_orthonormal by auto
-qed
-
-lemma (in euclidean_space) basis_eq_0_iff [simp]:
-  "basis i = 0 \<longleftrightarrow> DIM('a) \<le> i"
+lemma (in euclidean_space) euclidean_eqI:
+  assumes b: "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
 proof -
-  have "inner (basis i) (basis i) = 0 \<longleftrightarrow> DIM('a) \<le> i"
-    by (simp add: dot_basis)
-  thus ?thesis by simp
+  from b have "\<forall>b\<in>Basis. inner (x - y) b = 0"
+    by (simp add: inner_diff_left)
+  then show "x = y"
+    by (simp add: euclidean_all_zero_iff)
 qed
 
-lemma (in euclidean_space) norm_basis [simp]:
-  "norm (basis i) = (if i < DIM('a) then 1 else 0)"
-  unfolding norm_eq_sqrt_inner dot_basis by simp
-
-lemma (in euclidean_space) basis_neq_0 [intro]:
-  assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
-  using assms by simp
-
-subsubsection {* Projecting components *}
-
-definition (in euclidean_space) euclidean_component (infixl "$$" 90)
-  where "x $$ i = inner (basis i) x"
-
-lemma bounded_linear_euclidean_component:
-  "bounded_linear (\<lambda>x. euclidean_component x i)"
-  unfolding euclidean_component_def
-  by (rule bounded_linear_inner_right)
-
-lemmas tendsto_euclidean_component [tendsto_intros] =
-  bounded_linear.tendsto [OF bounded_linear_euclidean_component]
-
-lemmas isCont_euclidean_component [simp] =
-  bounded_linear.isCont [OF bounded_linear_euclidean_component]
-
-lemma euclidean_component_zero [simp]: "0 $$ i = 0"
-  unfolding euclidean_component_def by (rule inner_zero_right)
-
-lemma euclidean_component_add [simp]: "(x + y) $$ i = x $$ i + y $$ i"
-  unfolding euclidean_component_def by (rule inner_add_right)
-
-lemma euclidean_component_diff [simp]: "(x - y) $$ i = x $$ i - y $$ i"
-  unfolding euclidean_component_def by (rule inner_diff_right)
-
-lemma euclidean_component_minus [simp]: "(- x) $$ i = - (x $$ i)"
-  unfolding euclidean_component_def by (rule inner_minus_right)
-
-lemma euclidean_component_scaleR [simp]: "(scaleR a x) $$ i = a * (x $$ i)"
-  unfolding euclidean_component_def by (rule inner_scaleR_right)
-
-lemma euclidean_component_setsum [simp]: "(\<Sum>x\<in>A. f x) $$ i = (\<Sum>x\<in>A. f x $$ i)"
-  unfolding euclidean_component_def by (rule inner_setsum_right)
-
-lemma euclidean_eqI:
-  fixes x y :: "'a::euclidean_space"
-  assumes "\<And>i. i < DIM('a) \<Longrightarrow> x $$ i = y $$ i" shows "x = y"
-proof -
-  from assms have "\<forall>i<DIM('a). (x - y) $$ i = 0"
-    by simp
-  then show "x = y"
-    unfolding euclidean_component_def euclidean_all_zero by simp
-qed
-
-lemma euclidean_eq:
-  fixes x y :: "'a::euclidean_space"
-  shows "x = y \<longleftrightarrow> (\<forall>i<DIM('a). x $$ i = y $$ i)"
+lemma (in euclidean_space) euclidean_eq_iff:
+  "x = y \<longleftrightarrow> (\<forall>b\<in>Basis. inner x b = inner y b)"
   by (auto intro: euclidean_eqI)
 
-lemma (in euclidean_space) basis_component [simp]:
-  "basis i $$ j = (if i = j \<and> i < DIM('a) then 1 else 0)"
-  unfolding euclidean_component_def dot_basis by auto
-
-lemma (in euclidean_space) basis_at_neq_0 [intro]:
-  "i < DIM('a) \<Longrightarrow> basis i $$ i \<noteq> 0"
-  by simp
-
-lemma (in euclidean_space) euclidean_component_ge [simp]:
-  assumes "i \<ge> DIM('a)" shows "x $$ i = 0"
-  unfolding euclidean_component_def basis_zero[OF assms] by simp
+lemma (in euclidean_space) euclidean_representation_setsum:
+  "(\<Sum>i\<in>Basis. f i *\<^sub>R i) = b \<longleftrightarrow> (\<forall>i\<in>Basis. f i = inner b i)"
+  by (subst euclidean_eq_iff) simp
 
-lemmas euclidean_simps =
-  euclidean_component_add
-  euclidean_component_diff
-  euclidean_component_scaleR
-  euclidean_component_minus
-  euclidean_component_setsum
-  basis_component
-
-lemma euclidean_representation:
-  fixes x :: "'a::euclidean_space"
-  shows "x = (\<Sum>i<DIM('a). (x$$i) *\<^sub>R basis i)"
-  apply (rule euclidean_eqI)
-  apply (simp add: if_distrib setsum_delta cong: if_cong)
-  done
-
-subsubsection {* Binder notation for vectors *}
-
-definition (in euclidean_space) Chi (binder "\<chi>\<chi> " 10) where
-  "(\<chi>\<chi> i. f i) = (\<Sum>i<DIM('a). f i *\<^sub>R basis i)"
+lemma (in euclidean_space) euclidean_representation: "(\<Sum>b\<in>Basis. inner x b *\<^sub>R b) = x"
+  unfolding euclidean_representation_setsum by simp
 
-lemma euclidean_lambda_beta [simp]:
-  "((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = (if j < DIM('a) then f j else 0)"
-  by (auto simp: Chi_def if_distrib setsum_cases intro!: setsum_cong)
-
-lemma euclidean_lambda_beta':
-  "j < DIM('a) \<Longrightarrow> ((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = f j"
-  by simp
-
-lemma euclidean_lambda_beta'':"(\<forall>j < DIM('a::euclidean_space). P j (((\<chi>\<chi> i. f i)::'a) $$ j)) \<longleftrightarrow>
-  (\<forall>j < DIM('a::euclidean_space). P j (f j))" by auto
-
-lemma euclidean_beta_reduce[simp]:
-  "(\<chi>\<chi> i. x $$ i) = (x::'a::euclidean_space)"
-  by (simp add: euclidean_eq)
-
-lemma euclidean_lambda_beta_0[simp]:
-    "((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ 0 = f 0"
-  by (simp add: DIM_positive)
+lemma (in euclidean_space) choice_Basis_iff:
+  fixes P :: "'a \<Rightarrow> real \<Rightarrow> bool"
+  shows "(\<forall>i\<in>Basis. \<exists>x. P i x) \<longleftrightarrow> (\<exists>x. \<forall>i\<in>Basis. P i (inner x i))"
+  unfolding bchoice_iff
+proof safe
+  fix f assume "\<forall>i\<in>Basis. P i (f i)"
+  then show "\<exists>x. \<forall>i\<in>Basis. P i (inner x i)"
+    by (auto intro!: exI[of _ "\<Sum>i\<in>Basis. f i *\<^sub>R i"])
+qed auto
 
-lemma euclidean_inner:
-  "inner x (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) * (y $$ i))"
-  by (subst (1 2) euclidean_representation,
-    simp add: inner_setsum_left inner_setsum_right
-    dot_basis if_distrib setsum_cases mult_commute)
-
-lemma euclidean_dist_l2:
-  fixes x y :: "'a::euclidean_space"
-  shows "dist x y = setL2 (\<lambda>i. dist (x $$ i) (y $$ i)) {..<DIM('a)}"
-  unfolding dist_norm norm_eq_sqrt_inner setL2_def
-  by (simp add: euclidean_inner power2_eq_square)
-
-lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
-  unfolding euclidean_component_def
-  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
-
-lemma dist_nth_le: "dist (x $$ i) (y $$ i) \<le> dist x (y::'a::euclidean_space)"
-  unfolding euclidean_dist_l2 [where 'a='a]
-  by (cases "i < DIM('a)", rule member_le_setL2, auto)
+lemma DIM_positive: "0 < DIM('a::euclidean_space)"
+  by (simp add: card_gt_0_iff)
 
 subsection {* Subclass relationships *}
 
@@ -239,11 +100,13 @@
     assume "open {x}"
     then obtain e where "0 < e" and e: "\<forall>y. dist y x < e \<longrightarrow> y = x"
       unfolding open_dist by fast
-    def y \<equiv> "x + scaleR (e/2) (sgn (basis 0))"
+    def y \<equiv> "x + scaleR (e/2) (SOME b. b \<in> Basis)"
+    have [simp]: "(SOME b. b \<in> Basis) \<in> Basis"
+      by (rule someI_ex) (auto simp: ex_in_conv)
     from `0 < e` have "y \<noteq> x"
-      unfolding y_def by (simp add: sgn_zero_iff DIM_positive)
+      unfolding y_def by (auto intro!: nonzero_Basis)
     from `0 < e` have "dist y x < e"
-      unfolding y_def by (simp add: dist_norm norm_sgn)
+      unfolding y_def by (simp add: dist_norm norm_Basis)
     from `y \<noteq> x` and `dist y x < e` show "False"
       using e by simp
   qed
@@ -256,23 +119,17 @@
 instantiation real :: euclidean_space
 begin
 
-definition
-  "Basis = {1::real}"
-
-definition
-  "dimension (t::real itself) = 1"
-
-definition [simp]:
-  "basis i = (if i = 0 then 1 else (0::real))"
-
-lemma DIM_real [simp]: "DIM(real) = 1"
-  by (rule dimension_real_def)
+definition 
+  [simp]: "Basis = {1::real}"
 
 instance
   by default (auto simp add: Basis_real_def)
 
 end
 
+lemma DIM_real[simp]: "DIM(real) = 1"
+  by simp
+
 subsubsection {* Type @{typ complex} *}
 
 instantiation complex :: euclidean_space
@@ -281,20 +138,13 @@
 definition Basis_complex_def:
   "Basis = {1, ii}"
 
-definition
-  "dimension (t::complex itself) = 2"
-
-definition
-  "basis i = (if i = 0 then 1 else if i = 1 then ii else 0)"
-
 instance
-  by default (auto simp add: Basis_complex_def dimension_complex_def
-    basis_complex_def intro: complex_eqI split: split_if_asm)
+  by default (auto simp add: Basis_complex_def intro: complex_eqI split: split_if_asm)
 
 end
 
 lemma DIM_complex[simp]: "DIM(complex) = 2"
-  by (rule dimension_complex_def)
+  unfolding Basis_complex_def by simp
 
 subsubsection {* Type @{typ "'a \<times> 'b"} *}
 
@@ -304,12 +154,6 @@
 definition
   "Basis = (\<lambda>u. (u, 0)) ` Basis \<union> (\<lambda>v. (0, v)) ` Basis"
 
-definition
-  "dimension (t::('a \<times> 'b) itself) = DIM('a) + DIM('b)"
-
-definition
-  "basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
-
 instance proof
   show "(Basis :: ('a \<times> 'b) set) \<noteq> {}"
     unfolding Basis_prod_def by simp
@@ -327,20 +171,12 @@
   show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
     unfolding Basis_prod_def ball_Un ball_simps
     by (simp add: inner_prod_def prod_eq_iff euclidean_all_zero_iff)
-next
-  show "DIM('a \<times> 'b) = card (Basis :: ('a \<times> 'b) set)"
-    unfolding dimension_prod_def Basis_prod_def
-    by (simp add: card_Un_disjoint disjoint_iff_not_equal
-      card_image inj_on_def dimension_def)
-next
-  show "basis ` {..<DIM('a \<times> 'b)} = (Basis :: ('a \<times> 'b) set)"
-    by (auto simp add: Basis_prod_def dimension_prod_def basis_prod_def
-      image_def elim!: Basis_elim)
-next
-  show "basis ` {DIM('a \<times> 'b)..} = {0::('a \<times> 'b)}"
-    by (auto simp add: dimension_prod_def basis_prod_def prod_eq_iff image_def)
 qed
 
+lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('a) + DIM('b)"
+  unfolding Basis_prod_def
+  by (subst card_Un_disjoint) (auto intro!: card_image arg_cong2[where f="op +"] inj_onI)
+
 end
 
 end
--- a/src/HOL/Multivariate_Analysis/Fashoda.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Fashoda.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -7,6 +7,14 @@
 imports Brouwer_Fixpoint Path_Connected Cartesian_Euclidean_Space
 begin
 
+(* move *)
+
+lemma cart_eq_inner_axis: "a $ i = a \<bullet> axis i 1"
+  by (simp add: inner_axis)
+
+lemma axis_in_Basis: "a \<in> Basis \<Longrightarrow> axis i a \<in> Basis"
+  by (auto simp add: Basis_vec_def axis_eq_axis)
+
 subsection {*Fashoda meet theorem. *}
 
 lemma infnorm_2: "infnorm (x::real^2) = max (abs(x$1)) (abs(x$2))"
@@ -30,7 +38,7 @@
   have lem1:"\<forall>z::real^2. infnorm(negatex z) = infnorm z"
     unfolding negatex_def infnorm_2 vector_2 by auto
   have lem2:"\<forall>z. z\<noteq>0 \<longrightarrow> infnorm(sqprojection z) = 1" unfolding sqprojection_def
-    unfolding infnorm_mul[unfolded smult_conv_scaleR] unfolding abs_inverse real_abs_infnorm
+    unfolding infnorm_mul[unfolded scalar_mult_eq_scaleR] unfolding abs_inverse real_abs_infnorm
     apply(subst infnorm_eq_0[THEN sym]) by auto
   let ?F = "(\<lambda>w::real^2. (f \<circ> (\<lambda>x. x$1)) w - (g \<circ> (\<lambda>x. x$2)) w)"
   have *:"\<And>i. (\<lambda>x::real^2. x $ i) ` {- 1..1} = {- 1..1::real}"
@@ -133,12 +141,6 @@
     apply(rule_tac x="iscale s" in bexI) prefer 3 apply(rule_tac x="iscale t" in bexI)
     using isc[unfolded subset_eq, rule_format] by auto qed
 
-(* move *)
-lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
-  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
-  unfolding interval_bij_cart split_conv vec_eq_iff vec_lambda_beta
-  apply(rule,insert assms,erule_tac x=i in allE) by auto
-
 lemma fashoda: fixes b::"real^2"
   assumes "path f" "path g" "path_image f \<subseteq> {a..b}" "path_image g \<subseteq> {a..b}"
   "(pathstart f)$1 = a$1" "(pathfinish f)$1 = b$1"
@@ -184,8 +186,10 @@
       "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
       "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
       "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1"
-      unfolding interval_bij_cart vector_component_simps o_def split_conv
-      unfolding assms[unfolded pathstart_def pathfinish_def] using as by auto qed note z=this
+      using assms as 
+      by (simp_all add: axis_in_Basis cart_eq_inner_axis pathstart_def pathfinish_def interval_bij_def)
+         (simp_all add: inner_axis)
+  qed note z=this
   from z(1) guess zf unfolding image_iff .. note zf=this
   from z(2) guess zg unfolding image_iff .. note zg=this
   have *:"\<forall>i. (- 1) $ i < (1::real^2) $ i \<and> a $ i < b $ i" unfolding forall_2 using as by auto
@@ -201,7 +205,7 @@
 proof- 
   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
-      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+      unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[THEN sym] vector_component_simps by blast }
   { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
     { fix b a assume "b + u * a > a + u * b"
       hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
@@ -225,7 +229,7 @@
 proof- 
   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
-      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+      unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[THEN sym] vector_component_simps by blast }
   { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
     { fix b a assume "b + u * a > a + u * b"
       hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
--- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -504,96 +504,11 @@
   apply (simp add: axis_def)
   done
 
-text {* A bijection between @{text "'n::finite"} and @{text "{..<CARD('n)}"} *}
-
-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: 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
-
-lemma pi'_inj[intro]: "inj \<pi>'"
-  using bij_betw_pi' unfolding bij_betw_def by auto
-
-lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"
-  using bij_betw_pi' unfolding bij_betw_def by auto
-
-lemma pi_pi'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"
-  using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)
-
-lemma pi'_pi[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
-  using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)
-
-lemma pi_pi'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
-  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 vec :: (euclidean_space, finite) euclidean_space
 begin
 
 definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
 
-definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
-
-definition "basis i =
-  (if i < (CARD('b) * DIM('a))
-  then axis (\<pi>(i div DIM('a))) (basis (i mod DIM('a)))
-  else 0)"
-
-lemma basis_eq:
-  assumes "i < CARD('b)" and "j < DIM('a)"
-  shows "basis (j + i * DIM('a)) = axis (\<pi> i) (basis j)"
-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_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)"
-  apply (subst basis_eq)
-  using pi'_range assms by (simp_all add: axis_def)
-
-lemma split_times_into_modulo[consumes 1]:
-  fixes k :: nat
-  assumes "k < A * B"
-  obtains i j where "i < A" and "j < B" and "k = j + i * B"
-proof
-  have "A * B \<noteq> 0"
-  proof assume "A * B = 0" with assms show False by simp qed
-  hence "0 < B" by auto
-  thus "k mod B < B" using `0 < B` by auto
-next
-  have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)
-  also have "... < A * B" using assms by simp
-  finally show "k div B < A" by auto
-qed simp
-
-lemma linear_less_than_times:
-  fixes i j A B :: nat assumes "i < B" "j < A"
-  shows "j + i * A < B * A"
-proof -
-  have "i * A + j < (Suc i)*A" using `j < A` by simp
-  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_vec_def)
-
 instance proof
   show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
     unfolding Basis_vec_def by simp
@@ -611,27 +526,17 @@
   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)
-next
-  show "DIM('a ^ 'b) = card (Basis :: ('a ^ 'b) set)"
-    unfolding Basis_vec_def dimension_vec_def dimension_def
-    by (simp add: card_UN_disjoint [unfolded disjoint_iff_not_equal]
-      axis_eq_axis nonzero_Basis)
-next
-  show "basis ` {..<DIM('a ^ 'b)} = (Basis :: ('a ^ 'b) set)"
-    unfolding Basis_vec_def
-    apply auto
-    apply (erule split_times_into_modulo)
-    apply (simp add: basis_eq axis_eq_axis)
-    apply (erule Basis_elim)
-    apply (simp add: image_def basis_vec_def axis_eq_axis)
-    apply (rule rev_bexI, simp)
-    apply (erule linear_less_than_times [OF pi'_range])
+qed
+
+lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+  apply (simp add: Basis_vec_def)
+  apply (subst card_UN_disjoint)
+     apply simp
     apply simp
-    done
-next
-  show "basis ` {DIM('a ^ 'b)..} = {0::'a ^ 'b}"
-    by (auto simp add: image_def basis_vec_def)
-qed
+   apply (auto simp: axis_eq_axis) [1]
+  apply (subst card_UN_disjoint)
+     apply (auto simp: axis_eq_axis)
+  done
 
 end
 
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -71,12 +71,8 @@
   apply (auto simp: isUb_def setle_def)
   done
 
-lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x $$ k)"
-  apply (rule bounded_linearI[where K=1])
-  using component_le_norm[of _ k]
-  unfolding real_norm_def
-  apply auto
-  done
+lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \<bullet> k)"
+  by (rule bounded_linear_inner_left)
 
 lemma transitive_stepwise_lt_eq:
   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
@@ -162,16 +158,10 @@
 
 subsection {* Some useful lemmas about intervals. *}
 
-abbreviation One  where "One \<equiv> ((\<chi>\<chi> i. 1)::_::ordered_euclidean_space)"
-
-lemma empty_as_interval: "{} = {One..0}"
-  apply (rule set_eqI, rule)
-  defer
-  unfolding mem_interval
-  using UNIV_witness[where 'a='n]
-  apply (erule_tac exE, rule_tac x = x in allE)
-  apply auto
-  done
+abbreviation One where "One \<equiv> ((\<Sum>Basis)::_::euclidean_space)"
+
+lemma empty_as_interval: "{} = {One..(0::'a::ordered_euclidean_space)}"
+  by (auto simp: set_eq_iff eucl_le[where 'a='a] intro!: bexI[OF _ SOME_Basis])
 
 lemma interior_subset_union_intervals: 
   assumes "i = {a..b::'a::ordered_euclidean_space}" "j = {c..d}"
@@ -255,17 +245,17 @@
             done
         next
           case False
-          then obtain k where "x$$k \<le> a$$k \<or> x$$k \<ge> b$$k" and k:"k<DIM('a)"
+          then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k:"k\<in>Basis"
             unfolding mem_interval by (auto simp add: not_less)
-          hence "x$$k = a$$k \<or> x$$k = b$$k"
+          hence "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
             using True unfolding ab and mem_interval
-              apply (erule_tac x = k in allE)
+              apply (erule_tac x = k in ballE)
               apply auto
               done
           hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
           proof (erule_tac disjE)
-            let ?z = "x - (e/2) *\<^sub>R basis k"
-            assume as: "x$$k = a$$k"
+            let ?z = "x - (e/2) *\<^sub>R k"
+            assume as: "x\<bullet>k = a\<bullet>k"
             have "ball ?z (e / 2) \<inter> i = {}"
               apply (rule ccontr)
               unfolding ex_in_conv[THEN sym]
@@ -273,15 +263,12 @@
               fix y
               assume "y \<in> ball ?z (e / 2) \<inter> i"
               hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
-              hence "\<bar>(?z - y) $$ k\<bar> < e/2"
-                using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
-              hence "y$$k < a$$k"
-                using e[THEN conjunct1] k by (auto simp add: field_simps as)
+              hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+                using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
+              hence "y\<bullet>k < a\<bullet>k"
+                using e[THEN conjunct1] k by (auto simp add: field_simps as inner_Basis inner_simps)
               hence "y \<notin> i"
-                unfolding ab mem_interval not_all
-                apply (rule_tac x=k in exI)
-                using k apply auto
-                done
+                unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
               thus False using yi by auto
             qed
             moreover
@@ -290,10 +277,10 @@
             proof
               fix y
               assume as: "y\<in> ball ?z (e/2)"
-              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"
+              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
                 apply -
-                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
-                unfolding norm_scaleR norm_basis
+                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
+                unfolding norm_scaleR norm_Basis[OF k]
                 apply auto
                 done
               also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
@@ -310,8 +297,8 @@
               apply auto
               done
           next
-            let ?z = "x + (e/2) *\<^sub>R basis k"
-            assume as: "x$$k = b$$k"
+            let ?z = "x + (e/2) *\<^sub>R k"
+            assume as: "x\<bullet>k = b\<bullet>k"
             have "ball ?z (e / 2) \<inter> i = {}"
               apply (rule ccontr)
               unfolding ex_in_conv[THEN sym]
@@ -319,15 +306,12 @@
               fix y
               assume "y \<in> ball ?z (e / 2) \<inter> i"
               hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
-              hence "\<bar>(?z - y) $$ k\<bar> < e/2"
-                using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
-              hence "y$$k > b$$k"
-                using e[THEN conjunct1] k by(auto simp add:field_simps as)
+              hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+                using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
+              hence "y\<bullet>k > b\<bullet>k"
+                using e[THEN conjunct1] k by(auto simp add:field_simps inner_simps inner_Basis as)
               hence "y \<notin> i"
-                unfolding ab mem_interval not_all
-                using k apply (rule_tac x=k in exI)
-                apply auto
-                done
+                unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
               thus False using yi by auto
             qed
             moreover
@@ -336,11 +320,11 @@
             proof
               fix y
               assume as: "y\<in> ball ?z (e/2)"
-              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"
+              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
                 apply -
-                apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
+                apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
                 unfolding norm_scaleR
-                apply auto
+                apply (auto simp: k)
                 done
               also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
                 apply (rule add_strict_left_mono)
@@ -383,61 +367,25 @@
 
 subsection {* Bounds on intervals where they exist. *}
 
-definition "interval_upperbound (s::('a::ordered_euclidean_space) set) =
-  ((\<chi>\<chi> i. Sup {a. \<exists>x\<in>s. x$$i = a})::'a)"
-
-definition "interval_lowerbound (s::('a::ordered_euclidean_space) set) =
-  ((\<chi>\<chi> i. Inf {a. \<exists>x\<in>s. x$$i = a})::'a)"
+definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
+  "interval_upperbound s = (\<Sum>i\<in>Basis. Sup {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
+
+definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
+  "interval_lowerbound s = (\<Sum>i\<in>Basis. Inf {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
 
 lemma interval_upperbound[simp]:
-  assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i"
-  shows "interval_upperbound {a..b} = b"
-  using assms
-  unfolding interval_upperbound_def
-  apply (subst euclidean_eq[where 'a='a])
-  apply safe
-  unfolding euclidean_lambda_beta'
-  apply (erule_tac x=i in allE)
-  apply (rule Sup_unique)
-  unfolding setle_def
-  apply rule
-  unfolding mem_Collect_eq
-  apply (erule bexE)
-  unfolding mem_interval
-  defer
-  apply (rule, rule)
-  apply (rule_tac x="b$$i" in bexI)
-  defer
-  unfolding mem_Collect_eq
-  apply (rule_tac x=b in bexI)
-  unfolding mem_interval
-  using assms apply auto
-  done
+  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
+    interval_upperbound {a..b} = (b::'a::ordered_euclidean_space)"
+  unfolding interval_upperbound_def euclidean_representation_setsum
+  by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setle_def
+           intro!: Sup_unique)
 
 lemma interval_lowerbound[simp]:
-  assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i"
-  shows "interval_lowerbound {a..b} = a"
-  using assms
-  unfolding interval_lowerbound_def
-  apply (subst euclidean_eq[where 'a='a])
-  apply safe
-  unfolding euclidean_lambda_beta'
-  apply (erule_tac x=i in allE)
-  apply (rule Inf_unique)
-  unfolding setge_def
-  apply rule
-  unfolding mem_Collect_eq
-  apply (erule bexE)
-  unfolding mem_interval
-  defer
-  apply (rule, rule)
-  apply (rule_tac x = "a$$i" in bexI)
-  defer
-  unfolding mem_Collect_eq
-  apply (rule_tac x=a in bexI)
-  unfolding mem_interval
-  using assms apply auto
-  done
+  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
+    interval_lowerbound {a..b} = (a::'a::ordered_euclidean_space)"
+  unfolding interval_lowerbound_def euclidean_representation_setsum
+  by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setge_def
+           intro!: Inf_unique)
 
 lemmas interval_bounds = interval_upperbound interval_lowerbound
 
@@ -449,15 +397,15 @@
 subsection {* Content (length, area, volume...) of an interval. *}
 
 definition "content (s::('a::ordered_euclidean_space) set) =
-  (if s = {} then 0 else (\<Prod>i<DIM('a). (interval_upperbound s)$$i - (interval_lowerbound s)$$i))"
-
-lemma interval_not_empty:"\<forall>i<DIM('a). a$$i \<le> b$$i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
+  (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
+
+lemma interval_not_empty:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
   unfolding interval_eq_empty unfolding not_ex not_less by auto
 
 lemma content_closed_interval:
   fixes a::"'a::ordered_euclidean_space"
-  assumes "\<forall>i<DIM('a). a$$i \<le> b$$i"
-  shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
+  assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
+  shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   using interval_not_empty[OF assms]
   unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms]
   by auto
@@ -465,7 +413,7 @@
 lemma content_closed_interval':
   fixes a::"'a::ordered_euclidean_space"
   assumes "{a..b}\<noteq>{}"
-  shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
+  shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   apply (rule content_closed_interval)
   using assms unfolding interval_ne_empty
   apply assumption
@@ -482,13 +430,13 @@
 lemma content_singleton[simp]: "content {a} = 0"
 proof -
   have "content {a .. a} = 0"
-    by (subst content_closed_interval) auto
+    by (subst content_closed_interval) (auto simp: ex_in_conv)
   then show ?thesis by simp
 qed
 
 lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1"
 proof -
-  have *: "\<forall>i<DIM('a). (0::'a)$$i \<le> (One::'a)$$i" by auto
+  have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i" by auto
   have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto
 qed
@@ -498,12 +446,12 @@
   shows "0 \<le> content {a..b}"
 proof (cases "{a..b} = {}")
   case False
-  hence *: "\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty .
-  have "(\<Prod>i<DIM('a). interval_upperbound {a..b} $$ i - interval_lowerbound {a..b} $$ i) \<ge> 0"
+  hence *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty .
+  have "(\<Prod>i\<in>Basis. interval_upperbound {a..b} \<bullet> i - interval_lowerbound {a..b} \<bullet> i) \<ge> 0"
     apply (rule setprod_nonneg)
     unfolding interval_bounds[OF *]
     using *
-    apply (erule_tac x=x in allE)
+    apply (erule_tac x=x in ballE)
     apply auto
     done
   thus ?thesis unfolding content_def by (auto simp del:interval_bounds')
@@ -511,75 +459,59 @@
 
 lemma content_pos_lt:
   fixes a::"'a::ordered_euclidean_space"
-  assumes "\<forall>i<DIM('a). a$$i < b$$i"
+  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
   shows "0 < content {a..b}"
 proof -
-  have help_lemma1: "\<forall>i<DIM('a). a$$i < b$$i \<Longrightarrow> \<forall>i<DIM('a). a$$i \<le> ((b$$i)::real)"
-    apply (rule, erule_tac x=i in allE)
+  have help_lemma1: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> \<forall>i\<in>Basis. a\<bullet>i \<le> ((b\<bullet>i)::real)"
+    apply (rule, erule_tac x=i in ballE)
     apply auto
     done
   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]]
     apply(rule setprod_pos)
-    using assms apply (erule_tac x=x in allE)
+    using assms apply (erule_tac x=x in ballE)
     apply auto
     done
 qed
 
-lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i)"
+lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
 proof (cases "{a..b} = {}")
   case True
   thus ?thesis
     unfolding content_def if_P[OF True]
     unfolding interval_eq_empty
     apply -
-    apply (rule, erule exE)
-    apply (rule_tac x = i in exI)
+    apply (rule, erule bexE)
+    apply (rule_tac x = i in bexI)
     apply auto
     done
 next
   case False
   from this[unfolded interval_eq_empty not_ex not_less]
-  have as: "\<forall>i<DIM('a). b $$ i \<ge> a $$ i" by fastforce
+  have as: "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i" by fastforce
   show ?thesis
-    unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_lessThan]
-    apply rule
-    apply (erule_tac[!] exE bexE)
-    unfolding interval_bounds[OF as]
-    apply (rule_tac x=x in exI)
-    defer
-    apply (rule_tac x=i in bexI)
-    using as apply (erule_tac x=i in allE)
-    apply auto
-    done
+    unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_Basis]
+    using as
+    by (auto intro!: bexI)
 qed
 
 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
 
 lemma content_closed_interval_cases:
   "content {a..b::'a::ordered_euclidean_space} =
-    (if \<forall>i<DIM('a). a$$i \<le> b$$i then setprod (\<lambda>i. b$$i - a$$i) {..<DIM('a)} else 0)"
-  apply (rule cond_cases) 
-  apply (rule content_closed_interval)
-  unfolding content_eq_0 not_all not_le
-  defer
-  apply (erule exE,rule_tac x=x in exI)
-  apply auto
-  done
+    (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
+  by (auto simp: not_le content_eq_0 intro: less_imp_le content_closed_interval)
 
 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
 
-(*lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
-  unfolding content_eq_0 by auto*)
-
-lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
+lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
   apply rule
   defer
   apply (rule content_pos_lt, assumption)
 proof -
   assume "0 < content {a..b}"
   hence "content {a..b} \<noteq> 0" by auto
-  thus "\<forall>i<DIM('a). a$$i < b$$i"
+  thus "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
     unfolding content_eq_0 not_ex not_le by fastforce
 qed
 
@@ -593,20 +525,20 @@
   thus ?thesis using content_pos_le[of c d] by auto
 next
   case False
-  hence ab_ne:"\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty by auto
+  hence ab_ne:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty by auto
   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   have "{c..d} \<noteq> {}" using assms False by auto
-  hence cd_ne:"\<forall>i<DIM('a). c $$ i \<le> d $$ i" using assms unfolding interval_ne_empty by auto
+  hence cd_ne:"\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i" using assms unfolding interval_ne_empty by auto
   show ?thesis
     unfolding content_def
     unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`]
     apply(rule setprod_mono,rule)
   proof
-    fix i
-    assume i:"i\<in>{..<DIM('a)}"
-    show "0 \<le> b $$ i - a $$ i" using ab_ne[THEN spec[where x=i]] i by auto
-    show "b $$ i - a $$ i \<le> d $$ i - c $$ i"
+    fix i :: 'a
+    assume i:"i\<in>Basis"
+    show "0 \<le> b \<bullet> i - a \<bullet> i" using ab_ne[THEN bspec, OF i] i by auto
+    show "b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i]
       using i by auto
@@ -857,134 +789,84 @@
   show "k \<noteq> {}" using k d1(3) d2(3) by auto show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto qed
 
 lemma partial_division_extend_1:
-  assumes "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}" "{c..d} \<noteq> {}"
+  assumes incl: "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}" and nonempty: "{c..d} \<noteq> {}"
   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
-proof- def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def using DIM_positive[where 'a='a] by auto
-  guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_lessThan[of "DIM('a)"]] .. note \<pi>=this
-  def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
-  have \<pi>':"bij_betw \<pi>' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
-  hence \<pi>'_i:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
-  have \<pi>_i:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi> i <DIM('a)" using \<pi> unfolding bij_betw_def n_def by auto 
-  have \<pi>_\<pi>'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi> (\<pi>' i) = i" unfolding \<pi>'_def
-    apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
-  have \<pi>'_\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq)
-    using \<pi> unfolding n_def bij_betw_def by auto
-  have "{c..d} \<noteq> {}" using assms by auto
-  let ?p1 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else if \<pi>' i = l then c$$\<pi> l else b$$i)}"
-  let ?p2 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else if \<pi>' i = l then d$$\<pi> l else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else b$$i)}"
-  let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
-  have abcd:"\<And>i. i<DIM('a) \<Longrightarrow> a $$ i \<le> c $$ i \<and> c$$i \<le> d$$i \<and> d $$ i \<le> b $$ i" using assms
-    unfolding subset_interval interval_eq_empty by auto
-  show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
-  proof- have "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < Suc n"
-    proof(rule ccontr,unfold not_less) fix i assume i:"i<DIM('a)" and "Suc n \<le> \<pi>' i"
-      hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' i unfolding bij_betw_def by auto
-    qed hence "c = (\<chi>\<chi> i. if \<pi>' i < Suc n then c $$ i else a $$ i)"
-        "d = (\<chi>\<chi> i. if \<pi>' i < Suc n then d $$ i else if \<pi>' i = n + 1 then c $$ \<pi> (n + 1) else b $$ i)"
-      unfolding euclidean_eq[where 'a='a] using \<pi>' unfolding bij_betw_def by auto
-    thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
-    have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}"  "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"
-      unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
-    proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
-      then guess i unfolding mem_interval not_all not_imp .. note i=conjunctD2[OF this]
-      show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
-        apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
-    qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
-    proof- fix x assume x:"x\<in>{a..b}"
-      { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
-      let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $$ \<pi> i \<le> x $$ \<pi> i \<and> x $$ \<pi> i \<le> d $$ \<pi> i)}"
-      assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all not_imp ..
-      hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
-      hence M:"finite ?M" "?M \<noteq> {}" by auto
-      def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]
-        Min_gr_iff[OF M,unfolded l_def[symmetric]]
-      have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
-        apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
-      proof- assume as:"x $$ \<pi> l < c $$ \<pi> l"
-        show "x \<in> ?p1 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
-        proof- case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal1 by auto
-          thus ?case using as x[unfolded mem_interval,rule_format,of i]
-            apply auto using l(3)[of "\<pi>' i"] using goal1 by(auto elim!:ballE[where x="\<pi>' i"])
-        next case goal2 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal2 by auto
-          thus ?case using as x[unfolded mem_interval,rule_format,of i]
-            apply auto using l(3)[of "\<pi>' i"] using goal2 by(auto elim!:ballE[where x="\<pi>' i"])
-        qed
-      next assume as:"x $$ \<pi> l > d $$ \<pi> l"
-        show "x \<in> ?p2 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
-        proof- fix i assume i:"i<DIM('a)"
-          have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using i by auto
-          thus "(if \<pi>' i < l then c $$ i else if \<pi>' i = l then d $$ \<pi> l else a $$ i) \<le> x $$ i"
-            "x $$ i \<le> (if \<pi>' i < l then d $$ i else b $$ i)"
-            using as x[unfolded mem_interval,rule_format,of i]
-            apply auto using l(3)[of "\<pi>' i"] i by(auto elim!:ballE[where x="\<pi>' i"])
-        qed qed
-      thus "x \<in> \<Union>?p" using l(2) by blast 
-    qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
-    
-    show "finite ?p" by auto
-    fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto
-    show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
-    proof fix i x assume i:"i<DIM('a)" assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
-      ultimately show "a$$i \<le> x$$i" "x$$i \<le> b$$i" using abcd[of i] using l using i
-        by(auto elim!:allE[where x=i] simp add:eucl_le[where 'a='a]) (* FIXME: SLOW *)
-    qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
-    proof- case goal1 thus ?case using abcd[of x] by auto
-    next   case goal2 thus ?case using abcd[of x] by auto
-    qed thus "k \<noteq> {}" using k by auto
-    show "\<exists>a b. k = {a..b}" using k by auto
-    fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto
-    { fix k k' l l'
-      assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
-      assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
-      assume "l \<le> l'" fix x
-      have "x \<notin> interior k \<inter> interior k'" 
-      proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
-        case True hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l'" using \<pi>'_i using l' by(auto simp add:less_Suc_eq_le)
-        hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l' then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
-        hence k':"k' = {c..d}" using l'(1) unfolding * by auto
-        have ln:"l < n + 1" 
-        proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
-          hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l" using \<pi>'_i by(auto simp add:less_Suc_eq_le)
-          hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
-          hence "k = {c..d}" using l(1) \<pi>'_i unfolding * by(auto)
-          thus False using `k\<noteq>k'` k' by auto
-        qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'_\<pi>[of l] using l ln by auto
-        have "x $$ \<pi> l < c $$ \<pi> l \<or> d $$ \<pi> l < x $$ \<pi> l" using l(1) apply-
-        proof(erule disjE)
-          assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>_i[of l] by(auto simp add:** not_less)
-        next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>_i[of l] unfolding ** by auto
-        qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
-          by(auto elim!:allE[where x="\<pi> l"])
-      next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
-        hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
-        note \<pi>_l = \<pi>'_\<pi>[OF ln(1)] \<pi>'_\<pi>[OF ln(2)]
-        assume x:"x \<in> interior k \<inter> interior k'"
-        show False using l(1) l'(1) apply-
-        proof(erule_tac[!] disjE)+
-          assume as:"k = ?p1 l" "k' = ?p1 l'"
-          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          have "l \<noteq> l'" using k'(2)[unfolded as] by auto
-          thus False using *[of "\<pi> l'"] *[of "\<pi> l"] ln using \<pi>_i[OF ln(1)] \<pi>_i[OF ln(2)] apply(cases "l<l'")
-            by(auto simp add:euclidean_lambda_beta' \<pi>_l \<pi>_i n_def)
-        next assume as:"k = ?p2 l" "k' = ?p2 l'"
-          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
-          thus False using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln by(auto simp add:euclidean_lambda_beta' \<pi>_l \<pi>_i n_def)
-        next assume as:"k = ?p1 l" "k' = ?p2 l'"
-          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          show False using abcd[of "\<pi> l'"] using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln apply(cases "l=l'")
-            by(auto simp add:euclidean_lambda_beta' \<pi>_l \<pi>_i n_def)
-        next assume as:"k = ?p2 l" "k' = ?p1 l'"
-          note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
-          show False using *[of "\<pi> l"] *[of "\<pi> l'"] ln `l \<le> l'` apply(cases "l=l'") using abcd[of "\<pi> l'"] 
-            by(auto simp add:euclidean_lambda_beta' \<pi>_l \<pi>_i n_def)
-        qed qed } 
-    from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
-      apply - apply(cases "l' \<le> l") using k'(2) by auto            
-    thus "interior k \<inter> interior k' = {}" by auto        
-qed qed
+proof
+  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a. {(\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)}"
+  def p \<equiv> "?B ` (Basis \<rightarrow>\<^isub>E {(a, c), (c, d), (d, b)})"
+
+  show "{c .. d} \<in> p"
+    unfolding p_def
+    by (auto simp add: interval_eq_empty eucl_le[where 'a='a]
+             intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
+
+  {  fix i :: 'a assume "i \<in> Basis"
+    with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
+      unfolding interval_eq_empty subset_interval by (auto simp: not_le) }
+  note ord = this
+
+  show "p division_of {a..b}"
+  proof (rule division_ofI)
+    show "finite p"
+      unfolding p_def by (auto intro!: finite_PiE)
+    { fix k assume "k \<in> p"
+      then obtain f where f: "f \<in> Basis \<rightarrow>\<^isub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
+        by (auto simp: p_def)
+      then show "\<exists>a b. k = {a..b}" by auto
+      have "k \<subseteq> {a..b} \<and> k \<noteq> {}"
+      proof (simp add: k interval_eq_empty subset_interval not_less, safe)
+        fix i :: 'a assume i: "i \<in> Basis"
+        moreover with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
+          by (auto simp: PiE_iff)
+        moreover note ord[of i]
+        ultimately show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
+          by (auto simp: subset_iff eucl_le[where 'a='a])
+      qed
+      then show "k \<noteq> {}" "k \<subseteq> {a .. b}" by auto
+      { 
+      fix l assume "l \<in> p"
+      then obtain g where g: "g \<in> Basis \<rightarrow>\<^isub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
+        by (auto simp: p_def)
+      assume "l \<noteq> k"
+      have "\<exists>i\<in>Basis. f i \<noteq> g i"
+      proof (rule ccontr)
+        assume "\<not> (\<exists>i\<in>Basis. f i \<noteq> g i)"
+        with f g have "f = g"
+          by (auto simp: PiE_iff extensional_def intro!: ext)
+        with `l \<noteq> k` show False
+          by (simp add: l k)
+      qed
+      then guess i ..
+      moreover then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
+          "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
+        using f g by (auto simp: PiE_iff)
+      moreover note ord[of i]
+      ultimately show "interior l \<inter> interior k = {}"
+        by (auto simp add: l k interior_closed_interval disjoint_interval intro!: bexI[of _ i]) }
+      note `k \<subseteq> { a.. b}` }
+    moreover
+    { fix x assume x: "x \<in> {a .. b}"
+      have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
+      proof
+        fix i :: 'a assume "i \<in> Basis"
+        with x ord[of i] 
+        have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
+            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
+          by (auto simp: eucl_le[where 'a='a])
+        then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
+          by auto
+      qed
+      then guess f unfolding bchoice_iff .. note f = this
+      moreover then have "restrict f Basis \<in> Basis \<rightarrow>\<^isub>E {(a, c), (c, d), (d, b)}"
+        by auto
+      moreover from f have "x \<in> ?B (restrict f Basis)"
+        by (auto simp: mem_interval eucl_le[where 'a='a])
+      ultimately have "\<exists>k\<in>p. x \<in> k"
+        unfolding p_def by blast }
+    ultimately show "\<Union>p = {a..b}"
+      by auto
+  qed
+qed
 
 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}" proof(cases "p = {}")
@@ -1415,9 +1297,9 @@
 lemma interval_bisection_step:  fixes type::"'a::ordered_euclidean_space"
   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::'a})"
   obtains c d where "~(P{c..d})"
-  "\<forall>i<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$i"
+  "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
-  note ab=this[unfolded interval_eq_empty not_ex not_less]
+  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" by (auto simp: interval_eq_empty not_le)
   { fix f have "finite f \<Longrightarrow>
         (\<forall>s\<in>f. P s) \<Longrightarrow>
         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
@@ -1428,60 +1310,72 @@
         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
         using insert by auto
     qed } note * = this
-  let ?A = "{{c..d} | c d::'a. \<forall>i<DIM('a). (c$$i = a$$i) \<and> (d$$i = (a$$i + b$$i) / 2) \<or> (c$$i = (a$$i + b$$i) / 2) \<and> (d$$i = b$$i)}"
-  let ?PP = "\<lambda>c d. \<forall>i<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$i"
+  let ?A = "{{c..d} | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or> (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
+  let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
-    let ?B = "(\<lambda>s.{(\<chi>\<chi> i. if i \<in> s then a$$i else (a$$i + b$$i) / 2)::'a ..
-      (\<chi>\<chi> i. if i \<in> s then (a$$i + b$$i) / 2 else b$$i)}) ` {s. s \<subseteq> {..<DIM('a)}}"
+    let ?B = "(\<lambda>s.{(\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i)::'a ..
+      (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)}) ` {s. s \<subseteq> Basis}"
     have "?A \<subseteq> ?B" proof case goal1
       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
-      show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. i<DIM('a) \<and> c$$i = a$$i}" in bexI)
-        unfolding c_d apply(rule * ) unfolding euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta' mem_Collect_eq
-      proof- fix i assume "i<DIM('a)" thus " c $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then a $$ i else (a $$ i + b $$ i) / 2)"
-          "d $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then (a $$ i + b $$ i) / 2 else b $$ i)"
-          using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
+      show "x\<in>?B" unfolding image_iff
+        apply(rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
+        unfolding c_d
+        apply(rule *)
+        apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
+                        cong: ball_cong)
+        apply safe
+      proof-
+        fix i :: 'a assume i: "i\<in>Basis"
+        thus " c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
+          "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
+          using c_d(2)[of i] ab[OF i] by(auto simp add:field_simps)
       qed qed
     thus "finite ?A" apply(rule finite_subset) by auto
     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
     note c_d=this[rule_format]
     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 thus ?case 
-        using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
+        using c_d(2)[of i] using ab[OF `i \<in> Basis`] by auto qed
     show "\<exists>a b. s = {a..b}" unfolding c_d by auto
     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
     note e_f=this[rule_format]
     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
-    then obtain i where "c$$i \<noteq> e$$i \<or> d$$i \<noteq> f$$i" and i':"i<DIM('a)" unfolding de_Morgan_conj euclidean_eq[where 'a='a] by auto
-    hence i:"c$$i \<noteq> e$$i" "d$$i \<noteq> f$$i" apply- apply(erule_tac[!] disjE)
-    proof- assume "c$$i \<noteq> e$$i" thus "d$$i \<noteq> f$$i" using c_d(2)[of i] e_f(2)[of i] by fastforce
-    next   assume "d$$i \<noteq> f$$i" thus "c$$i \<noteq> e$$i" using c_d(2)[of i] e_f(2)[of i] by fastforce
+    then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i':"i\<in>Basis"
+      unfolding euclidean_eq_iff[where 'a='a] by auto
+    hence i:"c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i" apply- apply(erule_tac[!] disjE)
+    proof- assume "c\<bullet>i \<noteq> e\<bullet>i" thus "d\<bullet>i \<noteq> f\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
+    next   assume "d\<bullet>i \<noteq> f\<bullet>i" thus "c\<bullet>i \<noteq> e\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
     qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
     show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
-      hence x:"c$$i < d$$i" "e$$i < f$$i" "c$$i < f$$i" "e$$i < d$$i" unfolding mem_interval using i'
-        apply-apply(erule_tac[!] x=i in allE)+ by auto
+      hence x:"c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i" unfolding mem_interval using i'
+        apply-apply(erule_tac[!] x=i in ballE)+ by auto
       show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
-      proof(erule_tac[!] conjE) assume as:"c $$ i = a $$ i" "d $$ i = (a $$ i + b $$ i) / 2"
-        show False using e_f(2)[of i] and i x unfolding as by(fastforce simp add:field_simps)
-      next assume as:"c $$ i = (a $$ i + b $$ i) / 2" "d $$ i = b $$ i"
-        show False using e_f(2)[of i] and i x unfolding as by(fastforce simp add:field_simps)
+      proof(erule_tac[!] conjE) assume as:"c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
+        show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
+      next assume as:"c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
+        show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
       qed qed qed
   also have "\<Union> ?A = {a..b}" proof(rule set_eqI,rule)
     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
     show "x\<in>{a..b}" unfolding mem_interval proof safe
-      fix i assume "i<DIM('a)" thus "a $$ i \<le> x $$ i" "x $$ i \<le> b $$ i"
-        using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
+      fix i :: 'a assume i: "i\<in>Basis" thus "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
+        using c_d(1)[OF i] c_d(2)[unfolded mem_interval,THEN bspec, OF i] by auto qed
   next fix x assume x:"x\<in>{a..b}"
-    have "\<forall>i<DIM('a). \<exists>c d. (c = a$$i \<and> d = (a$$i + b$$i) / 2 \<or> c = (a$$i + b$$i) / 2 \<and> d = b$$i) \<and> c\<le>x$$i \<and> x$$i \<le> d"
-      (is "\<forall>i<DIM('a). \<exists>c d. ?P i c d") unfolding mem_interval proof(rule,rule) fix i
-      have "?P i (a$$i) ((a $$ i + b $$ i) / 2) \<or> ?P i ((a $$ i + b $$ i) / 2) (b$$i)"
-        using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
-    qed thus "x\<in>\<Union>?A" unfolding Union_iff unfolding lambda_skolem' unfolding Bex_def mem_Collect_eq
+    have "\<forall>i\<in>Basis. \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
+      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d") unfolding mem_interval
+    proof
+      fix i :: 'a assume i: "i \<in> Basis"
+      have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
+        using x[unfolded mem_interval,THEN bspec, OF i] by auto thus "\<exists>c d. ?P i c d" by blast
+    qed
+    thus "x\<in>\<Union>?A"
+      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
   qed finally show False using assms by auto qed
 
@@ -1490,8 +1384,8 @@
   obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
 proof-
   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
-    (\<forall>i<DIM('a). fst x$$i \<le> fst y$$i \<and> fst y$$i \<le> snd y$$i \<and> snd y$$i \<le> snd x$$i \<and>
-                           2 * (snd y$$i - fst y$$i) \<le> snd x$$i - fst x$$i))" proof case goal1 thus ?case proof-
+    (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
+                           2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" proof case goal1 thus ?case proof-
       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
       thus ?thesis apply(cases "P {fst x..snd x}") by auto
     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
@@ -1499,8 +1393,8 @@
     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
   def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
-    (\<forall>i<DIM('a). A(n)$$i \<le> A(Suc n)$$i \<and> A(Suc n)$$i \<le> B(Suc n)$$i \<and> B(Suc n)$$i \<le> B(n)$$i \<and> 
-    2 * (B(Suc n)$$i - A(Suc n)$$i) \<le> B(n)$$i - A(n)$$i)" (is "\<And>n. ?P n")
+    (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and> 
+    2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
@@ -1508,29 +1402,28 @@
     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
 
   have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
-  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$$i - a$$i) {..<DIM('a)}) / e"] .. note n=this
+  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] .. note n=this
     show ?case apply(rule_tac x=n in exI) proof(rule,rule)
       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
-      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$$i)) {..<DIM('a)}" unfolding dist_norm by(rule norm_le_l1)
-      also have "\<dots> \<le> setsum (\<lambda>i. B n$$i - A n$$i) {..<DIM('a)}"
-      proof(rule setsum_mono) fix i show "\<bar>(x - y) $$ i\<bar> \<le> B n $$ i - A n $$ i"
-          using xy[unfolded mem_interval,THEN spec[where x=i]] by auto qed
-      also have "\<dots> \<le> setsum (\<lambda>i. b$$i - a$$i) {..<DIM('a)} / 2^n" unfolding setsum_divide_distrib
+      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis" unfolding dist_norm by(rule norm_le_l1)
+      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
+      proof(rule setsum_mono)
+        fix i :: 'a assume i: "i \<in> Basis" show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
+          using xy[unfolded mem_interval,THEN bspec, OF i] by (auto simp: inner_diff_left) qed
+      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n" unfolding setsum_divide_distrib
       proof(rule setsum_mono) case goal1 thus ?case
         proof(induct n) case 0 thus ?case unfolding AB by auto
-        next case (Suc n) have "B (Suc n) $$ i - A (Suc n) $$ i \<le> (B n $$ i - A n $$ i) / 2"
+        next case (Suc n) have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
             using AB(4)[of i n] using goal1 by auto
-          also have "\<dots> \<le> (b $$ i - a $$ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
+          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
         qed qed
       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
     qed qed
-  { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
-    have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
-    proof(induct d) case 0 thus ?case by auto
-    next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
-        apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
-      proof- case goal1 thus ?case using AB(4)[of i "m + d"] by(auto simp add:field_simps)
-      qed qed } note ABsubset = this 
+  { fix n m :: nat assume "m \<le> n" then have "{A n..B n} \<subseteq> {A m..B m}"
+    proof(induct rule: inc_induct)
+      case (step i) show ?case
+        using AB(4) by (intro order_trans[OF step(2)] subset_interval_imp) auto
+    qed simp } note ABsubset = this 
   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
   proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
   then guess x0 .. note x0=this[rule_format]
@@ -1784,7 +1677,7 @@
   apply(rule integrable_linear) by assumption+
 
 lemma integral_component_eq[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
-  assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $$ k) = integral s f $$ k"
+  assumes "f integrable_on s" shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
   unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
 
 lemma has_integral_setsum:
@@ -1852,8 +1745,9 @@
   apply(rule integral_unique) using has_integral_empty .
 
 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a::'a::ordered_euclidean_space}"
-proof- have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq[where 'a='a]
-    apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps)
+proof-
+  have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq_iff[where 'a='a]
+    apply safe prefer 3 apply(erule_tac x=b in ballE) by(auto simp add: field_simps)
   show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
     apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
     unfolding interior_closed_interval using interval_sing by auto qed
@@ -1913,42 +1807,48 @@
 
 subsection {* Additivity of integral on abutting intervals. *}
 
-lemma interval_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
-  "{a..b} \<inter> {x. x$$k \<le> c} = {a .. (\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)}"
-  "{a..b} \<inter> {x. x$$k \<ge> c} = {(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i) .. b}"
-  apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval mem_Collect_eq using assms by auto
-
-lemma content_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
-  "content {a..b} = content({a..b} \<inter> {x. x$$k \<le> c}) + content({a..b} \<inter> {x. x$$k >= c})"
-proof- note simps = interval_split[OF assms] content_closed_interval_cases eucl_le[where 'a='a]
-  { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps using assms by auto }
-  have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" "\<And>x. finite ({..<DIM('a)}-{x})" "\<And>x. x\<notin>{..<DIM('a)}-{x}"
+lemma interval_split:
+  fixes a::"'a::ordered_euclidean_space" assumes "k \<in> Basis"
+  shows
+    "{a..b} \<inter> {x. x\<bullet>k \<le> c} = {a .. (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i) *\<^sub>R i)}"
+    "{a..b} \<inter> {x. x\<bullet>k \<ge> c} = {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i) *\<^sub>R i) .. b}"
+  apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval mem_Collect_eq using assms
+  by auto
+
+lemma content_split: fixes a::"'a::ordered_euclidean_space" assumes "k\<in>Basis" shows
+  "content {a..b} = content({a..b} \<inter> {x. x\<bullet>k \<le> c}) + content({a..b} \<inter> {x. x\<bullet>k >= c})"
+proof cases
+  note simps = interval_split[OF assms] content_closed_interval_cases eucl_le[where 'a='a]
+  have *:"Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
     using assms by auto
-  have *:"\<And>X Y Z. (\<Prod>i\<in>{..<DIM('a)}. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>{..<DIM('a)}-{k}. Z i (Y i))"
-    "(\<Prod>i\<in>{..<DIM('a)}. b$$i - a$$i) = (\<Prod>i\<in>{..<DIM('a)}-{k}. b$$i - a$$i) * (b$$k - a$$k)" 
+  have *:"\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
+    "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)" 
     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
-  assume as:"a\<le>b" moreover have "\<And>x. min (b $$ k) c = max (a $$ k) c
-    \<Longrightarrow> x* (b$$k - a$$k) = x*(max (a $$ k) c - a $$ k) + x*(b $$ k - max (a $$ k) c)"
+  assume as:"a\<le>b" moreover have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c
+    \<Longrightarrow> x* (b\<bullet>k - a\<bullet>k) = x*(max (a \<bullet> k) c - a \<bullet> k) + x*(b \<bullet> k - max (a \<bullet> k) c)"
     by  (auto simp add:field_simps)
-  moreover have **:"(\<Prod>i<DIM('a). ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i - a $$ i) = 
-    (\<Prod>i<DIM('a). (if i = k then min (b $$ k) c else b $$ i) - a $$ i)"
-    "(\<Prod>i<DIM('a). b $$ i - ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i) =
-    (\<Prod>i<DIM('a). b $$ i - (if i = k then max (a $$ k) c else a $$ i))"
-    apply(rule_tac[!] setprod.cong) by auto
-  have "\<not> a $$ k \<le> c \<Longrightarrow> \<not> c \<le> b $$ k \<Longrightarrow> False"
+  moreover have **:"(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) = 
+      (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
+    "(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
+      (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
+    by (auto intro!: setprod_cong)
+  have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
     unfolding not_le using as[unfolded eucl_le[where 'a='a],rule_format,of k] assms by auto
   ultimately show ?thesis using assms unfolding simps **
-    unfolding *(1)[of "\<lambda>i x. b$$i - x"] *(1)[of "\<lambda>i x. x - a$$i"] unfolding  *(2) 
-    apply(subst(2) euclidean_lambda_beta''[where 'a='a])
-    apply(subst(3) euclidean_lambda_beta''[where 'a='a]) by auto
+    unfolding *(1)[of "\<lambda>i x. b\<bullet>i - x"] *(1)[of "\<lambda>i x. x - a\<bullet>i"] unfolding *(2)
+    by auto
+next
+  assume "\<not> a \<le> b" then have "{a .. b} = {}"
+    unfolding interval_eq_empty by (auto simp: eucl_le[where 'a='a] not_le)
+  then show ?thesis by auto
 qed
 
 lemma division_split_left_inj: fixes type::"'a::ordered_euclidean_space"
   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2" 
-  "k1 \<inter> {x::'a. x$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}"and k:"k<DIM('a)"
-  shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
+  "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"and k:"k\<in>Basis"
+  shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof- note d=division_ofD[OF assms(1)]
-  have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k \<le> c}) = {})"
+  have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x\<bullet>k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x\<bullet>k \<le> c}) = {})"
     unfolding  interval_split[OF k] content_eq_0_interior by auto
   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
@@ -1958,10 +1858,10 @@
  
 lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
-  "k1 \<inter> {x::'a. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}" and k:"k<DIM('a)"
-  shows "content(k1 \<inter> {x. x$$k \<ge> c}) = 0"
+  "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" and k:"k\<in>Basis"
+  shows "content(k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
 proof- note d=division_ofD[OF assms(1)]
-  have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k >= c}) = {})"
+  have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x\<bullet>k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x\<bullet>k >= c}) = {})"
     unfolding interval_split[OF k] content_eq_0_interior by auto
   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
@@ -1970,25 +1870,25 @@
     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
 
 lemma tagged_division_split_left_inj: fixes x1::"'a::ordered_euclidean_space"
-  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}" 
-  and k:"k<DIM('a)"
-  shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
+  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}" 
+  and k:"k\<in>Basis"
+  shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
     apply(rule_tac[1-2] *) using assms(2-) by auto qed
 
 lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
-  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}" 
-  and k:"k<DIM('a)"
-  shows "content(k1 \<inter> {x. x$$k \<ge> c}) = 0"
+  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" 
+  and k:"k\<in>Basis"
+  shows "content(k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
     apply(rule_tac[1-2] *) using assms(2-) by auto qed
 
 lemma division_split: fixes a::"'a::ordered_euclidean_space"
-  assumes "p division_of {a..b}" and k:"k<DIM('a)"
-  shows "{l \<inter> {x. x$$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<le> c} = {})} division_of({a..b} \<inter> {x. x$$k \<le> c})" (is "?p1 division_of ?I1") and 
-        "{l \<inter> {x. x$$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$$k \<ge> c})" (is "?p2 division_of ?I2")
+  assumes "p division_of {a..b}" and k:"k\<in>Basis"
+  shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})} division_of({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is "?p1 division_of ?I1") and 
+        "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is "?p2 division_of ?I2")
 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms(1)]
   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
@@ -2006,34 +1906,34 @@
 qed
 
 lemma has_integral_split: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "(f has_integral i) ({a..b} \<inter> {x. x$$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$$k \<ge> c})" and k:"k<DIM('a)"
+  assumes "(f has_integral i) ({a..b} \<inter> {x. x\<bullet>k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" and k:"k\<in>Basis"
   shows "(f has_integral (i + j)) ({a..b})"
 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
   guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[THEN sym,OF k]]
   guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[THEN sym,OF k]]
-  let ?d = "\<lambda>x. if x$$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$$k - c)) \<inter> d1 x \<inter> d2 x"
+  let ?d = "\<lambda>x. if x\<bullet>k = c then (d1 x \<inter> d2 x) else ball x (abs(x\<bullet>k - c)) \<inter> d1 x \<inter> d2 x"
   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
-    have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
-         "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
+    have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\<bullet>k \<le> c} = {}) \<Longrightarrow> x\<bullet>k \<le> c"
+         "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\<bullet>k \<ge> c} = {}) \<Longrightarrow> x\<bullet>k \<ge> c"
     proof- fix x kk assume as:"(x,kk)\<in>p"
-      show "~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
+      show "~(kk \<inter> {x. x\<bullet>k \<le> c} = {}) \<Longrightarrow> x\<bullet>k \<le> c"
       proof(rule ccontr) case goal1
-        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ k - c\<bar>"
+        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
-        hence "\<exists>y. y \<in> ball x \<bar>x $$ k - c\<bar> \<inter> {x. x $$ k \<le> c}" using goal1(1) by blast 
-        then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<le> c" apply-apply(rule le_less_trans)
-          using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
+        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<le> c}" using goal1(1) by blast 
+        then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c" apply-apply(rule le_less_trans)
+          using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
       qed
-      show "~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
+      show "~(kk \<inter> {x. x\<bullet>k \<ge> c} = {}) \<Longrightarrow> x\<bullet>k \<ge> c"
       proof(rule ccontr) case goal1
-        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ k - c\<bar>"
+        from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
-        hence "\<exists>y. y \<in> ball x \<bar>x $$ k - c\<bar> \<inter> {x. x $$ k \<ge> c}" using goal1(1) by blast 
-        then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<ge> c" apply-apply(rule le_less_trans)
-          using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
+        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<ge> c}" using goal1(1) by blast 
+        then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c" apply-apply(rule le_less_trans)
+          using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
       qed
     qed
@@ -2053,15 +1953,15 @@
     qed auto
     have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
 
-    let ?M1 = "{(x,kk \<inter> {x. x$$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$k \<le> c} \<noteq> {}}"
+    let ?M1 = "{(x,kk \<inter> {x. x\<bullet>k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
-    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$$k \<le> c}" unfolding p(8)[THEN sym] by auto
+    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x\<bullet>k \<le> c}" unfolding p(8)[THEN sym] by auto
       fix x l assume xl:"(x,l)\<in>?M1"
       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
       thus "l \<subseteq> d1 x" unfolding xl' by auto
-      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
+      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x \<bullet> k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
         using lem0(1)[OF xl'(3-4)] by auto
       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastforce simp add: interval_split[OF k,where c=c])
       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
@@ -2073,15 +1973,15 @@
         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
       qed qed moreover
 
-    let ?M2 = "{(x,kk \<inter> {x. x$$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$k \<ge> c} \<noteq> {}}" 
+    let ?M2 = "{(x,kk \<inter> {x. x\<bullet>k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}" 
     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
-    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x$$k \<ge> c}" unfolding p(8)[THEN sym] by auto
+    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x\<bullet>k \<ge> c}" unfolding p(8)[THEN sym] by auto
       fix x l assume xl:"(x,l)\<in>?M2"
       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
       thus "l \<subseteq> d2 x" unfolding xl' by auto
-      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
+      show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x \<bullet> k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
         using lem0(2)[OF xl'(3-4)] by auto
       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastforce simp add: interval_split[OF k, where c=c])
       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
@@ -2098,98 +1998,95 @@
     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0" using scaleR_zero_left by auto
       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)
        = (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto
-      also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) +
-        (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $$ k}) *\<^sub>R f x) - (i + j)"
+      also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
+        (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) using k by auto
       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) using k by auto
       qed also note setsum_addf[THEN sym]
-      also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $$ k}) *\<^sub>R f x) x
+      also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x
         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
-        thus "content (b \<inter> {x. x $$ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $$ k}) *\<^sub>R f a = content b *\<^sub>R f a"
+        thus "content (b \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a = content b *\<^sub>R f a"
           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[OF k,of u v c] by auto
       qed note setsum_cong2[OF this]
-      finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $$ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $$ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
-        ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $$ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $$ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
+      finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x \<bullet> k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
+        ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x \<bullet> k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
 
-(*lemma has_integral_split_cart: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
-  assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"
-  shows "(f has_integral (i + j)) ({a..b})" *)
-
 subsection {* A sort of converse, integrability on subintervals. *}
 
 lemma tagged_division_union_interval: fixes a::"'a::ordered_euclidean_space"
-  assumes "p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c})"
-  and k:"k<DIM('a)"
+  assumes "p1 tagged_division_of ({a..b} \<inter> {x. x\<bullet>k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x\<bullet>k \<ge> c})"
+  and k:"k\<in>Basis"
   shows "(p1 \<union> p2) tagged_division_of ({a..b})"
-proof- have *:"{a..b} = ({a..b} \<inter> {x. x$$k \<le> c}) \<union> ({a..b} \<inter> {x. x$$k \<ge> c})" by auto
+proof- have *:"{a..b} = ({a..b} \<inter> {x. x\<bullet>k \<le> c}) \<union> ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" by auto
   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms(1-2)])
     unfolding interval_split[OF k] interior_closed_interval using k
-    by(auto simp add: eucl_less[where 'a='a] elim!:allE[where x=k]) qed
+    by(auto simp add: eucl_less[where 'a='a] elim!: ballE[where x=k]) qed
 
 lemma has_integral_separate_sides: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "(f has_integral i) ({a..b})" "e>0" and k:"k<DIM('a)"
-  obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> c}) \<and> d fine p1 \<and>
-                                p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c}) \<and> d fine p2
+  assumes "(f has_integral i) ({a..b})" "e>0" and k:"k\<in>Basis"
+  obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
+                                p2 tagged_division_of ({a..b} \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2
                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
 proof- guess d using has_integralD[OF assms(1-2)] . note d=this
   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
-  proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
-                   assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $$ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
+  proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
+                   assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x \<bullet> k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
     have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
-      have "b \<subseteq> {x. x$$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastforce
-      moreover have "interior {x::'a. x $$ k = c} = {}" 
-      proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x$$k = c}" by auto
+      have "b \<subseteq> {x. x\<bullet>k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastforce
+      moreover have "interior {x::'a. x \<bullet> k = c} = {}" 
+      proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x\<bullet>k = c}" by auto
         then guess e unfolding mem_interior .. note e=this
-        have x:"x$$k = c" using x interior_subset by fastforce
-        have *:"\<And>i. i<DIM('a) \<Longrightarrow> \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>
-          = (if i = k then e/2 else 0)" using e by auto
-        have "(\<Sum>i<DIM('a). \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>) =
-          (\<Sum>i<DIM('a). (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
+        have x:"x\<bullet>k = c" using x interior_subset by fastforce
+        have *:"\<And>i. i\<in>Basis \<Longrightarrow> \<bar>(x - (x + (e / 2) *\<^sub>R k)) \<bullet> i\<bar>
+          = (if i = k then e/2 else 0)" using e k by (auto simp: inner_simps inner_not_same_Basis)
+        have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
+          (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
         also have "... < e" apply(subst setsum_delta) using e by auto 
-        finally have "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball dist_norm
-          by(rule le_less_trans[OF norm_le_l1])
-        hence "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> {x. x$$k = c}" using e by auto
-        thus False unfolding mem_Collect_eq using e x k by auto
+        finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
+          unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
+        hence "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}" using e by auto
+        thus False unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule interior_mono) by auto
       thus "content b *\<^sub>R f a = 0" by auto
     qed auto
     also have "\<dots> < e" by(rule k d(2) p12 fine_union p1 p2)+
     finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
 
-lemma integrable_split[intro]: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
-  assumes "f integrable_on {a..b}" and k:"k<DIM('a)"
-  shows "f integrable_on ({a..b} \<inter> {x. x$$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$$k \<ge> c})" (is ?t2) 
+lemma integrable_split[intro]:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
+  assumes "f integrable_on {a..b}" and k:"k\<in>Basis"
+  shows "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2) 
 proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
-  def b' \<equiv> "(\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)::'a"
-  and a' \<equiv> "(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i)::'a"
+  def b' \<equiv> "\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i::'a"
+  def a' \<equiv> "\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i::'a"
   show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
       \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
-    show "?P {x. x $$ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
-    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c} \<and> d fine p1
-        \<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c} \<and> d fine p2"
+    show "?P {x. x \<bullet> k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
+    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1
+        \<and> p2 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           using p using assms by(auto simp add:algebra_simps)
       qed qed  
-    show "?P {x. x $$ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
-    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p1
-        \<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p2"
+    show "?P {x. x \<bullet> k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
+    proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1
+        \<and> p2 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
@@ -2203,13 +2100,13 @@
 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
   "operative opp f \<equiv> 
     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
-    (\<forall>a b c. \<forall>k<DIM('b). f({a..b}) =
-                   opp (f({a..b} \<inter> {x. x$$k \<le> c}))
-                       (f({a..b} \<inter> {x. x$$k \<ge> c})))"
+    (\<forall>a b c. \<forall>k\<in>Basis. f({a..b}) =
+                   opp (f({a..b} \<inter> {x. x\<bullet>k \<le> c}))
+                       (f({a..b} \<inter> {x. x\<bullet>k \<ge> c})))"
 
 lemma operativeD[dest]: fixes type::"'a::ordered_euclidean_space"  assumes "operative opp f"
   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b::'a} = neutral(opp)"
-  "\<And>a b c k. k<DIM('a) \<Longrightarrow> f({a..b}) = opp (f({a..b} \<inter> {x. x$$k \<le> c})) (f({a..b} \<inter> {x. x$$k \<ge> c}))"
+  "\<And>a b c k. k\<in>Basis \<Longrightarrow> f({a..b}) = opp (f({a..b} \<inter> {x. x\<bullet>k \<le> c})) (f({a..b} \<inter> {x. x\<bullet>k \<ge> c}))"
   using assms unfolding operative_def by auto
 
 lemma operative_trivial:
@@ -2342,18 +2239,20 @@
 lemma operative_integral: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
-  apply(rule,rule,rule,rule) defer apply(rule allI impI)+
-proof- fix a b c k assume k:"k<DIM('a)" show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
-    lifted op + (if f integrable_on {a..b} \<inter> {x. x $$ k \<le> c} then Some (integral ({a..b} \<inter> {x. x $$ k \<le> c}) f) else None)
-    (if f integrable_on {a..b} \<inter> {x. c \<le> x $$ k} then Some (integral ({a..b} \<inter> {x. c \<le> x $$ k}) f) else None)"
+  apply(rule,rule,rule,rule) defer apply(rule allI ballI)+
+proof-
+  fix a b c and k :: 'a assume k:"k\<in>Basis"
+  show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
+    lifted op + (if f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c} then Some (integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f) else None)
+    (if f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k} then Some (integral ({a..b} \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
   proof(cases "f integrable_on {a..b}") 
     case True show ?thesis unfolding if_P[OF True] using k apply-
       unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k]) 
       apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
-  next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x $$ k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x $$ k}))"
+  next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k}))"
     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
-        apply(rule_tac x="integral ({a..b} \<inter> {x. x $$ k \<le> c}) f + integral ({a..b} \<inter> {x. x $$ k \<ge> c}) f" in exI)
+        apply(rule_tac x="integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f + integral ({a..b} \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
         apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
       thus False using False by auto
     qed thus ?thesis using False by auto 
@@ -2365,91 +2264,110 @@
 subsection {* Points of division of a partition. *}
 
 definition "division_points (k::('a::ordered_euclidean_space) set) d = 
-    {(j,x). j<DIM('a) \<and> (interval_lowerbound k)$$j < x \<and> x < (interval_upperbound k)$$j \<and>
-           (\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
+    {(j,x). j\<in>Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
+           (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
 
 lemma division_points_finite: fixes i::"('a::ordered_euclidean_space) set"
   assumes "d division_of i" shows "finite (division_points i d)"
 proof- note assm = division_ofD[OF assms]
-  let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$$j < x \<and> x < (interval_upperbound i)$$j \<and>
-           (\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
-  have *:"division_points i d = \<Union>(?M ` {..<DIM('a)})"
+  let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and>
+           (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
+  have *:"division_points i d = \<Union>(?M ` Basis)"
     unfolding division_points_def by auto
   show ?thesis unfolding * using assm by auto qed
 
 lemma division_points_subset: fixes a::"'a::ordered_euclidean_space"
-  assumes "d division_of {a..b}" "\<forall>i<DIM('a). a$$i < b$$i"  "a$$k < c" "c < b$$k" and k:"k<DIM('a)"
-  shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<le> c} = {})}
+  assumes "d division_of {a..b}" "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k" and k:"k\<in>Basis"
+  shows "division_points ({a..b} \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})}
                   \<subseteq> division_points ({a..b}) d" (is ?t1) and
-        "division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})}
+        "division_points ({a..b} \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})}
                   \<subseteq> division_points ({a..b}) d" (is ?t2)
 proof- note assm = division_ofD[OF assms(1)]
-  have *:"\<forall>i<DIM('a). a$$i \<le> b$$i"   "\<forall>i<DIM('a). a$$i \<le> ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i"
-    "\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i \<le> b$$i"  "min (b $$ k) c = c" "max (a $$ k) c = c"
+  have *:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
+    "\<forall>i\<in>Basis. a\<bullet>i \<le> (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i"
+    "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i \<le> b\<bullet>i"
+    "min (b \<bullet> k) c = c" "max (a \<bullet> k) c = c"
     using assms using less_imp_le by auto
-  show ?t1 unfolding division_points_def interval_split[OF k, of a b]
-    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
-    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
-    unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta'
-  proof- fix i l x assume as:"a $$ fst x < snd x" "snd x < (if fst x = k then c else b $$ fst x)"
-      "interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x"
-      "i = l \<inter> {x. x $$ k \<le> c}" "l \<in> d" "l \<inter> {x. x $$ k \<le> c} \<noteq> {}" and fstx:"fst x <DIM('a)"
+  show ?t1
+    unfolding division_points_def interval_split[OF k, of a b]
+    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
+    unfolding *
+    unfolding subset_eq
+    apply(rule)
+    unfolding mem_Collect_eq split_beta
+    apply(erule bexE conjE)+
+    apply(simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
+    apply(erule exE conjE)+
+  proof
+    fix i l x assume as:"a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)"
+      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}" and fstx:"fst x \<in>Basis"
     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
-    have *:"\<forall>i<DIM('a). u $$ i \<le> ((\<chi>\<chi> i. if i = k then min (v $$ k) c else v $$ i)::'a) $$ i"
+    have *:"\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i"
       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
-    have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
-    show "fst x <DIM('a) \<and> a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x
-      \<or> interval_upperbound i $$ fst x = snd x)" apply(rule,rule fstx)
-      using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
-      apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
-      apply(case_tac[!] "fst x = k") using assms fstx apply- unfolding euclidean_lambda_beta by auto
+    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      apply (rule bexI[OF _ `l \<in> d`])
+      using as(1-3,5) fstx
+      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
+      by (auto split: split_if_asm)
+    show "snd x < b \<bullet> fst x"
+      using as(2) `c < b\<bullet>k` by (auto split: split_if_asm)
   qed
-  show ?t2 unfolding division_points_def interval_split[OF k, of a b]
+  show ?t2
+    unfolding division_points_def interval_split[OF k, of a b]
     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
-    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
-    unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta' apply(rule,assumption)
-  proof- fix i l x assume as:"(if fst x = k then c else a $$ fst x) < snd x" "snd x < b $$ fst x"
-      "interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x" 
-      "i = l \<inter> {x. c \<le> x $$ k}" "l \<in> d" "l \<inter> {x. c \<le> x $$ k} \<noteq> {}" and fstx:"fst x < DIM('a)"
+    unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta
+    apply(erule bexE conjE)+
+    apply(simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
+    apply(erule exE conjE)+
+  proof
+    fix i l x assume as:"(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
+      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" 
+      "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" and fstx:"fst x \<in> Basis"
     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
-    have *:"\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (u $$ k) c else u $$ i)::'a) $$ i \<le> v $$ i"
+    have *:"\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i"
       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
-    have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
-    show "a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x \<or>
-      interval_upperbound i $$ fst x = snd x)"
-      using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
-      apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
-      apply(case_tac[!] "fst x = k") using assms fstx apply-  by(auto simp add:euclidean_lambda_beta'[OF k]) qed qed
+    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      apply (rule bexI[OF _ `l \<in> d`])
+      using as(1-3,5) fstx
+      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
+      by (auto split: split_if_asm)
+    show "a \<bullet> fst x < snd x"
+      using as(1) `a\<bullet>k < c` by (auto split: split_if_asm)
+   qed
+qed
 
 lemma division_points_psubset: fixes a::"'a::ordered_euclidean_space"
-  assumes "d division_of {a..b}"  "\<forall>i<DIM('a). a$$i < b$$i"  "a$$k < c" "c < b$$k"
-  "l \<in> d" "interval_lowerbound l$$k = c \<or> interval_upperbound l$$k = c" and k:"k<DIM('a)"
-  shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}
+  assumes "d division_of {a..b}"  "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
+  "l \<in> d" "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c" and k:"k\<in>Basis"
+  shows "division_points ({a..b} \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}
               \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
-        "division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}
+        "division_points ({a..b} \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}
               \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 
-proof- have ab:"\<forall>i<DIM('a). a$$i \<le> b$$i" using assms(2) by(auto intro!:less_imp_le)
+proof- have ab:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" using assms(2) by(auto intro!:less_imp_le)
   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
-  have uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "\<forall>i<DIM('a). a$$i \<le> u$$i \<and> v$$i \<le> b$$i"
+  have uv:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i"
     using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
-  have *:"interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
-         "interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
+  have *:"interval_upperbound ({a..b} \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
+         "interval_upperbound ({a..b} \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
-    apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
-    apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
+    apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
+    apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
 
-  have *:"interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
-         "interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
+  have *:"interval_lowerbound ({a..b} \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
+         "interval_lowerbound ({a..b} \<inter> {x. x \<bullet> k \<ge> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
-    apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
-    apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
+    apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
+    apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4) k]) by auto qed
 
@@ -2548,42 +2466,47 @@
             using operativeD(1)[OF assms(2)] x by auto
         qed qed }
     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
-    hence ab':"\<forall>i<DIM('a). a$$i \<le> b$$i" by (auto intro!: less_imp_le) show ?case 
+    hence ab':"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" by (auto intro!: less_imp_le) show ?case 
     proof(cases "division_points {a..b} d = {}")
       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
-        (\<forall>j<DIM('a). u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j)"
+        (\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)"
         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
-        apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule,rule)
-      proof- fix u v j assume j:"j<DIM('a)" assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
-        hence uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "u$$j \<le> v$$j" using j unfolding interval_ne_empty by auto
-        have *:"\<And>p r Q. \<not> j<DIM('a) \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as j by auto
-        have "(j, u$$j) \<notin> division_points {a..b} d"
-          "(j, v$$j) \<notin> division_points {a..b} d" using True by auto
+        apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl)
+      proof
+        fix u v and j :: 'a assume j:"j\<in>Basis" assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
+        hence uv:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "u\<bullet>j \<le> v\<bullet>j" using j unfolding interval_ne_empty by auto
+        have *:"\<And>p r Q. \<not> j\<in>Basis \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as j by auto
+        have "(j, u\<bullet>j) \<notin> division_points {a..b} d"
+          "(j, v\<bullet>j) \<notin> division_points {a..b} d" using True by auto
         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
-        moreover have "a$$j \<le> u$$j" "v$$j \<le> b$$j" using division_ofD(2,2,3)[OF goal1(4) as] 
+        moreover have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" using division_ofD(2,2,3)[OF goal1(4) as] 
           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
           unfolding interval_ne_empty mem_interval using j by auto
-        ultimately show "u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j"
+        ultimately show "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j"
           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by auto
-      qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
+      qed
+      have "(1/2) *\<^sub>R (a+b) \<in> {a..b}"
+        unfolding mem_interval using ab by(auto intro!: less_imp_le simp: inner_simps)
       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
       have "{a..b} \<in> d"
       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
-        show "u = a" "v = b" unfolding euclidean_eq[where 'a='a]
-        proof(safe) fix j assume j:"j<DIM('a)" note i(2)[unfolded uv mem_interval,rule_format,of j]
-          thus "u $$ j = a $$ j" "v $$ j = b $$ j" using uv(2)[rule_format,of j] j by auto
+        show "u = a" "v = b" unfolding euclidean_eq_iff[where 'a='a]
+        proof(safe)
+          fix j :: 'a assume j:"j\<in>Basis"
+          note i(2)[unfolded uv mem_interval,rule_format,of j]
+          thus "u \<bullet> j = a \<bullet> j" "v \<bullet> j = b \<bullet> j" using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
         qed qed
       hence *:"d = insert {a..b} (d - {{a..b}})" by auto
       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
         then guess u v apply-by(erule exE conjE)+ note uv=this
         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
-        then obtain j where "u$$j \<noteq> a$$j \<or> v$$j \<noteq> b$$j" and j:"j<DIM('a)" unfolding euclidean_eq[where 'a='a] by auto
-        hence "u$$j = v$$j" using uv(2)[rule_format,OF j] by auto
-        hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) using j by auto
+        then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j:"j\<in>Basis" unfolding euclidean_eq_iff[where 'a='a] by auto
+        hence "u\<bullet>j = v\<bullet>j" using uv(2)[rule_format,OF j] by auto
+        hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in bexI) using j by auto
         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
@@ -2591,39 +2514,40 @@
       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
         by(erule exE conjE)+ note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
       from this(3) guess j .. note j=this
-      def d1 \<equiv> "{l \<inter> {x. x$$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}"
-      def d2 \<equiv> "{l \<inter> {x. x$$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}"
-      def cb \<equiv> "(\<chi>\<chi> i. if i = k then c else b$$i)::'a" and ca \<equiv> "(\<chi>\<chi> i. if i = k then c else a$$i)::'a"
+      def d1 \<equiv> "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
+      def d2 \<equiv> "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
+      def cb \<equiv> "(\<Sum>i\<in>Basis. (if i = k then c else b\<bullet>i) *\<^sub>R i)::'a"
+      def ca \<equiv> "(\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)::'a"
       note division_points_psubset[OF goal1(4) ab kc(1-2) j]
       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
-      hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x$$k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x$$k \<ge> c})"
+      hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x\<bullet>k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x\<bullet>k \<ge> c})"
         apply- unfolding interval_split[OF kc(4)] apply(rule_tac[!] goal1(1)[rule_format])
         using division_split[OF goal1(4), where k=k and c=c]
         unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
         using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
         unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto 
-      also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$k \<le> c}))"
+      also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<le> c}))"
         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
-      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x $$ k \<le> c} = y \<inter> {x. x $$ k \<le> c}" "l \<noteq> y" 
+      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y" 
         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
-        show "f (l \<inter> {x. x $$ k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
+        show "f (l \<inter> {x. x \<bullet> k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
-      qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$k \<ge> c}))"
+      qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<ge> c}))"
         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
-      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x $$ k} = y \<inter> {x. c \<le> x $$ k}" "l \<noteq> y" 
+      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y" 
         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
-        show "f (l \<inter> {x. x $$ k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
+        show "f (l \<inter> {x. x \<bullet> k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
-      qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x $$ k \<le> c})) (f (x \<inter> {x. c \<le> x $$ k}))"
+      qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x \<bullet> k \<le> c})) (f (x \<inter> {x. c \<le> x \<bullet> k}))"
         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto 
-      have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x $$ k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x $$ k})))
+      have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x \<bullet> k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x \<bullet> k})))
         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
         apply(rule iterate_op[THEN sym]) using goal1 by auto
       finally show ?thesis by auto
@@ -2754,31 +2678,34 @@
 subsection {* Similar theorems about relationship among components. *}
 
 lemma rsum_component_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$$i \<le> (g x)$$i"
-  shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$$i"
-  unfolding  euclidean_component_setsum apply(rule setsum_mono) apply safe
+  assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)\<bullet>i \<le> (g x)\<bullet>i"
+  shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
+  unfolding inner_setsum_left apply(rule setsum_mono) apply safe
 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
   from this(3) guess u v apply-by(erule exE)+ note b=this
-  show "(content b *\<^sub>R f a) $$ i \<le> (content b *\<^sub>R g a) $$ i" unfolding b
-    unfolding euclidean_simps real_scaleR_def apply(rule mult_left_mono)
+  show "(content b *\<^sub>R f a) \<bullet> i \<le> (content b *\<^sub>R g a) \<bullet> i" unfolding b
+    unfolding inner_simps real_scaleR_def apply(rule mult_left_mono)
     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
 
-lemma has_integral_component_le: fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
-  shows "i$$k \<le> j$$k"
+lemma has_integral_component_le:
+  fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes k: "k \<in> Basis"
+  assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
+  shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
   have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow> 
-    (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$$k \<le> (g x)$$k \<Longrightarrow> i$$k \<le> j$$k"
+    (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)\<bullet>k \<le> (g x)\<bullet>k \<Longrightarrow> i\<bullet>k \<le> j\<bullet>k"
   proof (rule ccontr)
     case goal1
-    then have *: "0 < (i$$k - j$$k) / 3" by auto
+    then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3" by auto
     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
-    note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k] term g
+    note p = this(1) conjunctD2[OF this(2)]
+    note le_less_trans[OF Basis_le_norm[OF k]]
     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
     thus False
-      unfolding euclidean_simps
+      unfolding inner_simps
       using rsum_component_le[OF p(1) goal1(3)]
       by (simp add: abs_real_def split: split_if_asm)
   qed let ?P = "\<exists>a b. s = {a..b}"
@@ -2787,8 +2714,9 @@
       show ?thesis apply(rule lem) using assms[unfolded s] by auto
     qed auto } assume as:"\<not> ?P"
   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
-  assume "\<not> i$$k \<le> j$$k" hence ij:"(i$$k - j$$k) / 3 > 0" by auto
-  note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
+  assume "\<not> i\<bullet>k \<le> j\<bullet>k" hence ij:"(i\<bullet>k - j\<bullet>k) / 3 > 0" by auto
+  note has_integral_altD[OF _ as this]
+  from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
   have "bounded (ball 0 B1 \<union> ball (0::'a) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
@@ -2796,69 +2724,50 @@
   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
   have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False"
     by (simp add: abs_real_def split: split_if_asm)
-  note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
-  have "w1$$k \<le> w2$$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
-  show False unfolding euclidean_simps by(rule *) qed
+  note le_less_trans[OF Basis_le_norm[OF k]] note this[OF w1(2)] this[OF w2(2)] moreover
+  have "w1\<bullet>k \<le> w2\<bullet>k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
+  show False unfolding inner_simps by(rule *)
+qed
 
 lemma integral_component_le: fixes g f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
-  shows "(integral s f)$$k \<le> (integral s g)$$k"
+  assumes "k\<in>Basis" "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
+  shows "(integral s f)\<bullet>k \<le> (integral s g)\<bullet>k"
   apply(rule has_integral_component_le) using integrable_integral assms by auto
 
-(*lemma has_integral_dest_vec1_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> real^1"
-  assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
-  shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
-  using assms(3) unfolding vector_le_def by auto
-
-lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
-  assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
-  shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
-  apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto*)
-
 lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> i$$k" 
-  using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2-) by auto
+  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> i\<bullet>k" 
+  using has_integral_component_le[OF assms(1) has_integral_0 assms(2)] using assms(3-) by auto
 
 lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> (integral s f)$$k"
+  assumes "k\<in>Basis" "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> (integral s f)\<bullet>k"
   apply(rule has_integral_component_nonneg) using assms by auto
 
-(*lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
-  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
-  using has_integral_component_nonneg[OF assms(1), of 1]
-  using assms(2) unfolding vector_le_def by auto
-
-lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
-  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
-  apply(rule has_integral_dest_vec1_nonneg) using assms by auto*)
-
 lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$$k \<le> 0"shows "i$$k \<le> 0" 
-  using has_integral_component_le[OF assms(1) has_integral_0] assms(2-) by auto
-
-(*lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
-  assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
-  using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto*)
-
-lemma has_integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)" shows "B * content {a..b} \<le> i$$k"
-  using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi>\<chi> i. B)::'b" k] assms(2-)
-  unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] by(auto simp add:field_simps)
-
-lemma has_integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$$k \<le> B" "k<DIM('b)"
-  shows "i$$k \<le> B * content({a..b})"
-  using has_integral_component_le[OF assms(1) has_integral_const, of k "\<chi>\<chi> i. B"]
-  unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] using assms(2) by(auto simp add:field_simps)
+  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"shows "i\<bullet>k \<le> 0" 
+  using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-) by auto
+
+lemma has_integral_component_lbound:
+  fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
+  assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)\<bullet>k" "k\<in>Basis"
+  shows "B * content {a..b} \<le> i\<bullet>k"
+  using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(\<Sum>i\<in>Basis. B *\<^sub>R i)::'b"] assms(2-)
+  by (auto simp add:field_simps)
+
+lemma has_integral_component_ubound:
+  fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
+  assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x\<bullet>k \<le> B" "k\<in>Basis"
+  shows "i\<bullet>k \<le> B * content({a..b})"
+  using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"]  assms(2-)
+  by(auto simp add:field_simps)
 
 lemma integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)"
-  shows "B * content({a..b}) \<le> (integral({a..b}) f)$$k"
+  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)\<bullet>k" "k\<in>Basis"
+  shows "B * content({a..b}) \<le> (integral({a..b}) f)\<bullet>k"
   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
 
 lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$$k \<le> B" "k<DIM('b)" 
-  shows "(integral({a..b}) f)$$k \<le> B * content({a..b})"
+  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\<bullet>k \<le> B" "k\<in>Basis" 
+  shows "(integral({a..b}) f)\<bullet>k \<le> B * content({a..b})"
   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
 
 subsection {* Uniform limit of integrable functions is integrable. *}
@@ -2949,68 +2858,76 @@
   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
   unfolding assms using neutral_add unfolding neutral_add using assms by auto 
 
-lemma interval_doublesplit:  fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)"
-  shows "{a..b} \<inter> {x . abs(x$$k - c) \<le> (e::real)} = 
-  {(\<chi>\<chi> i. if i = k then max (a$$k) (c - e) else a$$i) .. (\<chi>\<chi> i. if i = k then min (b$$k) (c + e) else b$$i)}"
+lemma interval_doublesplit:  fixes a::"'a::ordered_euclidean_space" assumes "k\<in>Basis"
+  shows "{a..b} \<inter> {x . abs(x\<bullet>k - c) \<le> (e::real)} = 
+  {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) .. 
+   (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)}"
 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
   show ?thesis unfolding * ** interval_split[OF assms] by(rule refl) qed
 
-lemma division_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "p division_of {a..b}" and k:"k<DIM('a)"
-  shows "{l \<inter> {x. abs(x$$k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x$$k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x$$k - c) \<le> e})"
+lemma division_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "p division_of {a..b}" and k:"k\<in>Basis"
+  shows "{l \<inter> {x. abs(x\<bullet>k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x\<bullet>k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x\<bullet>k - c) \<le> e})"
 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
   note division_split(2)[OF this, where c="c-e" and k=k,OF k] 
   thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
     apply(rule set_eqI) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
-    apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $$ k}" in exI) apply rule defer apply rule
+    apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI) apply rule defer apply rule
     apply(rule_tac x=l in exI) by blast+ qed
 
-lemma content_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "0 < e" and k:"k<DIM('a)"
-  obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$$k - c) \<le> d}) < e"
+lemma content_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "0 < e" and k:"k\<in>Basis"
+  obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x\<bullet>k - c) \<le> d}) < e"
 proof(cases "content {a..b} = 0")
   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
     unfolding interval_doublesplit[THEN sym,OF k] using assms by auto 
-next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})"
+next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})"
   note False[unfolded content_eq_0 not_ex not_le, rule_format]
-  hence "\<And>x. x<DIM('a) \<Longrightarrow> b$$x > a$$x" by(auto simp add:not_le)
-  hence prod0:"0 < setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
+  hence "\<And>x. x\<in>Basis \<Longrightarrow> b\<bullet>x > a\<bullet>x" by(auto simp add:not_le)
+  hence prod0:"0 < setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
-  proof(rule that[of d]) have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" using k by auto
-    have **:"{a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
-      (\<Prod>i\<in>{..<DIM('a)} - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i
-      - interval_lowerbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i)
-      = (\<Prod>i\<in>{..<DIM('a)} - {k}. b$$i - a$$i)" apply(rule setprod_cong,rule refl) 
+  proof(rule that[of d]) have *:"Basis = insert k (Basis - {k})" using k by auto
+    have **:"{a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
+      (\<Prod>i\<in>Basis - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i
+      - interval_lowerbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i)
+      = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)" apply(rule setprod_cong,rule refl) 
       unfolding interval_doublesplit[OF k] apply(subst interval_bounds) defer apply(subst interval_bounds)
       unfolding interval_eq_empty not_ex not_less by auto
-    show "content ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
+    show "content ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
       unfolding interval_doublesplit[OF k] interval_eq_empty not_ex not_less prefer 3
-      apply(subst interval_bounds) defer apply(subst interval_bounds) unfolding euclidean_lambda_beta'[OF k] if_P[OF refl]
-    proof- have "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) \<le> 2 * d" by auto
-      also have "... < e / (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
-      finally show "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) * (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i) < e"
-        unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
-
-lemma negligible_standard_hyperplane[intro]: fixes type::"'a::ordered_euclidean_space" assumes k:"k<DIM('a)"
-  shows "negligible {x::'a. x$$k = (c::real)}" 
+      apply(subst interval_bounds) defer apply(subst interval_bounds)
+      apply (simp_all only: k inner_setsum_left_Basis simp_thms if_P cong: bex_cong ball_cong)
+    proof -
+      have "(min (b \<bullet> k) (c + d) - max (a \<bullet> k) (c - d)) \<le> 2 * d" by auto
+      also have "... < e / (\<Prod>i\<in>Basis - {k}. b \<bullet> i - a \<bullet> i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
+      finally show "(min (b \<bullet> k) (c + d) - max (a \<bullet> k) (c - d)) * (\<Prod>i\<in>Basis - {k}. b \<bullet> i - a \<bullet> i) < e"
+        unfolding pos_less_divide_eq[OF prod0] .
+    qed auto
+  qed
+qed
+
+lemma negligible_standard_hyperplane[intro]: 
+  fixes k :: "'a::ordered_euclidean_space"
+  assumes k: "k \<in> Basis"
+  shows "negligible {x. x\<bullet>k = c}" 
   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
 proof-
   case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
-  let ?i = "indicator {x::'a. x$$k = c} :: 'a\<Rightarrow>real"
+  let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
-    have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$$k - c) \<le> d}) *\<^sub>R ?i x)"
+    have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x\<bullet>k - c) \<le> d}) *\<^sub>R ?i x)"
       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
       apply(cases,rule disjI1,assumption,rule disjI2)
-    proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
-      show "content l = content (l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
+    proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x\<bullet>k = c" unfolding indicator_def apply-by(rule ccontr,auto)
+      show "content l = content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
         apply(rule set_eqI,rule,rule) unfolding mem_Collect_eq
       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
-        note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
-        thus "\<bar>y $$ k - c\<bar> \<le> d" unfolding euclidean_simps xk by auto
+        note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF Basis_le_norm[OF k] this]
+        thus "\<bar>y \<bullet> k - c\<bar> \<le> d" unfolding inner_simps xk by auto
       qed auto qed
     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e" unfolding diff_0_right * unfolding real_scaleR_def real_norm_def
@@ -3018,33 +2935,33 @@
       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
       prefer 2 apply(subst(asm) eq_commute) apply assumption
       apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
-    proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}))"
+    proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
         apply(rule setsum_mono) unfolding split_paired_all split_conv 
         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k])
       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
-      proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<le> content {u..v}"
+      proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content {u..v}"
           unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
         thus ?case unfolding goal1 unfolding interval_doublesplit[OF k]
           by (blast intro: antisym)
-      next have *:"setsum content {l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
+      next have *:"setsum content {l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
-        proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
+        proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
         note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
-        from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})) < e"
+        from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e"
           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
-          assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}"
-          have "({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
+          assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
+          have "({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
           note interior_mono[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
-          hence "interior ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
-          thus "content ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
+          hence "interior ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
+          thus "content ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
         qed qed
-      finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) < e" .
+      finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
     qed qed qed
 
 subsection {* A technical lemma about "refinement" of division. *}
@@ -3224,7 +3141,7 @@
   using negligible_union by auto
 
 lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}" 
-  using negligible_standard_hyperplane[of 0 "a$$0"] by auto 
+  using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto 
 
 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
@@ -3276,11 +3193,20 @@
 subsection {* In particular, the boundary of an interval is negligible. *}
 
 lemma negligible_frontier_interval: "negligible({a::'a::ordered_euclidean_space..b} - {a<..<b})"
-proof- let ?A = "\<Union>((\<lambda>k. {x. x$$k = a$$k} \<union> {x::'a. x$$k = b$$k}) ` {..<DIM('a)})"
-  have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
-    apply(erule conjE exE)+ apply(rule_tac X="{x. x $$ xa = a $$ xa} \<union> {x. x $$ xa = b $$ xa}" in UnionI)
-    apply(erule_tac[!] x=xa in allE) by auto
-  thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
+proof-
+  let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
+  have "{a..b} - {a<..<b} \<subseteq> ?A"
+    apply rule unfolding Diff_iff mem_interval
+    apply simp
+    apply(erule conjE bexE)+
+    apply(rule_tac x=i in bexI)
+    by auto
+  thus ?thesis
+    apply-
+    apply(rule negligible_subset[of ?A])
+    apply(rule negligible_unions[OF finite_imageI])
+    by auto
+qed
 
 lemma has_integral_spike_interior:
   assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
@@ -3309,23 +3235,26 @@
 
 lemma operative_approximable: assumes "0 \<le> e" fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
   shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
-proof safe fix a b::"'b" { assume "content {a..b} = 0"
+proof safe
+  fix a b::"'b"
+  { assume "content {a..b} = 0"
     thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
-  { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k<DIM('b)"
-    show "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
-      "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
+  { fix c g and k :: 'b
+    assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k\<in>Basis"
+    show "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}"
+      "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k}"
       apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2) k] by auto }
-  fix c k g1 g2 assume as:"\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
-                          "\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g2 x) \<le> e" "g2 integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
-  assume k:"k<DIM('b)"
-  let ?g = "\<lambda>x. if x$$k = c then f x else if x$$k \<le> c then g1 x else g2 x"
+  fix c k g1 g2 assume as:"\<forall>x\<in>{a..b} \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}"
+                          "\<forall>x\<in>{a..b} \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e" "g2 integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k}"
+  assume k:"k\<in>Basis"
+  let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
   show "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" apply(rule_tac x="?g" in exI)
-  proof safe case goal1 thus ?case apply- apply(cases "x$$k=c", case_tac "x$$k < c") using as assms by auto
-  next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
+  proof safe case goal1 thus ?case apply- apply(cases "x\<bullet>k=c", case_tac "x\<bullet>k < c") using as assms by auto
+  next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
     then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k] 
     show ?case unfolding integrable_on_def by auto
-  next show "?g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
+  next show "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
 
 lemma approximable_on_division: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
@@ -3355,11 +3284,15 @@
 
 subsection {* Specialization of additivity to one dimension. *}
 
+lemma 
+  shows real_inner_1_left: "inner 1 x = x"
+  and real_inner_1_right: "inner x 1 = x"
+  by simp_all
+
 lemma operative_1_lt: assumes "monoidal opp"
   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
                 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
-  unfolding operative_def content_eq_0 DIM_real less_one simp_thms(39,41) Eucl_real_simps
-    (* The dnf_simps simplify "\<exists> x. x= _ \<and> _" and "\<forall>k. k = _ \<longrightarrow> _" *)
+  apply (simp add: operative_def content_eq_0 less_one)
 proof safe fix a b c::"real" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c}))
     (f ({a..b} \<inter> {x. c \<le> x}))" "a < c" "c < b"
     from this(2-) have "{a..b} \<inter> {x. x \<le> c} = {a..c}" "{a..b} \<inter> {x. x \<ge> c} = {c..b}" by auto
@@ -3376,10 +3309,11 @@
       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
     qed
   next case True hence *:"min (b) c = c" "max a c = c" by auto
-    have **:"0 < DIM(real)" by auto
-    have ***:"\<And>P Q. (\<chi>\<chi> i. if i = 0 then P i else Q i) = (P 0::real)" apply(subst euclidean_eq)
-      apply safe unfolding euclidean_lambda_beta' by auto
-    show ?thesis unfolding interval_split[OF **,unfolded Eucl_real_simps(1,3)] unfolding *** *
+    have **: "(1::real) \<in> Basis" by simp
+    have ***:"\<And>P Q. (\<Sum>i\<in>Basis. (if i = 1 then P i else Q i) *\<^sub>R i) = (P 1::real)" 
+      by simp
+    show ?thesis 
+      unfolding interval_split[OF **, unfolded real_inner_1_right] unfolding *** *
     proof(cases "c = a \<or> c = b")
       case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
         apply-apply(subst as(2)[rule_format]) using True by auto
@@ -3411,13 +3345,13 @@
 subsection {* Special case of additivity we need for the FCT. *}
 
 lemma interval_bound_sing[simp]: "interval_upperbound {a} = a"  "interval_lowerbound {a} = a"
-  unfolding interval_upperbound_def interval_lowerbound_def  by auto
+  unfolding interval_upperbound_def interval_lowerbound_def by (auto simp: euclidean_representation)
 
 lemma additive_tagged_division_1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b" "p tagged_division_of {a..b}"
   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
 proof- let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
-  have ***:"\<forall>i<DIM(real). a $$ i \<le> b $$ i" using assms by auto 
+  have ***:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" using assms by auto
   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty by auto
   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],THEN sym]
@@ -3518,38 +3452,30 @@
   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
     from k(2)[unfolded k content_eq_0] guess i .. 
-    hence i:"c$$i = d$$i" "i<DIM('a)" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
-    hence xi:"x$$i = d$$i" using as unfolding k mem_interval by (metis antisym)
-    def y \<equiv> "(\<chi>\<chi> j. if j = i then if c$$i \<le> (a$$i + b$$i) / 2 then c$$i +
-      min e (b$$i - c$$i) / 2 else c$$i - min e (c$$i - a$$i) / 2 else x$$j)::'a"
+    hence i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
+    hence xi:"x\<bullet>i = d\<bullet>i" using as unfolding k mem_interval by (metis antisym)
+    def y \<equiv> "(\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i) / 2 then c\<bullet>i +
+      min e (b\<bullet>i - c\<bullet>i) / 2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i) / 2 else x\<bullet>j) *\<^sub>R j)::'a"
     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
     proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastforce simp add: not_less)
-      hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
-      hence xyi:"y$$i \<noteq> x$$i" unfolding y_def unfolding i xi euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
-        apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2)
-        using assms(2)[unfolded content_eq_0] using i(2)
-        by (auto simp add: not_le min_def)
-      thus "y \<noteq> x" unfolding euclidean_eq[where 'a='a] using i by auto
-      have *:"{..<DIM('a)} = insert i ({..<DIM('a)} - {i})" using i by auto
-      have "norm (y - x) < e + setsum (\<lambda>i. 0) {..<DIM('a)}" apply(rule le_less_trans[OF norm_le_l1])
+      hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN bspec[where x=i]]
+      hence xyi:"y\<bullet>i \<noteq> x\<bullet>i"
+        unfolding y_def i xi using as(2) assms(2)[unfolded content_eq_0] i(2)
+        by (auto elim!: ballE[of _ _ i])
+      thus "y \<noteq> x" unfolding euclidean_eq_iff[where 'a='a] using i by auto
+      have *:"Basis = insert i (Basis - {i})" using i by auto
+      have "norm (y - x) < e + setsum (\<lambda>i. 0) Basis" apply(rule le_less_trans[OF norm_le_l1])
         apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
-      proof- show "\<bar>(y - x) $$ i\<bar> < e" unfolding y_def euclidean_simps euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
-          apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
-        show "(\<Sum>i\<in>{..<DIM('a)} - {i}. \<bar>(y - x) $$ i\<bar>) \<le> (\<Sum>i\<in>{..<DIM('a)}. 0)" unfolding y_def by auto 
+      proof-
+        show "\<bar>(y - x) \<bullet> i\<bar> < e"
+          using di as(2) y_def i xi by (auto simp: inner_simps)
+        show "(\<Sum>i\<in>Basis - {i}. \<bar>(y - x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
+          unfolding y_def by (auto simp: inner_simps)
       qed auto thus "dist y x < e" unfolding dist_norm by auto
-      have "y\<notin>k" unfolding k mem_interval apply rule apply(erule_tac x=i in allE) using xyi unfolding k i xi by auto
-      moreover have "y \<in> \<Union>s" unfolding s mem_interval
-      proof(rule,rule) note simps = y_def euclidean_lambda_beta' if_not_P
-        fix j assume j:"j<DIM('a)" show "a $$ j \<le> y $$ j \<and> y $$ j \<le> b $$ j" 
-        proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
-          thus ?thesis using j apply- unfolding simps if_not_P[OF False] unfolding mem_interval by auto
-        next case True note T = this show ?thesis
-          proof(cases "c $$ i \<le> (a $$ i + b $$ i) / 2")
-            case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
-              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
-          next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
-              using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
-          qed qed qed
+      have "y\<notin>k" unfolding k mem_interval apply rule apply(erule_tac x=i in ballE) using xyi k i xi by auto
+      moreover have "y \<in> \<Union>s"
+        using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i unfolding s mem_interval y_def
+        by (auto simp: field_simps elim!: ballE[of _ _ i])
       ultimately show "y \<in> \<Union>(s - {k})" by auto
     qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
@@ -3730,23 +3656,41 @@
  "content((\<lambda>x::'a::ordered_euclidean_space. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ DIM('a) * content {a..b}" (is "?l = ?r")
 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
       unfolding not_not using content_empty by auto }
-  have *:"DIM('a) = card {..<DIM('a)}" by auto
-  assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
-    case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
-      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
-      apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
-      apply(rule setprod_cong2) using True as unfolding interval_ne_empty euclidean_simps not_le  
-      by(auto simp add:field_simps intro:mult_left_mono)
-  next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
-      unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
-      apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
-      apply(rule setprod_cong2) using False as unfolding interval_ne_empty euclidean_simps not_le 
-      by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
+  assume as: "{a..b}\<noteq>{}" 
+  show ?thesis 
+  proof (cases "m \<ge> 0")
+    case True
+    with as have "{m *\<^sub>R a + c..m *\<^sub>R b + c} \<noteq> {}"
+      unfolding interval_ne_empty
+      apply (intro ballI)
+      apply (erule_tac x=i in ballE)
+      apply (auto simp: inner_simps intro!: mult_left_mono)
+      done
+    moreover from True have *: "\<And>i. (m *\<^sub>R b + c) \<bullet> i - (m *\<^sub>R a + c) \<bullet> i = m *\<^sub>R (b - a) \<bullet> i"
+      by (simp add: inner_simps field_simps)
+    ultimately show ?thesis
+      by (simp add: image_affinity_interval True content_closed_interval'
+                    setprod_timesf setprod_constant inner_diff_left)
+  next
+    case False
+    moreover with as have "{m *\<^sub>R b + c..m *\<^sub>R a + c} \<noteq> {}"
+      unfolding interval_ne_empty
+      apply (intro ballI)
+      apply (erule_tac x=i in ballE)
+      apply (auto simp: inner_simps intro!: mult_left_mono)
+      done
+    moreover from False have *: "\<And>i. (m *\<^sub>R a + c) \<bullet> i - (m *\<^sub>R b + c) \<bullet> i = (-m) *\<^sub>R (b - a) \<bullet> i"
+      by (simp add: inner_simps field_simps)
+    ultimately show ?thesis
+      by (simp add: image_affinity_interval content_closed_interval'
+                    setprod_timesf[symmetric] setprod_constant[symmetric] inner_diff_left)
+  qed
+qed
 
 lemma has_integral_affinity: fixes a::"'a::ordered_euclidean_space" assumes "(f has_integral i) {a..b}" "m \<noteq> 0"
   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
-  apply(rule has_integral_twiddle,safe) apply(rule zero_less_power) unfolding euclidean_eq[where 'a='a]
-  unfolding scaleR_right_distrib euclidean_simps scaleR_scaleR
+  apply(rule has_integral_twiddle,safe) apply(rule zero_less_power) unfolding euclidean_eq_iff[where 'a='a]
+  unfolding scaleR_right_distrib inner_simps scaleR_scaleR
   defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
   apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
 
@@ -3757,60 +3701,68 @@
 subsection {* Special case of stretching coordinate axes separately. *}
 
 lemma image_stretch_interval:
-  "(\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} =
-  (if {a..b} = {} then {} else {(\<chi>\<chi> k. min (m(k) * a$$k) (m(k) * b$$k))::'a ..  (\<chi>\<chi> k. max (m(k) * a$$k) (m(k) * b$$k))})"
-  (is "?l = ?r")
-proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
-next have *:"\<And>P Q. (\<forall>i<DIM('a). P i) \<and> (\<forall>i<DIM('a). Q i) \<longleftrightarrow> (\<forall>i<DIM('a). P i \<and> Q i)" by auto
-  case False note ab = this[unfolded interval_ne_empty]
-  show ?thesis apply-apply(rule set_eqI)
-  proof- fix x::"'a" have **:"\<And>P Q. (\<forall>i<DIM('a). P i = Q i) \<Longrightarrow> (\<forall>i<DIM('a). P i) = (\<forall>i<DIM('a). Q i)" by auto
-    show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
-      unfolding image_iff mem_interval Bex_def euclidean_simps euclidean_eq[where 'a='a] *
-      unfolding imp_conjR[THEN sym] apply(subst euclidean_lambda_beta'') apply(subst lambda_skolem'[THEN sym])
-      apply(rule **,rule,rule) unfolding euclidean_lambda_beta'
-    proof- fix i assume i:"i<DIM('a)" show "(\<exists>xa. (a $$ i \<le> xa \<and> xa \<le> b $$ i) \<and> x $$ i = m i * xa) =
-        (min (m i * a $$ i) (m i * b $$ i) \<le> x $$ i \<and> x $$ i \<le> max (m i * a $$ i) (m i * b $$ i))"
-      proof(cases "m i = 0") case True thus ?thesis using ab i by auto
-      next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
-        proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
-            "max (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab i unfolding min_def max_def by auto
-          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
-            using as by(auto simp add:field_simps)
-        next assume as:"0 > m i" hence *:"max (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
-            "min (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab as i unfolding min_def max_def 
-            by(auto simp add:field_simps mult_le_cancel_left_neg intro: order_antisym)
-          show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
-            using as by(auto simp add:field_simps) qed qed qed qed qed 
-
-lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
+  "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k)) *\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} =
+  (if {a..b} = {} then {} else
+    {(\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)::'a .. 
+     (\<Sum>k\<in>Basis. (max (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)})"
+proof cases
+  assume *: "{a..b} \<noteq> {}" 
+  show ?thesis
+    unfolding interval_ne_empty if_not_P[OF *]
+    apply (simp add: interval image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
+    apply (subst choice_Basis_iff[symmetric])
+  proof (intro allI ball_cong refl)
+    fix x i :: 'a assume "i \<in> Basis"
+    with * have a_le_b: "a \<bullet> i \<le> b \<bullet> i"
+      unfolding interval_ne_empty by auto
+    show "(\<exists>xa. x \<bullet> i = m i * xa \<and> a \<bullet> i \<le> xa \<and> xa \<le> b \<bullet> i) \<longleftrightarrow>
+        min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) \<le> x \<bullet> i \<and> x \<bullet> i \<le> max (m i * (a \<bullet> i)) (m i * (b \<bullet> i))"
+    proof cases
+      assume "m i \<noteq> 0"
+      moreover then have *: "\<And>a b. a = m i * b \<longleftrightarrow> b = a / m i"
+        by (auto simp add: field_simps)
+      moreover have
+          "min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (a \<bullet> i) else m i * (b \<bullet> i))"
+          "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (b \<bullet> i) else m i * (a \<bullet> i))"
+        using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
+      ultimately show ?thesis using a_le_b
+        unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps) 
+    qed (insert a_le_b, auto)
+  qed
+qed simp
+
+lemma interval_image_stretch_interval: 
+    "\<exists>u v. (\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
   unfolding image_stretch_interval by auto 
 
 lemma content_image_stretch_interval:
-  "content((\<lambda>x::'a::ordered_euclidean_space. (\<chi>\<chi> k. m k * x$$k)::'a) ` {a..b}) = abs(setprod m {..<DIM('a)}) * content({a..b})"
+  "content((\<lambda>x::'a::ordered_euclidean_space. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b}) = abs(setprod m Basis) * content({a..b})"
 proof(cases "{a..b} = {}") case True thus ?thesis
     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
-next case False hence "(\<lambda>x. (\<chi>\<chi> k. m k * x $$ k)::'a) ` {a..b} \<noteq> {}" by auto
+next case False hence "(\<lambda>x. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b} \<noteq> {}" by auto
   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
-    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff euclidean_lambda_beta'
-  proof- fix i assume i:"i<DIM('a)" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
-    thus "max (m i * a $$ i) (m i * b $$ i) - min (m i * a $$ i) (m i * b $$ i) = \<bar>m i\<bar> * (b $$ i - a $$ i)"
+    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff
+  proof (simp only: inner_setsum_left_Basis)
+    fix i :: 'a assume i:"i\<in>Basis" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
+    thus "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) - min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = 
+        \<bar>m i\<bar> * (b \<bullet> i - a \<bullet> i)"
       apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i 
       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
 
 lemma has_integral_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
-  assumes "(f has_integral i) {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
-  shows "((\<lambda>x. f(\<chi>\<chi> k. m k * x$$k)) has_integral
-             ((1/(abs(setprod m {..<DIM('a)}))) *\<^sub>R i)) ((\<lambda>x. (\<chi>\<chi> k. 1/(m k) * x$$k)::'a) ` {a..b})"
+  assumes "(f has_integral i) {a..b}" "\<forall>k\<in>Basis. ~(m k = 0)"
+  shows "((\<lambda>x. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) has_integral
+             ((1/(abs(setprod m Basis))) *\<^sub>R i)) ((\<lambda>x. (\<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k)) ` {a..b})"
   apply(rule has_integral_twiddle[where f=f]) unfolding zero_less_abs_iff content_image_stretch_interval
-  unfolding image_stretch_interval empty_as_interval euclidean_eq[where 'a='a] using assms
-proof- show "\<forall>y::'a. continuous (at y) (\<lambda>x. (\<chi>\<chi> k. m k * x $$ k)::'a)"
+  unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a] using assms
+proof- show "\<forall>y::'a. continuous (at y) (\<lambda>x. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k))"
    apply(rule,rule linear_continuous_at) unfolding linear_linear
-   unfolding linear_def euclidean_simps euclidean_eq[where 'a='a] by(auto simp add:field_simps) qed auto
+   unfolding linear_def inner_simps euclidean_eq_iff[where 'a='a] by(auto simp add:field_simps)
+qed auto
 
 lemma integrable_stretch:  fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
-  assumes "f integrable_on {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
-  shows "(\<lambda>x::'a. f(\<chi>\<chi> k. m k * x$$k)) integrable_on ((\<lambda>x. \<chi>\<chi> k. 1/(m k) * x$$k) ` {a..b})"
+  assumes "f integrable_on {a..b}" "\<forall>k\<in>Basis. ~(m k = 0)"
+  shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` {a..b})"
   using assms unfolding integrable_on_def apply-apply(erule exE) 
   apply(drule has_integral_stretch,assumption) by auto
 
@@ -3856,7 +3808,8 @@
     show ?thesis proof(cases,rule *,assumption)
       assume "\<not> a < b" hence "a = b" using assms(1) by auto
       hence *:"{a .. b} = {b}" "f b - f a = 0" by(auto simp add:  order_antisym)
-      show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0 using * `a=b` by auto
+      show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0 using * `a=b`
+        by (auto simp: ex_in_conv)
     qed } assume ab:"a < b"
   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
@@ -4106,11 +4059,14 @@
   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
 
   let ?d = "min k (c - a)/2" show ?thesis apply(rule that[of ?d])
-  proof safe show "?d > 0" using k(1) using assms(2) by auto
-    fix t assume as:"c - ?d < t" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
+  proof safe
+    show "?d > 0" using k(1) using assms(2) by auto
+    fix t assume as:"c - ?d < t" "t \<le> c"
+    let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
     { presume *:"t < c \<Longrightarrow> ?thesis"
       show ?thesis apply(cases "t = c") defer apply(rule *)
-        apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
+        apply(subst less_le) using `e>0` as(2) by auto }
+    assume "t < c"
 
     have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
     from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
@@ -4123,10 +4079,10 @@
     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
     
-    have *:"{a..c} \<inter> {x. x $$0 \<le> t} = {a..t}" "{a..c} \<inter> {x. x$$0 \<ge> t} = {t..c}"
+    have *:"{a..c} \<inter> {x. x \<bullet> 1 \<le> t} = {a..t}" "{a..c} \<inter> {x. x \<bullet> 1 \<ge> t} = {t..c}"
       using assms(2-3) as by(auto simp add:field_simps)
     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
-      apply(rule tagged_division_union_interval[of _ _ _ 0 "t"]) unfolding * apply(rule p)
+      apply(rule tagged_division_union_interval[of _ _ _ 1 "t"]) unfolding * apply(rule p)
       apply(rule tagged_division_of_self) unfolding fine_def
     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
@@ -4178,7 +4134,7 @@
   { presume *:"a<b \<Longrightarrow> ?thesis"
     show ?thesis apply(cases,rule *,assumption)
     proof- case goal1 hence "{a..b} = {x}" using as(1) apply-apply(rule set_eqI)
-        unfolding atLeastAtMost_iff by(auto simp only:field_simps not_less DIM_real)
+        unfolding atLeastAtMost_iff by(auto simp only:field_simps not_less)
       thus ?case using `e>0` by auto
     qed } assume "a<b"
   have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add:)
@@ -4355,65 +4311,41 @@
     qed(insert B `e>0`, auto)
   next assume as:"\<forall>e>0. ?r e" 
     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
-    def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
+    def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space" 
+    def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
-    proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
-        by(auto simp add:field_simps) qed
+    proof
+      case goal1 thus ?case using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def
+        by(auto simp add:field_simps setsum_negf)
+    qed
     have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
-    proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
+    proof
+      case goal1 thus ?case
+        using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def by (auto simp: setsum_negf)
+    qed
     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
     then guess y .. note y=this
 
     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
-      def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
+      def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space" 
+      def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
-      proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
-          by(auto simp add:field_simps) qed
+      proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def
+          by(auto simp add:field_simps setsum_negf) qed
       have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
-      proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
+      proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by (auto simp: setsum_negf) qed
       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
       thus False by auto qed
     thus ?l using y unfolding s by auto qed qed 
 
-(*lemma has_integral_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
-  "((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s"
-  unfolding has_integral'[unfolded has_integral] 
-proof case goal1 thus ?case apply safe
-  apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
-  apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
-  apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
-  apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
-  apply(subst(asm)(2) norm_vector_1) unfolding split_def
-  unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
-    Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
-  apply(subst(asm)(2) norm_vector_1) by auto
-next case goal2 thus ?case apply safe
-  apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
-  apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
-  apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
-  apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
-  apply(subst norm_vector_1) unfolding split_def
-  unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
-    Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
-  apply(subst norm_vector_1) by auto qed
-
-lemma integral_trans[simp]: assumes "(f::'n::ordered_euclidean_space \<Rightarrow> real) integrable_on s"
-  shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)"
-  apply(rule integral_unique) using assms by auto
-
-lemma integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
-  "(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)"
-  unfolding integrable_on_def
-  apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans
-  apply safe defer apply(rule_tac x="vec1 y" in exI) by auto *)
-
 lemma has_integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x) \<le> (g x)"
-  shows "i \<le> j" using has_integral_component_le[OF assms(1-2), of 0] using assms(3) by auto
+  shows "i \<le> j"
+  using has_integral_component_le[OF _ assms(1-2), of 1] using assms(3) by auto
 
 lemma integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
@@ -4422,7 +4354,7 @@
 
 lemma has_integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" 
-  using has_integral_component_nonneg[of "f" "i" s 0]
+  using has_integral_component_nonneg[of 1 f i s]
   unfolding o_def using assms by auto
 
 lemma integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
@@ -4519,19 +4451,20 @@
 subsection {* More lemmas that are useful later. *}
 
 lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
-  assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)$$k"
-  shows "i$$k \<le> j$$k"
+  assumes k: "k\<in>Basis" and as: "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\<bullet>k"
+  shows "i\<bullet>k \<le> j\<bullet>k"
 proof- note has_integral_restrict_univ[THEN sym, of f]
-  note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
-  show ?thesis apply(rule *) using assms(1,4) by auto qed
+  note as(2-3)[unfolded this] note * = has_integral_component_le[OF k this]
+  show ?thesis apply(rule *) using as(1,4) by auto qed
 
 lemma has_integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
-  assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
-  shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "f" "i" "j" 0] using assms by auto
+  assumes as: "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
+  shows "i \<le> j"
+  using has_integral_subset_component_le[OF _ assms(1), of 1 f i j] using assms by auto
 
 lemma integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
-  assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)$$k"
-  shows "(integral s f)$$k \<le> (integral t f)$$k"
+  assumes "k\<in>Basis" "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)\<bullet>k"
+  shows "(integral s f)\<bullet>k \<le> (integral t f)\<bullet>k"
   apply(rule has_integral_subset_component_le) using assms by auto
 
 lemma integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
@@ -4553,13 +4486,13 @@
   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
   show ?r proof safe fix a b::"'n::ordered_euclidean_space"
     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
-    let ?a = "(\<chi>\<chi> i. min (a$$i) (-B))::'n::ordered_euclidean_space" and ?b = "(\<chi>\<chi> i. max (b$$i) B)::'n::ordered_euclidean_space"
+    let ?a = "\<Sum>i\<in>Basis. min (a\<bullet>i) (-B) *\<^sub>R i::'n" and ?b = "\<Sum>i\<in>Basis. max (b\<bullet>i) B *\<^sub>R i::'n"
     show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
     proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval dist_norm
-      proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
+      proof case goal1 thus ?case using Basis_le_norm[of i x] by(auto simp add:field_simps) qed
       from B(2)[OF this] guess z .. note conjunct1[OF this]
       thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
-      show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
+      show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in ballE) by auto qed
 
     fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
@@ -4584,14 +4517,15 @@
         using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
         
 next assume ?r note as = conjunctD2[OF this,rule_format]
-  have "Cauchy (\<lambda>n. integral ({(\<chi>\<chi> i. - real n)::'n .. (\<chi>\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))"
+  let ?cube = "\<lambda>n. {(\<Sum>i\<in>Basis. - real n *\<^sub>R i)::'n .. (\<Sum>i\<in>Basis. real n *\<^sub>R i)} :: 'n set"
+  have "Cauchy (\<lambda>n. integral (?cube n) (\<lambda>x. if x \<in> s then f x else 0))"
   proof(unfold Cauchy_def,safe) case goal1
     from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
     from real_arch_simple[of B] guess N .. note N = this
-    { fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {(\<chi>\<chi> i. - real n)::'n..\<chi>\<chi> i. real n}" apply safe
+    { fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> ?cube n" apply safe
         unfolding mem_ball mem_interval dist_norm
-      proof case goal1 thus ?case using component_le_norm[of x i]
-          using n N by(auto simp add:field_simps) qed }
+      proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
+          using n N by(auto simp add:field_simps setsum_negf) qed }
     thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto
   qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
   note i = this[THEN LIMSEQ_D]
@@ -4611,9 +4545,9 @@
       proof safe show "N \<le> n" using n by auto
         fix x::"'n::ordered_euclidean_space" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
         thus "x\<in>{a..b}" using ab by blast 
-        show "x\<in>{\<chi>\<chi> i. - real n..\<chi>\<chi> i. real n}" using x unfolding mem_interval mem_ball dist_norm apply-
-        proof case goal1 thus ?case using component_le_norm[of x i]
-            using n by(auto simp add:field_simps) qed qed qed qed qed
+        show "x\<in>?cube n" using x unfolding mem_interval mem_ball dist_norm apply-
+        proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
+            using n by(auto simp add:field_simps setsum_negf) qed qed qed qed qed
 
 lemma integrable_altD: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on s"
@@ -4676,11 +4610,13 @@
     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
-    def c \<equiv> "(\<chi>\<chi> i. min (a$$i) (- (max B1 B2)))::'n" and d \<equiv> "(\<chi>\<chi> i. max (b$$i) (max B1 B2))::'n"
-    have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval dist_norm
-    proof(rule_tac[!] allI)
-      case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next
-      case goal2 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
+    def c \<equiv> "\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max B1 B2)) *\<^sub>R i::'n"
+    def d \<equiv> "\<Sum>i\<in>Basis. max (b\<bullet>i) (max B1 B2) *\<^sub>R i::'n"
+    have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}"
+      apply safe unfolding mem_ball mem_interval dist_norm
+    proof(rule_tac[!] ballI)
+      case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by auto next
+      case goal2 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by auto qed
     have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow>
       norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e"
       using obt(3) unfolding real_norm_def by arith 
@@ -4700,7 +4636,7 @@
         unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
         apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+
       proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
-          apply - apply rule apply(erule_tac x=i in allE) by auto
+          apply - apply rule apply(erule_tac x=i in ballE) by auto
       qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this
 
   show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv)
@@ -5019,7 +4955,7 @@
 proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
   show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using tendsto_const by auto
 next assume ab:"content {a..b} \<noteq> 0"
-  have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) $$ 0 \<le> (g x) $$ 0"
+  have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) \<bullet> 1 \<le> (g x) \<bullet> 1"
   proof safe case goal1 note assms(3)[rule_format,OF this]
     note * = Lim_component_ge[OF this trivial_limit_sequentially]
     show ?case apply(rule *) unfolding eventually_sequentially
@@ -5030,10 +4966,10 @@
     apply rule apply(rule integral_le) apply safe
     apply(rule assms(1-2)[rule_format])+ using assms(4) by auto
   then guess i .. note i=this
-  have i':"\<And>k. (integral({a..b}) (f k)) \<le> i$$0"
+  have i':"\<And>k. (integral({a..b}) (f k)) \<le> i\<bullet>1"
     apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially)
     unfolding eventually_sequentially apply(rule_tac x=k in exI)
-    apply(rule transitive_stepwise_le) prefer 3 unfolding Eucl_real_simps apply(rule integral_le)
+    apply(rule transitive_stepwise_le) prefer 3 unfolding inner_simps real_inner_1_right apply(rule integral_le)
     apply(rule assms(1-2)[rule_format])+ using assms(2) by auto
 
   have "(g has_integral i) {a..b}" unfolding has_integral
@@ -5044,7 +4980,7 @@
       apply(rule divide_pos_pos) by auto
     from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
 
-    have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i$$0 - (integral {a..b} (f k)) \<and> i$$0 - (integral {a..b} (f k)) < e / 4"
+    have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i\<bullet>1 - (integral {a..b} (f k)) \<and> i\<bullet>1 - (integral {a..b} (f k)) < e / 4"
     proof- case goal1 have "e/4 > 0" using e by auto
       from LIMSEQ_D [OF i this] guess r ..
       thus ?case apply(rule_tac x=r in exI) apply rule
@@ -5052,14 +4988,15 @@
       proof- case goal1 thus ?case using i'[of k] by auto qed qed
     then guess r .. note r=conjunctD2[OF this[rule_format]]
 
-    have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)$$0 - (f k x)$$0 \<and>
-           (g x)$$0 - (f k x)$$0 < e / (4 * content({a..b}))"
+    have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)\<bullet>1 - (f k x)\<bullet>1 \<and>
+           (g x)\<bullet>1 - (f k x)\<bullet>1 < e / (4 * content({a..b}))"
     proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact)
         using ab content_pos_le[of a b] by auto
       from assms(3)[rule_format, OF goal1, THEN LIMSEQ_D, OF this]
       guess n .. note n=this
       thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE)
-        unfolding dist_real_def using fg[rule_format,OF goal1] by(auto simp add:field_simps) qed
+        unfolding dist_real_def using fg[rule_format,OF goal1]
+        by (auto simp add:field_simps) qed
     from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
     def d \<equiv> "\<lambda>x. c (m x) x" 
 
@@ -5118,14 +5055,14 @@
 
        next case goal3
          note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
-         have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i$$0 - kr$$0
-           \<and> i$$0 - kr$$0 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto 
+         have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i\<bullet>1 - kr\<bullet>1
+           \<and> i\<bullet>1 - kr\<bullet>1 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto 
          show ?case unfolding real_norm_def apply(rule *[rule_format],safe)
-           apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded Eucl_real_simps]) 
+           apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded real_inner_1_right]) 
            apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
            apply(rule_tac[1-2] integral_le[OF ])
-         proof safe show "0 \<le> i$$0 - (integral {a..b} (f r))$$0" using r(1) by auto
-           show "i$$0 - (integral {a..b} (f r))$$0 < e / 4" using r(2) by auto
+         proof safe show "0 \<le> i\<bullet>1 - (integral {a..b} (f r))\<bullet>1" using r(1) by auto
+           show "i\<bullet>1 - (integral {a..b} (f r))\<bullet>1 < e / 4" using r(2) by auto
            fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
            show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k" 
              unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
@@ -5147,7 +5084,7 @@
     \<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x) \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially  \<Longrightarrow>
     bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
   proof- case goal1 note assms=this[rule_format]
-    have "\<forall>x\<in>s. \<forall>k. (f k x)$$0 \<le> (g x)$$0" apply safe apply(rule Lim_component_ge)
+    have "\<forall>x\<in>s. \<forall>k. (f k x)\<bullet>1 \<le> (g x)\<bullet>1" apply safe apply(rule Lim_component_ge)
       apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially)
       unfolding eventually_sequentially apply(rule_tac x=k in exI)
       apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format]
@@ -5156,9 +5093,10 @@
       apply(rule goal1(5)) apply(rule,rule integral_le) apply(rule goal1(2)[rule_format])+
       using goal1(3) by auto then guess i .. note i=this
     have "\<And>k. \<forall>x\<in>s. \<forall>n\<ge>k. f k x \<le> f n x" apply(rule,rule transitive_stepwise_le) using goal1(3) by auto
-    hence i':"\<forall>k. (integral s (f k))$$0 \<le> i$$0" apply-apply(rule,rule Lim_component_ge)
+    hence i':"\<forall>k. (integral s (f k))\<bullet>1 \<le> i\<bullet>1" apply-apply(rule,rule Lim_component_ge)
       apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
       apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
+      apply simp
       apply(rule goal1(2)[rule_format])+ by auto 
 
     note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
@@ -5199,14 +5137,14 @@
           apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N]
           apply-defer apply(subst norm_minus_commute) by auto
         have *:"\<And>f1 f2 g. abs(f1 - i) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i
-          \<longrightarrow> abs(g - i) < e" unfolding Eucl_real_simps by arith
+          \<longrightarrow> abs(g - i) < e" unfolding real_inner_1_right by arith
         show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e" 
           unfolding real_norm_def apply(rule *[rule_format])
           apply(rule **[unfolded real_norm_def])
           apply(rule M[rule_format,of "M + N",unfolded real_norm_def]) apply(rule le_add1)
           apply(rule integral_le[OF int int]) defer
-          apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded Eucl_real_simps]])
-        proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)$$0 \<le> (f n x)$$0"
+          apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded real_inner_1_right]])
+        proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)\<bullet>1 \<le> (f n x)\<bullet>1"
             apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto
         next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int]) 
             unfolding ifif integral_restrict_univ[OF int']
@@ -5323,9 +5261,9 @@
 
 lemma integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   fixes g::"'n => 'b::ordered_euclidean_space"
-  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
-  shows "norm(integral s f) \<le> (integral s g)$$k"
-proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x $$ k) o g)"
+  assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. norm(f x) \<le> (g x)\<bullet>k"
+  shows "norm(integral s f) \<le> (integral s g)\<bullet>k"
+proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x \<bullet> k) o g)"
     apply(rule integral_norm_bound_integral[OF assms(1)])
     apply(rule integrable_linear[OF assms(2)],rule)
     unfolding o_def by(rule assms)
@@ -5333,8 +5271,8 @@
 
 lemma has_integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   fixes g::"'n => 'b::ordered_euclidean_space"
-  assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
-  shows "norm(i) \<le> j$$k" using integral_norm_bound_integral_component[of f s g k]
+  assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)\<bullet>k"
+  shows "norm(i) \<le> j\<bullet>k" using integral_norm_bound_integral_component[of f s g k]
   unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
   using assms by auto
 
@@ -5742,97 +5680,126 @@
   shows "(\<lambda>x. setsum (\<lambda>a. f a x) t) absolutely_integrable_on s"
   using assms(1,2) apply induct defer apply(subst setsum.insert) apply assumption+ by(rule,auto)
 
-lemma absolutely_integrable_vector_abs: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "f absolutely_integrable_on s"
-  shows "(\<lambda>x. (\<chi>\<chi> i. abs(f x$$i))::'c::ordered_euclidean_space) absolutely_integrable_on s"
-proof- have *:"\<And>x. ((\<chi>\<chi> i. abs(f x$$i))::'c::ordered_euclidean_space) = (setsum (\<lambda>i.
-    (((\<lambda>y. (\<chi>\<chi> j. if j = i then y else 0)) o
-    (((\<lambda>x. (norm((\<chi>\<chi> j. if j = i then x$$i else 0)::'c::ordered_euclidean_space))) o f))) x)) {..<DIM('c)})"
-    unfolding euclidean_eq[where 'a='c] euclidean_component_setsum apply safe
-    unfolding euclidean_lambda_beta'
-  proof- case goal1 have *:"\<And>i xa. ((if i = xa then f x $$ xa else 0) * (if i = xa then f x $$ xa else 0)) =
-      (if i = xa then (f x $$ xa) * (f x $$ xa) else 0)" by auto
-    have *:"\<And>xa. norm ((\<chi>\<chi> j. if j = xa then f x $$ xa else 0)::'c) = (if xa<DIM('c) then abs (f x $$ xa) else 0)"
-      unfolding norm_eq_sqrt_inner euclidean_inner[where 'a='c]
-      by(auto simp add:setsum_delta[OF finite_lessThan] *)
-    have "\<bar>f x $$ i\<bar> = (setsum (\<lambda>k. if k = i then abs ((f x)$$i) else 0) {..<DIM('c)})"
-      unfolding setsum_delta[OF finite_lessThan] using goal1 by auto
-    also have "... = (\<Sum>xa<DIM('c). ((\<lambda>y. (\<chi>\<chi> j. if j = xa then y else 0)::'c) \<circ>
-                      (\<lambda>x. (norm ((\<chi>\<chi> j. if j = xa then x $$ xa else 0)::'c))) \<circ> f) x $$ i)"
-      unfolding o_def * apply(rule setsum_cong2)
-      unfolding euclidean_lambda_beta'[OF goal1 ] by auto
-    finally show ?case unfolding o_def . qed
-  show ?thesis unfolding * apply(rule absolutely_integrable_setsum) apply(rule finite_lessThan)
-    apply(rule absolutely_integrable_linear) unfolding o_def apply(rule absolutely_integrable_norm)
-    apply(rule absolutely_integrable_linear[OF assms,unfolded o_def]) unfolding linear_linear
-    apply(rule_tac[!] linearI) unfolding euclidean_eq[where 'a='c]
-    by(auto simp:euclidean_component_scaleR[where 'a=real,unfolded real_scaleR_def])
+lemma bounded_linear_setsum:
+  fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+  assumes f: "\<And>i. i\<in>I \<Longrightarrow> bounded_linear (f i)"
+  shows "bounded_linear (\<lambda>x. \<Sum>i\<in>I. f i x)"
+proof cases
+  assume "finite I" from this f show ?thesis
+    by (induct I) (auto intro!: bounded_linear_add bounded_linear_zero)
+qed (simp add: bounded_linear_zero)
+
+lemma absolutely_integrable_vector_abs:
+  fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
+  fixes T :: "'c::ordered_euclidean_space \<Rightarrow> 'b"
+  assumes f: "f absolutely_integrable_on s"
+  shows "(\<lambda>x. (\<Sum>i\<in>Basis. abs(f x\<bullet>T i) *\<^sub>R i)) absolutely_integrable_on s"
+    (is "?Tf absolutely_integrable_on s")
+proof -
+  have if_distrib: "\<And>P A B x. (if P then A else B) *\<^sub>R x = (if P then A *\<^sub>R x else B *\<^sub>R x)"
+    by simp
+  have *: "\<And>x. ?Tf x = (\<Sum>i\<in>Basis.
+    ((\<lambda>y. (\<Sum>j\<in>Basis. (if j = i then y else 0) *\<^sub>R j)) o
+     (\<lambda>x. (norm (\<Sum>j\<in>Basis. (if j = i then f x\<bullet>T i else 0) *\<^sub>R j)))) x)"
+    by (simp add: comp_def if_distrib setsum_cases)
+  show ?thesis
+    unfolding *
+    apply (rule absolutely_integrable_setsum[OF finite_Basis])
+    apply (rule absolutely_integrable_linear)
+    apply (rule absolutely_integrable_norm)
+    apply (rule absolutely_integrable_linear[OF f, unfolded o_def])
+    apply (auto simp: linear_linear euclidean_eq_iff[where 'a='c] inner_simps intro!: linearI)
+    done
 qed
 
-lemma absolutely_integrable_max: fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
+lemma absolutely_integrable_max:
+  fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
   assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
-  shows "(\<lambda>x. (\<chi>\<chi> i. max (f(x)$$i) (g(x)$$i))::'n::ordered_euclidean_space) absolutely_integrable_on s"
-proof- have *:"\<And>x. (1 / 2) *\<^sub>R (((\<chi>\<chi> i. \<bar>(f x - g x) $$ i\<bar>)::'n) + (f x + g x)) = (\<chi>\<chi> i. max (f(x)$$i) (g(x)$$i))"
-    unfolding euclidean_eq[where 'a='n] by auto
+  shows "(\<lambda>x. (\<Sum>i\<in>Basis. max (f(x)\<bullet>i) (g(x)\<bullet>i) *\<^sub>R i)::'n) absolutely_integrable_on s"
+proof -
+  have *:"\<And>x. (1 / 2) *\<^sub>R (((\<Sum>i\<in>Basis. \<bar>(f x - g x) \<bullet> i\<bar> *\<^sub>R i)::'n) + (f x + g x)) =
+      (\<Sum>i\<in>Basis. max (f(x)\<bullet>i) (g(x)\<bullet>i) *\<^sub>R i)"
+    unfolding euclidean_eq_iff[where 'a='n] by (auto simp: inner_simps)
   note absolutely_integrable_sub[OF assms] absolutely_integrable_add[OF assms]
-  note absolutely_integrable_vector_abs[OF this(1)] this(2)
-  note absolutely_integrable_add[OF this] note absolutely_integrable_cmul[OF this,of "1/2"]
-  thus ?thesis unfolding * . qed
-
-lemma absolutely_integrable_min: fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
+  note absolutely_integrable_vector_abs[OF this(1), where T="\<lambda>x. x"] this(2)
+  note absolutely_integrable_add[OF this]
+  note absolutely_integrable_cmul[OF this, of "1/2"]
+  thus ?thesis unfolding * .
+qed
+
+lemma absolutely_integrable_min:
+  fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
   assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
-  shows "(\<lambda>x. (\<chi>\<chi> i. min (f(x)$$i) (g(x)$$i))::'n::ordered_euclidean_space) absolutely_integrable_on s"
-proof- have *:"\<And>x. (1 / 2) *\<^sub>R ((f x + g x) - ((\<chi>\<chi> i. \<bar>(f x - g x) $$ i\<bar>)::'n)) = (\<chi>\<chi> i. min (f(x)$$i) (g(x)$$i))"
-    unfolding euclidean_eq[where 'a='n] by auto
+  shows "(\<lambda>x. (\<Sum>i\<in>Basis. min (f(x)\<bullet>i) (g(x)\<bullet>i) *\<^sub>R i)::'n) absolutely_integrable_on s"
+proof -
+  have *:"\<And>x. (1 / 2) *\<^sub>R ((f x + g x) - (\<Sum>i\<in>Basis. \<bar>(f x - g x) \<bullet> i\<bar> *\<^sub>R i::'n)) =
+      (\<Sum>i\<in>Basis. min (f(x)\<bullet>i) (g(x)\<bullet>i) *\<^sub>R i)"
+    unfolding euclidean_eq_iff[where 'a='n] by (auto simp: inner_simps)
+
   note absolutely_integrable_add[OF assms] absolutely_integrable_sub[OF assms]
-  note this(1) absolutely_integrable_vector_abs[OF this(2)]
-  note absolutely_integrable_sub[OF this] note absolutely_integrable_cmul[OF this,of "1/2"]
-  thus ?thesis unfolding * . qed
-
-lemma absolutely_integrable_abs_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
+  note this(1) absolutely_integrable_vector_abs[OF this(2), where T="\<lambda>x. x"]
+  note absolutely_integrable_sub[OF this]
+  note absolutely_integrable_cmul[OF this,of "1/2"]
+  thus ?thesis unfolding * .
+qed
+
+lemma absolutely_integrable_abs_eq:
+  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
   shows "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and>
-          (\<lambda>x. (\<chi>\<chi> i. abs(f x$$i))::'m) integrable_on s" (is "?l = ?r")
-proof assume ?l thus ?r apply-apply rule defer
+          (\<lambda>x. (\<Sum>i\<in>Basis. abs(f x\<bullet>i) *\<^sub>R i)::'m) integrable_on s" (is "?l = ?r")
+proof
+  assume ?l thus ?r apply-apply rule defer
     apply(drule absolutely_integrable_vector_abs) by auto
-next assume ?r { presume lem:"\<And>f::'n \<Rightarrow> 'm. f integrable_on UNIV \<Longrightarrow>
-    (\<lambda>x. (\<chi>\<chi> i. abs(f(x)$$i))::'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
-    have *:"\<And>x. (\<chi>\<chi> i. \<bar>(if x \<in> s then f x else 0) $$ i\<bar>) = (if x\<in>s then (\<chi>\<chi> i. \<bar>f x $$ i\<bar>) else (0::'m))"
-      unfolding euclidean_eq[where 'a='m] by auto
+next 
+  assume ?r
+  { presume lem:"\<And>f::'n \<Rightarrow> 'm. f integrable_on UNIV \<Longrightarrow>
+      (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
+    have *:"\<And>x. (\<Sum>i\<in>Basis. \<bar>(if x \<in> s then f x else 0) \<bullet> i\<bar> *\<^sub>R i) =
+        (if x\<in>s then (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) else (0::'m))"
+      unfolding euclidean_eq_iff[where 'a='m] by auto
     show ?l apply(subst absolutely_integrable_restrict_univ[THEN sym]) apply(rule lem)
       unfolding integrable_restrict_univ * using `?r` by auto }
-  fix f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assume assms:"f integrable_on UNIV"
-    "(\<lambda>x. (\<chi>\<chi> i. abs(f(x)$$i))::'m::ordered_euclidean_space) integrable_on UNIV"
-  let ?B = "setsum (\<lambda>i. integral UNIV (\<lambda>x. (\<chi>\<chi> j. abs(f x$$j)) ::'m::ordered_euclidean_space) $$ i) {..<DIM('m)}"
+  fix f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
+  assume assms:"f integrable_on UNIV" "(\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV"
+  let ?B = "\<Sum>i\<in>Basis. integral UNIV (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) \<bullet> i"
   show "f absolutely_integrable_on UNIV"
     apply(rule bounded_variation_absolutely_integrable[OF assms(1), where B="?B"],safe)
   proof- case goal1 note d=this and d'=division_ofD[OF this]
     have "(\<Sum>k\<in>d. norm (integral k f)) \<le>
-      (\<Sum>k\<in>d. setsum (op $$ (integral k (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m))) {..<DIM('m)})" apply(rule setsum_mono)
+      (\<Sum>k\<in>d. setsum (op \<bullet> (integral k (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m))) Basis)"
+      apply(rule setsum_mono)
       apply(rule order_trans[OF norm_le_l1]) apply(rule setsum_mono) unfolding lessThan_iff
-    proof- fix k and i assume "k\<in>d" and i:"i<DIM('m)"
+    proof- fix k and i :: 'm assume "k\<in>d" and i:"i\<in>Basis"
       from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this
-      show "\<bar>integral k f $$ i\<bar> \<le> integral k (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ i" apply(rule abs_leI)
-        unfolding euclidean_component_minus[THEN sym] defer apply(subst integral_neg[THEN sym])
-        defer apply(rule_tac[1-2] integral_component_le) apply(rule integrable_neg)
+      show "\<bar>integral k f \<bullet> i\<bar> \<le> integral k (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) \<bullet> i"
+        apply (rule abs_leI)
+        unfolding inner_minus_left[THEN sym] defer apply(subst integral_neg[THEN sym])
+        defer apply(rule_tac[1-2] integral_component_le[OF i]) apply(rule integrable_neg)
         using integrable_on_subinterval[OF assms(1),of a b]
-          integrable_on_subinterval[OF assms(2),of a b] unfolding ab by auto
-    qed also have "... \<le> setsum (op $$ (integral UNIV (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>))::'m)) {..<DIM('m)}"
+          integrable_on_subinterval[OF assms(2),of a b] i unfolding ab by auto
+    qed also have "... \<le> setsum (op \<bullet> (integral UNIV (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m))) Basis"
       apply(subst setsum_commute,rule setsum_mono)
-    proof- case goal1 have *:"(\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) integrable_on \<Union>d"
+    proof- case goal1 have *:"(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i::'m) integrable_on \<Union>d"
         using integrable_on_subdivision[OF d assms(2)] by auto
-      have "(\<Sum>i\<in>d. integral i (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ j)
-        = integral (\<Union>d) (\<lambda>x. (\<chi>\<chi> j. abs(f x$$j)) ::'m::ordered_euclidean_space) $$ j"
-        unfolding euclidean_component_setsum[THEN sym] integral_combine_division_topdown[OF * d] ..
-      also have "... \<le> integral UNIV (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ j"
-        apply(rule integral_subset_component_le) using assms * by auto
+      have "(\<Sum>i\<in>d. integral i (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i::'m) \<bullet> j)
+        = integral (\<Union>d) (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i::'m) \<bullet> j"
+        unfolding inner_setsum_left[symmetric] integral_combine_division_topdown[OF * d] ..
+      also have "... \<le> integral UNIV (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i::'m) \<bullet> j"
+        apply(rule integral_subset_component_le) using assms * `j\<in>Basis` by auto
       finally show ?case .
     qed finally show ?case . qed qed
 
-lemma nonnegative_absolutely_integrable: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
-  assumes "\<forall>x\<in>s. \<forall>i<DIM('m). 0 \<le> f(x)$$i" "f integrable_on s"
+lemma nonnegative_absolutely_integrable:
+  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
+  assumes "\<forall>x\<in>s. \<forall>i\<in>Basis. 0 \<le> f(x)\<bullet>i" "f integrable_on s"
   shows "f absolutely_integrable_on s"
-  unfolding absolutely_integrable_abs_eq apply rule defer
-  apply(rule integrable_eq[of _ f]) using assms apply-apply(subst euclidean_eq) by auto
+  unfolding absolutely_integrable_abs_eq
+  apply rule
+  apply (rule assms)
+  apply (rule integrable_eq[of _ f])
+  using assms
+  apply (auto simp: euclidean_eq_iff[where 'a='m])
+  done
 
 lemma absolutely_integrable_integrable_bound: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
   assumes "\<forall>x\<in>s. norm(f x) \<le> g x" "f integrable_on s" "g integrable_on s"
@@ -5881,8 +5848,8 @@
     apply(subst Inf_insert_finite) apply(rule finite_imageI[OF insert(1)])
   proof(cases "k={}") case True
     thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto
-  next case False thus ?P apply(subst if_not_P) defer      
-      apply(rule absolutely_integrable_min[where 'n=real,unfolded Eucl_real_simps])
+  next case False thus ?P apply(subst if_not_P) defer
+      apply (rule absolutely_integrable_min[where 'n=real, unfolded Basis_real_def, simplified])
       defer apply(rule insert(3)[OF False]) using insert(5) by auto
   qed qed auto
 
@@ -5897,7 +5864,7 @@
   proof(cases "k={}") case True
     thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto
   next case False thus ?P apply(subst if_not_P) defer
-      apply(rule absolutely_integrable_max[where 'n=real,unfolded Eucl_real_simps]) 
+      apply (rule absolutely_integrable_max[where 'n=real, unfolded Basis_real_def, simplified])
       defer apply(rule insert(3)[OF False]) using insert(5) by auto
   qed qed auto
 
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -398,6 +398,53 @@
   then show "h = g" by (simp add: ext)
 qed
 
+text {* TODO: The following lemmas about adjoints should hold for any
+Hilbert space (i.e. complete inner product space).
+(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})
+*}
+
+lemma adjoint_works:
+  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+  assumes lf: "linear f"
+  shows "x \<bullet> adjoint f y = f x \<bullet> y"
+proof -
+  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
+  proof (intro allI exI)
+    fix y :: "'m" and x
+    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
+    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
+      by (simp add: euclidean_representation)
+    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
+      unfolding linear_setsum[OF lf finite_Basis]
+      by (simp add: linear_cmul[OF lf])
+    finally show "f x \<bullet> y = x \<bullet> ?w"
+        by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+  qed
+  then show ?thesis
+    unfolding adjoint_def choice_iff
+    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
+qed
+
+lemma adjoint_clauses:
+  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+  assumes lf: "linear f"
+  shows "x \<bullet> adjoint f y = f x \<bullet> y"
+    and "adjoint f y \<bullet> x = y \<bullet> f x"
+  by (simp_all add: adjoint_works[OF lf] inner_commute)
+
+lemma adjoint_linear:
+  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+  assumes lf: "linear f"
+  shows "linear (adjoint f)"
+  by (simp add: lf linear_def euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
+    adjoint_clauses[OF lf] inner_simps)
+
+lemma adjoint_adjoint:
+  fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+  assumes lf: "linear f"
+  shows "adjoint (adjoint f) = f"
+  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
+
 subsection {* Interlude: Some properties of real sets *}
 
 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
@@ -1261,77 +1308,37 @@
 
 subsection{* Euclidean Spaces as Typeclass*}
 
-lemma independent_eq_inj_on:
-  fixes D :: nat
-    and f :: "nat \<Rightarrow> 'c::real_vector"
-  assumes "inj_on f {..<D}"
-  shows "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"
-proof -
-  from assms have eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"
-    and inj: "\<And>i. inj_on f ({..<D} - {i})"
-    by (auto simp: inj_on_def)
-  have *: "\<And>i. finite (f ` {..<D} - {i})" by simp
-  show ?thesis unfolding dependent_def span_finite[OF *]
-    by (auto simp: eq setsum_reindex[OF inj])
-qed
-
-lemma independent_basis: "independent (basis ` {..<DIM('a)} :: 'a::euclidean_space set)"
-  unfolding independent_eq_inj_on [OF basis_inj]
+lemma independent_Basis: "independent Basis"
+  unfolding dependent_def
+  apply (subst span_finite)
+  apply simp
   apply clarify
-  apply (drule_tac f="inner (basis a)" in arg_cong)
-  apply (simp add: inner_setsum_right dot_basis)
+  apply (drule_tac f="inner a" in arg_cong)
+  apply (simp add: inner_Basis inner_setsum_right eq_commute)
+  done
+
+lemma span_Basis[simp]: "span Basis = (UNIV :: 'a::euclidean_space set)"
+  apply (subst span_finite)
+  apply simp
+  apply (safe intro!: UNIV_I)
+  apply (rule_tac x="inner x" in exI)
+  apply (simp add: euclidean_representation)
   done
 
-lemma (in euclidean_space) range_basis: "range basis = insert 0 (basis ` {..<DIM('a)})"
-proof -
-  have *: "UNIV = {..<DIM('a)} \<union> {DIM('a)..}" by auto
-  show ?thesis unfolding * image_Un basis_finite by auto
-qed
-
-lemma (in euclidean_space) range_basis_finite[intro]: "finite (range basis)"
-  unfolding range_basis by auto
-
-lemma span_basis: "span (range basis) = (UNIV :: 'a::euclidean_space set)"
-proof -
-  { fix x :: 'a
-    have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
-      by (simp add: span_setsum span_mul span_superset)
-    then have "x \<in> span (range basis)"
-      by (simp only: euclidean_representation [symmetric])
-  } then show ?thesis by auto
-qed
-
-lemma basis_representation:
-  "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
-proof -
-  have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
-  have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
-    unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
-  then show ?thesis by fastforce
-qed
-
-lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
-  apply(subst span_basis[symmetric])
-  unfolding range_basis
-  apply auto
-  done
-
-lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
-  apply (subst card_image)
-  using basis_inj apply auto
-  done
-
-lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
-  unfolding span_basis' ..
-
-lemma norm_bound_component_le: "norm (x::'a::euclidean_space) \<le> e \<Longrightarrow> \<bar>x$$i\<bar> <= e"
-  by (metis component_le_norm order_trans)
-
-lemma norm_bound_component_lt: "norm (x::'a::euclidean_space) < e \<Longrightarrow> \<bar>x$$i\<bar> < e"
-  by (metis component_le_norm basic_trans_rules(21))
-
-lemma norm_le_l1: "norm (x::'a::euclidean_space) \<le> (\<Sum>i<DIM('a). \<bar>x $$ i\<bar>)"
-  apply (subst euclidean_representation[of x])
+lemma in_span_Basis: "x \<in> span Basis"
+  unfolding span_Basis ..
+
+lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
+  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
+
+lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
+  by (metis Basis_le_norm order_trans)
+
+lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
+  by (metis Basis_le_norm basic_trans_rules(21))
+
+lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
+  apply (subst euclidean_representation[of x, symmetric])
   apply (rule order_trans[OF norm_setsum])
   apply (auto intro!: setsum_mono)
   done
@@ -1339,61 +1346,29 @@
 lemma setsum_norm_allsubsets_bound:
   fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
-  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real DIM('n) *  e"
+  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
 proof -
-  let ?d = "real DIM('n)"
-  let ?nf = "\<lambda>x. norm (f x)"
-  let ?U = "{..<DIM('n)}"
-  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P) ?U"
+  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
+    by (rule setsum_mono) (rule norm_le_l1)
+  also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
     by (rule setsum_commute)
-  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
-  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P"
-    by (rule setsum_mono) (rule norm_le_l1)
-  also have "\<dots> \<le> 2 * ?d * e"
-    unfolding th0 th1
+  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
   proof (rule setsum_bounded)
-    fix i assume i: "i \<in> ?U"
-    let ?Pp = "{x. x\<in> P \<and> f x $$ i \<ge> 0}"
-    let ?Pn = "{x. x \<in> P \<and> f x $$ i < 0}"
-    have thp: "P = ?Pp \<union> ?Pn" by auto
-    have thp0: "?Pp \<inter> ?Pn ={}" by auto
-    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
-    have Ppe:"setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pp \<le> e"
-      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
-      by(auto intro: abs_le_D1)
-    have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
-      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
-      by(auto simp add: setsum_negf intro: abs_le_D1)
-    have "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn"
-      apply (subst thp)
-      apply (rule setsum_Un_zero)
-      using fP thp0 apply auto
-      done
-    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
-    finally show "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P \<le> 2*e" .
+    fix i :: 'n assume i: "i \<in> Basis"
+    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 
+      norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
+      by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf uminus_add_conv_diff
+                    norm_triangle_ineq4 inner_setsum_left
+          del: real_norm_def)
+    also have "\<dots> \<le> e + e" unfolding real_norm_def
+      by (intro add_mono norm_bound_Basis_le i fPs) auto
+    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
   qed
+  also have "\<dots> = 2 * real DIM('n) * e"
+    by (simp add: real_of_nat_def)
   finally show ?thesis .
 qed
 
-lemma lambda_skolem': "(\<forall>i<DIM('a::euclidean_space). \<exists>x. P i x) \<longleftrightarrow>
-   (\<exists>x::'a. \<forall>i<DIM('a). P i (x$$i))" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  let ?S = "{..<DIM('a)}"
-  { assume H: "?rhs"
-    then have ?lhs by auto }
-  moreover
-  { assume H: "?lhs"
-    then obtain f where f:"\<forall>i<DIM('a). P i (f i)" unfolding choice_iff' by metis
-    let ?x = "(\<chi>\<chi> i. (f i)) :: 'a"
-    { fix i assume i:"i<DIM('a)"
-      with f have "P i (f i)" by metis
-      then have "P i (?x$$i)" using i by auto }
-    then have "\<forall>i<DIM('a). P i (?x$$i)" by metis
-    then have ?rhs by metis }
-  ultimately show ?thesis by metis
-qed
-
-
 subsection {* Linearity and Bilinearity continued *}
 
 lemma linear_bounded:
@@ -1401,29 +1376,25 @@
   assumes lf: "linear f"
   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
 proof -
-  let ?S = "{..<DIM('a)}"
-  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
-  have fS: "finite ?S" by simp
+  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
   { fix x:: "'a"
-    let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
-    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
-      apply (subst euclidean_representation[of x])
-      apply rule
-      done
-    also have "\<dots> = norm (setsum (\<lambda> i. (x$$i) *\<^sub>R f (basis i)) ?S)"
-      using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] by auto
-    finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$$i) *\<^sub>R f (basis i))?S)" .
-    { fix i assume i: "i \<in> ?S"
-      from component_le_norm[of x i]
-      have "norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
+    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
+    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
+      unfolding euclidean_representation ..
+    also have "\<dots> = norm (setsum ?g Basis)"
+      using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf] by auto
+    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
+    { fix i :: 'a assume i: "i \<in> Basis"
+      from Basis_le_norm[OF i, of x]
+      have "norm (?g i) \<le> norm (f i) * norm x"
         unfolding norm_scaleR
-        apply (simp only: mult_commute)
+        apply (subst mult_commute)
         apply (rule mult_mono)
         apply (auto simp add: field_simps)
         done }
-    then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
+    then have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
       by metis
-    from setsum_norm_le[of _ "\<lambda>i. (x$$i) *\<^sub>R (f (basis i))", OF th]
+    from setsum_norm_le[of _ ?g, OF th]
     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
   then show ?thesis by blast
 qed
@@ -1438,12 +1409,15 @@
   let ?K = "\<bar>B\<bar> + 1"
   have Kp: "?K > 0" by arith
   { assume C: "B < 0"
-    have "((\<chi>\<chi> i. 1)::'a) \<noteq> 0" unfolding euclidean_eq[where 'a='a]
-      by(auto intro!:exI[where x=0])
-    then have "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
-    with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 0"
+    def One \<equiv> "\<Sum>Basis ::'a"
+    then have "One \<noteq> 0"
+      unfolding euclidean_eq_iff[where 'a='a]
+      by (simp add: inner_setsum_left inner_Basis setsum_cases)
+    then have "norm One > 0" by auto
+    with C have "B * norm One < 0"
       by (simp add: mult_less_0_iff)
-    with B[rule_format, of "(\<chi>\<chi> i. 1)::'a"] norm_ge_zero[of "f ((\<chi>\<chi> i. 1)::'a)"] have False by simp
+    with B[rule_format, of One] norm_ge_zero[of "f One"]
+    have False by simp
   }
   then have Bp: "B \<ge> 0" by (metis not_leE)
   { fix x::"'a"
@@ -1492,33 +1466,27 @@
   fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
   assumes bh: "bilinear h"
   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof -
-  let ?M = "{..<DIM('m)}"
-  let ?N = "{..<DIM('n)}"
-  let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
-  have fM: "finite ?M" and fN: "finite ?N" by simp_all
-  { fix x:: "'m" and  y :: "'n"
-    have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$$i) *\<^sub>R basis i) ?M) (setsum (\<lambda>i. (y$$i) *\<^sub>R basis i) ?N))" 
-      apply(subst euclidean_representation[where 'a='m])
-      apply(subst euclidean_representation[where 'a='n])
-      apply rule
-      done
-    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$i) *\<^sub>R basis i) ((y$$j) *\<^sub>R basis j)) (?M \<times> ?N))"  
-      unfolding bilinear_setsum[OF bh fM fN] ..
-    finally have th: "norm (h x y) = \<dots>" .
-    have "norm (h x y) \<le> ?B * norm x * norm y"
-      apply (simp add: setsum_left_distrib th)
+proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
+  fix x:: "'m" and  y :: "'n"
+  have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))" 
+    apply(subst euclidean_representation[where 'a='m])
+    apply(subst euclidean_representation[where 'a='n])
+    apply rule
+    done
+  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"  
+    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
+  finally have th: "norm (h x y) = \<dots>" .
+  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
+      apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
       apply (rule setsum_norm_le)
-      using fN fM
       apply simp
       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
         field_simps simp del: scaleR_scaleR)
       apply (rule mult_mono)
-      apply (auto simp add: zero_le_mult_iff component_le_norm)
+      apply (auto simp add: zero_le_mult_iff Basis_le_norm)
       apply (rule mult_mono)
-      apply (auto simp add: zero_le_mult_iff component_le_norm)
-      done }
-  then show ?thesis by metis
+      apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+      done
 qed
 
 lemma bilinear_bounded_pos:
@@ -1582,8 +1550,8 @@
 
 lemma independent_bound:
   fixes S:: "('a::euclidean_space) set"
-  shows "independent S \<Longrightarrow> finite S \<and> card S <= DIM('a::euclidean_space)"
-  using independent_span_bound[of "(basis::nat=>'a) ` {..<DIM('a)}" S] by auto
+  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a::euclidean_space)"
+  using independent_span_bound[OF finite_Basis, of S] by auto
 
 lemma dependent_biggerset:
   "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
@@ -1666,9 +1634,8 @@
 text {* More lemmas about dimension. *}
 
 lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
-  apply (rule dim_unique[of "(basis::nat=>'a) ` {..<DIM('a)}"])
-  using independent_basis apply auto
-  done
+  using independent_Basis
+  by (intro dim_unique[of Basis]) auto
 
 lemma dim_subset:
   "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
@@ -2256,20 +2223,9 @@
 lemma linear_eq_stdbasis:
   assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)"
     and lg: "linear g"
-    and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
+    and fg: "\<forall>b\<in>Basis. f b = g b"
   shows "f = g"
-proof -
-  let ?U = "{..<DIM('a)}"
-  let ?I = "(basis::nat=>'a) ` {..<DIM('a)}"
-  { fix x assume x: "x \<in> (UNIV :: 'a set)"
-    from equalityD2[OF span_basis'[where 'a='a]]
-    have IU: " (UNIV :: 'a set) \<subseteq> span ?I" by blast
-    have "f x = g x"
-      apply (rule linear_eq[OF lf lg IU,rule_format])
-      using fg x apply auto
-      done
-  } then show ?thesis by auto
-qed
+  using linear_eq[OF lf lg, of _ Basis] fg by auto
 
 text {* Similar results for bilinear functions. *}
 
@@ -2303,14 +2259,9 @@
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
   assumes bf: "bilinear f"
     and bg: "bilinear g"
-    and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
+    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
   shows "f = g"
-proof -
-  from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y"
-    by blast
-  from bilinear_eq[OF bf bg equalityD2[OF span_basis'] equalityD2[OF span_basis'] th]
-  show ?thesis by blast
-qed
+  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
 
 text {* Detailed theorems about left and right invertibility in general case. *}
 
@@ -2319,10 +2270,10 @@
   assumes lf: "linear f" and fi: "inj f"
   shows "\<exists>g. linear g \<and> g o f = id"
 proof -
-  from linear_independent_extend[OF independent_injective_image, OF independent_basis, OF lf fi]
+  from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
   obtain h:: "'b => 'a" where
-    h: "linear h" "\<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
-  from h(2) have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
+    h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x" by blast
+  from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
     using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
     by auto
 
@@ -2336,12 +2287,12 @@
   assumes lf: "linear f" and sf: "surj f"
   shows "\<exists>g. linear g \<and> f o g = id"
 proof -
-  from linear_independent_extend[OF independent_basis[where 'a='b],of "inv f"]
+  from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
   obtain h:: "'b \<Rightarrow> 'a" where
-    h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. h x = inv f x" by blast
+    h: "linear h" "\<forall>x\<in>Basis. h x = inv f x" by blast
   from h(2)
-  have th: "\<forall>i<DIM('b). (f o h) (basis i) = id (basis i)"
-    using sf by(auto simp add: surj_iff_all)
+  have th: "\<forall>i\<in>Basis. (f o h) i = id i"
+    using sf by (auto simp add: surj_iff_all)
   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
   have "f o h = id" .
   then show ?thesis using h(1) by blast
@@ -2538,18 +2489,18 @@
 
 subsection {* Infinity norm *}
 
-definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
+definition "infnorm (x::'a::euclidean_space) = Sup { abs (x \<bullet> b) |b. b \<in> Basis}"
 
 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
   by auto
 
 lemma infnorm_set_image:
-  "{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
-  (\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
+  "{ abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
+  by blast
 
 lemma infnorm_set_lemma:
-  shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
-  and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
+  shows "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
+  and "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
   unfolding infnorm_set_image
   by auto
 
@@ -2557,25 +2508,26 @@
   unfolding infnorm_def
   unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
   unfolding infnorm_set_image
-  by auto
+  by (auto simp: ex_in_conv)
 
 lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
 proof -
   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
-  have *:"\<And>i. i \<in> {..<DIM('a)} \<longleftrightarrow> i <DIM('a)" by auto
   show ?thesis
-  unfolding infnorm_def unfolding  Sup_finite_le_iff[ OF infnorm_set_lemma[where 'a='a]]
-  apply (subst diff_le_eq[symmetric])
-  unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
-  unfolding infnorm_set_image bex_simps
-  apply (subst th)
-  unfolding th1 *
-  unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma[where 'a='a]]
-  unfolding infnorm_set_image ball_simps bex_simps
-  unfolding euclidean_simps apply (metis th2)
-  done
+    unfolding infnorm_def 
+    unfolding Sup_finite_le_iff[ OF infnorm_set_lemma[where 'a='a]]
+    apply (subst diff_le_eq[symmetric])
+    unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
+    unfolding infnorm_set_image bex_simps
+    apply (subst th)
+    unfolding th1
+    unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma[where 'a='a]]
+    unfolding infnorm_set_image ball_simps bex_simps
+    apply (simp add: inner_add_left)
+    apply (metis th2)
+    done
 qed
 
 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
@@ -2584,7 +2536,7 @@
     unfolding infnorm_def
     unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
     unfolding infnorm_set_image ball_simps
-    apply (subst (1) euclidean_eq)
+    apply (subst (1) euclidean_eq_iff)
     apply auto
     done
   then show ?thesis using infnorm_pos_le[of x] by simp
@@ -2620,29 +2572,22 @@
 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
   using infnorm_pos_le[of x] by arith
 
-lemma component_le_infnorm: "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
-proof (cases "i<DIM('a)")
-  case False
-  then show ?thesis using infnorm_pos_le by auto
-next
-  case True
-  let ?U = "{..<DIM('a)}"
-  let ?S = "{\<bar>x$$i\<bar> |i. i<DIM('a)}"
-  have fS: "finite ?S" unfolding image_Collect[symmetric]
-    apply (rule finite_imageI) apply simp done
-  have S0: "?S \<noteq> {}" by blast
-  have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
-  show ?thesis unfolding infnorm_def  
-    apply(subst Sup_finite_ge_iff) using Sup_finite_in[OF fS S0]
-    using infnorm_set_image using True apply auto
-    done
+lemma Basis_le_infnorm:
+  assumes b: "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> infnorm (x::'a::euclidean_space)"
+  unfolding infnorm_def
+proof (subst Sup_finite_ge_iff)
+  let ?S = "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
+  show "finite ?S" by (rule infnorm_set_lemma)
+  show "?S \<noteq> {}" by auto
+  show "Bex ?S (op \<le> \<bar>x \<bullet> b\<bar>)"
+     using b by (auto intro!: exI[of _ b])
 qed
 
 lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) <= \<bar>a\<bar> * infnorm x"
   apply (subst infnorm_def)
   unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
-  unfolding infnorm_set_image ball_simps euclidean_component_scaleR abs_mult
-  using component_le_infnorm[of x]
+  unfolding infnorm_set_image ball_simps inner_scaleR abs_mult
+  using Basis_le_infnorm[of _ x]
   apply (auto intro: mult_mono)
   done
 
@@ -2671,9 +2616,13 @@
 lemma infnorm_le_norm: "infnorm x \<le> norm x"
   unfolding infnorm_def Sup_finite_le_iff[OF infnorm_set_lemma]
   unfolding infnorm_set_image  ball_simps
-  by (metis component_le_norm)
-
-lemma norm_le_infnorm: "norm(x) <= sqrt(real DIM('a)) * infnorm(x::'a::euclidean_space)"
+  by (metis Basis_le_norm)
+
+lemma euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
+  by (subst (1 2) euclidean_representation[symmetric, where 'a='a])
+     (simp add: inner_setsum_left inner_setsum_right setsum_cases inner_Basis ac_simps if_distrib)
+
+lemma norm_le_infnorm: "norm(x) <= sqrt DIM('a) * infnorm(x::'a::euclidean_space)"
 proof -
   let ?d = "DIM('a)"
   have "real ?d \<ge> 0" by simp
@@ -2683,13 +2632,14 @@
     by (simp add: zero_le_mult_iff infnorm_pos_le)
   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)^2"
     unfolding power_mult_distrib d2
-    unfolding real_of_nat_def apply(subst euclidean_inner)
+    unfolding real_of_nat_def
+    apply(subst euclidean_inner)
     apply (subst power2_abs[symmetric])
     apply (rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<twosuperior>"]])
     apply (auto simp add: power2_eq_square[symmetric])
     apply (subst power2_abs[symmetric])
     apply (rule power_mono)
-    unfolding infnorm_def  Sup_finite_ge_iff[OF infnorm_set_lemma]
+    unfolding infnorm_def Sup_finite_ge_iff[OF infnorm_set_lemma]
     unfolding infnorm_set_image bex_simps
     apply (rule_tac x=i in bexI)
     apply auto
@@ -2872,8 +2822,8 @@
 subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
 
 class ordered_euclidean_space = ord + euclidean_space +
-  assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
-    and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
+  assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
+    and eucl_less: "x < y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"
 
 lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
   unfolding eucl_less[where 'a='a] by auto
@@ -2889,35 +2839,16 @@
 
 lemma atLeastAtMost_singleton_euclidean[simp]:
   fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
-  by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
-
-lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
-
-instance real::ordered_euclidean_space
-  by default (auto simp add: euclidean_component_def)
-
-lemma Eucl_real_simps[simp]:
-  "(x::real) $$ 0 = x"
-  "(\<chi>\<chi> i. f i) = ((f 0)::real)"
-  "\<And>i. i > 0 \<Longrightarrow> x $$ i = 0"
-  defer apply(subst euclidean_eq) apply safe
-  unfolding euclidean_lambda_beta'
-  unfolding euclidean_component_def apply auto
-  done
-
-lemma complex_basis[simp]:
-  shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
-  unfolding basis_complex_def by auto
-
-lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::euclidean_space) + DIM('a::euclidean_space)"
-  (* FIXME: why this orientation? Why not "DIM('a) + DIM('b)" ? *)
-  unfolding dimension_prod_def by (rule add_commute)
+  by (force simp: eucl_le[where 'a='a] euclidean_eq_iff[where 'a='a])
+
+instance real :: ordered_euclidean_space
+  by default (auto simp add: Basis_real_def)
 
 instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
 begin
 
-definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
-definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
+definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
+definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"
 
 instance
   by default (auto simp: less_prod_def less_eq_prod_def)
--- a/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -1,5 +1,5 @@
 theory Multivariate_Analysis
-imports Fashoda Extended_Real_Limits
+imports Fashoda Extended_Real_Limits Determinants
 begin
 
 end
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -22,8 +22,11 @@
 
   moreover
   {assume H: ?lhs
-    have bp: "b \<ge> 0" apply-apply(rule order_trans [OF norm_ge_zero])
-      apply(rule H[rule_format, of "basis 0::'a"]) by auto 
+    have bp: "b \<ge> 0"
+      apply -
+      apply(rule order_trans [OF norm_ge_zero])
+      apply(rule H[rule_format, of "SOME x::'a. x \<in> Basis"])
+      by (auto intro: SOME_Basis norm_Basis)
     {fix x :: "'a"
       {assume "x = 0"
         then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
@@ -50,8 +53,8 @@
 proof-
   {
     let ?S = "{norm (f x) |x. norm x = 1}"
-    have "norm (f (basis 0)) \<in> ?S" unfolding mem_Collect_eq
-      apply(rule_tac x="basis 0" in exI) by auto
+    have "norm (f (SOME i. i \<in> Basis)) \<in> ?S"
+      by (auto intro!: exI[of _ "SOME i. i \<in> Basis"] norm_Basis SOME_Basis)
     hence Se: "?S \<noteq> {}" by auto
     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
@@ -70,8 +73,8 @@
 qed
 
 lemma onorm_pos_le: assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)" shows "0 <= onorm f"
-  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 0"]] 
-  using DIM_positive[where 'a='n] by auto
+  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "SOME i. i \<in> Basis"]] 
+  by (simp add: SOME_Basis)
 
 lemma onorm_eq_0: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
@@ -87,7 +90,7 @@
 proof-
   let ?f = "\<lambda>x::'a. (y::'b)"
   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
-    apply safe apply(rule_tac x="basis 0" in exI) by auto
+    by (auto simp: SOME_Basis intro!: exI[of _ "SOME i. i \<in> Basis"])
   show ?thesis
     unfolding onorm_def th
     apply (rule Sup_unique) by (simp_all  add: setle_def)
--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -819,35 +819,40 @@
   assumes "2 \<le> DIM('a::euclidean_space)"
   shows "path_connected((UNIV::'a::euclidean_space set) - {a})"
 proof -
-  let ?A = "{x::'a. \<exists>i\<in>{..<DIM('a)}. x $$ i < a $$ i}"
-  let ?B = "{x::'a. \<exists>i\<in>{..<DIM('a)}. a $$ i < x $$ i}"
+  let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
+  let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
 
   have A: "path_connected ?A"
     unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
-    fix i
-    assume "i \<in> {..<DIM('a)}"
-    then show "(\<chi>\<chi> i. a $$ i - 1) \<in> {x::'a. x $$ i < a $$ i}" by simp
-    show "path_connected {x. x $$ i < a $$ i}" unfolding euclidean_component_def
-      by (rule convex_imp_path_connected [OF convex_halfspace_lt])
+    fix i :: 'a
+    assume "i \<in> Basis"
+    then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" by simp
+    show "path_connected {x. x \<bullet> i < a \<bullet> i}"
+      using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
+      by (simp add: inner_commute)
   qed
   have B: "path_connected ?B" unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
-    fix i
-    assume "i \<in> {..<DIM('a)}"
-    then show "(\<chi>\<chi> i. a $$ i + 1) \<in> {x::'a. a $$ i < x $$ i}" by simp
-    show "path_connected {x. a $$ i < x $$ i}" unfolding euclidean_component_def
-      by (rule convex_imp_path_connected [OF convex_halfspace_gt])
+    fix i :: 'a
+    assume "i \<in> Basis"
+    then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" by simp
+    show "path_connected {x. a \<bullet> i < x \<bullet> i}"
+      using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
+      by (simp add: inner_commute)
   qed
-  from assms have "1 < DIM('a)" by auto
-  then have "a + basis 0 - basis 1 \<in> ?A \<inter> ?B" by auto
+  obtain S :: "'a set" where "S \<subseteq> Basis" "card S = Suc (Suc 0)"
+    using ex_card[OF assms] by auto
+  then obtain b0 b1 :: 'a where "b0 \<in> Basis" "b1 \<in> Basis" "b0 \<noteq> b1"
+    unfolding card_Suc_eq by auto
+  then have "a + b0 - b1 \<in> ?A \<inter> ?B" by (auto simp: inner_simps inner_Basis)
   then have "?A \<inter> ?B \<noteq> {}" by fast
   with A B have "path_connected (?A \<union> ?B)"
     by (rule path_connected_Un)
-  also have "?A \<union> ?B = {x. \<exists>i\<in>{..<DIM('a)}. x $$ i \<noteq> a $$ i}"
+  also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
     unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
   also have "\<dots> = {x. x \<noteq> a}"
-    unfolding Bex_def euclidean_eq [where 'a='a] by simp
+    unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def)
   also have "\<dots> = UNIV - {a}" by auto
   finally show ?thesis .
 qed
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Dec 14 14:46:01 2012 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Dec 14 15:46:01 2012 +0100
@@ -13,9 +13,17 @@
   "~~/src/HOL/Library/Countable_Set"
   Linear_Algebra
   "~~/src/HOL/Library/Glbs"
+  "~~/src/HOL/Library/FuncSet"
   Norm_Arith
 begin
 
+lemma countable_PiE: 
+  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
+  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
+
+lemma countable_rat: "countable \<rat>"
+  unfolding Rats_def by auto
+
 subsection {* Topological Basis *}
 
 context topological_space
@@ -593,86 +601,74 @@
 
 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
 
+lemma euclidean_dist_l2:
+  fixes x y :: "'a :: euclidean_space"
+  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
+  unfolding dist_norm norm_eq_sqrt_inner setL2_def
+  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
+
+definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
+
 lemma rational_boxes:
-  fixes x :: "'a\<Colon>ordered_euclidean_space"
+  fixes x :: "'a\<Colon>euclidean_space"
   assumes "0 < e"
-  shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"
+  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
 proof -
   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
-  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
-  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")
+  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
+  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
   proof
-    fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e
-    show "?th i" by auto
+    fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto
   qed
   from choice[OF this] guess a .. note a = this
-  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")
+  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
   proof
-    fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e
-    show "?th i" by auto
+    fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto
   qed
   from choice[OF this] guess b .. note b = this
-  { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
-    have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"
+  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
+  show ?thesis
+  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
+    fix y :: 'a assume *: "y \<in> box ?a ?b"
+    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"
       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
-    also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
+    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
-      fix i assume i: "i \<in> {..<DIM('a)}"
-      have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto
-      moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto
-      moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto
-      ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto
-      then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"
+      fix i :: "'a" assume i: "i \<in> Basis"
+      have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)
+      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto
+      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto
+      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto
+      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
         unfolding e'_def by (auto simp: dist_real_def)
-      then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
+      then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
         by (rule power_strict_mono) auto
-      then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
+      then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
         by (simp add: power_divide)
     qed auto
-    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
-    finally have "dist x y < e" . }
-  with a b show ?thesis
-    apply (rule_tac exI[of _ "Chi a"])
-    apply (rule_tac exI[of _ "Chi b"])
-    using eucl_less[where 'a='a] by auto
-qed
-
-lemma ex_rat_list:
-  fixes x :: "'a\<Colon>ordered_euclidean_space"
-  assumes "\<And> i. x $$ i \<in> \<rat>"
-  shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"
-proof -
-  have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast
-  from choice[OF this] guess r ..
-  then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
-qed
-
-lemma open_UNION:
-  fixes M :: "'a\<Colon>ordered_euclidean_space set"
-  assumes "open M"
-  shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
-                   (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
-    (is "M = UNION ?idx ?box")
+    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat)
+    finally show "y \<in> ball x e" by (auto simp: ball_def)
+  qed (insert a b, auto simp: box_def)
+qed
+ 
+lemma open_UNION_box:
+  fixes M :: "'a\<Colon>euclidean_space set"
+  assumes "open M" 
+  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
+  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
+  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
+  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
 proof safe
   fix x assume "x \<in> M"
   obtain e where e: "e > 0" "ball x e \<subseteq> M"
-    using openE[OF assms `x \<in> M`] by auto
-  then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"
-    using rational_boxes[OF e(1)] by blast
-  then obtain p q where pq: "length p = DIM ('a)"
-                            "length q = DIM ('a)"
-                            "\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"
-    using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
-  hence p: "Chi (of_rat \<circ> op ! p) = a"
-    using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
-    unfolding o_def by auto
-  from pq have q: "Chi (of_rat \<circ> op ! q) = b"
-    using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
-    unfolding o_def by auto
-  have "x \<in> ?box (p, q)"
-    using p q ab by auto
-  thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
-qed auto
+    using openE[OF `open M` `x \<in> M`] by auto
+  moreover then obtain a b where ab: "x \<in> box a b"
+    "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
+    using rational_boxes[OF e(1)] by metis
+  ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
+     by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
+        (auto simp: euclidean_representation I_def a'_def b'_def)
+qed (auto simp: I_def)
 
 subsection{* Connectedness *}
 
@@ -1156,14 +1152,10 @@
   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
 unfolding eventually_at dist_nz by auto
 
-lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
+lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *)
+  "eventually P (at a within S) \<longleftrightarrow>
         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
-unfolding eventually_within eventually_at dist_nz by auto
-
-lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
-        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
-unfolding eventually_within
-by auto (metis dense order_le_less_trans)
+  by (rule eventually_within_less)
 
 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
   unfolding trivial_limit_def
@@ -1721,7 +1713,7 @@
     assume "\<not> (\<exists>y\<in>A. dist y x < e)"
     hence "infdist x A \<ge> e" using `a \<in> A`
       unfolding infdist_def
-      by (force intro: Inf_greatest simp: dist_commute)
+      by (force simp: dist_commute)
     with x `0 < e` show False by auto
   qed
 qed
@@ -2374,56 +2366,41 @@
     by auto
 qed
 
-lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
-  apply (erule bounded_linear_image)
-  apply (rule bounded_linear_euclidean_component)
-  done
-
 lemma compact_lemma:
   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
   assumes "bounded s" and "\<forall>n. f n \<in> s"
-  shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
-        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
-proof
-  fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
-  have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
-  hence "\<exists>l::'a. \<exists>r. subseq r \<and>
-      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
+  shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>
+        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
+proof safe
+  fix d :: "'a set" assume d: "d \<subseteq> Basis" 
+  with finite_Basis have "finite d" by (blast intro: finite_subset)
+  from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
+      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
   proof(induct d) case empty thus ?case unfolding subseq_def by auto
-  next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
-    have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
+  next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto
+    have s': "bounded ((\<lambda>x. x \<bullet> k) ` s)" using `bounded s`
+      by (auto intro!: bounded_linear_image bounded_linear_inner_left)
     obtain l1::"'a" and r1 where r1:"subseq r1" and
-      lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
+      lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
       using insert(3) using insert(4) by auto
-    have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
-    obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
+    have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` s" using `\<forall>n. f n \<in> s` by simp
+    obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
       using r1 and r2 unfolding r_def o_def subseq_def by auto
     moreover
-    def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
+    def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
     { fix e::real assume "e>0"
-      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
-      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
-      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
+      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast
+      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)
+      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
         by (rule eventually_subseq)
-      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
-        using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
-        using insert.prems by auto
+      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
+        using N1' N2 
+        by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
     }
     ultimately show ?case by auto
   qed
-  thus "\<exists>l::'a. \<exists>r. subseq r \<and>
-      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
-    apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
-    apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe 
-    apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
-    apply(erule_tac x=i in ballE) 
-  proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
-    assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
-    hence *:"i\<ge>DIM('a)" by auto
-    thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
-  qed
 qed