# HG changeset patch
# User huffman
# Date 1312993422 25200
# Node ID 691c52e900cab050b18f1958a21f5e6571d756a3
# Parent  0f35a870ecf16263c295ca32815c03b3071329b4
split Linear_Algebra.thy from Euclidean_Space.thy

diff -r 0f35a870ecf1 -r 691c52e900ca src/HOL/Multivariate_Analysis/Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Aug 10 09:23:42 2011 -0700
@@ -1,1586 +1,18 @@
 (*  Title:      HOL/Multivariate_Analysis/Euclidean_Space.thy
-    Author:     Amine Chaieb, University of Cambridge
+    Author:     Johannes Hölzl, TU München
+    Author:     Brian Huffman, Portland State University
 *)
 
-header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
+header {* Finite-Dimensional Inner Product Spaces *}
 
 theory Euclidean_Space
 imports
   Complex_Main
-  "~~/src/HOL/Library/Infinite_Set"
   "~~/src/HOL/Library/Inner_Product"
-  L2_Norm
-  "~~/src/HOL/Library/Convex"
-uses
-  "~~/src/HOL/Library/positivstellensatz.ML"  (* FIXME duplicate use!? *)
-  ("normarith.ML")
-begin
-
-lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
-  by auto
-
-notation inner (infix "\<bullet>" 70)
-
-subsection {* A connectedness or intermediate value lemma with several applications. *}
-
-lemma connected_real_lemma:
-  fixes f :: "real \<Rightarrow> 'a::metric_space"
-  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
-  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
-  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
-  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
-  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
-  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
-proof-
-  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
-  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
-  have Sub: "\<exists>y. isUb UNIV ?S y"
-    apply (rule exI[where x= b])
-    using ab fb e12 by (auto simp add: isUb_def setle_def)
-  from reals_complete[OF Se Sub] obtain l where
-    l: "isLub UNIV ?S l"by blast
-  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
-    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
-    by (metis linorder_linear)
-  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
-    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
-    by (metis linorder_linear not_le)
-    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
-    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
-    have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
-    then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
-    {assume le2: "f l \<in> e2"
-      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
-      hence lap: "l - a > 0" using alb by arith
-      from e2[rule_format, OF le2] obtain e where
-        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
-      from dst[OF alb e(1)] obtain d where
-        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
-      let ?d' = "min (d/2) ((l - a)/2)"
-      have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
-        by (simp add: min_max.less_infI2)
-      then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
-      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
-      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
-      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
-      moreover
-      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
-      ultimately have False using e12 alb d' by auto}
-    moreover
-    {assume le1: "f l \<in> e1"
-    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
-      hence blp: "b - l > 0" using alb by arith
-      from e1[rule_format, OF le1] obtain e where
-        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
-      from dst[OF alb e(1)] obtain d where
-        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
-      have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
-      then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
-      then obtain d' where d': "d' > 0" "d' < d" by metis
-      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
-      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
-      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
-      with l d' have False
-        by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
-    ultimately show ?thesis using alb by metis
-qed
-
-text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case *}
-
-lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
-proof-
-  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
-  thus ?thesis by (simp add: field_simps power2_eq_square)
-qed
-
-lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
-  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
-  apply (rule_tac x="s" in exI)
-  apply auto
-  apply (erule_tac x=y in allE)
-  apply auto
-  done
-
-lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
-  using real_sqrt_le_iff[of x "y^2"] by simp
-
-lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
-  using real_sqrt_le_mono[of "x^2" y] by simp
-
-lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
-  using real_sqrt_less_mono[of "x^2" y] by simp
-
-lemma sqrt_even_pow2: assumes n: "even n"
-  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
-proof-
-  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
-  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
-    by (simp only: power_mult[symmetric] mult_commute)
-  then show ?thesis  using m by simp
-qed
-
-lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
-  apply (cases "x = 0", simp_all)
-  using sqrt_divide_self_eq[of x]
-  apply (simp add: inverse_eq_divide field_simps)
-  done
-
-text{* Hence derive more interesting properties of the norm. *}
-
-(* FIXME: same as norm_scaleR
-lemma norm_mul[simp]: "norm(a *\<^sub>R x) = abs(a) * norm x"
-  by (simp add: norm_vector_def setL2_right_distrib abs_mult)
-*)
-
-lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
-  by (simp add: setL2_def power2_eq_square)
-
-lemma norm_cauchy_schwarz:
-  shows "inner x y <= norm x * norm y"
-  using Cauchy_Schwarz_ineq2[of x y] by auto
-
-lemma norm_cauchy_schwarz_abs:
-  shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
-  by (rule Cauchy_Schwarz_ineq2)
-
-lemma norm_triangle_sub:
-  fixes x y :: "'a::real_normed_vector"
-  shows "norm x \<le> norm y  + norm (x - y)"
-  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
-
-lemma real_abs_norm: "\<bar>norm x\<bar> = norm x"
-  by (rule abs_norm_cancel)
-lemma real_abs_sub_norm: "\<bar>norm x - norm y\<bar> <= norm(x - y)"
-  by (rule norm_triangle_ineq3)
-lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
-  by (simp add: norm_eq_sqrt_inner) 
-lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
-  by (simp add: norm_eq_sqrt_inner)
-lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
-  apply(subst order_eq_iff) unfolding norm_le by auto
-lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
-  unfolding norm_eq_sqrt_inner by auto
-
-text{* Squaring equations and inequalities involving norms.  *}
-
-lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
-  by (simp add: norm_eq_sqrt_inner)
-
-lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
-  by (auto simp add: norm_eq_sqrt_inner)
-
-lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
-proof
-  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
-  then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
-  then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
-next
-  assume "x\<twosuperior> \<le> y\<twosuperior>"
-  then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
-  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
-qed
-
-lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
-  using norm_ge_zero[of x]
-  apply arith
-  done
-
-lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
-  using norm_ge_zero[of x]
-  apply arith
-  done
-
-lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
-  by (metis not_le norm_ge_square)
-lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
-  by (metis norm_le_square not_less)
-
-text{* Dot product in terms of the norm rather than conversely. *}
-
-lemmas inner_simps = inner.add_left inner.add_right inner.diff_right inner.diff_left 
-inner.scaleR_left inner.scaleR_right
-
-lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
-  unfolding power2_norm_eq_inner inner_simps inner_commute by auto 
-
-lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
-  unfolding power2_norm_eq_inner inner_simps inner_commute by(auto simp add:algebra_simps)
-
-text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
-
-lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
-proof
-  assume ?lhs then show ?rhs by simp
-next
-  assume ?rhs
-  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
-  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_simps inner_commute)
-  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_simps inner_commute)
-  then show "x = y" by (simp)
-qed
-
-subsection{* General linear decision procedure for normed spaces. *}
-
-lemma norm_cmul_rule_thm:
-  fixes x :: "'a::real_normed_vector"
-  shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
-  unfolding norm_scaleR
-  apply (erule mult_left_mono)
-  apply simp
-  done
-
-  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
-lemma norm_add_rule_thm:
-  fixes x1 x2 :: "'a::real_normed_vector"
-  shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
-  by (rule order_trans [OF norm_triangle_ineq add_mono])
-
-lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
-  by (simp add: field_simps)
-
-lemma pth_1:
-  fixes x :: "'a::real_normed_vector"
-  shows "x == scaleR 1 x" by simp
-
-lemma pth_2:
-  fixes x :: "'a::real_normed_vector"
-  shows "x - y == x + -y" by (atomize (full)) simp
-
-lemma pth_3:
-  fixes x :: "'a::real_normed_vector"
-  shows "- x == scaleR (-1) x" by simp
-
-lemma pth_4:
-  fixes x :: "'a::real_normed_vector"
-  shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
-
-lemma pth_5:
-  fixes x :: "'a::real_normed_vector"
-  shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
-
-lemma pth_6:
-  fixes x :: "'a::real_normed_vector"
-  shows "scaleR c (x + y) == scaleR c x + scaleR c y"
-  by (simp add: scaleR_right_distrib)
-
-lemma pth_7:
-  fixes x :: "'a::real_normed_vector"
-  shows "0 + x == x" and "x + 0 == x" by simp_all
-
-lemma pth_8:
-  fixes x :: "'a::real_normed_vector"
-  shows "scaleR c x + scaleR d x == scaleR (c + d) x"
-  by (simp add: scaleR_left_distrib)
-
-lemma pth_9:
-  fixes x :: "'a::real_normed_vector" shows
-  "(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
-  "scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
-  "(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
-  by (simp_all add: algebra_simps)
-
-lemma pth_a:
-  fixes x :: "'a::real_normed_vector"
-  shows "scaleR 0 x + y == y" by simp
-
-lemma pth_b:
-  fixes x :: "'a::real_normed_vector" shows
-  "scaleR c x + scaleR d y == scaleR c x + scaleR d y"
-  "(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
-  "scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
-  "(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
-  by (simp_all add: algebra_simps)
-
-lemma pth_c:
-  fixes x :: "'a::real_normed_vector" shows
-  "scaleR c x + scaleR d y == scaleR d y + scaleR c x"
-  "(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
-  "scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
-  "(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
-  by (simp_all add: algebra_simps)
-
-lemma pth_d:
-  fixes x :: "'a::real_normed_vector"
-  shows "x + 0 == x" by simp
-
-lemma norm_imp_pos_and_ge:
-  fixes x :: "'a::real_normed_vector"
-  shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
-  by atomize auto
-
-lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
-
-lemma norm_pths:
-  fixes x :: "'a::real_normed_vector" shows
-  "x = y \<longleftrightarrow> norm (x - y) \<le> 0"
-  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
-  using norm_ge_zero[of "x - y"] by auto
-
-use "normarith.ML"
-
-method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
-*} "prove simple linear statements about vector norms"
-
-
-text{* Hence more metric properties. *}
-
-lemma norm_triangle_half_r:
-  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
-  using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
-
-lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
-  shows "norm (x - x') < e"
-  using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
-  unfolding dist_norm[THEN sym] .
-
-lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
-  by (metis order_trans norm_triangle_ineq)
-
-lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
-  by (metis basic_trans_rules(21) norm_triangle_ineq)
-
-lemma dist_triangle_add:
-  fixes x y x' y' :: "'a::real_normed_vector"
-  shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
-  by norm
-
-lemma dist_triangle_add_half:
-  fixes x x' y y' :: "'a::real_normed_vector"
-  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
-  by norm
-
-lemma setsum_clauses:
-  shows "setsum f {} = 0"
-  and "finite S \<Longrightarrow> setsum f (insert x S) =
-                 (if x \<in> S then setsum f S else f x + setsum f S)"
-  by (auto simp add: insert_absorb)
-
-lemma setsum_norm:
-  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
-  assumes fS: "finite S"
-  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 x S)
-  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
-  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
-    using "2.hyps" by simp
-  finally  show ?case  using "2.hyps" by simp
-qed
-
-lemma setsum_norm_le:
-  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
-  assumes fS: "finite S"
-  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
-  shows "norm (setsum f S) \<le> setsum g S"
-proof-
-  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
-    by - (rule setsum_mono, simp)
-  then show ?thesis using setsum_norm[OF fS, of f] fg
-    by arith
-qed
-
-lemma setsum_norm_bound:
-  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
-  assumes fS: "finite S"
-  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
-  shows "norm (setsum f S) \<le> of_nat (card S) * K"
-  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
-  by simp
-
-lemma setsum_group:
-  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
-  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
-  apply (subst setsum_image_gen[OF fS, of g f])
-  apply (rule setsum_mono_zero_right[OF fT fST])
-  by (auto intro: setsum_0')
-
-lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> y = setsum (\<lambda>x. f x \<bullet> y) S "
-  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
-
-lemma dot_rsum: "finite S \<Longrightarrow> y \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
-  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
-
-lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
-proof
-  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
-  hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_simps)
-  hence "(y - z) \<bullet> (y - z) = 0" ..
-  thus "y = z" by simp
-qed simp
-
-lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
-proof
-  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
-  hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_simps)
-  hence "(x - y) \<bullet> (x - y) = 0" ..
-  thus "x = y" by simp
-qed simp
-
-subsection{* Orthogonality. *}
-
-context real_inner
+  "~~/src/HOL/Library/Product_Vector"
 begin
 
-definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
-
-lemma orthogonal_clauses:
-  "orthogonal a 0"
-  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
-  "orthogonal a x \<Longrightarrow> orthogonal a (-x)"
-  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
-  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
-  "orthogonal 0 a"
-  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
-  "orthogonal x a \<Longrightarrow> orthogonal (-x) a"
-  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
-  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
-  unfolding orthogonal_def inner_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
- 
-end
-
-lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
-  by (simp add: orthogonal_def inner_commute)
-
-subsection{* Linear functions. *}
-
-definition
-  linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
-  "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
-
-lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
-  shows "linear f" using assms unfolding linear_def by auto
-
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
-  by (simp add: linear_def algebra_simps)
-
-lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
-  by (simp add: linear_def)
-
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
-  by (simp add: linear_def algebra_simps)
-
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
-  by (simp add: linear_def algebra_simps)
-
-lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
-  by (simp add: linear_def)
-
-lemma linear_id: "linear id" by (simp add: linear_def id_def)
-
-lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
-
-lemma linear_compose_setsum:
-  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
-  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
-  using lS
-  apply (induct rule: finite_induct[OF fS])
-  by (auto simp add: linear_zero intro: linear_compose_add)
-
-lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
-  unfolding linear_def
-  apply clarsimp
-  apply (erule allE[where x="0::'a"])
-  apply simp
-  done
-
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
-
-lemma linear_neg: "linear f ==> f (-x) = - f x"
-  using linear_cmul [where c="-1"] by simp
-
-lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
-
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
-  by (simp add: diff_minus linear_add linear_neg)
-
-lemma linear_setsum:
-  assumes lf: "linear f" and fS: "finite S"
-  shows "f (setsum g S) = setsum (f o g) S"
-proof (induct rule: finite_induct[OF fS])
-  case 1 thus ?case by (simp add: linear_0[OF lf])
-next
-  case (2 x F)
-  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
-    by simp
-  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
-  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
-  finally show ?case .
-qed
-
-lemma linear_setsum_mul:
-  assumes lf: "linear f" and fS: "finite S"
-  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
-  using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
-  linear_cmul[OF lf] by simp
-
-lemma linear_injective_0:
-  assumes lf: "linear f"
-  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
-proof-
-  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
-  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
-  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
-    by (simp add: linear_sub[OF lf])
-  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
-  finally show ?thesis .
-qed
-
-subsection{* Bilinear functions. *}
-
-definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
-
-lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
-  by (simp add: bilinear_def linear_def)
-lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
-  by (simp add: bilinear_def linear_def)
-
-lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
-  by (simp add: bilinear_def linear_def)
-
-lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
-  by (simp add: bilinear_def linear_def)
-
-lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
-  by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
-
-lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
-  by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
-
-lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
-  using add_imp_eq[of x y 0] by auto
-
-lemma bilinear_lzero:
-  assumes bh: "bilinear h" shows "h 0 x = 0"
-  using bilinear_ladd[OF bh, of 0 0 x]
-    by (simp add: eq_add_iff field_simps)
-
-lemma bilinear_rzero:
-  assumes bh: "bilinear h" shows "h x 0 = 0"
-  using bilinear_radd[OF bh, of x 0 0 ]
-    by (simp add: eq_add_iff field_simps)
-
-lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
-  by (simp  add: diff_minus bilinear_ladd bilinear_lneg)
-
-lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
-  by (simp  add: diff_minus bilinear_radd bilinear_rneg)
-
-lemma bilinear_setsum:
-  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
-  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
-proof-
-  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
-    apply (rule linear_setsum[unfolded o_def])
-    using bh fS by (auto simp add: bilinear_def)
-  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
-    apply (rule setsum_cong, simp)
-    apply (rule linear_setsum[unfolded o_def])
-    using bh fT by (auto simp add: bilinear_def)
-  finally show ?thesis unfolding setsum_cartesian_product .
-qed
-
-subsection{* Adjoints. *}
-
-definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
-
-lemma adjoint_unique:
-  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
-  shows "adjoint f = g"
-unfolding adjoint_def
-proof (rule some_equality)
-  show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
-next
-  fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
-  hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
-  hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
-  hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
-  hence "\<forall>y. h y = g y" by simp
-  thus "h = g" by (simp add: ext)
-qed
-
-lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
-
-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"
-  shows "\<forall>n \<ge> m. d n < e m"
-  using assms apply auto
-  apply (erule_tac x="n" in allE)
-  apply (erule_tac x="n" in allE)
-  apply auto
-  done
-
-
-lemma infinite_enumerate: assumes fS: "infinite S"
-  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
-unfolding subseq_def
-using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
-
-lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
-apply auto
-apply (rule_tac x="d/2" in exI)
-apply auto
-done
-
-
-lemma triangle_lemma:
-  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
-  shows "x <= y + z"
-proof-
-  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
-  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
-  from y z have yz: "y + z \<ge> 0" by arith
-  from power2_le_imp_le[OF th yz] show ?thesis .
-qed
-
-text {* TODO: move to NthRoot *}
-lemma sqrt_add_le_add_sqrt:
-  assumes x: "0 \<le> x" and y: "0 \<le> y"
-  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
-apply (rule power2_le_imp_le)
-apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
-apply (simp add: mult_nonneg_nonneg x y)
-apply (simp add: add_nonneg_nonneg x y)
-done
-
-subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
-
-definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
-  "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
-
-lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
-  unfolding hull_def by auto
-
-lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
-unfolding hull_def subset_iff by auto
-
-lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
-using hull_same[of s S] hull_in[of S s] by metis
-
-
-lemma hull_hull: "S hull (S hull s) = S hull s"
-  unfolding hull_def by blast
-
-lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
-  unfolding hull_def by blast
-
-lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
-  unfolding hull_def by blast
-
-lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
-  unfolding hull_def by blast
-
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
-  unfolding hull_def by blast
-
-lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
-  unfolding hull_def by blast
-
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
-           ==> (S hull s = t)"
-unfolding hull_def by auto
-
-lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
-  using hull_minimal[of S "{x. P x}" Q]
-  by (auto simp add: subset_eq Collect_def mem_def)
-
-lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
-
-lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
-unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
-
-lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
-  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
-apply rule
-apply (rule hull_mono)
-unfolding Un_subset_iff
-apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
-apply (rule hull_minimal)
-apply (metis hull_union_subset)
-apply (metis hull_in T)
-done
-
-lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
-  unfolding hull_def by blast
-
-lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
-by (metis hull_redundant_eq)
-
-text{* Archimedian properties and useful consequences. *}
-
-lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
-  using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
-lemmas real_arch_lt = reals_Archimedean2
-
-lemmas real_arch = reals_Archimedean3
-
-lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
-  using reals_Archimedean
-  apply (auto simp add: field_simps)
-  apply (subgoal_tac "inverse (real n) > 0")
-  apply arith
-  apply simp
-  done
-
-lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
-proof(induct n)
-  case 0 thus ?case by simp
-next
-  case (Suc n)
-  hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
-  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
-  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
-  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
-    apply (simp add: field_simps)
-    using mult_left_mono[OF p Suc.prems] by simp
-  finally show ?case  by (simp add: real_of_nat_Suc field_simps)
-qed
-
-lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
-proof-
-  from x have x0: "x - 1 > 0" by arith
-  from real_arch[OF x0, rule_format, of y]
-  obtain n::nat where n:"y < real n * (x - 1)" by metis
-  from x0 have x00: "x- 1 \<ge> 0" by arith
-  from real_pow_lbound[OF x00, of n] n
-  have "y < x^n" by auto
-  then show ?thesis by metis
-qed
-
-lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
-  using real_arch_pow[of 2 x] by simp
-
-lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
-  shows "\<exists>n. x^n < y"
-proof-
-  {assume x0: "x > 0"
-    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
-    from real_arch_pow[OF ix, of "1/y"]
-    obtain n where n: "1/y < (1/x)^n" by blast
-    then
-    have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
-  moreover
-  {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
-  ultimately show ?thesis by metis
-qed
-
-lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
-  by (metis real_arch_inv)
-
-lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
-  apply (rule forall_pos_mono)
-  apply auto
-  apply (atomize)
-  apply (erule_tac x="n - 1" in allE)
-  apply auto
-  done
-
-lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
-  shows "x = 0"
-proof-
-  {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
-    from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
-    with xc[rule_format, of n] have "n = 0" by arith
-    with n c have False by simp}
-  then show ?thesis by blast
-qed
-
-subsection {* Geometric progression *}
-
-lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
-  (is "?lhs = ?rhs")
-proof-
-  {assume x1: "x = 1" hence ?thesis by simp}
-  moreover
-  {assume x1: "x\<noteq>1"
-    hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
-    from geometric_sum[OF x1, of "Suc n", unfolded x1']
-    have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
-      unfolding atLeastLessThanSuc_atLeastAtMost
-      using x1' apply (auto simp only: field_simps)
-      apply (simp add: field_simps)
-      done
-    then have ?thesis by (simp add: field_simps) }
-  ultimately show ?thesis by metis
-qed
-
-lemma sum_gp_multiplied: assumes mn: "m <= n"
-  shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
-  (is "?lhs = ?rhs")
-proof-
-  let ?S = "{0..(n - m)}"
-  from mn have mn': "n - m \<ge> 0" by arith
-  let ?f = "op + m"
-  have i: "inj_on ?f ?S" unfolding inj_on_def by auto
-  have f: "?f ` ?S = {m..n}"
-    using mn apply (auto simp add: image_iff Bex_def) by arith
-  have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
-    by (rule ext, simp add: power_add power_mult)
-  from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
-  have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
-  then show ?thesis unfolding sum_gp_basic using mn
-    by (simp add: field_simps power_add[symmetric])
-qed
-
-lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
-   (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
-                    else (x^ m - x^ (Suc n)) / (1 - x))"
-proof-
-  {assume nm: "n < m" hence ?thesis by simp}
-  moreover
-  {assume "\<not> n < m" hence nm: "m \<le> n" by arith
-    {assume x: "x = 1"  hence ?thesis by simp}
-    moreover
-    {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
-      from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
-    ultimately have ?thesis by metis
-  }
-  ultimately show ?thesis by metis
-qed
-
-lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
-  (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
-  unfolding sum_gp[of x m "m + n"] power_Suc
-  by (simp add: field_simps power_add)
-
-
-subsection{* A bit of linear algebra. *}
-
-definition (in real_vector)
-  subspace :: "'a set \<Rightarrow> bool" where
-  "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
-
-definition (in real_vector) "span S = (subspace hull S)"
-definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
-abbreviation (in real_vector) "independent s == ~(dependent s)"
-
-text {* Closure properties of subspaces. *}
-
-lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
-  by (metis subspace_def)
-
-lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
-  by (metis subspace_def)
-
-lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
-  by (metis scaleR_minus1_left subspace_mul)
-
-lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
-  by (metis diff_minus subspace_add subspace_neg)
-
-lemma (in real_vector) subspace_setsum:
-  assumes sA: "subspace A" and fB: "finite B"
-  and f: "\<forall>x\<in> B. f x \<in> A"
-  shows "setsum f B \<in> A"
-  using  fB f sA
-  apply(induct rule: finite_induct[OF fB])
-  by (simp add: subspace_def sA, auto simp add: sA subspace_add)
-
-lemma subspace_linear_image:
-  assumes lf: "linear f" and sS: "subspace S"
-  shows "subspace(f ` S)"
-  using lf sS linear_0[OF lf]
-  unfolding linear_def subspace_def
-  apply (auto simp add: image_iff)
-  apply (rule_tac x="x + y" in bexI, auto)
-  apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
-  done
-
-lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
-  by (auto simp add: subspace_def linear_def linear_0[of f])
-
-lemma subspace_trivial: "subspace {0}"
-  by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
-  by (simp add: subspace_def)
-
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
-  by (metis span_def hull_mono)
-
-lemma (in real_vector) subspace_span: "subspace(span S)"
-  unfolding span_def
-  apply (rule hull_in[unfolded mem_def])
-  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
-  apply auto
-  apply (erule_tac x="X" in ballE)
-  apply (simp add: mem_def)
-  apply blast
-  apply (erule_tac x="X" in ballE)
-  apply (erule_tac x="X" in ballE)
-  apply (erule_tac x="X" in ballE)
-  apply (clarsimp simp add: mem_def)
-  apply simp
-  apply simp
-  apply simp
-  apply (erule_tac x="X" in ballE)
-  apply (erule_tac x="X" in ballE)
-  apply (simp add: mem_def)
-  apply simp
-  apply simp
-  done
-
-lemma (in real_vector) span_clauses:
-  "a \<in> S ==> a \<in> span S"
-  "0 \<in> span S"
-  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
-  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
-  by (metis span_def hull_subset subset_eq)
-     (metis subspace_span subspace_def)+
-
-lemma (in real_vector) span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
-  and P: "subspace P" and x: "x \<in> span S" shows "P x"
-proof-
-  from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
-  from P have P': "P \<in> subspace" by (simp add: mem_def)
-  from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
-  show "P x" by (metis mem_def subset_eq)
-qed
-
-lemma span_empty[simp]: "span {} = {0}"
-  apply (simp add: span_def)
-  apply (rule hull_unique)
-  apply (auto simp add: mem_def subspace_def)
-  unfolding mem_def[of "0::'a", symmetric]
-  apply simp
-  done
-
-lemma (in real_vector) independent_empty[intro]: "independent {}"
-  by (simp add: dependent_def)
-
-lemma dependent_single[simp]:
-  "dependent {x} \<longleftrightarrow> x = 0"
-  unfolding dependent_def by auto
-
-lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
-  apply (clarsimp simp add: dependent_def span_mono)
-  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
-  apply force
-  apply (rule span_mono)
-  apply auto
-  done
-
-lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
-  by (metis order_antisym span_def hull_minimal mem_def)
-
-lemma (in real_vector) span_induct': assumes SP: "\<forall>x \<in> S. P x"
-  and P: "subspace P" shows "\<forall>x \<in> span S. P x"
-  using span_induct SP P by blast
-
-inductive (in real_vector) span_induct_alt_help for S:: "'a \<Rightarrow> bool"
-  where
-  span_induct_alt_help_0: "span_induct_alt_help S 0"
-  | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *\<^sub>R x + z)"
-
-lemma span_induct_alt':
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
-proof-
-  {fix x:: "'a" assume x: "span_induct_alt_help S x"
-    have "h x"
-      apply (rule span_induct_alt_help.induct[OF x])
-      apply (rule h0)
-      apply (rule hS, assumption, assumption)
-      done}
-  note th0 = this
-  {fix x assume x: "x \<in> span S"
-
-    have "span_induct_alt_help S x"
-      proof(rule span_induct[where x=x and S=S])
-        show "x \<in> span S" using x .
-      next
-        fix x assume xS : "x \<in> S"
-          from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
-          show "span_induct_alt_help S x" by simp
-        next
-        have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
-        moreover
-        {fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
-          from h
-          have "span_induct_alt_help S (x + y)"
-            apply (induct rule: span_induct_alt_help.induct)
-            apply simp
-            unfolding add_assoc
-            apply (rule span_induct_alt_help_S)
-            apply assumption
-            apply simp
-            done}
-        moreover
-        {fix c x assume xt: "span_induct_alt_help S x"
-          then have "span_induct_alt_help S (c *\<^sub>R x)"
-            apply (induct rule: span_induct_alt_help.induct)
-            apply (simp add: span_induct_alt_help_0)
-            apply (simp add: scaleR_right_distrib)
-            apply (rule span_induct_alt_help_S)
-            apply assumption
-            apply simp
-            done
-        }
-        ultimately show "subspace (span_induct_alt_help S)"
-          unfolding subspace_def mem_def Ball_def by blast
-      qed}
-  with th0 show ?thesis by blast
-qed
-
-lemma span_induct_alt:
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
-  shows "h x"
-using span_induct_alt'[of h S] h0 hS x by blast
-
-text {* Individual closure properties. *}
-
-lemma span_span: "span (span A) = span A"
-  unfolding span_def hull_hull ..
-
-lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
-
-lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
-
-lemma span_inc: "S \<subseteq> span S"
-  by (metis subset_eq span_superset)
-
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
-  unfolding dependent_def apply(rule_tac x=0 in bexI)
-  using assms span_0 by auto
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
-  by (metis subspace_add subspace_span)
-
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
-  by (metis subspace_span subspace_mul)
-
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
-  by (metis subspace_neg subspace_span)
-
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
-  by (metis subspace_span subspace_sub)
-
-lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
-  by (rule subspace_setsum, rule subspace_span)
-
-lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
-  apply (auto simp only: span_add span_sub)
-  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
-  by (simp only: span_add span_sub)
-
-text {* Mapping under linear image. *}
-
-lemma span_linear_image: assumes lf: "linear f"
-  shows "span (f ` S) = f ` (span S)"
-proof-
-  {fix x
-    assume x: "x \<in> span (f ` S)"
-    have "x \<in> f ` span S"
-      apply (rule span_induct[where x=x and S = "f ` S"])
-      apply (clarsimp simp add: image_iff)
-      apply (frule span_superset)
-      apply blast
-      apply (simp only: mem_def)
-      apply (rule subspace_linear_image[OF lf])
-      apply (rule subspace_span)
-      apply (rule x)
-      done}
-  moreover
-  {fix x assume x: "x \<in> span S"
-    have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_eqI)
-      unfolding mem_def Collect_def ..
-    have "f x \<in> span (f ` S)"
-      apply (rule span_induct[where S=S])
-      apply (rule span_superset)
-      apply simp
-      apply (subst th0)
-      apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
-      apply (rule x)
-      done}
-  ultimately show ?thesis by blast
-qed
-
-text {* The key breakdown property. *}
-
-lemma span_breakdown:
-  assumes bS: "b \<in> S" and aS: "a \<in> span S"
-  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})" (is "?P a")
-proof-
-  {fix x assume xS: "x \<in> S"
-    {assume ab: "x = b"
-      then have "?P x"
-        apply simp
-        apply (rule exI[where x="1"], simp)
-        by (rule span_0)}
-    moreover
-    {assume ab: "x \<noteq> b"
-      then have "?P x"  using xS
-        apply -
-        apply (rule exI[where x=0])
-        apply (rule span_superset)
-        by simp}
-    ultimately have "?P x" by blast}
-  moreover have "subspace ?P"
-    unfolding subspace_def
-    apply auto
-    apply (simp add: mem_def)
-    apply (rule exI[where x=0])
-    using span_0[of "S - {b}"]
-    apply (simp add: mem_def)
-    apply (clarsimp simp add: mem_def)
-    apply (rule_tac x="k + ka" in exI)
-    apply (subgoal_tac "x + y - (k + ka) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
-    apply (simp only: )
-    apply (rule span_add[unfolded mem_def])
-    apply assumption+
-    apply (simp add: algebra_simps)
-    apply (clarsimp simp add: mem_def)
-    apply (rule_tac x= "c*k" in exI)
-    apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
-    apply (simp only: )
-    apply (rule span_mul[unfolded mem_def])
-    apply assumption
-    by (simp add: algebra_simps)
-  ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
-qed
-
-lemma span_breakdown_eq:
-  "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume x: "x \<in> span (insert a S)"
-    from x span_breakdown[of "a" "insert a S" "x"]
-    have ?rhs apply clarsimp
-      apply (rule_tac x= "k" in exI)
-      apply (rule set_rev_mp[of _ "span (S - {a})" _])
-      apply assumption
-      apply (rule span_mono)
-      apply blast
-      done}
-  moreover
-  { fix k assume k: "x - k *\<^sub>R a \<in> span S"
-    have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
-    have "(x - k *\<^sub>R a) + k *\<^sub>R a \<in> span (insert a S)"
-      apply (rule span_add)
-      apply (rule set_rev_mp[of _ "span S" _])
-      apply (rule k)
-      apply (rule span_mono)
-      apply blast
-      apply (rule span_mul)
-      apply (rule span_superset)
-      apply blast
-      done
-    then have ?lhs using eq by metis}
-  ultimately show ?thesis by blast
-qed
-
-text {* Hence some "reversal" results. *}
-
-lemma in_span_insert:
-  assumes a: "a \<in> span (insert b S)" and na: "a \<notin> span S"
-  shows "b \<in> span (insert a S)"
-proof-
-  from span_breakdown[of b "insert b S" a, OF insertI1 a]
-  obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
-  {assume k0: "k = 0"
-    with k have "a \<in> span S"
-      apply (simp)
-      apply (rule set_rev_mp)
-      apply assumption
-      apply (rule span_mono)
-      apply blast
-      done
-    with na  have ?thesis by blast}
-  moreover
-  {assume k0: "k \<noteq> 0"
-    have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
-    from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
-      by (simp add: algebra_simps)
-    from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
-      by (rule span_mul)
-    hence th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
-      unfolding eq' .
-
-    from k
-    have ?thesis
-      apply (subst eq)
-      apply (rule span_sub)
-      apply (rule span_mul)
-      apply (rule span_superset)
-      apply blast
-      apply (rule set_rev_mp)
-      apply (rule th)
-      apply (rule span_mono)
-      using na by blast}
-  ultimately show ?thesis by blast
-qed
-
-lemma in_span_delete:
-  assumes a: "a \<in> span S"
-  and na: "a \<notin> span (S-{b})"
-  shows "b \<in> span (insert a (S - {b}))"
-  apply (rule in_span_insert)
-  apply (rule set_rev_mp)
-  apply (rule a)
-  apply (rule span_mono)
-  apply blast
-  apply (rule na)
-  done
-
-text {* Transitivity property. *}
-
-lemma span_trans:
-  assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
-  shows "y \<in> span S"
-proof-
-  from span_breakdown[of x "insert x S" y, OF insertI1 y]
-  obtain k where k: "y -k*\<^sub>R x \<in> span (S - {x})" by auto
-  have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
-  show ?thesis
-    apply (subst eq)
-    apply (rule span_add)
-    apply (rule set_rev_mp)
-    apply (rule k)
-    apply (rule span_mono)
-    apply blast
-    apply (rule span_mul)
-    by (rule x)
-qed
-
-lemma span_insert_0[simp]: "span (insert 0 S) = span S"
-  using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
-
-text {* An explicit expansion is sometimes needed. *}
-
-lemma span_explicit:
-  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
-  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
-proof-
-  {fix x assume x: "x \<in> ?E"
-    then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
-      by blast
-    have "x \<in> span P"
-      unfolding u[symmetric]
-      apply (rule span_setsum[OF fS])
-      using span_mono[OF SP]
-      by (auto intro: span_superset span_mul)}
-  moreover
-  have "\<forall>x \<in> span P. x \<in> ?E"
-    unfolding mem_def Collect_def
-  proof(rule span_induct_alt')
-    show "?h 0"
-      apply (rule exI[where x="{}"]) by simp
-  next
-    fix c x y
-    assume x: "x \<in> P" and hy: "?h y"
-    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
-      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
-    let ?S = "insert x S"
-    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
-                  else u y"
-    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
-    {assume xS: "x \<in> S"
-      have S1: "S = (S - {x}) \<union> {x}"
-        and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
-      have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
-        using xS
-        by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
-          setsum_clauses(2)[OF fS] cong del: if_weak_cong)
-      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
-        apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
-        by (simp add: algebra_simps)
-      also have "\<dots> = c*\<^sub>R x + y"
-        by (simp add: add_commute u)
-      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
-    then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast}
-  moreover
-  {assume xS: "x \<notin> S"
-    have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
-      unfolding u[symmetric]
-      apply (rule setsum_cong2)
-      using xS by auto
-    have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
-      by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
-  ultimately have "?Q ?S ?u (c*\<^sub>R x + y)"
-    by (cases "x \<in> S", simp, simp)
-    then show "?h (c*\<^sub>R x + y)"
-      apply -
-      apply (rule exI[where x="?S"])
-      apply (rule exI[where x="?u"]) by metis
-  qed
-  ultimately show ?thesis by blast
-qed
-
-lemma dependent_explicit:
-  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))" (is "?lhs = ?rhs")
-proof-
-  {assume dP: "dependent P"
-    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
-      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
-      unfolding dependent_def span_explicit by blast
-    let ?S = "insert a S"
-    let ?u = "\<lambda>y. if y = a then - 1 else u y"
-    let ?v = a
-    from aP SP have aS: "a \<notin> S" by blast
-    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
-    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
-      using fS aS
-      apply (simp add: setsum_clauses field_simps)
-      apply (subst (2) ua[symmetric])
-      apply (rule setsum_cong2)
-      by auto
-    with th0 have ?rhs
-      apply -
-      apply (rule exI[where x= "?S"])
-      apply (rule exI[where x= "?u"])
-      by clarsimp}
-  moreover
-  {fix S u v assume fS: "finite S"
-      and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
-    and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
-    let ?a = v
-    let ?S = "S - {v}"
-    let ?u = "\<lambda>i. (- u i) / u v"
-    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
-    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
-      using fS vS uv
-      by (simp add: setsum_diff1 divide_inverse field_simps)
-    also have "\<dots> = ?a"
-      unfolding scaleR_right.setsum [symmetric] u
-      using uv by simp
-    finally  have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
-    with th0 have ?lhs
-      unfolding dependent_def span_explicit
-      apply -
-      apply (rule bexI[where x= "?a"])
-      apply (simp_all del: scaleR_minus_left)
-      apply (rule exI[where x= "?S"])
-      by (auto simp del: scaleR_minus_left)}
-  ultimately show ?thesis by blast
-qed
-
-
-lemma span_finite:
-  assumes fS: "finite S"
-  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
-  (is "_ = ?rhs")
-proof-
-  {fix y assume y: "y \<in> span S"
-    from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
-      u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" unfolding span_explicit by blast
-    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
-    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
-      using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
-    hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
-    hence "y \<in> ?rhs" by auto}
-  moreover
-  {fix y u assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
-    then have "y \<in> span S" using fS unfolding span_explicit by auto}
-  ultimately show ?thesis by blast
-qed
-
-lemma Int_Un_cancel: "(A \<union> B) \<inter> A = A" "(A \<union> B) \<inter> B = B" by auto
-
-lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
-proof safe
-  fix x assume "x \<in> span (A \<union> B)"
-  then obtain S u where S: "finite S" "S \<subseteq> A \<union> B" and x: "x = (\<Sum>v\<in>S. u v *\<^sub>R v)"
-    unfolding span_explicit by auto
-
-  let ?Sa = "\<Sum>v\<in>S\<inter>A. u v *\<^sub>R v"
-  let ?Sb = "(\<Sum>v\<in>S\<inter>(B - A). u v *\<^sub>R v)"
-  show "x \<in> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
-  proof
-    show "x = (case (?Sa, ?Sb) of (a, b) \<Rightarrow> a + b)"
-      unfolding x using S
-      by (simp, subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
-
-    from S have "?Sa \<in> span A" unfolding span_explicit
-      by (auto intro!: exI[of _ "S \<inter> A"])
-    moreover from S have "?Sb \<in> span B" unfolding span_explicit
-      by (auto intro!: exI[of _ "S \<inter> (B - A)"])
-    ultimately show "(?Sa, ?Sb) \<in> span A \<times> span B" by simp
-  qed
-next
-  fix a b assume "a \<in> span A" and "b \<in> span B"
-  then obtain Sa ua Sb ub where span:
-    "finite Sa" "Sa \<subseteq> A" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
-    "finite Sb" "Sb \<subseteq> B" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
-    unfolding span_explicit by auto
-  let "?u v" = "(if v \<in> Sa then ua v else 0) + (if v \<in> Sb then ub v else 0)"
-  from span have "finite (Sa \<union> Sb)" "Sa \<union> Sb \<subseteq> A \<union> B"
-    and "a + b = (\<Sum>v\<in>(Sa\<union>Sb). ?u v *\<^sub>R v)"
-    unfolding setsum_addf scaleR_left_distrib
-    by (auto simp add: if_distrib cond_application_beta setsum_cases Int_Un_cancel)
-  thus "a + b \<in> span (A \<union> B)"
-    unfolding span_explicit by (auto intro!: exI[of _ ?u])
-qed
-
-text {* This is useful for building a basis step-by-step. *}
-
-lemma independent_insert:
-  "independent(insert a S) \<longleftrightarrow>
-      (if a \<in> S then independent S
-                else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume aS: "a \<in> S"
-    hence ?thesis using insert_absorb[OF aS] by simp}
-  moreover
-  {assume aS: "a \<notin> S"
-    {assume i: ?lhs
-      then have ?rhs using aS
-        apply simp
-        apply (rule conjI)
-        apply (rule independent_mono)
-        apply assumption
-        apply blast
-        by (simp add: dependent_def)}
-    moreover
-    {assume i: ?rhs
-      have ?lhs using i aS
-        apply simp
-        apply (auto simp add: dependent_def)
-        apply (case_tac "aa = a", auto)
-        apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
-        apply simp
-        apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
-        apply (subgoal_tac "insert aa (S - {aa}) = S")
-        apply simp
-        apply blast
-        apply (rule in_span_insert)
-        apply assumption
-        apply blast
-        apply blast
-        done}
-    ultimately have ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-text {* The degenerate case of the Exchange Lemma. *}
-
-lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
-  by blast
-
-lemma spanning_subset_independent:
-  assumes BA: "B \<subseteq> A" and iA: "independent A"
-  and AsB: "A \<subseteq> span B"
-  shows "A = B"
-proof
-  from BA show "B \<subseteq> A" .
-next
-  from span_mono[OF BA] span_mono[OF AsB]
-  have sAB: "span A = span B" unfolding span_span by blast
-
-  {fix x assume x: "x \<in> A"
-    from iA have th0: "x \<notin> span (A - {x})"
-      unfolding dependent_def using x by blast
-    from x have xsA: "x \<in> span A" by (blast intro: span_superset)
-    have "A - {x} \<subseteq> A" by blast
-    hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
-    {assume xB: "x \<notin> B"
-      from xB BA have "B \<subseteq> A -{x}" by blast
-      hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
-      with th1 th0 sAB have "x \<notin> span A" by blast
-      with x have False by (metis span_superset)}
-    then have "x \<in> B" by blast}
-  then show "A \<subseteq> B" by blast
-qed
-
-text {* The general case of the Exchange Lemma, the key to what follows. *}
-
-lemma exchange_lemma:
-  assumes f:"finite t" and i: "independent s"
-  and sp:"s \<subseteq> span t"
-  shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
-using f i sp
-proof(induct "card (t - s)" arbitrary: s t rule: less_induct)
-  case less
-  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
-  let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
-  let ?ths = "\<exists>t'. ?P t'"
-  {assume st: "s \<subseteq> t"
-    from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
-      by (auto intro: span_superset)}
-  moreover
-  {assume st: "t \<subseteq> s"
-
-    from spanning_subset_independent[OF st s sp]
-      st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
-      by (auto intro: span_superset)}
-  moreover
-  {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
-    from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
-      from b have "t - {b} - s \<subset> t - s" by blast
-      then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
-        by (auto intro: psubset_card_mono)
-      from b ft have ct0: "card t \<noteq> 0" by auto
-    {assume stb: "s \<subseteq> span(t -{b})"
-      from ft have ftb: "finite (t -{b})" by auto
-      from less(1)[OF cardlt ftb s stb]
-      obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" and fu: "finite u" by blast
-      let ?w = "insert b u"
-      have th0: "s \<subseteq> insert b u" using u by blast
-      from u(3) b have "u \<subseteq> s \<union> t" by blast
-      then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
-      have bu: "b \<notin> u" using b u by blast
-      from u(1) ft b have "card u = (card t - 1)" by auto
-      then
-      have th2: "card (insert b u) = card t"
-        using card_insert_disjoint[OF fu bu] ct0 by auto
-      from u(4) have "s \<subseteq> span u" .
-      also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
-      finally have th3: "s \<subseteq> span (insert b u)" .
-      from th0 th1 th2 th3 fu have th: "?P ?w"  by blast
-      from th have ?ths by blast}
-    moreover
-    {assume stb: "\<not> s \<subseteq> span(t -{b})"
-      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
-      have ab: "a \<noteq> b" using a b by blast
-      have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
-      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
-        using cardlt ft a b by auto
-      have ft': "finite (insert a (t - {b}))" using ft by auto
-      {fix x assume xs: "x \<in> s"
-        have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
-        from b(1) have "b \<in> span t" by (simp add: span_superset)
-        have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
-          using  a sp unfolding subset_eq by auto
-        from xs sp have "x \<in> span t" by blast
-        with span_mono[OF t]
-        have x: "x \<in> span (insert b (insert a (t - {b})))" ..
-        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
-      then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
-
-      from less(1)[OF mlt ft' s sp'] obtain u where
-        u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
-        "s \<subseteq> span u" by blast
-      from u a b ft at ct0 have "?P u" by auto
-      then have ?ths by blast }
-    ultimately have ?ths by blast
-  }
-  ultimately
-  show ?ths  by blast
-qed
-
-text {* This implies corresponding size bounds. *}
-
-lemma independent_span_bound:
-  assumes f: "finite t" and i: "independent s" and sp:"s \<subseteq> span t"
-  shows "finite s \<and> card s \<le> card t"
-  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
-
-
-lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
-proof-
-  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
-  show ?thesis unfolding eq
-    apply (rule finite_imageI)
-    apply (rule finite)
-    done
-qed
-
-subsection{* Euclidean Spaces as Typeclass*}
+subsection {* Type class of Euclidean spaces *}
 
 class euclidean_space = real_inner +
   fixes dimension :: "'a itself \<Rightarrow> nat"
@@ -1621,67 +53,6 @@
   "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_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
-  by (rule inj_onI, rule ccontr, cut_tac i=x and j=y in dot_basis, simp)
-
-lemma (in euclidean_space) basis_finite: "basis ` {DIM('a)..} = {0}"
-  by (auto intro: image_eqI [where x="DIM('a)"])
-
-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 * 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]
-  apply clarify
-  apply (drule_tac f="inner (basis a)" in arg_cong)
-  apply (simp add: inner_right.setsum dot_basis)
-  done
-
-lemma dimensionI:
-  assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0::'a::euclidean_space};
-    independent (basis ` {..<d} :: 'a set);
-    inj_on (basis :: nat \<Rightarrow> 'a) {..<d} \<rbrakk> \<Longrightarrow> P d"
-  shows "P DIM('a::euclidean_space)"
-  using DIM_positive basis_finite independent_basis basis_inj
-  by (rule assms)
-
-lemma (in euclidean_space) dimension_eq:
-  assumes "\<And>i. i < d \<Longrightarrow> basis i \<noteq> 0"
-  assumes "\<And>i. d \<le> i \<Longrightarrow> basis i = 0"
-  shows "DIM('a) = d"
-proof (rule linorder_cases [of "DIM('a)" d])
-  assume "DIM('a) < d"
-  hence "basis DIM('a) \<noteq> 0" by (rule assms)
-  thus ?thesis by simp
-next
-  assume "d < DIM('a)"
-  hence "basis d \<noteq> 0" by simp
-  thus ?thesis by (simp add: assms)
-next
-  assume "DIM('a) = d" thus ?thesis .
-qed
-
-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 (in euclidean_space) basis_neq_0 [intro]:
   assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
   using assms by simp
@@ -1773,1522 +144,18 @@
   by (simp add: DIM_positive)
 
 lemma euclidean_inner:
-  "x \<bullet> (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
-proof -
-  have "x \<bullet> y = (\<Sum>i<DIM('a). x $$ i *\<^sub>R basis i) \<bullet>
-                (\<Sum>i<DIM('a). y $$ i *\<^sub>R (basis i :: 'a))"
-    by (subst (1 2) euclidean_representation) simp
-  also have "\<dots> = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
-    unfolding inner_left.setsum inner_right.setsum
-    by (auto simp add: dot_basis if_distrib setsum_cases intro!: setsum_cong)
-  finally show ?thesis .
-qed
-
-lemma span_basis: "span (range basis) = (UNIV :: 'a::euclidean_space set)"
-proof -
-  { fix x :: 'a
-    have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
-      by (simp add: span_setsum span_mul span_superset)
-    hence "x \<in> span (range basis)"
-      by (simp only: euclidean_representation [symmetric])
-  } thus ?thesis by auto
-qed
-
-lemma basis_representation:
-  "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
-proof -
-  have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
-  have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
-    unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
-  thus ?thesis by fastsimp
-qed
-
-lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
-  apply(subst span_basis[symmetric]) unfolding range_basis by auto
-
-lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
-  apply(subst card_image) using basis_inj by auto
-
-lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
-  unfolding span_basis' ..
-
-lemma norm_basis[simp]:"norm (basis i::'a::euclidean_space) = (if i<DIM('a) then 1 else 0)"
-  unfolding norm_eq_sqrt_inner dot_basis by auto
+  "inner x (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) * (y $$ i))"
+  by (subst (1 2) euclidean_representation,
+    simp add: inner_left.setsum inner_right.setsum
+    dot_basis if_distrib setsum_cases mult_commute)
 
 lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
   unfolding euclidean_component_def
-  apply(rule order_trans[OF real_inner_class.Cauchy_Schwarz_ineq2]) by auto
-
-lemma norm_bound_component_le: "norm (x::'a::euclidean_space) \<le> e \<Longrightarrow> \<bar>x$$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])
-  apply (rule order_trans[OF setsum_norm])
-  by (auto intro!: setsum_mono)
-
-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"
-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"
-    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 $$ i\<bar>) ?U) P"
-    apply (rule setsum_mono)    by (rule norm_le_l1)
-  also have "\<dots> \<le> 2 * ?d * e"
-    unfolding th0 th1
-  proof(rule setsum_bounded)
-    fix i assume i: "i \<in> ?U"
-    let ?Pp = "{x. x\<in> P \<and> f x $$ 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[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
-      unfolding euclidean_component.setsum by(auto intro: abs_le_D1)
-    have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
-      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
-      unfolding euclidean_component.setsum euclidean_component.minus
-      by(auto simp add: setsum_negf intro: abs_le_D1)
-    have "setsum (\<lambda>x. \<bar>f x $$ 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 by auto
-    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
-    finally show "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P \<le> 2*e" .
-  qed
-  finally show ?thesis .
-qed
-
-lemma choice_iff': "(\<forall>x<d. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x<d. P x (f x))" by metis
-
-lemma lambda_skolem': "(\<forall>i<DIM('a::euclidean_space). \<exists>x. P i x) \<longleftrightarrow>
-   (\<exists>x::'a. \<forall>i<DIM('a). P i (x$$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$$i)" using i by auto
-    }
-    hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
-    hence ?rhs by metis }
-  ultimately show ?thesis by metis
-qed
-
-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)"
-
-lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
-  unfolding eucl_less[where 'a='a] by auto
-
-lemma euclidean_trans[trans]:
-  fixes x y z :: "'a::ordered_euclidean_space"
-  shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
-  and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
-  and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
-  by (force simp: eucl_less[where 'a='a] eucl_le[where 'a='a])+
-
-subsection {* Linearity and Bilinearity continued *}
-
-lemma linear_bounded:
-  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  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
-  {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]) ..
-    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$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
-      unfolding norm_scaleR
-      apply (simp only: mult_commute)
-      apply (rule mult_mono)
-      by (auto simp add: field_simps) }
-    then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
-    from setsum_norm_le[OF fS, of "\<lambda>i. (x$$i) *\<^sub>R (f (basis i))", OF th]
-    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
-  then show ?thesis by blast
-qed
-
-lemma linear_bounded_pos:
-  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes lf: "linear f"
-  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
-proof-
-  from linear_bounded[OF lf] obtain B where
-    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
-  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] simp add:euclidean_component.zero)
-      hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
-      with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 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
-    }
-    then have Bp: "B \<ge> 0" by (metis not_leE)
-    {fix x::"'a"
-      have "norm (f x) \<le> ?K *  norm x"
-      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
-      apply (auto simp add: field_simps split add: abs_split)
-      apply (erule order_trans, simp)
-      done
-  }
-  then show ?thesis using Kp by blast
-qed
-
-lemma linear_conv_bounded_linear:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  shows "linear f \<longleftrightarrow> bounded_linear f"
-proof
-  assume "linear f"
-  show "bounded_linear f"
-  proof
-    fix x y show "f (x + y) = f x + f y"
-      using `linear f` unfolding linear_def by simp
-  next
-    fix r x show "f (scaleR r x) = scaleR r (f x)"
-      using `linear f` unfolding linear_def by simp
-  next
-    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
-      using `linear f` by (rule linear_bounded)
-    thus "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
-      by (simp add: mult_commute)
-  qed
-next
-  assume "bounded_linear f"
-  then interpret f: bounded_linear f .
-  show "linear f"
-    by (simp add: f.add f.scaleR linear_def)
-qed
-
-lemma bounded_linearI': fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
-  shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
-  by(rule linearI[OF assms])
-
-
-lemma bilinear_bounded:
-  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$$i) *\<^sub>R basis i) ?N))" 
-      apply(subst euclidean_representation[where 'a='m])
-      apply(subst euclidean_representation[where 'a='n]) ..
-    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$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)
-      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 (rule mult_mono)
-      apply (auto simp add: zero_le_mult_iff component_le_norm)
-      done}
-  then show ?thesis by metis
-qed
-
-lemma bilinear_bounded_pos:
-  fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
-  assumes bh: "bilinear h"
-  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof-
-  from bilinear_bounded[OF bh] obtain B where
-    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
-  let ?K = "\<bar>B\<bar> + 1"
-  have Kp: "?K > 0" by arith
-  have KB: "B < ?K" by arith
-  {fix x::'a and y::'b
-    from KB Kp
-    have "B * norm x * norm y \<le> ?K * norm x * norm y"
-      apply -
-      apply (rule mult_right_mono, rule mult_right_mono)
-      by auto
-    then have "norm (h x y) \<le> ?K * norm x * norm y"
-      using B[rule_format, of x y] by simp}
-  with Kp show ?thesis by blast
-qed
-
-lemma bilinear_conv_bounded_bilinear:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
-  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
-proof
-  assume "bilinear h"
-  show "bounded_bilinear h"
-  proof
-    fix x y z show "h (x + y) z = h x z + h y z"
-      using `bilinear h` unfolding bilinear_def linear_def by simp
-  next
-    fix x y z show "h x (y + z) = h x y + h x z"
-      using `bilinear h` unfolding bilinear_def linear_def by simp
-  next
-    fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
-      using `bilinear h` unfolding bilinear_def linear_def
-      by simp
-  next
-    fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
-      using `bilinear h` unfolding bilinear_def linear_def
-      by simp
-  next
-    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-      using `bilinear h` by (rule bilinear_bounded)
-    thus "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
-      by (simp add: mult_ac)
-  qed
-next
-  assume "bounded_bilinear h"
-  then interpret h: bounded_bilinear h .
-  show "bilinear h"
-    unfolding bilinear_def linear_conv_bounded_linear
-    using h.bounded_linear_left h.bounded_linear_right
-    by simp
-qed
-
-subsection {* We continue. *}
-
-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
-
-lemma dependent_biggerset: "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
-  by (metis independent_bound not_less)
-
-text {* Hence we can create a maximal independent subset. *}
-
-lemma maximal_independent_subset_extend:
-  assumes sv: "(S::('a::euclidean_space) set) \<subseteq> V" and iS: "independent S"
-  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
-  using sv iS
-proof(induct "DIM('a) - card S" arbitrary: S rule: less_induct)
-  case less
-  note sv = `S \<subseteq> V` and i = `independent S`
-  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
-  let ?ths = "\<exists>x. ?P x"
-  let ?d = "DIM('a)"
-  {assume "V \<subseteq> span S"
-    then have ?ths  using sv i by blast }
-  moreover
-  {assume VS: "\<not> V \<subseteq> span S"
-    from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
-    from a have aS: "a \<notin> S" by (auto simp add: span_superset)
-    have th0: "insert a S \<subseteq> V" using a sv by blast
-    from independent_insert[of a S]  i a
-    have th1: "independent (insert a S)" by auto
-    have mlt: "?d - card (insert a S) < ?d - card S"
-      using aS a independent_bound[OF th1]
-      by auto
-
-    from less(1)[OF mlt th0 th1]
-    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
-      by blast
-    from B have "?P B" by auto
-    then have ?ths by blast}
-  ultimately show ?ths by blast
-qed
-
-lemma maximal_independent_subset:
-  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
-  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty)
-
-
-text {* Notion of dimension. *}
-
-definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
-
-lemma basis_exists:  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
-unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
-using maximal_independent_subset[of V] independent_bound
-by auto
-
-text {* Consequences of independence or spanning for cardinality. *}
-
-lemma independent_card_le_dim: 
-  assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B" shows "card B \<le> dim V"
-proof -
-  from basis_exists[of V] `B \<subseteq> V`
-  obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
-  with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
-  show ?thesis by auto
-qed
-
-lemma span_card_ge_dim:  "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
-  by (metis basis_exists[of V] independent_span_bound subset_trans)
-
-lemma basis_card_eq_dim:
-  "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
-  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
-
-lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
-  by (metis basis_card_eq_dim)
-
-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 by auto
-
-lemma dim_subset:
-  "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
-  using basis_exists[of T] basis_exists[of S]
-  by (metis independent_card_le_dim subset_trans)
-
-lemma dim_subset_UNIV: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
-  by (metis dim_subset subset_UNIV dim_UNIV)
-
-text {* Converses to those. *}
-
-lemma card_ge_dim_independent:
-  assumes BV:"(B::('a::euclidean_space) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
-  shows "V \<subseteq> span B"
-proof-
-  {fix a assume aV: "a \<in> V"
-    {assume aB: "a \<notin> span B"
-      then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
-      from aV BV have th0: "insert a B \<subseteq> V" by blast
-      from aB have "a \<notin>B" by (auto simp add: span_superset)
-      with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
-    then have "a \<in> span B"  by blast}
-  then show ?thesis by blast
-qed
-
-lemma card_le_dim_spanning:
-  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V" and VB: "V \<subseteq> span B"
-  and fB: "finite B" and dVB: "dim V \<ge> card B"
-  shows "independent B"
-proof-
-  {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
-    from a fB have c0: "card B \<noteq> 0" by auto
-    from a fB have cb: "card (B -{a}) = card B - 1" by auto
-    from BV a have th0: "B -{a} \<subseteq> V" by blast
-    {fix x assume x: "x \<in> V"
-      from a have eq: "insert a (B -{a}) = B" by blast
-      from x VB have x': "x \<in> span B" by blast
-      from span_trans[OF a(2), unfolded eq, OF x']
-      have "x \<in> span (B -{a})" . }
-    then have th1: "V \<subseteq> span (B -{a})" by blast
-    have th2: "finite (B -{a})" using fB by auto
-    from span_card_ge_dim[OF th0 th1 th2]
-    have c: "dim V \<le> card (B -{a})" .
-    from c c0 dVB cb have False by simp}
-  then show ?thesis unfolding dependent_def by blast
-qed
-
-lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
-  by (metis order_eq_iff card_le_dim_spanning
-    card_ge_dim_independent)
-
-text {* More general size bound lemmas. *}
-
-lemma independent_bound_general:
-  "independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
-  by (metis independent_card_le_dim independent_bound subset_refl)
-
-lemma dependent_biggerset_general: "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
-  using independent_bound_general[of S] by (metis linorder_not_le)
-
-lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
-proof-
-  have th0: "dim S \<le> dim (span S)"
-    by (auto simp add: subset_eq intro: dim_subset span_superset)
-  from basis_exists[of S]
-  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
-  have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
-  have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
-  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
-    using fB(2)  by arith
-qed
-
-lemma subset_le_dim: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
-  by (metis dim_span dim_subset)
-
-lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
-  by (metis dim_span)
-
-lemma spans_image:
-  assumes lf: "linear f" and VB: "V \<subseteq> span B"
-  shows "f ` V \<subseteq> span (f ` B)"
-  unfolding span_linear_image[OF lf]
-  by (metis VB image_mono)
-
-lemma dim_image_le:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S)"
-proof-
-  from basis_exists[of S] obtain B where
-    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
-  have "dim (f ` S) \<le> card (f ` B)"
-    apply (rule span_card_ge_dim)
-    using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
-  also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
-  finally show ?thesis .
-qed
-
-text {* Relation between bases and injectivity/surjectivity of map. *}
-
-lemma spanning_surjective_image:
-  assumes us: "UNIV \<subseteq> span S"
-  and lf: "linear f" and sf: "surj f"
-  shows "UNIV \<subseteq> span (f ` S)"
-proof-
-  have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
-  also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
-finally show ?thesis .
-qed
-
-lemma independent_injective_image:
-  assumes iS: "independent S" and lf: "linear f" and fi: "inj f"
-  shows "independent (f ` S)"
-proof-
-  {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
-    have eq: "f ` S - {f a} = f ` (S - {a})" using fi
-      by (auto simp add: inj_on_def)
-    from a have "f a \<in> f ` span (S -{a})"
-      unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
-    hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
-    with a(1) iS  have False by (simp add: dependent_def) }
-  then show ?thesis unfolding dependent_def by blast
-qed
-
-text {* Picking an orthogonal replacement for a spanning set. *}
-
-    (* FIXME : Move to some general theory ?*)
-definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
-
-lemma vector_sub_project_orthogonal: "(b::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
-  unfolding inner_simps by auto
-
-lemma basis_orthogonal:
-  fixes B :: "('a::euclidean_space) set"
-  assumes fB: "finite B"
-  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
-  (is " \<exists>C. ?P B C")
-proof(induct rule: finite_induct[OF fB])
-  case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
-next
-  case (2 a B)
-  note fB = `finite B` and aB = `a \<notin> B`
-  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
-  obtain C where C: "finite C" "card C \<le> card B"
-    "span C = span B" "pairwise orthogonal C" by blast
-  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
-  let ?C = "insert ?a C"
-  from C(1) have fC: "finite ?C" by simp
-  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
-  {fix x k
-    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
-    have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
-      apply (simp only: scaleR_right_diff_distrib th0)
-      apply (rule span_add_eq)
-      apply (rule span_mul)
-      apply (rule span_setsum[OF C(1)])
-      apply clarify
-      apply (rule span_mul)
-      by (rule span_superset)}
-  then have SC: "span ?C = span (insert a B)"
-    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
-  thm pairwise_def
-  {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
-    {assume xa: "x = ?a" and ya: "y = ?a"
-      have "orthogonal x y" using xa ya xy by blast}
-    moreover
-    {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
-      from ya have Cy: "C = insert y (C - {y})" by blast
-      have fth: "finite (C - {y})" using C by simp
-      have "orthogonal x y"
-        using xa ya
-        unfolding orthogonal_def xa inner_simps diff_eq_0_iff_eq
-        apply simp
-        apply (subst Cy)
-        using C(1) fth
-        apply (simp only: setsum_clauses)
-        apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
-        apply (rule setsum_0')
-        apply clarsimp
-        apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
-        by auto}
-    moreover
-    {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
-      from xa have Cx: "C = insert x (C - {x})" by blast
-      have fth: "finite (C - {x})" using C by simp
-      have "orthogonal x y"
-        using xa ya
-        unfolding orthogonal_def ya inner_simps diff_eq_0_iff_eq
-        apply simp
-        apply (subst Cx)
-        using C(1) fth
-        apply (simp only: setsum_clauses)
-        apply (subst inner_commute[of x])
-        apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
-        apply (rule setsum_0')
-        apply clarsimp
-        apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
-        by auto}
-    moreover
-    {assume xa: "x \<in> C" and ya: "y \<in> C"
-      have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
-    ultimately have "orthogonal x y" using xC yC by blast}
-  then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
-  from fC cC SC CPO have "?P (insert a B) ?C" by blast
-  then show ?case by blast
-qed
-
-lemma orthogonal_basis_exists:
-  fixes V :: "('a::euclidean_space) set"
-  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
-proof-
-  from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
-  from B have fB: "finite B" "card B = dim V" using independent_bound by auto
-  from basis_orthogonal[OF fB(1)] obtain C where
-    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
-  from C B
-  have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
-  from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
-  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
-  have iC: "independent C" by (simp add: dim_span)
-  from C fB have "card C \<le> dim V" by simp
-  moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
-    by (simp add: dim_span)
-  ultimately have CdV: "card C = dim V" using C(1) by simp
-  from C B CSV CdV iC show ?thesis by auto
-qed
-
-lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
-  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
-  by(auto simp add: span_span)
-
-text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
-
-lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
-  assumes sU: "span S \<noteq> UNIV"
-  shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
-proof-
-  from sU obtain a where a: "a \<notin> span S" by blast
-  from orthogonal_basis_exists obtain B where
-    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
-    by blast
-  from B have fB: "finite B" "card B = dim S" using independent_bound by auto
-  from span_mono[OF B(2)] span_mono[OF B(3)]
-  have sSB: "span S = span B" by (simp add: span_span)
-  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
-  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
-    unfolding sSB
-    apply (rule span_setsum[OF fB(1)])
-    apply clarsimp
-    apply (rule span_mul)
-    by (rule span_superset)
-  with a have a0:"?a  \<noteq> 0" by auto
-  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
-  proof(rule span_induct')
-    show "subspace (\<lambda>x. ?a \<bullet> x = 0)" by (auto simp add: subspace_def mem_def inner_simps)
-next
-    {fix x assume x: "x \<in> B"
-      from x have B': "B = insert x (B - {x})" by blast
-      have fth: "finite (B - {x})" using fB by simp
-      have "?a \<bullet> x = 0"
-        apply (subst B') using fB fth
-        unfolding setsum_clauses(2)[OF fth]
-        apply simp unfolding inner_simps
-        apply (clarsimp simp add: inner_simps dot_lsum)
-        apply (rule setsum_0', rule ballI)
-        unfolding inner_commute
-        by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
-    then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
-  qed
-  with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
-qed
-
-lemma span_not_univ_subset_hyperplane:
-  assumes SU: "span S \<noteq> (UNIV ::('a::euclidean_space) set)"
-  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
-  using span_not_univ_orthogonal[OF SU] by auto
-
-lemma lowdim_subset_hyperplane: fixes S::"('a::euclidean_space) set"
-  assumes d: "dim S < DIM('a)"
-  shows "\<exists>(a::'a). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
-proof-
-  {assume "span S = UNIV"
-    hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
-    hence "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
-    with d have False by arith}
-  hence th: "span S \<noteq> UNIV" by blast
-  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
-qed
-
-text {* We can extend a linear basis-basis injection to the whole set. *}
-
-lemma linear_indep_image_lemma:
-  assumes lf: "linear f" and fB: "finite B"
-  and ifB: "independent (f ` B)"
-  and fi: "inj_on f B" and xsB: "x \<in> span B"
-  and fx: "f x = 0"
-  shows "x = 0"
-  using fB ifB fi xsB fx
-proof(induct arbitrary: x rule: finite_induct[OF fB])
-  case 1 thus ?case by (auto simp add:  span_empty)
-next
-  case (2 a b x)
-  have fb: "finite b" using "2.prems" by simp
-  have th0: "f ` b \<subseteq> f ` (insert a b)"
-    apply (rule image_mono) by blast
-  from independent_mono[ OF "2.prems"(2) th0]
-  have ifb: "independent (f ` b)"  .
-  have fib: "inj_on f b"
-    apply (rule subset_inj_on [OF "2.prems"(3)])
-    by blast
-  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
-  obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
-  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
-    unfolding span_linear_image[OF lf]
-    apply (rule imageI)
-    using k span_mono[of "b-{a}" b] by blast
-  hence "f x - k*\<^sub>R f a \<in> span (f ` b)"
-    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
-  hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
-    using "2.prems"(5) by simp
-  {assume k0: "k = 0"
-    from k0 k have "x \<in> span (b -{a})" by simp
-    then have "x \<in> span b" using span_mono[of "b-{a}" b]
-      by blast}
-  moreover
-  {assume k0: "k \<noteq> 0"
-    from span_mul[OF th, of "- 1/ k"] k0
-    have th1: "f a \<in> span (f ` b)"
-      by auto
-    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
-    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
-    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
-    have "f a \<notin> span (f ` b)" using tha
-      using "2.hyps"(2)
-      "2.prems"(3) by auto
-    with th1 have False by blast
-    then have "x \<in> span b" by blast}
-  ultimately have xsb: "x \<in> span b" by blast
-  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
-  show "x = 0" .
-qed
-
-text {* We can extend a linear mapping from basis. *}
+  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
 
-lemma linear_independent_extend_lemma:
-  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
-  assumes fi: "finite B" and ib: "independent B"
-  shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
-           \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
-           \<and> (\<forall>x\<in> B. g x = f x)"
-using ib fi
-proof(induct rule: finite_induct[OF fi])
-  case 1 thus ?case by (auto simp add: span_empty)
-next
-  case (2 a b)
-  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
-    by (simp_all add: independent_insert)
-  from "2.hyps"(3)[OF ibf] obtain g where
-    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
-    "\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
-  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
-  {fix z assume z: "z \<in> span (insert a b)"
-    have th0: "z - ?h z *\<^sub>R a \<in> span b"
-      apply (rule someI_ex)
-      unfolding span_breakdown_eq[symmetric]
-      using z .
-    {fix k assume k: "z - k *\<^sub>R a \<in> span b"
-      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
-        by (simp add: field_simps scaleR_left_distrib [symmetric])
-      from span_sub[OF th0 k]
-      have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
-      {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
-        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
-        have "a \<in> span b" by simp
-        with "2.prems"(1) "2.hyps"(2) have False
-          by (auto simp add: dependent_def)}
-      then have "k = ?h z" by blast}
-    with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
-  note h = this
-  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
-  {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
-    have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
-      by (simp add: algebra_simps)
-    have addh: "?h (x + y) = ?h x + ?h y"
-      apply (rule conjunct2[OF h, rule_format, symmetric])
-      apply (rule span_add[OF x y])
-      unfolding tha
-      by (metis span_add x y conjunct1[OF h, rule_format])
-    have "?g (x + y) = ?g x + ?g y"
-      unfolding addh tha
-      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
-      by (simp add: scaleR_left_distrib)}
-  moreover
-  {fix x:: "'a" and c:: real  assume x: "x \<in> span (insert a b)"
-    have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
-      by (simp add: algebra_simps)
-    have hc: "?h (c *\<^sub>R x) = c * ?h x"
-      apply (rule conjunct2[OF h, rule_format, symmetric])
-      apply (metis span_mul x)
-      by (metis tha span_mul x conjunct1[OF h])
-    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
-      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
-      by (simp add: algebra_simps)}
-  moreover
-  {fix x assume x: "x \<in> (insert a b)"
-    {assume xa: "x = a"
-      have ha1: "1 = ?h a"
-        apply (rule conjunct2[OF h, rule_format])
-        apply (metis span_superset insertI1)
-        using conjunct1[OF h, OF span_superset, OF insertI1]
-        by (auto simp add: span_0)
-
-      from xa ha1[symmetric] have "?g x = f x"
-        apply simp
-        using g(2)[rule_format, OF span_0, of 0]
-        by simp}
-    moreover
-    {assume xb: "x \<in> b"
-      have h0: "0 = ?h x"
-        apply (rule conjunct2[OF h, rule_format])
-        apply (metis  span_superset x)
-        apply simp
-        apply (metis span_superset xb)
-        done
-      have "?g x = f x"
-        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
-    ultimately have "?g x = f x" using x by blast }
-  ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
-qed
-
-lemma linear_independent_extend:
-  assumes iB: "independent (B:: ('a::euclidean_space) set)"
-  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
-proof-
-  from maximal_independent_subset_extend[of B UNIV] iB
-  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
-
-  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
-  obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
-           \<and> (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
-           \<and> (\<forall>x\<in> C. g x = f x)" by blast
-  from g show ?thesis unfolding linear_def using C
-    apply clarsimp by blast
-qed
-
-text {* Can construct an isomorphism between spaces of same dimension. *}
-
-lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
-  and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
-using fB c
-proof(induct arbitrary: B rule: finite_induct[OF fA])
-  case 1 thus ?case by simp
-next
-  case (2 x s t)
-  thus ?case
-  proof(induct rule: finite_induct[OF "2.prems"(1)])
-    case 1    then show ?case by simp
-  next
-    case (2 y t)
-    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
-    from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
-      f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
-    from f "2.prems"(2) "2.hyps"(2) show ?case
-      apply -
-      apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
-      by (auto simp add: inj_on_def)
-  qed
-qed
-
-lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
-  c: "card A = card B"
-  shows "A = B"
-proof-
-  from fB AB have fA: "finite A" by (auto intro: finite_subset)
-  from fA fB have fBA: "finite (B - A)" by auto
-  have e: "A \<inter> (B - A) = {}" by blast
-  have eq: "A \<union> (B - A) = B" using AB by blast
-  from card_Un_disjoint[OF fA fBA e, unfolded eq c]
-  have "card (B - A) = 0" by arith
-  hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
-  with AB show "A = B" by blast
-qed
-
-lemma subspace_isomorphism:
-  assumes s: "subspace (S:: ('a::euclidean_space) set)"
-  and t: "subspace (T :: ('b::euclidean_space) set)"
-  and d: "dim S = dim T"
-  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
-proof-
-  from basis_exists[of S] independent_bound obtain B where
-    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
-  from basis_exists[of T] independent_bound obtain C where
-    C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
-  from B(4) C(4) card_le_inj[of B C] d obtain f where
-    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
-  from linear_independent_extend[OF B(2)] obtain g where
-    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
-  from inj_on_iff_eq_card[OF fB, of f] f(2)
-  have "card (f ` B) = card B" by simp
-  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
-    by simp
-  have "g ` B = f ` B" using g(2)
-    by (auto simp add: image_iff)
-  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
-  finally have gBC: "g ` B = C" .
-  have gi: "inj_on g B" using f(2) g(2)
-    by (auto simp add: inj_on_def)
-  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
-  {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
-    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
-    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
-    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
-    have "x=y" using g0[OF th1 th0] by simp }
-  then have giS: "inj_on g S"
-    unfolding inj_on_def by blast
-  from span_subspace[OF B(1,3) s]
-  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
-  also have "\<dots> = span C" unfolding gBC ..
-  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
-  finally have gS: "g ` S = T" .
-  from g(1) gS giS show ?thesis by blast
-qed
-
-text {* Linear functions are equal on a subspace if they are on a spanning set. *}
-
-lemma subspace_kernel:
-  assumes lf: "linear f"
-  shows "subspace {x. f x = 0}"
-apply (simp add: subspace_def)
-by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
-
-lemma linear_eq_0_span:
-  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
-  shows "\<forall>x \<in> span B. f x = 0"
-proof
-  fix x assume x: "x \<in> span B"
-  let ?P = "\<lambda>x. f x = 0"
-  from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
-  with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
-qed
-
-lemma linear_eq_0:
-  assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
-  shows "\<forall>x \<in> S. f x = 0"
-  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
-
-lemma linear_eq:
-  assumes lf: "linear f" and lg: "linear g" and S: "S \<subseteq> span B"
-  and fg: "\<forall> x\<in> B. f x = g x"
-  shows "\<forall>x\<in> S. f x = g x"
-proof-
-  let ?h = "\<lambda>x. f x - g x"
-  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
-  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
-  show ?thesis by simp
-qed
-
-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)"
-  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 by auto }
-  then show ?thesis by (auto intro: ext)
-qed
-
-text {* Similar results for bilinear functions. *}
-
-lemma bilinear_eq:
-  assumes bf: "bilinear f"
-  and bg: "bilinear g"
-  and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
-  and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
-  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
-proof-
-  let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
-  from bf bg have sp: "subspace ?P"
-    unfolding bilinear_def linear_def subspace_def bf bg
-    by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
-
-  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
-    apply -
-    apply (rule ballI)
-    apply (rule span_induct[of B ?P])
-    defer
-    apply (rule sp)
-    apply assumption
-    apply (clarsimp simp add: Ball_def)
-    apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
-    using fg
-    apply (auto simp add: subspace_def)
-    using bf bg unfolding bilinear_def linear_def
-    by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
-  then show ?thesis using SB TC by (auto intro: ext)
-qed
-
-lemma bilinear_eq_stdbasis: 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)"
-  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 intro: ext)
-qed
-
-text {* Detailed theorems about left and right invertibility in general case. *}
-
-lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
-  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]
-  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)"
-    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
-    by auto
-
-  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
-  have "h o f = id" .
-  then show ?thesis using h(1) by blast
-qed
-
-lemma linear_surjective_right_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
-  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"]
-  obtain h:: "'b \<Rightarrow> 'a" where
-    h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. 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)
-  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
-qed
-
-text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
-
-lemma linear_injective_imp_surjective:  fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and fi: "inj f"
-  shows "surj f"
-proof-
-  let ?U = "UNIV :: 'a set"
-  from basis_exists[of ?U] obtain B
-    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
-    by blast
-  from B(4) have d: "dim ?U = card B" by simp
-  have th: "?U \<subseteq> span (f ` B)"
-    apply (rule card_ge_dim_independent)
-    apply blast
-    apply (rule independent_injective_image[OF B(2) lf fi])
-    apply (rule order_eq_refl)
-    apply (rule sym)
-    unfolding d
-    apply (rule card_image)
-    apply (rule subset_inj_on[OF fi])
-    by blast
-  from th show ?thesis
-    unfolding span_linear_image[OF lf] surj_def
-    using B(3) by blast
-qed
-
-text {* And vice versa. *}
-
-lemma surjective_iff_injective_gen:
-  assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
-  and ST: "f ` S \<subseteq> T"
-  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume h: "?lhs"
-    {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
-      from x fS have S0: "card S \<noteq> 0" by auto
-      {assume xy: "x \<noteq> y"
-        have th: "card S \<le> card (f ` (S - {y}))"
-          unfolding c
-          apply (rule card_mono)
-          apply (rule finite_imageI)
-          using fS apply simp
-          using h xy x y f unfolding subset_eq image_iff
-          apply auto
-          apply (case_tac "xa = f x")
-          apply (rule bexI[where x=x])
-          apply auto
-          done
-        also have " \<dots> \<le> card (S -{y})"
-          apply (rule card_image_le)
-          using fS by simp
-        also have "\<dots> \<le> card S - 1" using y fS by simp
-        finally have False  using S0 by arith }
-      then have "x = y" by blast}
-    then have ?rhs unfolding inj_on_def by blast}
-  moreover
-  {assume h: ?rhs
-    have "f ` S = T"
-      apply (rule card_subset_eq[OF fT ST])
-      unfolding card_image[OF h] using c .
-    then have ?lhs by blast}
-  ultimately show ?thesis by blast
-qed
+subsection {* Class instances *}
 
-lemma linear_surjective_imp_injective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and sf: "surj f"
-  shows "inj f"
-proof-
-  let ?U = "UNIV :: 'a set"
-  from basis_exists[of ?U] obtain B
-    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
-    by blast
-  {fix x assume x: "x \<in> span B" and fx: "f x = 0"
-    from B(2) have fB: "finite B" using independent_bound by auto
-    have fBi: "independent (f ` B)"
-      apply (rule card_le_dim_spanning[of "f ` B" ?U])
-      apply blast
-      using sf B(3)
-      unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
-      apply blast
-      using fB apply blast
-      unfolding d[symmetric]
-      apply (rule card_image_le)
-      apply (rule fB)
-      done
-    have th0: "dim ?U \<le> card (f ` B)"
-      apply (rule span_card_ge_dim)
-      apply blast
-      unfolding span_linear_image[OF lf]
-      apply (rule subset_trans[where B = "f ` UNIV"])
-      using sf unfolding surj_def apply blast
-      apply (rule image_mono)
-      apply (rule B(3))
-      apply (metis finite_imageI fB)
-      done
-
-    moreover have "card (f ` B) \<le> card B"
-      by (rule card_image_le, rule fB)
-    ultimately have th1: "card B = card (f ` B)" unfolding d by arith
-    have fiB: "inj_on f B"
-      unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
-    from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
-    have "x = 0" by blast}
-  note th = this
-  from th show ?thesis unfolding linear_injective_0[OF lf]
-    using B(3) by blast
-qed
-
-text {* Hence either is enough for isomorphism. *}
-
-lemma left_right_inverse_eq:
-  assumes fg: "f o g = id" and gh: "g o h = id"
-  shows "f = h"
-proof-
-  have "f = f o (g o h)" unfolding gh by simp
-  also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
-  finally show "f = h" unfolding fg by simp
-qed
-
-lemma isomorphism_expand:
-  "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
-  by (simp add: fun_eq_iff o_def id_def)
-
-lemma linear_injective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and fi: "inj f"
-  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
-unfolding isomorphism_expand[symmetric]
-using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
-by (metis left_right_inverse_eq)
-
-lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and sf: "surj f"
-  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
-unfolding isomorphism_expand[symmetric]
-using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
-by (metis left_right_inverse_eq)
-
-text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
-
-lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and lf': "linear f'"
-  shows "f o f' = id \<longleftrightarrow> f' o f = id"
-proof-
-  {fix f f':: "'a => 'a"
-    assume lf: "linear f" "linear f'" and f: "f o f' = id"
-    from f have sf: "surj f"
-      apply (auto simp add: o_def id_def surj_def)
-      by metis
-    from linear_surjective_isomorphism[OF lf(1) sf] lf f
-    have "f' o f = id" unfolding fun_eq_iff o_def id_def
-      by metis}
-  then show ?thesis using lf lf' by metis
-qed
-
-text {* Moreover, a one-sided inverse is automatically linear. *}
-
-lemma left_inverse_linear: fixes f::"'a::euclidean_space => 'a::euclidean_space"
-  assumes lf: "linear f" and gf: "g o f = id"
-  shows "linear g"
-proof-
-  from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
-    by metis
-  from linear_injective_isomorphism[OF lf fi]
-  obtain h:: "'a \<Rightarrow> 'a" where
-    h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
-  have "h = g" apply (rule ext) using gf h(2,3)
-    apply (simp add: o_def id_def fun_eq_iff)
-    by metis
-  with h(1) show ?thesis by blast
-qed
-
-subsection {* Infinity norm *}
-
-definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
-
-lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
-  by auto
-
-lemma infnorm_set_image:
-  "{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
-  (\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
-
-lemma infnorm_set_lemma:
-  shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
-  and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
-  unfolding infnorm_set_image
-  by auto
-
-lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
-  unfolding infnorm_def
-  unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
-  unfolding infnorm_set_image
-  by auto
-
-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 by (metis th2)
-qed
-
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
-proof-
-  have "infnorm x <= 0 \<longleftrightarrow> x = 0"
-    unfolding infnorm_def
-    unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
-    unfolding infnorm_set_image ball_simps
-    apply(subst (1) euclidean_eq) unfolding euclidean_component.zero
-    by auto
-  then show ?thesis using infnorm_pos_le[of x] by simp
-qed
-
-lemma infnorm_0: "infnorm 0 = 0"
-  by (simp add: infnorm_eq_0)
-
-lemma infnorm_neg: "infnorm (- x) = infnorm x"
-  unfolding infnorm_def
-  apply (rule cong[of "Sup" "Sup"])
-  apply blast by(auto simp add: euclidean_simps)
-
-lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
-proof-
-  have "y - x = - (x - y)" by simp
-  then show ?thesis  by (metis infnorm_neg)
-qed
-
-lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
-proof-
-  have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
-    by arith
-  from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
-  have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
-    "infnorm y \<le> infnorm (x - y) + infnorm x"
-    by (simp_all add: field_simps infnorm_neg diff_minus[symmetric])
-  from th[OF ths]  show ?thesis .
-qed
-
-lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
-  using infnorm_pos_le[of x] by arith
-
-lemma component_le_infnorm:
-  shows "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
-proof(cases "i<DIM('a)")
-  case False thus ?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) by simp
-  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 by auto
-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_scaleR abs_mult
-  using component_le_infnorm[of x] by(auto intro: mult_mono) 
-
-lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
-proof-
-  {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
-  moreover
-  {assume a0: "a \<noteq> 0"
-    from a0 have th: "(1/a) *\<^sub>R (a *\<^sub>R x) = x" by simp
-    from a0 have ap: "\<bar>a\<bar> > 0" by arith
-    from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
-    have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*\<^sub>R x)"
-      unfolding th by simp
-    with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *\<^sub>R x))" by (simp add: field_simps)
-    then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*\<^sub>R x)"
-      using ap by (simp add: field_simps)
-    with infnorm_mul_lemma[of a x] have ?thesis by arith }
-  ultimately show ?thesis by blast
-qed
-
-lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
-  using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
-
-text {* Prove that it differs only up to a bound from Euclidean norm. *}
-
-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 card_enum: "card {1 .. n} = n" by auto
-
-lemma norm_le_infnorm: "norm(x) <= sqrt(real DIM('a)) * infnorm(x::'a::euclidean_space)"
-proof-
-  let ?d = "DIM('a)"
-  have "real ?d \<ge> 0" by simp
-  hence d2: "(sqrt (real ?d))^2 = real ?d"
-    by (auto intro: real_sqrt_pow2)
-  have th: "sqrt (real ?d) * infnorm x \<ge> 0"
-    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)
-    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_set_image bex_simps apply(rule_tac x=i in bexI) by auto
-  from real_le_lsqrt[OF inner_ge_zero th th1]
-  show ?thesis unfolding norm_eq_sqrt_inner id_def .
-qed
-
-text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
-
-lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume h: "x = 0"
-    hence ?thesis by simp}
-  moreover
-  {assume h: "y = 0"
-    hence ?thesis by simp}
-  moreover
-  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
-    have "?rhs \<longleftrightarrow> (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
-      using x y
-      unfolding inner_simps
-      unfolding power2_norm_eq_inner[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: inner_commute)
-      apply (simp add: field_simps) by metis
-    also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
-      by (simp add: field_simps inner_commute)
-    also have "\<dots> \<longleftrightarrow> ?lhs" using x y
-      apply simp
-      by metis
-    finally have ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-lemma norm_cauchy_schwarz_abs_eq:
-  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
-                norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
-  have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
-    by simp
-  also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
-     (-x) \<bullet> y = norm x * norm y)"
-    unfolding norm_cauchy_schwarz_eq[symmetric]
-    unfolding norm_minus_cancel norm_scaleR ..
-  also have "\<dots> \<longleftrightarrow> ?lhs"
-    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps by auto
-  finally show ?thesis ..
-qed
-
-lemma norm_triangle_eq:
-  fixes x y :: "'a::real_inner"
-  shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-proof-
-  {assume x: "x =0 \<or> y =0"
-    hence ?thesis by (cases "x=0", simp_all)}
-  moreover
-  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    hence "norm x \<noteq> 0" "norm y \<noteq> 0"
-      by simp_all
-    hence n: "norm x > 0" "norm y > 0"
-      using norm_ge_zero[of x] norm_ge_zero[of y]
-      by arith+
-    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
-    have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
-      apply (rule th) using n norm_ge_zero[of "x + y"]
-      by arith
-    also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-      unfolding norm_cauchy_schwarz_eq[symmetric]
-      unfolding power2_norm_eq_inner inner_simps
-      by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
-    finally have ?thesis .}
-  ultimately show ?thesis by blast
-qed
-
-subsection {* Collinearity *}
-
-definition
-  collinear :: "'a::real_vector set \<Rightarrow> bool" where
-  "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
-
-lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
-
-lemma collinear_sing: "collinear {x}"
-  by (simp add: collinear_def)
-
-lemma collinear_2: "collinear {x, y}"
-  apply (simp add: collinear_def)
-  apply (rule exI[where x="x - y"])
-  apply auto
-  apply (rule exI[where x=1], simp)
-  apply (rule exI[where x="- 1"], simp)
-  done
-
-lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume "x=0 \<or> y = 0" hence ?thesis
-      by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
-  moreover
-  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    {assume h: "?lhs"
-      then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u" unfolding collinear_def by blast
-      from u[rule_format, of x 0] u[rule_format, of y 0]
-      obtain cx and cy where
-        cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
-        by auto
-      from cx x have cx0: "cx \<noteq> 0" by auto
-      from cy y have cy0: "cy \<noteq> 0" by auto
-      let ?d = "cy / cx"
-      from cx cy cx0 have "y = ?d *\<^sub>R x"
-        by simp
-      hence ?rhs using x y by blast}
-    moreover
-    {assume h: "?rhs"
-      then obtain c where c: "y = c *\<^sub>R x" using x y by blast
-      have ?lhs unfolding collinear_def c
-        apply (rule exI[where x=x])
-        apply auto
-        apply (rule exI[where x="- 1"], simp)
-        apply (rule exI[where x= "-c"], simp)
-        apply (rule exI[where x=1], simp)
-        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
-        apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
-        done}
-    ultimately have ?thesis by blast}
-  ultimately show ?thesis by blast
-qed
-
-lemma norm_cauchy_schwarz_equal:
-  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
-unfolding norm_cauchy_schwarz_abs_eq
-apply (cases "x=0", simp_all add: collinear_2)
-apply (cases "y=0", simp_all add: collinear_2 insert_commute)
-unfolding collinear_lemma
-apply simp
-apply (subgoal_tac "norm x \<noteq> 0")
-apply (subgoal_tac "norm y \<noteq> 0")
-apply (rule iffI)
-apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
-apply (rule exI[where x="(1/norm x) * norm y"])
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (rule exI[where x="(1/norm x) * - norm y"])
-apply clarify
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (erule exE)
-apply (erule ssubst)
-unfolding scaleR_scaleR
-unfolding norm_scaleR
-apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
-apply (case_tac "c <= 0", simp add: field_simps)
-apply (simp add: field_simps)
-apply (case_tac "c <= 0", simp add: field_simps)
-apply (simp add: field_simps)
-apply simp
-apply simp
-done
-
-subsection "Instantiate @{typ real} and @{typ complex} as typeclass @{text ordered_euclidean_space}."
+subsubsection {* Type @{typ real} *}
 
 instantiation real :: euclidean_space
 begin
@@ -3307,18 +174,7 @@
 
 end
 
-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 by auto
+subsubsection {* Type @{typ complex} *}
 
 instantiation complex :: euclidean_space
 begin
@@ -3353,14 +209,10 @@
 
 end
 
-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_complex[simp]: "DIM(complex) = 2"
   by (rule dimension_complex_def)
 
-section {* Products Spaces *}
+subsubsection {* Type @{typ "'a \<times> 'b"} *}
 
 instantiation prod :: (euclidean_space, euclidean_space) euclidean_space
 begin
@@ -3400,18 +252,4 @@
 
 end
 
-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)
-
-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)"
-
-instance proof qed (auto simp: less_prod_def less_eq_prod_def)
 end
-
-
-end
diff -r 0f35a870ecf1 -r 691c52e900ca src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Aug 10 09:23:42 2011 -0700
@@ -0,0 +1,3181 @@
+(*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
+    Author:     Amine Chaieb, University of Cambridge
+*)
+
+header {* Elementary linear algebra on Euclidean spaces *}
+
+theory Linear_Algebra
+imports
+  Euclidean_Space
+  "~~/src/HOL/Library/Infinite_Set"
+  L2_Norm
+  "~~/src/HOL/Library/Convex"
+uses
+  "~~/src/HOL/Library/positivstellensatz.ML"  (* FIXME duplicate use!? *)
+  ("normarith.ML")
+begin
+
+lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
+  by auto
+
+notation inner (infix "\<bullet>" 70)
+
+subsection {* A connectedness or intermediate value lemma with several applications. *}
+
+lemma connected_real_lemma:
+  fixes f :: "real \<Rightarrow> 'a::metric_space"
+  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
+  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
+  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
+  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
+  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
+  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
+proof-
+  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
+  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
+  have Sub: "\<exists>y. isUb UNIV ?S y"
+    apply (rule exI[where x= b])
+    using ab fb e12 by (auto simp add: isUb_def setle_def)
+  from reals_complete[OF Se Sub] obtain l where
+    l: "isLub UNIV ?S l"by blast
+  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
+    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
+    by (metis linorder_linear)
+  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
+    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
+    by (metis linorder_linear not_le)
+    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
+    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
+    have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
+    then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
+    {assume le2: "f l \<in> e2"
+      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
+      hence lap: "l - a > 0" using alb by arith
+      from e2[rule_format, OF le2] obtain e where
+        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
+      from dst[OF alb e(1)] obtain d where
+        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
+      let ?d' = "min (d/2) ((l - a)/2)"
+      have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
+        by (simp add: min_max.less_infI2)
+      then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
+      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
+      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
+      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
+      moreover
+      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
+      ultimately have False using e12 alb d' by auto}
+    moreover
+    {assume le1: "f l \<in> e1"
+    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
+      hence blp: "b - l > 0" using alb by arith
+      from e1[rule_format, OF le1] obtain e where
+        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
+      from dst[OF alb e(1)] obtain d where
+        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
+      have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
+      then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
+      then obtain d' where d': "d' > 0" "d' < d" by metis
+      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
+      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
+      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
+      with l d' have False
+        by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
+    ultimately show ?thesis using alb by metis
+qed
+
+text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case *}
+
+lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
+proof-
+  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
+  thus ?thesis by (simp add: field_simps power2_eq_square)
+qed
+
+lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
+  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
+  apply (rule_tac x="s" in exI)
+  apply auto
+  apply (erule_tac x=y in allE)
+  apply auto
+  done
+
+lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
+  using real_sqrt_le_iff[of x "y^2"] by simp
+
+lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
+  using real_sqrt_le_mono[of "x^2" y] by simp
+
+lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
+  using real_sqrt_less_mono[of "x^2" y] by simp
+
+lemma sqrt_even_pow2: assumes n: "even n"
+  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
+proof-
+  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
+  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
+    by (simp only: power_mult[symmetric] mult_commute)
+  then show ?thesis  using m by simp
+qed
+
+lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
+  apply (cases "x = 0", simp_all)
+  using sqrt_divide_self_eq[of x]
+  apply (simp add: inverse_eq_divide field_simps)
+  done
+
+text{* Hence derive more interesting properties of the norm. *}
+
+(* FIXME: same as norm_scaleR
+lemma norm_mul[simp]: "norm(a *\<^sub>R x) = abs(a) * norm x"
+  by (simp add: norm_vector_def setL2_right_distrib abs_mult)
+*)
+
+lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
+  by (simp add: setL2_def power2_eq_square)
+
+lemma norm_cauchy_schwarz:
+  shows "inner x y <= norm x * norm y"
+  using Cauchy_Schwarz_ineq2[of x y] by auto
+
+lemma norm_cauchy_schwarz_abs:
+  shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
+  by (rule Cauchy_Schwarz_ineq2)
+
+lemma norm_triangle_sub:
+  fixes x y :: "'a::real_normed_vector"
+  shows "norm x \<le> norm y  + norm (x - y)"
+  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
+
+lemma real_abs_norm: "\<bar>norm x\<bar> = norm x"
+  by (rule abs_norm_cancel)
+lemma real_abs_sub_norm: "\<bar>norm x - norm y\<bar> <= norm(x - y)"
+  by (rule norm_triangle_ineq3)
+lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
+  by (simp add: norm_eq_sqrt_inner) 
+lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+  by (simp add: norm_eq_sqrt_inner)
+lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+  apply(subst order_eq_iff) unfolding norm_le by auto
+lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+  unfolding norm_eq_sqrt_inner by auto
+
+text{* Squaring equations and inequalities involving norms.  *}
+
+lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
+  by (simp add: norm_eq_sqrt_inner)
+
+lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
+  by (auto simp add: norm_eq_sqrt_inner)
+
+lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
+proof
+  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+  then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
+  then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
+next
+  assume "x\<twosuperior> \<le> y\<twosuperior>"
+  then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
+  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
+qed
+
+lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
+  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
+  using norm_ge_zero[of x]
+  apply arith
+  done
+
+lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
+  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
+  using norm_ge_zero[of x]
+  apply arith
+  done
+
+lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
+  by (metis not_le norm_ge_square)
+lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
+  by (metis norm_le_square not_less)
+
+text{* Dot product in terms of the norm rather than conversely. *}
+
+lemmas inner_simps = inner.add_left inner.add_right inner.diff_right inner.diff_left 
+inner.scaleR_left inner.scaleR_right
+
+lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
+  unfolding power2_norm_eq_inner inner_simps inner_commute by auto 
+
+lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
+  unfolding power2_norm_eq_inner inner_simps inner_commute by(auto simp add:algebra_simps)
+
+text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
+
+lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?lhs then show ?rhs by simp
+next
+  assume ?rhs
+  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
+  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_simps inner_commute)
+  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_simps inner_commute)
+  then show "x = y" by (simp)
+qed
+
+subsection{* General linear decision procedure for normed spaces. *}
+
+lemma norm_cmul_rule_thm:
+  fixes x :: "'a::real_normed_vector"
+  shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
+  unfolding norm_scaleR
+  apply (erule mult_left_mono)
+  apply simp
+  done
+
+  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
+lemma norm_add_rule_thm:
+  fixes x1 x2 :: "'a::real_normed_vector"
+  shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
+  by (rule order_trans [OF norm_triangle_ineq add_mono])
+
+lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
+  by (simp add: field_simps)
+
+lemma pth_1:
+  fixes x :: "'a::real_normed_vector"
+  shows "x == scaleR 1 x" by simp
+
+lemma pth_2:
+  fixes x :: "'a::real_normed_vector"
+  shows "x - y == x + -y" by (atomize (full)) simp
+
+lemma pth_3:
+  fixes x :: "'a::real_normed_vector"
+  shows "- x == scaleR (-1) x" by simp
+
+lemma pth_4:
+  fixes x :: "'a::real_normed_vector"
+  shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
+
+lemma pth_5:
+  fixes x :: "'a::real_normed_vector"
+  shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
+
+lemma pth_6:
+  fixes x :: "'a::real_normed_vector"
+  shows "scaleR c (x + y) == scaleR c x + scaleR c y"
+  by (simp add: scaleR_right_distrib)
+
+lemma pth_7:
+  fixes x :: "'a::real_normed_vector"
+  shows "0 + x == x" and "x + 0 == x" by simp_all
+
+lemma pth_8:
+  fixes x :: "'a::real_normed_vector"
+  shows "scaleR c x + scaleR d x == scaleR (c + d) x"
+  by (simp add: scaleR_left_distrib)
+
+lemma pth_9:
+  fixes x :: "'a::real_normed_vector" shows
+  "(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
+  "scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
+  "(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
+  by (simp_all add: algebra_simps)
+
+lemma pth_a:
+  fixes x :: "'a::real_normed_vector"
+  shows "scaleR 0 x + y == y" by simp
+
+lemma pth_b:
+  fixes x :: "'a::real_normed_vector" shows
+  "scaleR c x + scaleR d y == scaleR c x + scaleR d y"
+  "(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
+  "scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
+  "(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
+  by (simp_all add: algebra_simps)
+
+lemma pth_c:
+  fixes x :: "'a::real_normed_vector" shows
+  "scaleR c x + scaleR d y == scaleR d y + scaleR c x"
+  "(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
+  "scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
+  "(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
+  by (simp_all add: algebra_simps)
+
+lemma pth_d:
+  fixes x :: "'a::real_normed_vector"
+  shows "x + 0 == x" by simp
+
+lemma norm_imp_pos_and_ge:
+  fixes x :: "'a::real_normed_vector"
+  shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
+  by atomize auto
+
+lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
+
+lemma norm_pths:
+  fixes x :: "'a::real_normed_vector" shows
+  "x = y \<longleftrightarrow> norm (x - y) \<le> 0"
+  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
+  using norm_ge_zero[of "x - y"] by auto
+
+use "normarith.ML"
+
+method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
+*} "prove simple linear statements about vector norms"
+
+
+text{* Hence more metric properties. *}
+
+lemma norm_triangle_half_r:
+  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
+  using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
+
+lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
+  shows "norm (x - x') < e"
+  using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
+  unfolding dist_norm[THEN sym] .
+
+lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
+  by (metis order_trans norm_triangle_ineq)
+
+lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
+  by (metis basic_trans_rules(21) norm_triangle_ineq)
+
+lemma dist_triangle_add:
+  fixes x y x' y' :: "'a::real_normed_vector"
+  shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
+  by norm
+
+lemma dist_triangle_add_half:
+  fixes x x' y y' :: "'a::real_normed_vector"
+  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
+  by norm
+
+lemma setsum_clauses:
+  shows "setsum f {} = 0"
+  and "finite S \<Longrightarrow> setsum f (insert x S) =
+                 (if x \<in> S then setsum f S else f x + setsum f S)"
+  by (auto simp add: insert_absorb)
+
+lemma setsum_norm:
+  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+  assumes fS: "finite S"
+  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
+proof(induct rule: finite_induct[OF fS])
+  case 1 thus ?case by simp
+next
+  case (2 x S)
+  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
+  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
+    using "2.hyps" by simp
+  finally  show ?case  using "2.hyps" by simp
+qed
+
+lemma setsum_norm_le:
+  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+  assumes fS: "finite S"
+  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
+  shows "norm (setsum f S) \<le> setsum g S"
+proof-
+  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
+    by - (rule setsum_mono, simp)
+  then show ?thesis using setsum_norm[OF fS, of f] fg
+    by arith
+qed
+
+lemma setsum_norm_bound:
+  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+  assumes fS: "finite S"
+  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
+  shows "norm (setsum f S) \<le> of_nat (card S) * K"
+  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
+  by simp
+
+lemma setsum_group:
+  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
+  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
+  apply (subst setsum_image_gen[OF fS, of g f])
+  apply (rule setsum_mono_zero_right[OF fT fST])
+  by (auto intro: setsum_0')
+
+lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> y = setsum (\<lambda>x. f x \<bullet> y) S "
+  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
+
+lemma dot_rsum: "finite S \<Longrightarrow> y \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
+  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
+
+lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
+proof
+  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
+  hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_simps)
+  hence "(y - z) \<bullet> (y - z) = 0" ..
+  thus "y = z" by simp
+qed simp
+
+lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
+proof
+  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
+  hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_simps)
+  hence "(x - y) \<bullet> (x - y) = 0" ..
+  thus "x = y" by simp
+qed simp
+
+subsection{* Orthogonality. *}
+
+context real_inner
+begin
+
+definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
+
+lemma orthogonal_clauses:
+  "orthogonal a 0"
+  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
+  "orthogonal a x \<Longrightarrow> orthogonal a (-x)"
+  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
+  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
+  "orthogonal 0 a"
+  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
+  "orthogonal x a \<Longrightarrow> orthogonal (-x) a"
+  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
+  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
+  unfolding orthogonal_def inner_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
+ 
+end
+
+lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
+  by (simp add: orthogonal_def inner_commute)
+
+subsection{* Linear functions. *}
+
+definition
+  linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
+  "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
+
+lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+  shows "linear f" using assms unfolding linear_def by auto
+
+lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+  by (simp add: linear_def algebra_simps)
+
+lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
+  by (simp add: linear_def)
+
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
+  by (simp add: linear_def algebra_simps)
+
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
+  by (simp add: linear_def algebra_simps)
+
+lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
+  by (simp add: linear_def)
+
+lemma linear_id: "linear id" by (simp add: linear_def id_def)
+
+lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
+
+lemma linear_compose_setsum:
+  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
+  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
+  using lS
+  apply (induct rule: finite_induct[OF fS])
+  by (auto simp add: linear_zero intro: linear_compose_add)
+
+lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
+  unfolding linear_def
+  apply clarsimp
+  apply (erule allE[where x="0::'a"])
+  apply simp
+  done
+
+lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
+
+lemma linear_neg: "linear f ==> f (-x) = - f x"
+  using linear_cmul [where c="-1"] by simp
+
+lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
+
+lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+  by (simp add: diff_minus linear_add linear_neg)
+
+lemma linear_setsum:
+  assumes lf: "linear f" and fS: "finite S"
+  shows "f (setsum g S) = setsum (f o g) S"
+proof (induct rule: finite_induct[OF fS])
+  case 1 thus ?case by (simp add: linear_0[OF lf])
+next
+  case (2 x F)
+  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
+    by simp
+  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
+  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
+  finally show ?case .
+qed
+
+lemma linear_setsum_mul:
+  assumes lf: "linear f" and fS: "finite S"
+  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
+  using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
+  linear_cmul[OF lf] by simp
+
+lemma linear_injective_0:
+  assumes lf: "linear f"
+  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
+proof-
+  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
+  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
+  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
+    by (simp add: linear_sub[OF lf])
+  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
+  finally show ?thesis .
+qed
+
+subsection{* Bilinear functions. *}
+
+definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
+
+lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
+  by (simp add: bilinear_def linear_def)
+lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
+  by (simp add: bilinear_def linear_def)
+
+lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
+  by (simp add: bilinear_def linear_def)
+
+lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
+  by (simp add: bilinear_def linear_def)
+
+lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
+  by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
+
+lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
+  by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
+
+lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
+  using add_imp_eq[of x y 0] by auto
+
+lemma bilinear_lzero:
+  assumes bh: "bilinear h" shows "h 0 x = 0"
+  using bilinear_ladd[OF bh, of 0 0 x]
+    by (simp add: eq_add_iff field_simps)
+
+lemma bilinear_rzero:
+  assumes bh: "bilinear h" shows "h x 0 = 0"
+  using bilinear_radd[OF bh, of x 0 0 ]
+    by (simp add: eq_add_iff field_simps)
+
+lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
+  by (simp  add: diff_minus bilinear_ladd bilinear_lneg)
+
+lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
+  by (simp  add: diff_minus bilinear_radd bilinear_rneg)
+
+lemma bilinear_setsum:
+  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
+  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
+proof-
+  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
+    apply (rule linear_setsum[unfolded o_def])
+    using bh fS by (auto simp add: bilinear_def)
+  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
+    apply (rule setsum_cong, simp)
+    apply (rule linear_setsum[unfolded o_def])
+    using bh fT by (auto simp add: bilinear_def)
+  finally show ?thesis unfolding setsum_cartesian_product .
+qed
+
+subsection{* Adjoints. *}
+
+definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
+
+lemma adjoint_unique:
+  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
+  shows "adjoint f = g"
+unfolding adjoint_def
+proof (rule some_equality)
+  show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
+next
+  fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
+  hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
+  hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
+  hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
+  hence "\<forall>y. h y = g y" by simp
+  thus "h = g" by (simp add: ext)
+qed
+
+lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
+
+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"
+  shows "\<forall>n \<ge> m. d n < e m"
+  using assms apply auto
+  apply (erule_tac x="n" in allE)
+  apply (erule_tac x="n" in allE)
+  apply auto
+  done
+
+
+lemma infinite_enumerate: assumes fS: "infinite S"
+  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
+unfolding subseq_def
+using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
+
+lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
+apply auto
+apply (rule_tac x="d/2" in exI)
+apply auto
+done
+
+
+lemma triangle_lemma:
+  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
+  shows "x <= y + z"
+proof-
+  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
+  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
+  from y z have yz: "y + z \<ge> 0" by arith
+  from power2_le_imp_le[OF th yz] show ?thesis .
+qed
+
+text {* TODO: move to NthRoot *}
+lemma sqrt_add_le_add_sqrt:
+  assumes x: "0 \<le> x" and y: "0 \<le> y"
+  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
+apply (rule power2_le_imp_le)
+apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
+apply (simp add: mult_nonneg_nonneg x y)
+apply (simp add: add_nonneg_nonneg x y)
+done
+
+subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
+
+definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
+  "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
+
+lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
+  unfolding hull_def by auto
+
+lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
+unfolding hull_def subset_iff by auto
+
+lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
+using hull_same[of s S] hull_in[of S s] by metis
+
+
+lemma hull_hull: "S hull (S hull s) = S hull s"
+  unfolding hull_def by blast
+
+lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
+  unfolding hull_def by blast
+
+lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
+  unfolding hull_def by blast
+
+lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
+  unfolding hull_def by blast
+
+lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
+  unfolding hull_def by blast
+
+lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
+  unfolding hull_def by blast
+
+lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
+           ==> (S hull s = t)"
+unfolding hull_def by auto
+
+lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
+  using hull_minimal[of S "{x. P x}" Q]
+  by (auto simp add: subset_eq Collect_def mem_def)
+
+lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
+
+lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
+unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
+
+lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
+  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
+apply rule
+apply (rule hull_mono)
+unfolding Un_subset_iff
+apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
+apply (rule hull_minimal)
+apply (metis hull_union_subset)
+apply (metis hull_in T)
+done
+
+lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
+  unfolding hull_def by blast
+
+lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
+by (metis hull_redundant_eq)
+
+text{* Archimedian properties and useful consequences. *}
+
+lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
+  using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
+lemmas real_arch_lt = reals_Archimedean2
+
+lemmas real_arch = reals_Archimedean3
+
+lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
+  using reals_Archimedean
+  apply (auto simp add: field_simps)
+  apply (subgoal_tac "inverse (real n) > 0")
+  apply arith
+  apply simp
+  done
+
+lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
+proof(induct n)
+  case 0 thus ?case by simp
+next
+  case (Suc n)
+  hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
+  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
+  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
+  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
+    apply (simp add: field_simps)
+    using mult_left_mono[OF p Suc.prems] by simp
+  finally show ?case  by (simp add: real_of_nat_Suc field_simps)
+qed
+
+lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
+proof-
+  from x have x0: "x - 1 > 0" by arith
+  from real_arch[OF x0, rule_format, of y]
+  obtain n::nat where n:"y < real n * (x - 1)" by metis
+  from x0 have x00: "x- 1 \<ge> 0" by arith
+  from real_pow_lbound[OF x00, of n] n
+  have "y < x^n" by auto
+  then show ?thesis by metis
+qed
+
+lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
+  using real_arch_pow[of 2 x] by simp
+
+lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
+  shows "\<exists>n. x^n < y"
+proof-
+  {assume x0: "x > 0"
+    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
+    from real_arch_pow[OF ix, of "1/y"]
+    obtain n where n: "1/y < (1/x)^n" by blast
+    then
+    have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
+  moreover
+  {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
+  ultimately show ?thesis by metis
+qed
+
+lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+  by (metis real_arch_inv)
+
+lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+  apply (rule forall_pos_mono)
+  apply auto
+  apply (atomize)
+  apply (erule_tac x="n - 1" in allE)
+  apply auto
+  done
+
+lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
+  shows "x = 0"
+proof-
+  {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
+    from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
+    with xc[rule_format, of n] have "n = 0" by arith
+    with n c have False by simp}
+  then show ?thesis by blast
+qed
+
+subsection {* Geometric progression *}
+
+lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
+  (is "?lhs = ?rhs")
+proof-
+  {assume x1: "x = 1" hence ?thesis by simp}
+  moreover
+  {assume x1: "x\<noteq>1"
+    hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
+    from geometric_sum[OF x1, of "Suc n", unfolded x1']
+    have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
+      unfolding atLeastLessThanSuc_atLeastAtMost
+      using x1' apply (auto simp only: field_simps)
+      apply (simp add: field_simps)
+      done
+    then have ?thesis by (simp add: field_simps) }
+  ultimately show ?thesis by metis
+qed
+
+lemma sum_gp_multiplied: assumes mn: "m <= n"
+  shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
+  (is "?lhs = ?rhs")
+proof-
+  let ?S = "{0..(n - m)}"
+  from mn have mn': "n - m \<ge> 0" by arith
+  let ?f = "op + m"
+  have i: "inj_on ?f ?S" unfolding inj_on_def by auto
+  have f: "?f ` ?S = {m..n}"
+    using mn apply (auto simp add: image_iff Bex_def) by arith
+  have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
+    by (rule ext, simp add: power_add power_mult)
+  from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
+  have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
+  then show ?thesis unfolding sum_gp_basic using mn
+    by (simp add: field_simps power_add[symmetric])
+qed
+
+lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
+   (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
+                    else (x^ m - x^ (Suc n)) / (1 - x))"
+proof-
+  {assume nm: "n < m" hence ?thesis by simp}
+  moreover
+  {assume "\<not> n < m" hence nm: "m \<le> n" by arith
+    {assume x: "x = 1"  hence ?thesis by simp}
+    moreover
+    {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
+      from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
+    ultimately have ?thesis by metis
+  }
+  ultimately show ?thesis by metis
+qed
+
+lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
+  (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
+  unfolding sum_gp[of x m "m + n"] power_Suc
+  by (simp add: field_simps power_add)
+
+
+subsection{* A bit of linear algebra. *}
+
+definition (in real_vector)
+  subspace :: "'a set \<Rightarrow> bool" where
+  "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
+
+definition (in real_vector) "span S = (subspace hull S)"
+definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
+abbreviation (in real_vector) "independent s == ~(dependent s)"
+
+text {* Closure properties of subspaces. *}
+
+lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+  by (metis subspace_def)
+
+lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
+  by (metis subspace_def)
+
+lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
+  by (metis scaleR_minus1_left subspace_mul)
+
+lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
+  by (metis diff_minus subspace_add subspace_neg)
+
+lemma (in real_vector) subspace_setsum:
+  assumes sA: "subspace A" and fB: "finite B"
+  and f: "\<forall>x\<in> B. f x \<in> A"
+  shows "setsum f B \<in> A"
+  using  fB f sA
+  apply(induct rule: finite_induct[OF fB])
+  by (simp add: subspace_def sA, auto simp add: sA subspace_add)
+
+lemma subspace_linear_image:
+  assumes lf: "linear f" and sS: "subspace S"
+  shows "subspace(f ` S)"
+  using lf sS linear_0[OF lf]
+  unfolding linear_def subspace_def
+  apply (auto simp add: image_iff)
+  apply (rule_tac x="x + y" in bexI, auto)
+  apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
+  done
+
+lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
+  by (auto simp add: subspace_def linear_def linear_0[of f])
+
+lemma subspace_trivial: "subspace {0}"
+  by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
+  by (simp add: subspace_def)
+
+lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+  by (metis span_def hull_mono)
+
+lemma (in real_vector) subspace_span: "subspace(span S)"
+  unfolding span_def
+  apply (rule hull_in[unfolded mem_def])
+  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
+  apply auto
+  apply (erule_tac x="X" in ballE)
+  apply (simp add: mem_def)
+  apply blast
+  apply (erule_tac x="X" in ballE)
+  apply (erule_tac x="X" in ballE)
+  apply (erule_tac x="X" in ballE)
+  apply (clarsimp simp add: mem_def)
+  apply simp
+  apply simp
+  apply simp
+  apply (erule_tac x="X" in ballE)
+  apply (erule_tac x="X" in ballE)
+  apply (simp add: mem_def)
+  apply simp
+  apply simp
+  done
+
+lemma (in real_vector) span_clauses:
+  "a \<in> S ==> a \<in> span S"
+  "0 \<in> span S"
+  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
+  by (metis span_def hull_subset subset_eq)
+     (metis subspace_span subspace_def)+
+
+lemma (in real_vector) span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
+  and P: "subspace P" and x: "x \<in> span S" shows "P x"
+proof-
+  from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
+  from P have P': "P \<in> subspace" by (simp add: mem_def)
+  from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
+  show "P x" by (metis mem_def subset_eq)
+qed
+
+lemma span_empty[simp]: "span {} = {0}"
+  apply (simp add: span_def)
+  apply (rule hull_unique)
+  apply (auto simp add: mem_def subspace_def)
+  unfolding mem_def[of "0::'a", symmetric]
+  apply simp
+  done
+
+lemma (in real_vector) independent_empty[intro]: "independent {}"
+  by (simp add: dependent_def)
+
+lemma dependent_single[simp]:
+  "dependent {x} \<longleftrightarrow> x = 0"
+  unfolding dependent_def by auto
+
+lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
+  apply (clarsimp simp add: dependent_def span_mono)
+  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
+  apply force
+  apply (rule span_mono)
+  apply auto
+  done
+
+lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
+  by (metis order_antisym span_def hull_minimal mem_def)
+
+lemma (in real_vector) span_induct': assumes SP: "\<forall>x \<in> S. P x"
+  and P: "subspace P" shows "\<forall>x \<in> span S. P x"
+  using span_induct SP P by blast
+
+inductive (in real_vector) span_induct_alt_help for S:: "'a \<Rightarrow> bool"
+  where
+  span_induct_alt_help_0: "span_induct_alt_help S 0"
+  | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *\<^sub>R x + z)"
+
+lemma span_induct_alt':
+  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
+proof-
+  {fix x:: "'a" assume x: "span_induct_alt_help S x"
+    have "h x"
+      apply (rule span_induct_alt_help.induct[OF x])
+      apply (rule h0)
+      apply (rule hS, assumption, assumption)
+      done}
+  note th0 = this
+  {fix x assume x: "x \<in> span S"
+
+    have "span_induct_alt_help S x"
+      proof(rule span_induct[where x=x and S=S])
+        show "x \<in> span S" using x .
+      next
+        fix x assume xS : "x \<in> S"
+          from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+          show "span_induct_alt_help S x" by simp
+        next
+        have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
+        moreover
+        {fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
+          from h
+          have "span_induct_alt_help S (x + y)"
+            apply (induct rule: span_induct_alt_help.induct)
+            apply simp
+            unfolding add_assoc
+            apply (rule span_induct_alt_help_S)
+            apply assumption
+            apply simp
+            done}
+        moreover
+        {fix c x assume xt: "span_induct_alt_help S x"
+          then have "span_induct_alt_help S (c *\<^sub>R x)"
+            apply (induct rule: span_induct_alt_help.induct)
+            apply (simp add: span_induct_alt_help_0)
+            apply (simp add: scaleR_right_distrib)
+            apply (rule span_induct_alt_help_S)
+            apply assumption
+            apply simp
+            done
+        }
+        ultimately show "subspace (span_induct_alt_help S)"
+          unfolding subspace_def mem_def Ball_def by blast
+      qed}
+  with th0 show ?thesis by blast
+qed
+
+lemma span_induct_alt:
+  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
+  shows "h x"
+using span_induct_alt'[of h S] h0 hS x by blast
+
+text {* Individual closure properties. *}
+
+lemma span_span: "span (span A) = span A"
+  unfolding span_def hull_hull ..
+
+lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
+
+lemma span_inc: "S \<subseteq> span S"
+  by (metis subset_eq span_superset)
+
+lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
+  unfolding dependent_def apply(rule_tac x=0 in bexI)
+  using assms span_0 by auto
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+  by (metis subspace_add subspace_span)
+
+lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+  by (metis subspace_span subspace_mul)
+
+lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+  by (metis subspace_neg subspace_span)
+
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+  by (metis subspace_span subspace_sub)
+
+lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
+  by (rule subspace_setsum, rule subspace_span)
+
+lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
+  apply (auto simp only: span_add span_sub)
+  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
+  by (simp only: span_add span_sub)
+
+text {* Mapping under linear image. *}
+
+lemma span_linear_image: assumes lf: "linear f"
+  shows "span (f ` S) = f ` (span S)"
+proof-
+  {fix x
+    assume x: "x \<in> span (f ` S)"
+    have "x \<in> f ` span S"
+      apply (rule span_induct[where x=x and S = "f ` S"])
+      apply (clarsimp simp add: image_iff)
+      apply (frule span_superset)
+      apply blast
+      apply (simp only: mem_def)
+      apply (rule subspace_linear_image[OF lf])
+      apply (rule subspace_span)
+      apply (rule x)
+      done}
+  moreover
+  {fix x assume x: "x \<in> span S"
+    have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_eqI)
+      unfolding mem_def Collect_def ..
+    have "f x \<in> span (f ` S)"
+      apply (rule span_induct[where S=S])
+      apply (rule span_superset)
+      apply simp
+      apply (subst th0)
+      apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
+      apply (rule x)
+      done}
+  ultimately show ?thesis by blast
+qed
+
+text {* The key breakdown property. *}
+
+lemma span_breakdown:
+  assumes bS: "b \<in> S" and aS: "a \<in> span S"
+  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})" (is "?P a")
+proof-
+  {fix x assume xS: "x \<in> S"
+    {assume ab: "x = b"
+      then have "?P x"
+        apply simp
+        apply (rule exI[where x="1"], simp)
+        by (rule span_0)}
+    moreover
+    {assume ab: "x \<noteq> b"
+      then have "?P x"  using xS
+        apply -
+        apply (rule exI[where x=0])
+        apply (rule span_superset)
+        by simp}
+    ultimately have "?P x" by blast}
+  moreover have "subspace ?P"
+    unfolding subspace_def
+    apply auto
+    apply (simp add: mem_def)
+    apply (rule exI[where x=0])
+    using span_0[of "S - {b}"]
+    apply (simp add: mem_def)
+    apply (clarsimp simp add: mem_def)
+    apply (rule_tac x="k + ka" in exI)
+    apply (subgoal_tac "x + y - (k + ka) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
+    apply (simp only: )
+    apply (rule span_add[unfolded mem_def])
+    apply assumption+
+    apply (simp add: algebra_simps)
+    apply (clarsimp simp add: mem_def)
+    apply (rule_tac x= "c*k" in exI)
+    apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
+    apply (simp only: )
+    apply (rule span_mul[unfolded mem_def])
+    apply assumption
+    by (simp add: algebra_simps)
+  ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
+qed
+
+lemma span_breakdown_eq:
+  "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  {assume x: "x \<in> span (insert a S)"
+    from x span_breakdown[of "a" "insert a S" "x"]
+    have ?rhs apply clarsimp
+      apply (rule_tac x= "k" in exI)
+      apply (rule set_rev_mp[of _ "span (S - {a})" _])
+      apply assumption
+      apply (rule span_mono)
+      apply blast
+      done}
+  moreover
+  { fix k assume k: "x - k *\<^sub>R a \<in> span S"
+    have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
+    have "(x - k *\<^sub>R a) + k *\<^sub>R a \<in> span (insert a S)"
+      apply (rule span_add)
+      apply (rule set_rev_mp[of _ "span S" _])
+      apply (rule k)
+      apply (rule span_mono)
+      apply blast
+      apply (rule span_mul)
+      apply (rule span_superset)
+      apply blast
+      done
+    then have ?lhs using eq by metis}
+  ultimately show ?thesis by blast
+qed
+
+text {* Hence some "reversal" results. *}
+
+lemma in_span_insert:
+  assumes a: "a \<in> span (insert b S)" and na: "a \<notin> span S"
+  shows "b \<in> span (insert a S)"
+proof-
+  from span_breakdown[of b "insert b S" a, OF insertI1 a]
+  obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
+  {assume k0: "k = 0"
+    with k have "a \<in> span S"
+      apply (simp)
+      apply (rule set_rev_mp)
+      apply assumption
+      apply (rule span_mono)
+      apply blast
+      done
+    with na  have ?thesis by blast}
+  moreover
+  {assume k0: "k \<noteq> 0"
+    have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
+    from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
+      by (simp add: algebra_simps)
+    from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
+      by (rule span_mul)
+    hence th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
+      unfolding eq' .
+
+    from k
+    have ?thesis
+      apply (subst eq)
+      apply (rule span_sub)
+      apply (rule span_mul)
+      apply (rule span_superset)
+      apply blast
+      apply (rule set_rev_mp)
+      apply (rule th)
+      apply (rule span_mono)
+      using na by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma in_span_delete:
+  assumes a: "a \<in> span S"
+  and na: "a \<notin> span (S-{b})"
+  shows "b \<in> span (insert a (S - {b}))"
+  apply (rule in_span_insert)
+  apply (rule set_rev_mp)
+  apply (rule a)
+  apply (rule span_mono)
+  apply blast
+  apply (rule na)
+  done
+
+text {* Transitivity property. *}
+
+lemma span_trans:
+  assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
+  shows "y \<in> span S"
+proof-
+  from span_breakdown[of x "insert x S" y, OF insertI1 y]
+  obtain k where k: "y -k*\<^sub>R x \<in> span (S - {x})" by auto
+  have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
+  show ?thesis
+    apply (subst eq)
+    apply (rule span_add)
+    apply (rule set_rev_mp)
+    apply (rule k)
+    apply (rule span_mono)
+    apply blast
+    apply (rule span_mul)
+    by (rule x)
+qed
+
+lemma span_insert_0[simp]: "span (insert 0 S) = span S"
+  using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
+
+text {* An explicit expansion is sometimes needed. *}
+
+lemma span_explicit:
+  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
+  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
+proof-
+  {fix x assume x: "x \<in> ?E"
+    then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
+      by blast
+    have "x \<in> span P"
+      unfolding u[symmetric]
+      apply (rule span_setsum[OF fS])
+      using span_mono[OF SP]
+      by (auto intro: span_superset span_mul)}
+  moreover
+  have "\<forall>x \<in> span P. x \<in> ?E"
+    unfolding mem_def Collect_def
+  proof(rule span_induct_alt')
+    show "?h 0"
+      apply (rule exI[where x="{}"]) by simp
+  next
+    fix c x y
+    assume x: "x \<in> P" and hy: "?h y"
+    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
+      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
+    let ?S = "insert x S"
+    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
+                  else u y"
+    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
+    {assume xS: "x \<in> S"
+      have S1: "S = (S - {x}) \<union> {x}"
+        and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
+      have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
+        using xS
+        by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
+          setsum_clauses(2)[OF fS] cong del: if_weak_cong)
+      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
+        apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
+        by (simp add: algebra_simps)
+      also have "\<dots> = c*\<^sub>R x + y"
+        by (simp add: add_commute u)
+      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
+    then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast}
+  moreover
+  {assume xS: "x \<notin> S"
+    have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
+      unfolding u[symmetric]
+      apply (rule setsum_cong2)
+      using xS by auto
+    have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
+      by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
+  ultimately have "?Q ?S ?u (c*\<^sub>R x + y)"
+    by (cases "x \<in> S", simp, simp)
+    then show "?h (c*\<^sub>R x + y)"
+      apply -
+      apply (rule exI[where x="?S"])
+      apply (rule exI[where x="?u"]) by metis
+  qed
+  ultimately show ?thesis by blast
+qed
+
+lemma dependent_explicit:
+  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))" (is "?lhs = ?rhs")
+proof-
+  {assume dP: "dependent P"
+    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
+      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
+      unfolding dependent_def span_explicit by blast
+    let ?S = "insert a S"
+    let ?u = "\<lambda>y. if y = a then - 1 else u y"
+    let ?v = a
+    from aP SP have aS: "a \<notin> S" by blast
+    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
+    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
+      using fS aS
+      apply (simp add: setsum_clauses field_simps)
+      apply (subst (2) ua[symmetric])
+      apply (rule setsum_cong2)
+      by auto
+    with th0 have ?rhs
+      apply -
+      apply (rule exI[where x= "?S"])
+      apply (rule exI[where x= "?u"])
+      by clarsimp}
+  moreover
+  {fix S u v assume fS: "finite S"
+      and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
+    and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
+    let ?a = v
+    let ?S = "S - {v}"
+    let ?u = "\<lambda>i. (- u i) / u v"
+    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
+    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
+      using fS vS uv
+      by (simp add: setsum_diff1 divide_inverse field_simps)
+    also have "\<dots> = ?a"
+      unfolding scaleR_right.setsum [symmetric] u
+      using uv by simp
+    finally  have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
+    with th0 have ?lhs
+      unfolding dependent_def span_explicit
+      apply -
+      apply (rule bexI[where x= "?a"])
+      apply (simp_all del: scaleR_minus_left)
+      apply (rule exI[where x= "?S"])
+      by (auto simp del: scaleR_minus_left)}
+  ultimately show ?thesis by blast
+qed
+
+
+lemma span_finite:
+  assumes fS: "finite S"
+  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
+  (is "_ = ?rhs")
+proof-
+  {fix y assume y: "y \<in> span S"
+    from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
+      u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" unfolding span_explicit by blast
+    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
+    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
+      using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
+    hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
+    hence "y \<in> ?rhs" by auto}
+  moreover
+  {fix y u assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
+    then have "y \<in> span S" using fS unfolding span_explicit by auto}
+  ultimately show ?thesis by blast
+qed
+
+lemma Int_Un_cancel: "(A \<union> B) \<inter> A = A" "(A \<union> B) \<inter> B = B" by auto
+
+lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
+proof safe
+  fix x assume "x \<in> span (A \<union> B)"
+  then obtain S u where S: "finite S" "S \<subseteq> A \<union> B" and x: "x = (\<Sum>v\<in>S. u v *\<^sub>R v)"
+    unfolding span_explicit by auto
+
+  let ?Sa = "\<Sum>v\<in>S\<inter>A. u v *\<^sub>R v"
+  let ?Sb = "(\<Sum>v\<in>S\<inter>(B - A). u v *\<^sub>R v)"
+  show "x \<in> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
+  proof
+    show "x = (case (?Sa, ?Sb) of (a, b) \<Rightarrow> a + b)"
+      unfolding x using S
+      by (simp, subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
+
+    from S have "?Sa \<in> span A" unfolding span_explicit
+      by (auto intro!: exI[of _ "S \<inter> A"])
+    moreover from S have "?Sb \<in> span B" unfolding span_explicit
+      by (auto intro!: exI[of _ "S \<inter> (B - A)"])
+    ultimately show "(?Sa, ?Sb) \<in> span A \<times> span B" by simp
+  qed
+next
+  fix a b assume "a \<in> span A" and "b \<in> span B"
+  then obtain Sa ua Sb ub where span:
+    "finite Sa" "Sa \<subseteq> A" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
+    "finite Sb" "Sb \<subseteq> B" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
+    unfolding span_explicit by auto
+  let "?u v" = "(if v \<in> Sa then ua v else 0) + (if v \<in> Sb then ub v else 0)"
+  from span have "finite (Sa \<union> Sb)" "Sa \<union> Sb \<subseteq> A \<union> B"
+    and "a + b = (\<Sum>v\<in>(Sa\<union>Sb). ?u v *\<^sub>R v)"
+    unfolding setsum_addf scaleR_left_distrib
+    by (auto simp add: if_distrib cond_application_beta setsum_cases Int_Un_cancel)
+  thus "a + b \<in> span (A \<union> B)"
+    unfolding span_explicit by (auto intro!: exI[of _ ?u])
+qed
+
+text {* This is useful for building a basis step-by-step. *}
+
+lemma independent_insert:
+  "independent(insert a S) \<longleftrightarrow>
+      (if a \<in> S then independent S
+                else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  {assume aS: "a \<in> S"
+    hence ?thesis using insert_absorb[OF aS] by simp}
+  moreover
+  {assume aS: "a \<notin> S"
+    {assume i: ?lhs
+      then have ?rhs using aS
+        apply simp
+        apply (rule conjI)
+        apply (rule independent_mono)
+        apply assumption
+        apply blast
+        by (simp add: dependent_def)}
+    moreover
+    {assume i: ?rhs
+      have ?lhs using i aS
+        apply simp
+        apply (auto simp add: dependent_def)
+        apply (case_tac "aa = a", auto)
+        apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
+        apply simp
+        apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
+        apply (subgoal_tac "insert aa (S - {aa}) = S")
+        apply simp
+        apply blast
+        apply (rule in_span_insert)
+        apply assumption
+        apply blast
+        apply blast
+        done}
+    ultimately have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+text {* The degenerate case of the Exchange Lemma. *}
+
+lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
+  by blast
+
+lemma spanning_subset_independent:
+  assumes BA: "B \<subseteq> A" and iA: "independent A"
+  and AsB: "A \<subseteq> span B"
+  shows "A = B"
+proof
+  from BA show "B \<subseteq> A" .
+next
+  from span_mono[OF BA] span_mono[OF AsB]
+  have sAB: "span A = span B" unfolding span_span by blast
+
+  {fix x assume x: "x \<in> A"
+    from iA have th0: "x \<notin> span (A - {x})"
+      unfolding dependent_def using x by blast
+    from x have xsA: "x \<in> span A" by (blast intro: span_superset)
+    have "A - {x} \<subseteq> A" by blast
+    hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
+    {assume xB: "x \<notin> B"
+      from xB BA have "B \<subseteq> A -{x}" by blast
+      hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
+      with th1 th0 sAB have "x \<notin> span A" by blast
+      with x have False by (metis span_superset)}
+    then have "x \<in> B" by blast}
+  then show "A \<subseteq> B" by blast
+qed
+
+text {* The general case of the Exchange Lemma, the key to what follows. *}
+
+lemma exchange_lemma:
+  assumes f:"finite t" and i: "independent s"
+  and sp:"s \<subseteq> span t"
+  shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+using f i sp
+proof(induct "card (t - s)" arbitrary: s t rule: less_induct)
+  case less
+  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
+  let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+  let ?ths = "\<exists>t'. ?P t'"
+  {assume st: "s \<subseteq> t"
+    from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+      by (auto intro: span_superset)}
+  moreover
+  {assume st: "t \<subseteq> s"
+
+    from spanning_subset_independent[OF st s sp]
+      st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+      by (auto intro: span_superset)}
+  moreover
+  {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
+    from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
+      from b have "t - {b} - s \<subset> t - s" by blast
+      then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
+        by (auto intro: psubset_card_mono)
+      from b ft have ct0: "card t \<noteq> 0" by auto
+    {assume stb: "s \<subseteq> span(t -{b})"
+      from ft have ftb: "finite (t -{b})" by auto
+      from less(1)[OF cardlt ftb s stb]
+      obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" and fu: "finite u" by blast
+      let ?w = "insert b u"
+      have th0: "s \<subseteq> insert b u" using u by blast
+      from u(3) b have "u \<subseteq> s \<union> t" by blast
+      then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
+      have bu: "b \<notin> u" using b u by blast
+      from u(1) ft b have "card u = (card t - 1)" by auto
+      then
+      have th2: "card (insert b u) = card t"
+        using card_insert_disjoint[OF fu bu] ct0 by auto
+      from u(4) have "s \<subseteq> span u" .
+      also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
+      finally have th3: "s \<subseteq> span (insert b u)" .
+      from th0 th1 th2 th3 fu have th: "?P ?w"  by blast
+      from th have ?ths by blast}
+    moreover
+    {assume stb: "\<not> s \<subseteq> span(t -{b})"
+      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
+      have ab: "a \<noteq> b" using a b by blast
+      have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
+      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
+        using cardlt ft a b by auto
+      have ft': "finite (insert a (t - {b}))" using ft by auto
+      {fix x assume xs: "x \<in> s"
+        have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
+        from b(1) have "b \<in> span t" by (simp add: span_superset)
+        have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
+          using  a sp unfolding subset_eq by auto
+        from xs sp have "x \<in> span t" by blast
+        with span_mono[OF t]
+        have x: "x \<in> span (insert b (insert a (t - {b})))" ..
+        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
+      then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
+
+      from less(1)[OF mlt ft' s sp'] obtain u where
+        u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
+        "s \<subseteq> span u" by blast
+      from u a b ft at ct0 have "?P u" by auto
+      then have ?ths by blast }
+    ultimately have ?ths by blast
+  }
+  ultimately
+  show ?ths  by blast
+qed
+
+text {* This implies corresponding size bounds. *}
+
+lemma independent_span_bound:
+  assumes f: "finite t" and i: "independent s" and sp:"s \<subseteq> span t"
+  shows "finite s \<and> card s \<le> card t"
+  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
+
+
+lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
+proof-
+  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
+  show ?thesis unfolding eq
+    apply (rule finite_imageI)
+    apply (rule finite)
+    done
+qed
+
+subsection{* Euclidean Spaces as Typeclass*}
+
+lemma (in euclidean_space) basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
+  by (rule inj_onI, rule ccontr, cut_tac i=x and j=y in dot_basis, simp)
+
+lemma (in euclidean_space) basis_finite: "basis ` {DIM('a)..} = {0}"
+  by (auto intro: image_eqI [where x="DIM('a)"])
+
+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 * 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]
+  apply clarify
+  apply (drule_tac f="inner (basis a)" in arg_cong)
+  apply (simp add: inner_right.setsum dot_basis)
+  done
+
+lemma dimensionI:
+  assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0::'a::euclidean_space};
+    independent (basis ` {..<d} :: 'a set);
+    inj_on (basis :: nat \<Rightarrow> 'a) {..<d} \<rbrakk> \<Longrightarrow> P d"
+  shows "P DIM('a::euclidean_space)"
+  using DIM_positive basis_finite independent_basis basis_inj
+  by (rule assms)
+
+lemma (in euclidean_space) dimension_eq:
+  assumes "\<And>i. i < d \<Longrightarrow> basis i \<noteq> 0"
+  assumes "\<And>i. d \<le> i \<Longrightarrow> basis i = 0"
+  shows "DIM('a) = d"
+proof (rule linorder_cases [of "DIM('a)" d])
+  assume "DIM('a) < d"
+  hence "basis DIM('a) \<noteq> 0" by (rule assms)
+  thus ?thesis by simp
+next
+  assume "d < DIM('a)"
+  hence "basis d \<noteq> 0" by simp
+  thus ?thesis by (simp add: assms)
+next
+  assume "DIM('a) = d" thus ?thesis .
+qed
+
+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 $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
+      by (simp add: span_setsum span_mul span_superset)
+    hence "x \<in> span (range basis)"
+      by (simp only: euclidean_representation [symmetric])
+  } thus ?thesis by auto
+qed
+
+lemma basis_representation:
+  "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
+proof -
+  have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
+  have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
+    unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
+  thus ?thesis by fastsimp
+qed
+
+lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
+  apply(subst span_basis[symmetric]) unfolding range_basis by auto
+
+lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
+  apply(subst card_image) using basis_inj by auto
+
+lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
+  unfolding span_basis' ..
+
+lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
+  unfolding euclidean_component_def
+  apply(rule order_trans[OF real_inner_class.Cauchy_Schwarz_ineq2]) by auto
+
+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 $$ i\<bar>)"
+  apply (subst euclidean_representation[of x])
+  apply (rule order_trans[OF setsum_norm])
+  by (auto intro!: setsum_mono)
+
+lemma setsum_norm_allsubsets_bound:
+  fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
+  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
+  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real DIM('n) *  e"
+proof-
+  let ?d = "real DIM('n)"
+  let ?nf = "\<lambda>x. norm (f x)"
+  let ?U = "{..<DIM('n)}"
+  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P) ?U"
+    by (rule setsum_commute)
+  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
+  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P"
+    apply (rule setsum_mono)    by (rule norm_le_l1)
+  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 $$ i\<bar>) ?Pp \<le> e"
+      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
+      unfolding euclidean_component.setsum by(auto intro: abs_le_D1)
+    have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
+      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
+      unfolding euclidean_component.setsum euclidean_component.minus
+      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 $$ i\<bar>) ?Pn"
+      apply (subst thp)
+      apply (rule setsum_Un_zero)
+      using fP thp0 by auto
+    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
+    finally show "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P \<le> 2*e" .
+  qed
+  finally show ?thesis .
+qed
+
+lemma choice_iff': "(\<forall>x<d. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x<d. P x (f x))" by metis
+
+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
+    }
+    hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
+    hence ?rhs by metis }
+  ultimately show ?thesis by metis
+qed
+
+subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
+
+class ordered_euclidean_space = ord + euclidean_space +
+  assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
+  and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
+
+lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
+  unfolding eucl_less[where 'a='a] by auto
+
+lemma euclidean_trans[trans]:
+  fixes x y z :: "'a::ordered_euclidean_space"
+  shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+  and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+  and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
+  by (force simp: eucl_less[where 'a='a] eucl_le[where 'a='a])+
+
+subsection {* Linearity and Bilinearity continued *}
+
+lemma linear_bounded:
+  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+  assumes lf: "linear f"
+  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
+proof-
+  let ?S = "{..<DIM('a)}"
+  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
+  have fS: "finite ?S" by simp
+  {fix x:: "'a"
+    let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
+    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
+      apply(subst euclidean_representation[of x]) ..
+    also have "\<dots> = norm (setsum (\<lambda> i. (x$$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"
+      unfolding norm_scaleR
+      apply (simp only: mult_commute)
+      apply (rule mult_mono)
+      by (auto simp add: field_simps) }
+    then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
+    from setsum_norm_le[OF fS, of "\<lambda>i. (x$$i) *\<^sub>R (f (basis i))", OF th]
+    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
+  then show ?thesis by blast
+qed
+
+lemma linear_bounded_pos:
+  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+  assumes lf: "linear f"
+  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
+proof-
+  from linear_bounded[OF lf] obtain B where
+    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
+  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] simp add:euclidean_component.zero)
+      hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
+      with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 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
+    }
+    then have Bp: "B \<ge> 0" by (metis not_leE)
+    {fix x::"'a"
+      have "norm (f x) \<le> ?K *  norm x"
+      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
+      apply (auto simp add: field_simps split add: abs_split)
+      apply (erule order_trans, simp)
+      done
+  }
+  then show ?thesis using Kp by blast
+qed
+
+lemma linear_conv_bounded_linear:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+  shows "linear f \<longleftrightarrow> bounded_linear f"
+proof
+  assume "linear f"
+  show "bounded_linear f"
+  proof
+    fix x y show "f (x + y) = f x + f y"
+      using `linear f` unfolding linear_def by simp
+  next
+    fix r x show "f (scaleR r x) = scaleR r (f x)"
+      using `linear f` unfolding linear_def by simp
+  next
+    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
+      using `linear f` by (rule linear_bounded)
+    thus "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
+      by (simp add: mult_commute)
+  qed
+next
+  assume "bounded_linear f"
+  then interpret f: bounded_linear f .
+  show "linear f"
+    by (simp add: f.add f.scaleR linear_def)
+qed
+
+lemma bounded_linearI': fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+  assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+  shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
+  by(rule linearI[OF assms])
+
+
+lemma bilinear_bounded:
+  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]) ..
+    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)
+      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 (rule mult_mono)
+      apply (auto simp add: zero_le_mult_iff component_le_norm)
+      done}
+  then show ?thesis by metis
+qed
+
+lemma bilinear_bounded_pos:
+  fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
+  assumes bh: "bilinear h"
+  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
+proof-
+  from bilinear_bounded[OF bh] obtain B where
+    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
+  let ?K = "\<bar>B\<bar> + 1"
+  have Kp: "?K > 0" by arith
+  have KB: "B < ?K" by arith
+  {fix x::'a and y::'b
+    from KB Kp
+    have "B * norm x * norm y \<le> ?K * norm x * norm y"
+      apply -
+      apply (rule mult_right_mono, rule mult_right_mono)
+      by auto
+    then have "norm (h x y) \<le> ?K * norm x * norm y"
+      using B[rule_format, of x y] by simp}
+  with Kp show ?thesis by blast
+qed
+
+lemma bilinear_conv_bounded_bilinear:
+  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
+  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
+proof
+  assume "bilinear h"
+  show "bounded_bilinear h"
+  proof
+    fix x y z show "h (x + y) z = h x z + h y z"
+      using `bilinear h` unfolding bilinear_def linear_def by simp
+  next
+    fix x y z show "h x (y + z) = h x y + h x z"
+      using `bilinear h` unfolding bilinear_def linear_def by simp
+  next
+    fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
+      using `bilinear h` unfolding bilinear_def linear_def
+      by simp
+  next
+    fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
+      using `bilinear h` unfolding bilinear_def linear_def
+      by simp
+  next
+    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
+      using `bilinear h` by (rule bilinear_bounded)
+    thus "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
+      by (simp add: mult_ac)
+  qed
+next
+  assume "bounded_bilinear h"
+  then interpret h: bounded_bilinear h .
+  show "bilinear h"
+    unfolding bilinear_def linear_conv_bounded_linear
+    using h.bounded_linear_left h.bounded_linear_right
+    by simp
+qed
+
+subsection {* We continue. *}
+
+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
+
+lemma dependent_biggerset: "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
+  by (metis independent_bound not_less)
+
+text {* Hence we can create a maximal independent subset. *}
+
+lemma maximal_independent_subset_extend:
+  assumes sv: "(S::('a::euclidean_space) set) \<subseteq> V" and iS: "independent S"
+  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+  using sv iS
+proof(induct "DIM('a) - card S" arbitrary: S rule: less_induct)
+  case less
+  note sv = `S \<subseteq> V` and i = `independent S`
+  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+  let ?ths = "\<exists>x. ?P x"
+  let ?d = "DIM('a)"
+  {assume "V \<subseteq> span S"
+    then have ?ths  using sv i by blast }
+  moreover
+  {assume VS: "\<not> V \<subseteq> span S"
+    from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
+    from a have aS: "a \<notin> S" by (auto simp add: span_superset)
+    have th0: "insert a S \<subseteq> V" using a sv by blast
+    from independent_insert[of a S]  i a
+    have th1: "independent (insert a S)" by auto
+    have mlt: "?d - card (insert a S) < ?d - card S"
+      using aS a independent_bound[OF th1]
+      by auto
+
+    from less(1)[OF mlt th0 th1]
+    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
+      by blast
+    from B have "?P B" by auto
+    then have ?ths by blast}
+  ultimately show ?ths by blast
+qed
+
+lemma maximal_independent_subset:
+  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty)
+
+
+text {* Notion of dimension. *}
+
+definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
+
+lemma basis_exists:  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
+unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
+using maximal_independent_subset[of V] independent_bound
+by auto
+
+text {* Consequences of independence or spanning for cardinality. *}
+
+lemma independent_card_le_dim: 
+  assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B" shows "card B \<le> dim V"
+proof -
+  from basis_exists[of V] `B \<subseteq> V`
+  obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
+  with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
+  show ?thesis by auto
+qed
+
+lemma span_card_ge_dim:  "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
+  by (metis basis_exists[of V] independent_span_bound subset_trans)
+
+lemma basis_card_eq_dim:
+  "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
+  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
+
+lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
+  by (metis basis_card_eq_dim)
+
+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 by auto
+
+lemma dim_subset:
+  "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
+  using basis_exists[of T] basis_exists[of S]
+  by (metis independent_card_le_dim subset_trans)
+
+lemma dim_subset_UNIV: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
+  by (metis dim_subset subset_UNIV dim_UNIV)
+
+text {* Converses to those. *}
+
+lemma card_ge_dim_independent:
+  assumes BV:"(B::('a::euclidean_space) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
+  shows "V \<subseteq> span B"
+proof-
+  {fix a assume aV: "a \<in> V"
+    {assume aB: "a \<notin> span B"
+      then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
+      from aV BV have th0: "insert a B \<subseteq> V" by blast
+      from aB have "a \<notin>B" by (auto simp add: span_superset)
+      with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
+    then have "a \<in> span B"  by blast}
+  then show ?thesis by blast
+qed
+
+lemma card_le_dim_spanning:
+  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V" and VB: "V \<subseteq> span B"
+  and fB: "finite B" and dVB: "dim V \<ge> card B"
+  shows "independent B"
+proof-
+  {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
+    from a fB have c0: "card B \<noteq> 0" by auto
+    from a fB have cb: "card (B -{a}) = card B - 1" by auto
+    from BV a have th0: "B -{a} \<subseteq> V" by blast
+    {fix x assume x: "x \<in> V"
+      from a have eq: "insert a (B -{a}) = B" by blast
+      from x VB have x': "x \<in> span B" by blast
+      from span_trans[OF a(2), unfolded eq, OF x']
+      have "x \<in> span (B -{a})" . }
+    then have th1: "V \<subseteq> span (B -{a})" by blast
+    have th2: "finite (B -{a})" using fB by auto
+    from span_card_ge_dim[OF th0 th1 th2]
+    have c: "dim V \<le> card (B -{a})" .
+    from c c0 dVB cb have False by simp}
+  then show ?thesis unfolding dependent_def by blast
+qed
+
+lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
+  by (metis order_eq_iff card_le_dim_spanning
+    card_ge_dim_independent)
+
+text {* More general size bound lemmas. *}
+
+lemma independent_bound_general:
+  "independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
+  by (metis independent_card_le_dim independent_bound subset_refl)
+
+lemma dependent_biggerset_general: "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+  using independent_bound_general[of S] by (metis linorder_not_le)
+
+lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
+proof-
+  have th0: "dim S \<le> dim (span S)"
+    by (auto simp add: subset_eq intro: dim_subset span_superset)
+  from basis_exists[of S]
+  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
+  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
+  have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
+  have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
+  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
+    using fB(2)  by arith
+qed
+
+lemma subset_le_dim: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
+  by (metis dim_span dim_subset)
+
+lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
+  by (metis dim_span)
+
+lemma spans_image:
+  assumes lf: "linear f" and VB: "V \<subseteq> span B"
+  shows "f ` V \<subseteq> span (f ` B)"
+  unfolding span_linear_image[OF lf]
+  by (metis VB image_mono)
+
+lemma dim_image_le:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S)"
+proof-
+  from basis_exists[of S] obtain B where
+    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
+  from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
+  have "dim (f ` S) \<le> card (f ` B)"
+    apply (rule span_card_ge_dim)
+    using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
+  also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
+  finally show ?thesis .
+qed
+
+text {* Relation between bases and injectivity/surjectivity of map. *}
+
+lemma spanning_surjective_image:
+  assumes us: "UNIV \<subseteq> span S"
+  and lf: "linear f" and sf: "surj f"
+  shows "UNIV \<subseteq> span (f ` S)"
+proof-
+  have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
+  also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
+finally show ?thesis .
+qed
+
+lemma independent_injective_image:
+  assumes iS: "independent S" and lf: "linear f" and fi: "inj f"
+  shows "independent (f ` S)"
+proof-
+  {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
+    have eq: "f ` S - {f a} = f ` (S - {a})" using fi
+      by (auto simp add: inj_on_def)
+    from a have "f a \<in> f ` span (S -{a})"
+      unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
+    hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
+    with a(1) iS  have False by (simp add: dependent_def) }
+  then show ?thesis unfolding dependent_def by blast
+qed
+
+text {* Picking an orthogonal replacement for a spanning set. *}
+
+    (* FIXME : Move to some general theory ?*)
+definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
+
+lemma vector_sub_project_orthogonal: "(b::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
+  unfolding inner_simps by auto
+
+lemma basis_orthogonal:
+  fixes B :: "('a::euclidean_space) set"
+  assumes fB: "finite B"
+  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
+  (is " \<exists>C. ?P B C")
+proof(induct rule: finite_induct[OF fB])
+  case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
+next
+  case (2 a B)
+  note fB = `finite B` and aB = `a \<notin> B`
+  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
+  obtain C where C: "finite C" "card C \<le> card B"
+    "span C = span B" "pairwise orthogonal C" by blast
+  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
+  let ?C = "insert ?a C"
+  from C(1) have fC: "finite ?C" by simp
+  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
+  {fix x k
+    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
+    have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
+      apply (simp only: scaleR_right_diff_distrib th0)
+      apply (rule span_add_eq)
+      apply (rule span_mul)
+      apply (rule span_setsum[OF C(1)])
+      apply clarify
+      apply (rule span_mul)
+      by (rule span_superset)}
+  then have SC: "span ?C = span (insert a B)"
+    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
+  thm pairwise_def
+  {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
+    {assume xa: "x = ?a" and ya: "y = ?a"
+      have "orthogonal x y" using xa ya xy by blast}
+    moreover
+    {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
+      from ya have Cy: "C = insert y (C - {y})" by blast
+      have fth: "finite (C - {y})" using C by simp
+      have "orthogonal x y"
+        using xa ya
+        unfolding orthogonal_def xa inner_simps diff_eq_0_iff_eq
+        apply simp
+        apply (subst Cy)
+        using C(1) fth
+        apply (simp only: setsum_clauses)
+        apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
+        apply (rule setsum_0')
+        apply clarsimp
+        apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
+        by auto}
+    moreover
+    {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
+      from xa have Cx: "C = insert x (C - {x})" by blast
+      have fth: "finite (C - {x})" using C by simp
+      have "orthogonal x y"
+        using xa ya
+        unfolding orthogonal_def ya inner_simps diff_eq_0_iff_eq
+        apply simp
+        apply (subst Cx)
+        using C(1) fth
+        apply (simp only: setsum_clauses)
+        apply (subst inner_commute[of x])
+        apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
+        apply (rule setsum_0')
+        apply clarsimp
+        apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
+        by auto}
+    moreover
+    {assume xa: "x \<in> C" and ya: "y \<in> C"
+      have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
+    ultimately have "orthogonal x y" using xC yC by blast}
+  then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
+  from fC cC SC CPO have "?P (insert a B) ?C" by blast
+  then show ?case by blast
+qed
+
+lemma orthogonal_basis_exists:
+  fixes V :: "('a::euclidean_space) set"
+  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
+proof-
+  from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
+  from B have fB: "finite B" "card B = dim V" using independent_bound by auto
+  from basis_orthogonal[OF fB(1)] obtain C where
+    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
+  from C B
+  have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
+  from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
+  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
+  have iC: "independent C" by (simp add: dim_span)
+  from C fB have "card C \<le> dim V" by simp
+  moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
+    by (simp add: dim_span)
+  ultimately have CdV: "card C = dim V" using C(1) by simp
+  from C B CSV CdV iC show ?thesis by auto
+qed
+
+lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
+  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
+  by(auto simp add: span_span)
+
+text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
+
+lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
+  assumes sU: "span S \<noteq> UNIV"
+  shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
+proof-
+  from sU obtain a where a: "a \<notin> span S" by blast
+  from orthogonal_basis_exists obtain B where
+    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
+    by blast
+  from B have fB: "finite B" "card B = dim S" using independent_bound by auto
+  from span_mono[OF B(2)] span_mono[OF B(3)]
+  have sSB: "span S = span B" by (simp add: span_span)
+  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
+  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
+    unfolding sSB
+    apply (rule span_setsum[OF fB(1)])
+    apply clarsimp
+    apply (rule span_mul)
+    by (rule span_superset)
+  with a have a0:"?a  \<noteq> 0" by auto
+  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
+  proof(rule span_induct')
+    show "subspace (\<lambda>x. ?a \<bullet> x = 0)" by (auto simp add: subspace_def mem_def inner_simps)
+next
+    {fix x assume x: "x \<in> B"
+      from x have B': "B = insert x (B - {x})" by blast
+      have fth: "finite (B - {x})" using fB by simp
+      have "?a \<bullet> x = 0"
+        apply (subst B') using fB fth
+        unfolding setsum_clauses(2)[OF fth]
+        apply simp unfolding inner_simps
+        apply (clarsimp simp add: inner_simps dot_lsum)
+        apply (rule setsum_0', rule ballI)
+        unfolding inner_commute
+        by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
+    then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
+  qed
+  with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
+qed
+
+lemma span_not_univ_subset_hyperplane:
+  assumes SU: "span S \<noteq> (UNIV ::('a::euclidean_space) set)"
+  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
+  using span_not_univ_orthogonal[OF SU] by auto
+
+lemma lowdim_subset_hyperplane: fixes S::"('a::euclidean_space) set"
+  assumes d: "dim S < DIM('a)"
+  shows "\<exists>(a::'a). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
+proof-
+  {assume "span S = UNIV"
+    hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
+    hence "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
+    with d have False by arith}
+  hence th: "span S \<noteq> UNIV" by blast
+  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
+qed
+
+text {* We can extend a linear basis-basis injection to the whole set. *}
+
+lemma linear_indep_image_lemma:
+  assumes lf: "linear f" and fB: "finite B"
+  and ifB: "independent (f ` B)"
+  and fi: "inj_on f B" and xsB: "x \<in> span B"
+  and fx: "f x = 0"
+  shows "x = 0"
+  using fB ifB fi xsB fx
+proof(induct arbitrary: x rule: finite_induct[OF fB])
+  case 1 thus ?case by (auto simp add:  span_empty)
+next
+  case (2 a b x)
+  have fb: "finite b" using "2.prems" by simp
+  have th0: "f ` b \<subseteq> f ` (insert a b)"
+    apply (rule image_mono) by blast
+  from independent_mono[ OF "2.prems"(2) th0]
+  have ifb: "independent (f ` b)"  .
+  have fib: "inj_on f b"
+    apply (rule subset_inj_on [OF "2.prems"(3)])
+    by blast
+  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
+  obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
+  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
+    unfolding span_linear_image[OF lf]
+    apply (rule imageI)
+    using k span_mono[of "b-{a}" b] by blast
+  hence "f x - k*\<^sub>R f a \<in> span (f ` b)"
+    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
+  hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
+    using "2.prems"(5) by simp
+  {assume k0: "k = 0"
+    from k0 k have "x \<in> span (b -{a})" by simp
+    then have "x \<in> span b" using span_mono[of "b-{a}" b]
+      by blast}
+  moreover
+  {assume k0: "k \<noteq> 0"
+    from span_mul[OF th, of "- 1/ k"] k0
+    have th1: "f a \<in> span (f ` b)"
+      by auto
+    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
+    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
+    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
+    have "f a \<notin> span (f ` b)" using tha
+      using "2.hyps"(2)
+      "2.prems"(3) by auto
+    with th1 have False by blast
+    then have "x \<in> span b" by blast}
+  ultimately have xsb: "x \<in> span b" by blast
+  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
+  show "x = 0" .
+qed
+
+text {* We can extend a linear mapping from basis. *}
+
+lemma linear_independent_extend_lemma:
+  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
+  assumes fi: "finite B" and ib: "independent B"
+  shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
+           \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
+           \<and> (\<forall>x\<in> B. g x = f x)"
+using ib fi
+proof(induct rule: finite_induct[OF fi])
+  case 1 thus ?case by (auto simp add: span_empty)
+next
+  case (2 a b)
+  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
+    by (simp_all add: independent_insert)
+  from "2.hyps"(3)[OF ibf] obtain g where
+    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
+    "\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
+  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
+  {fix z assume z: "z \<in> span (insert a b)"
+    have th0: "z - ?h z *\<^sub>R a \<in> span b"
+      apply (rule someI_ex)
+      unfolding span_breakdown_eq[symmetric]
+      using z .
+    {fix k assume k: "z - k *\<^sub>R a \<in> span b"
+      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
+        by (simp add: field_simps scaleR_left_distrib [symmetric])
+      from span_sub[OF th0 k]
+      have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
+      {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
+        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
+        have "a \<in> span b" by simp
+        with "2.prems"(1) "2.hyps"(2) have False
+          by (auto simp add: dependent_def)}
+      then have "k = ?h z" by blast}
+    with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
+  note h = this
+  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
+  {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
+    have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
+      by (simp add: algebra_simps)
+    have addh: "?h (x + y) = ?h x + ?h y"
+      apply (rule conjunct2[OF h, rule_format, symmetric])
+      apply (rule span_add[OF x y])
+      unfolding tha
+      by (metis span_add x y conjunct1[OF h, rule_format])
+    have "?g (x + y) = ?g x + ?g y"
+      unfolding addh tha
+      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
+      by (simp add: scaleR_left_distrib)}
+  moreover
+  {fix x:: "'a" and c:: real  assume x: "x \<in> span (insert a b)"
+    have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
+      by (simp add: algebra_simps)
+    have hc: "?h (c *\<^sub>R x) = c * ?h x"
+      apply (rule conjunct2[OF h, rule_format, symmetric])
+      apply (metis span_mul x)
+      by (metis tha span_mul x conjunct1[OF h])
+    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
+      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
+      by (simp add: algebra_simps)}
+  moreover
+  {fix x assume x: "x \<in> (insert a b)"
+    {assume xa: "x = a"
+      have ha1: "1 = ?h a"
+        apply (rule conjunct2[OF h, rule_format])
+        apply (metis span_superset insertI1)
+        using conjunct1[OF h, OF span_superset, OF insertI1]
+        by (auto simp add: span_0)
+
+      from xa ha1[symmetric] have "?g x = f x"
+        apply simp
+        using g(2)[rule_format, OF span_0, of 0]
+        by simp}
+    moreover
+    {assume xb: "x \<in> b"
+      have h0: "0 = ?h x"
+        apply (rule conjunct2[OF h, rule_format])
+        apply (metis  span_superset x)
+        apply simp
+        apply (metis span_superset xb)
+        done
+      have "?g x = f x"
+        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
+    ultimately have "?g x = f x" using x by blast }
+  ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
+qed
+
+lemma linear_independent_extend:
+  assumes iB: "independent (B:: ('a::euclidean_space) set)"
+  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
+proof-
+  from maximal_independent_subset_extend[of B UNIV] iB
+  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
+
+  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
+  obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
+           \<and> (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
+           \<and> (\<forall>x\<in> C. g x = f x)" by blast
+  from g show ?thesis unfolding linear_def using C
+    apply clarsimp by blast
+qed
+
+text {* Can construct an isomorphism between spaces of same dimension. *}
+
+lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
+  and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
+using fB c
+proof(induct arbitrary: B rule: finite_induct[OF fA])
+  case 1 thus ?case by simp
+next
+  case (2 x s t)
+  thus ?case
+  proof(induct rule: finite_induct[OF "2.prems"(1)])
+    case 1    then show ?case by simp
+  next
+    case (2 y t)
+    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
+    from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
+      f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
+    from f "2.prems"(2) "2.hyps"(2) show ?case
+      apply -
+      apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
+      by (auto simp add: inj_on_def)
+  qed
+qed
+
+lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
+  c: "card A = card B"
+  shows "A = B"
+proof-
+  from fB AB have fA: "finite A" by (auto intro: finite_subset)
+  from fA fB have fBA: "finite (B - A)" by auto
+  have e: "A \<inter> (B - A) = {}" by blast
+  have eq: "A \<union> (B - A) = B" using AB by blast
+  from card_Un_disjoint[OF fA fBA e, unfolded eq c]
+  have "card (B - A) = 0" by arith
+  hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
+  with AB show "A = B" by blast
+qed
+
+lemma subspace_isomorphism:
+  assumes s: "subspace (S:: ('a::euclidean_space) set)"
+  and t: "subspace (T :: ('b::euclidean_space) set)"
+  and d: "dim S = dim T"
+  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
+proof-
+  from basis_exists[of S] independent_bound obtain B where
+    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
+  from basis_exists[of T] independent_bound obtain C where
+    C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
+  from B(4) C(4) card_le_inj[of B C] d obtain f where
+    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
+  from linear_independent_extend[OF B(2)] obtain g where
+    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
+  from inj_on_iff_eq_card[OF fB, of f] f(2)
+  have "card (f ` B) = card B" by simp
+  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
+    by simp
+  have "g ` B = f ` B" using g(2)
+    by (auto simp add: image_iff)
+  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
+  finally have gBC: "g ` B = C" .
+  have gi: "inj_on g B" using f(2) g(2)
+    by (auto simp add: inj_on_def)
+  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
+  {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
+    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
+    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
+    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
+    have "x=y" using g0[OF th1 th0] by simp }
+  then have giS: "inj_on g S"
+    unfolding inj_on_def by blast
+  from span_subspace[OF B(1,3) s]
+  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
+  also have "\<dots> = span C" unfolding gBC ..
+  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
+  finally have gS: "g ` S = T" .
+  from g(1) gS giS show ?thesis by blast
+qed
+
+text {* Linear functions are equal on a subspace if they are on a spanning set. *}
+
+lemma subspace_kernel:
+  assumes lf: "linear f"
+  shows "subspace {x. f x = 0}"
+apply (simp add: subspace_def)
+by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
+
+lemma linear_eq_0_span:
+  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
+  shows "\<forall>x \<in> span B. f x = 0"
+proof
+  fix x assume x: "x \<in> span B"
+  let ?P = "\<lambda>x. f x = 0"
+  from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
+  with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
+qed
+
+lemma linear_eq_0:
+  assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
+  shows "\<forall>x \<in> S. f x = 0"
+  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
+
+lemma linear_eq:
+  assumes lf: "linear f" and lg: "linear g" and S: "S \<subseteq> span B"
+  and fg: "\<forall> x\<in> B. f x = g x"
+  shows "\<forall>x\<in> S. f x = g x"
+proof-
+  let ?h = "\<lambda>x. f x - g x"
+  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
+  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
+  show ?thesis by simp
+qed
+
+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)"
+  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 by auto }
+  then show ?thesis by (auto intro: ext)
+qed
+
+text {* Similar results for bilinear functions. *}
+
+lemma bilinear_eq:
+  assumes bf: "bilinear f"
+  and bg: "bilinear g"
+  and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
+  and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
+  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
+proof-
+  let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
+  from bf bg have sp: "subspace ?P"
+    unfolding bilinear_def linear_def subspace_def bf bg
+    by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
+
+  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
+    apply -
+    apply (rule ballI)
+    apply (rule span_induct[of B ?P])
+    defer
+    apply (rule sp)
+    apply assumption
+    apply (clarsimp simp add: Ball_def)
+    apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
+    using fg
+    apply (auto simp add: subspace_def)
+    using bf bg unfolding bilinear_def linear_def
+    by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
+  then show ?thesis using SB TC by (auto intro: ext)
+qed
+
+lemma bilinear_eq_stdbasis: 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)"
+  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 intro: ext)
+qed
+
+text {* Detailed theorems about left and right invertibility in general case. *}
+
+lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+  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]
+  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)"
+    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
+    by auto
+
+  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
+  have "h o f = id" .
+  then show ?thesis using h(1) by blast
+qed
+
+lemma linear_surjective_right_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+  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"]
+  obtain h:: "'b \<Rightarrow> 'a" where
+    h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. 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)
+  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
+qed
+
+text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
+
+lemma linear_injective_imp_surjective:  fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and fi: "inj f"
+  shows "surj f"
+proof-
+  let ?U = "UNIV :: 'a set"
+  from basis_exists[of ?U] obtain B
+    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
+    by blast
+  from B(4) have d: "dim ?U = card B" by simp
+  have th: "?U \<subseteq> span (f ` B)"
+    apply (rule card_ge_dim_independent)
+    apply blast
+    apply (rule independent_injective_image[OF B(2) lf fi])
+    apply (rule order_eq_refl)
+    apply (rule sym)
+    unfolding d
+    apply (rule card_image)
+    apply (rule subset_inj_on[OF fi])
+    by blast
+  from th show ?thesis
+    unfolding span_linear_image[OF lf] surj_def
+    using B(3) by blast
+qed
+
+text {* And vice versa. *}
+
+lemma surjective_iff_injective_gen:
+  assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
+  and ST: "f ` S \<subseteq> T"
+  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  {assume h: "?lhs"
+    {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
+      from x fS have S0: "card S \<noteq> 0" by auto
+      {assume xy: "x \<noteq> y"
+        have th: "card S \<le> card (f ` (S - {y}))"
+          unfolding c
+          apply (rule card_mono)
+          apply (rule finite_imageI)
+          using fS apply simp
+          using h xy x y f unfolding subset_eq image_iff
+          apply auto
+          apply (case_tac "xa = f x")
+          apply (rule bexI[where x=x])
+          apply auto
+          done
+        also have " \<dots> \<le> card (S -{y})"
+          apply (rule card_image_le)
+          using fS by simp
+        also have "\<dots> \<le> card S - 1" using y fS by simp
+        finally have False  using S0 by arith }
+      then have "x = y" by blast}
+    then have ?rhs unfolding inj_on_def by blast}
+  moreover
+  {assume h: ?rhs
+    have "f ` S = T"
+      apply (rule card_subset_eq[OF fT ST])
+      unfolding card_image[OF h] using c .
+    then have ?lhs by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma linear_surjective_imp_injective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and sf: "surj f"
+  shows "inj f"
+proof-
+  let ?U = "UNIV :: 'a set"
+  from basis_exists[of ?U] obtain B
+    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
+    by blast
+  {fix x assume x: "x \<in> span B" and fx: "f x = 0"
+    from B(2) have fB: "finite B" using independent_bound by auto
+    have fBi: "independent (f ` B)"
+      apply (rule card_le_dim_spanning[of "f ` B" ?U])
+      apply blast
+      using sf B(3)
+      unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
+      apply blast
+      using fB apply blast
+      unfolding d[symmetric]
+      apply (rule card_image_le)
+      apply (rule fB)
+      done
+    have th0: "dim ?U \<le> card (f ` B)"
+      apply (rule span_card_ge_dim)
+      apply blast
+      unfolding span_linear_image[OF lf]
+      apply (rule subset_trans[where B = "f ` UNIV"])
+      using sf unfolding surj_def apply blast
+      apply (rule image_mono)
+      apply (rule B(3))
+      apply (metis finite_imageI fB)
+      done
+
+    moreover have "card (f ` B) \<le> card B"
+      by (rule card_image_le, rule fB)
+    ultimately have th1: "card B = card (f ` B)" unfolding d by arith
+    have fiB: "inj_on f B"
+      unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
+    from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
+    have "x = 0" by blast}
+  note th = this
+  from th show ?thesis unfolding linear_injective_0[OF lf]
+    using B(3) by blast
+qed
+
+text {* Hence either is enough for isomorphism. *}
+
+lemma left_right_inverse_eq:
+  assumes fg: "f o g = id" and gh: "g o h = id"
+  shows "f = h"
+proof-
+  have "f = f o (g o h)" unfolding gh by simp
+  also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
+  finally show "f = h" unfolding fg by simp
+qed
+
+lemma isomorphism_expand:
+  "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
+  by (simp add: fun_eq_iff o_def id_def)
+
+lemma linear_injective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and fi: "inj f"
+  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
+unfolding isomorphism_expand[symmetric]
+using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
+by (metis left_right_inverse_eq)
+
+lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and sf: "surj f"
+  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
+unfolding isomorphism_expand[symmetric]
+using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
+by (metis left_right_inverse_eq)
+
+text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
+
+lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and lf': "linear f'"
+  shows "f o f' = id \<longleftrightarrow> f' o f = id"
+proof-
+  {fix f f':: "'a => 'a"
+    assume lf: "linear f" "linear f'" and f: "f o f' = id"
+    from f have sf: "surj f"
+      apply (auto simp add: o_def id_def surj_def)
+      by metis
+    from linear_surjective_isomorphism[OF lf(1) sf] lf f
+    have "f' o f = id" unfolding fun_eq_iff o_def id_def
+      by metis}
+  then show ?thesis using lf lf' by metis
+qed
+
+text {* Moreover, a one-sided inverse is automatically linear. *}
+
+lemma left_inverse_linear: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+  assumes lf: "linear f" and gf: "g o f = id"
+  shows "linear g"
+proof-
+  from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
+    by metis
+  from linear_injective_isomorphism[OF lf fi]
+  obtain h:: "'a \<Rightarrow> 'a" where
+    h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
+  have "h = g" apply (rule ext) using gf h(2,3)
+    apply (simp add: o_def id_def fun_eq_iff)
+    by metis
+  with h(1) show ?thesis by blast
+qed
+
+subsection {* Infinity norm *}
+
+definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
+
+lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
+  by auto
+
+lemma infnorm_set_image:
+  "{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
+  (\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
+
+lemma infnorm_set_lemma:
+  shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
+  and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
+  unfolding infnorm_set_image
+  by auto
+
+lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
+  unfolding infnorm_def
+  unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
+  unfolding infnorm_set_image
+  by auto
+
+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 by (metis th2)
+qed
+
+lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
+proof-
+  have "infnorm x <= 0 \<longleftrightarrow> x = 0"
+    unfolding infnorm_def
+    unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
+    unfolding infnorm_set_image ball_simps
+    apply(subst (1) euclidean_eq) unfolding euclidean_component.zero
+    by auto
+  then show ?thesis using infnorm_pos_le[of x] by simp
+qed
+
+lemma infnorm_0: "infnorm 0 = 0"
+  by (simp add: infnorm_eq_0)
+
+lemma infnorm_neg: "infnorm (- x) = infnorm x"
+  unfolding infnorm_def
+  apply (rule cong[of "Sup" "Sup"])
+  apply blast by(auto simp add: euclidean_simps)
+
+lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
+proof-
+  have "y - x = - (x - y)" by simp
+  then show ?thesis  by (metis infnorm_neg)
+qed
+
+lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
+proof-
+  have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
+    by arith
+  from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
+  have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
+    "infnorm y \<le> infnorm (x - y) + infnorm x"
+    by (simp_all add: field_simps infnorm_neg diff_minus[symmetric])
+  from th[OF ths]  show ?thesis .
+qed
+
+lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
+  using infnorm_pos_le[of x] by arith
+
+lemma component_le_infnorm:
+  shows "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
+proof(cases "i<DIM('a)")
+  case False thus ?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) by simp
+  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 by auto
+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_scaleR abs_mult
+  using component_le_infnorm[of x] by(auto intro: mult_mono) 
+
+lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
+proof-
+  {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
+  moreover
+  {assume a0: "a \<noteq> 0"
+    from a0 have th: "(1/a) *\<^sub>R (a *\<^sub>R x) = x" by simp
+    from a0 have ap: "\<bar>a\<bar> > 0" by arith
+    from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
+    have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*\<^sub>R x)"
+      unfolding th by simp
+    with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *\<^sub>R x))" by (simp add: field_simps)
+    then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*\<^sub>R x)"
+      using ap by (simp add: field_simps)
+    with infnorm_mul_lemma[of a x] have ?thesis by arith }
+  ultimately show ?thesis by blast
+qed
+
+lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
+  using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
+
+text {* Prove that it differs only up to a bound from Euclidean norm. *}
+
+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 card_enum: "card {1 .. n} = n" by auto
+
+lemma norm_le_infnorm: "norm(x) <= sqrt(real DIM('a)) * infnorm(x::'a::euclidean_space)"
+proof-
+  let ?d = "DIM('a)"
+  have "real ?d \<ge> 0" by simp
+  hence d2: "(sqrt (real ?d))^2 = real ?d"
+    by (auto intro: real_sqrt_pow2)
+  have th: "sqrt (real ?d) * infnorm x \<ge> 0"
+    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)
+    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_set_image bex_simps apply(rule_tac x=i in bexI) by auto
+  from real_le_lsqrt[OF inner_ge_zero th th1]
+  show ?thesis unfolding norm_eq_sqrt_inner id_def .
+qed
+
+text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
+
+lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  {assume h: "x = 0"
+    hence ?thesis by simp}
+  moreover
+  {assume h: "y = 0"
+    hence ?thesis by simp}
+  moreover
+  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+    from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
+    have "?rhs \<longleftrightarrow> (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
+      using x y
+      unfolding inner_simps
+      unfolding power2_norm_eq_inner[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: inner_commute)
+      apply (simp add: field_simps) by metis
+    also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
+      by (simp add: field_simps inner_commute)
+    also have "\<dots> \<longleftrightarrow> ?lhs" using x y
+      apply simp
+      by metis
+    finally have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma norm_cauchy_schwarz_abs_eq:
+  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
+                norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
+  have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
+    by simp
+  also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
+     (-x) \<bullet> y = norm x * norm y)"
+    unfolding norm_cauchy_schwarz_eq[symmetric]
+    unfolding norm_minus_cancel norm_scaleR ..
+  also have "\<dots> \<longleftrightarrow> ?lhs"
+    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps by auto
+  finally show ?thesis ..
+qed
+
+lemma norm_triangle_eq:
+  fixes x y :: "'a::real_inner"
+  shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+proof-
+  {assume x: "x =0 \<or> y =0"
+    hence ?thesis by (cases "x=0", simp_all)}
+  moreover
+  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+    hence "norm x \<noteq> 0" "norm y \<noteq> 0"
+      by simp_all
+    hence n: "norm x > 0" "norm y > 0"
+      using norm_ge_zero[of x] norm_ge_zero[of y]
+      by arith+
+    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
+    have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
+      apply (rule th) using n norm_ge_zero[of "x + y"]
+      by arith
+    also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+      unfolding norm_cauchy_schwarz_eq[symmetric]
+      unfolding power2_norm_eq_inner inner_simps
+      by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
+    finally have ?thesis .}
+  ultimately show ?thesis by blast
+qed
+
+subsection {* Collinearity *}
+
+definition
+  collinear :: "'a::real_vector set \<Rightarrow> bool" where
+  "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
+
+lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
+
+lemma collinear_sing: "collinear {x}"
+  by (simp add: collinear_def)
+
+lemma collinear_2: "collinear {x, y}"
+  apply (simp add: collinear_def)
+  apply (rule exI[where x="x - y"])
+  apply auto
+  apply (rule exI[where x=1], simp)
+  apply (rule exI[where x="- 1"], simp)
+  done
+
+lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  {assume "x=0 \<or> y = 0" hence ?thesis
+      by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
+  moreover
+  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+    {assume h: "?lhs"
+      then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u" unfolding collinear_def by blast
+      from u[rule_format, of x 0] u[rule_format, of y 0]
+      obtain cx and cy where
+        cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
+        by auto
+      from cx x have cx0: "cx \<noteq> 0" by auto
+      from cy y have cy0: "cy \<noteq> 0" by auto
+      let ?d = "cy / cx"
+      from cx cy cx0 have "y = ?d *\<^sub>R x"
+        by simp
+      hence ?rhs using x y by blast}
+    moreover
+    {assume h: "?rhs"
+      then obtain c where c: "y = c *\<^sub>R x" using x y by blast
+      have ?lhs unfolding collinear_def c
+        apply (rule exI[where x=x])
+        apply auto
+        apply (rule exI[where x="- 1"], simp)
+        apply (rule exI[where x= "-c"], simp)
+        apply (rule exI[where x=1], simp)
+        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
+        apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
+        done}
+    ultimately have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma norm_cauchy_schwarz_equal:
+  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
+unfolding norm_cauchy_schwarz_abs_eq
+apply (cases "x=0", simp_all add: collinear_2)
+apply (cases "y=0", simp_all add: collinear_2 insert_commute)
+unfolding collinear_lemma
+apply simp
+apply (subgoal_tac "norm x \<noteq> 0")
+apply (subgoal_tac "norm y \<noteq> 0")
+apply (rule iffI)
+apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
+apply (rule exI[where x="(1/norm x) * norm y"])
+apply (drule sym)
+unfolding scaleR_scaleR[symmetric]
+apply (simp add: field_simps)
+apply (rule exI[where x="(1/norm x) * - norm y"])
+apply clarify
+apply (drule sym)
+unfolding scaleR_scaleR[symmetric]
+apply (simp add: field_simps)
+apply (erule exE)
+apply (erule ssubst)
+unfolding scaleR_scaleR
+unfolding norm_scaleR
+apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
+apply (case_tac "c <= 0", simp add: field_simps)
+apply (simp add: field_simps)
+apply (case_tac "c <= 0", simp add: field_simps)
+apply (simp add: field_simps)
+apply simp
+apply simp
+done
+
+subsection "Instantiate @{typ real} and @{typ complex} as typeclass @{text ordered_euclidean_space}."
+
+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 $$ i = 0"
+  defer apply(subst euclidean_eq) apply safe
+  unfolding euclidean_lambda_beta'
+  unfolding euclidean_component_def by auto
+
+lemma complex_basis[simp]:
+  shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
+  unfolding basis_complex_def by auto
+
+section {* Products Spaces *}
+
+lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::euclidean_space) + DIM('a::euclidean_space)"
+  (* FIXME: why this orientation? Why not "DIM('a) + DIM('b)" ? *)
+  unfolding dimension_prod_def by (rule add_commute)
+
+instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
+begin
+
+definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
+definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
+
+instance proof qed (auto simp: less_prod_def less_eq_prod_def)
+end
+
+
+end
diff -r 0f35a870ecf1 -r 691c52e900ca src/HOL/Multivariate_Analysis/Operator_Norm.thy
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 10 09:23:42 2011 -0700
@@ -5,7 +5,7 @@
 header {* Operator Norm *}
 
 theory Operator_Norm
-imports Euclidean_Space
+imports Linear_Algebra
 begin
 
 definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
diff -r 0f35a870ecf1 -r 691c52e900ca src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 10 09:23:42 2011 -0700
@@ -7,7 +7,7 @@
 header {* Elementary topology in Euclidean space. *}
 
 theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space "~~/src/HOL/Library/Glbs"
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
 begin
 
 (* to be moved elsewhere *)