(*  Title:      Library/Euclidean_Space

     Author:     Amine Chaieb, University of Cambridge

 *)

 header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}

 theory Euclidean_Space

 imports

   Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"

   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type

   Inner_Product

 uses "positivstellensatz.ML" ("normarith.ML")

 begin

 text{* Some common special cases.*}

 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"

   by (metis num1_eq_iff)

 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"

   by auto (metis num1_eq_iff)

 lemma exhaust_2:

   fixes x :: 2 shows "x = 1 \<or> x = 2"

 proof (induct x)

   case (of_int z)

   then have "0 <= z" and "z < 2" by simp_all

   then have "z = 0 | z = 1" by arith

   then show ?case by auto

 qed

 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"

   by (metis exhaust_2)

 lemma exhaust_3:

   fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"

 proof (induct x)

   case (of_int z)

   then have "0 <= z" and "z < 3" by simp_all

   then have "z = 0 \<or> z = 1 \<or> z = 2" by arith

   then show ?case by auto

 qed

 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"

   by (metis exhaust_3)

 lemma UNIV_1: "UNIV = {1::1}"

   by (auto simp add: num1_eq_iff)

 lemma UNIV_2: "UNIV = {1::2, 2::2}"

   using exhaust_2 by auto

 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"

   using exhaust_3 by auto

 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"

   unfolding UNIV_1 by simp

 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"

   unfolding UNIV_2 by simp

 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"

   unfolding UNIV_3 by (simp add: add_ac)

 subsection{* Basic componentwise operations on vectors. *}

 instantiation cart :: (plus,finite) plus

 begin

 definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))"

 instance ..

 end

 instantiation cart :: (times,finite) times

 begin

   definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"

   instance ..

 end

 instantiation cart :: (minus,finite) minus begin

   definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"

 instance ..

 end

 instantiation cart :: (uminus,finite) uminus begin

   definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"

 instance ..

 end

 instantiation cart :: (zero,finite) zero begin

   definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"

 instance ..

 end

 instantiation cart :: (one,finite) one begin

   definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"

 instance ..

 end

 instantiation cart :: (ord,finite) ord

  begin

 definition vector_le_def:

   "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"

 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"

 instance by (intro_classes)

 end

 instantiation cart :: (scaleR, finite) scaleR

 begin

 definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))"

 instance ..

 end

 text{* Also the scalar-vector multiplication. *}

 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)

   where "c *s x = (\<chi> i. c * (x$i))"

 text{* Constant Vectors *} 

 definition "vec x = (\<chi> i. x)"

 text{* Dot products. *}

 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where

   "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) UNIV"

 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"

   by (simp add: dot_def setsum_1)

 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"

   by (simp add: dot_def setsum_2)

 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"

   by (simp add: dot_def setsum_3)

 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}

 method_setup vector = {*

 let

   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,

   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},

   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]

   val ss2 = @{simpset} addsimps

              [@{thm vector_add_def}, @{thm vector_mult_def},

               @{thm vector_minus_def}, @{thm vector_uminus_def},

               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},

               @{thm vector_scaleR_def},

               @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]

  fun vector_arith_tac ths =

    simp_tac ss1

    THEN' (fn i => rtac @{thm setsum_cong2} i

          ORELSE rtac @{thm setsum_0'} i

          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)

    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)

    THEN' asm_full_simp_tac (ss2 addsimps ths)

  in

   Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))

  end

 *} "Lifts trivial vector statements to real arith statements"

 lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)

 lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)

 text{* Obvious "component-pushing". *}

 lemma vec_component [simp]: "vec x $ i = x"

   by (vector vec_def)

 lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"

   by vector

 lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"

   by vector

 lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"

   by vector

 lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"

   by vector

 lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"

   by vector

 lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"

   by vector

 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector

 lemmas vector_component =

   vec_component vector_add_component vector_mult_component

   vector_smult_component vector_minus_component vector_uminus_component

   vector_scaleR_component cond_component

 subsection {* Some frequently useful arithmetic lemmas over vectors. *}

 instance cart :: (semigroup_add,finite) semigroup_add

   apply (intro_classes) by (vector add_assoc)

 instance cart :: (monoid_add,finite) monoid_add

   apply (intro_classes) by vector+

 instance cart :: (group_add,finite) group_add

   apply (intro_classes) by (vector algebra_simps)+

 instance cart :: (ab_semigroup_add,finite) ab_semigroup_add

   apply (intro_classes) by (vector add_commute)

 instance cart :: (comm_monoid_add,finite) comm_monoid_add

   apply (intro_classes) by vector

 instance cart :: (ab_group_add,finite) ab_group_add

   apply (intro_classes) by vector+

 instance cart :: (cancel_semigroup_add,finite) cancel_semigroup_add

   apply (intro_classes)

   by (vector Cart_eq)+

 instance cart :: (cancel_ab_semigroup_add,finite) cancel_ab_semigroup_add

   apply (intro_classes)

   by (vector Cart_eq)

 instance cart :: (real_vector, finite) real_vector

   by default (vector scaleR_left_distrib scaleR_right_distrib)+

 instance cart :: (semigroup_mult,finite) semigroup_mult

   apply (intro_classes) by (vector mult_assoc)

 instance cart :: (monoid_mult,finite) monoid_mult

   apply (intro_classes) by vector+

 instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult

   apply (intro_classes) by (vector mult_commute)

 instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult

   apply (intro_classes) by (vector mult_idem)

 instance cart :: (comm_monoid_mult,finite) comm_monoid_mult

   apply (intro_classes) by vector

 fun vector_power where

   "vector_power x 0 = 1"

   | "vector_power x (Suc n) = x * vector_power x n"

 instance cart :: (semiring,finite) semiring

   apply (intro_classes) by (vector ring_simps)+

 instance cart :: (semiring_0,finite) semiring_0

   apply (intro_classes) by (vector ring_simps)+

 instance cart :: (semiring_1,finite) semiring_1

   apply (intro_classes) by vector

 instance cart :: (comm_semiring,finite) comm_semiring

   apply (intro_classes) by (vector ring_simps)+

 instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes)

 instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..

 instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes)

 instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes)

 instance cart :: (ring,finite) ring by (intro_classes)

 instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes)

 instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes)

 instance cart :: (ring_1,finite) ring_1 ..

 instance cart :: (real_algebra,finite) real_algebra

   apply intro_classes

   apply (simp_all add: vector_scaleR_def ring_simps)

   apply vector

   apply vector

   done

 instance cart :: (real_algebra_1,finite) real_algebra_1 ..

 lemma of_nat_index:

   "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"

   apply (induct n)

   apply vector

   apply vector

   done

 lemma zero_index[simp]:

   "(0 :: 'a::zero ^'n)$i = 0" by vector

 lemma one_index[simp]:

   "(1 :: 'a::one ^'n)$i = 1" by vector

 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"

 proof-

   have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp

   also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)

   finally show ?thesis by simp

 qed

 instance cart :: (semiring_char_0,finite) semiring_char_0

 proof (intro_classes)

   fix m n ::nat

   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"

     by (simp add: Cart_eq of_nat_index)

 qed

 instance cart :: (comm_ring_1,finite) comm_ring_1 by intro_classes

 instance cart :: (ring_char_0,finite) ring_char_0 by intro_classes

 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"

   by (vector mult_assoc)

 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"

   by (vector ring_simps)

 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"

   by (vector ring_simps)

 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector

 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector

 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"

   by (vector ring_simps)

 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector

 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector

 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector

 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector

 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"

   by (vector ring_simps)

 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"

   by (simp add: Cart_eq)

 subsection {* Topological space *}

 instantiation cart :: (topological_space, finite) topological_space

 begin

 definition open_vector_def:

   "open (S :: ('a ^ 'b) set) \<longleftrightarrow>

     (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>

       (\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"

 instance proof

   show "open (UNIV :: ('a ^ 'b) set)"

     unfolding open_vector_def by auto

 next

   fix S T :: "('a ^ 'b) set"

   assume "open S" "open T" thus "open (S \<inter> T)"

     unfolding open_vector_def

     apply clarify

     apply (drule (1) bspec)+

     apply (clarify, rename_tac Sa Ta)

     apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)

     apply (simp add: open_Int)

     done

 next

   fix K :: "('a ^ 'b) set set"

   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"

     unfolding open_vector_def

     apply clarify

     apply (drule (1) bspec)

     apply (drule (1) bspec)

     apply clarify

     apply (rule_tac x=A in exI)

     apply fast

     done

 qed

 end

 lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"

 unfolding open_vector_def by auto

 lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"

 unfolding open_vector_def

 apply clarify

 apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)

 done

 lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"

 unfolding closed_open vimage_Compl [symmetric]

 by (rule open_vimage_Cart_nth)

 lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"

 proof -

   have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto

   thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"

     by (simp add: closed_INT closed_vimage_Cart_nth)

 qed

 lemma tendsto_Cart_nth [tendsto_intros]:

   assumes "((\<lambda>x. f x) ---> a) net"

   shows "((\<lambda>x. f x $ i) ---> a $ i) net"

 proof (rule topological_tendstoI)

   fix S assume "open S" "a $ i \<in> S"

   then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"

     by (simp_all add: open_vimage_Cart_nth)

   with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"

     by (rule topological_tendstoD)

   then show "eventually (\<lambda>x. f x $ i \<in> S) net"

     by simp

 qed

 subsection {* Square root of sum of squares *}

 definition

   "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"

 lemma setL2_cong:

   "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"

   unfolding setL2_def by simp

 lemma strong_setL2_cong:

   "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"

   unfolding setL2_def simp_implies_def by simp

 lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"

   unfolding setL2_def by simp

 lemma setL2_empty [simp]: "setL2 f {} = 0"

   unfolding setL2_def by simp

 lemma setL2_insert [simp]:

   "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>

     setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"

   unfolding setL2_def by (simp add: setsum_nonneg)

 lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"

   unfolding setL2_def by (simp add: setsum_nonneg)

 lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"

   unfolding setL2_def by simp

 lemma setL2_constant: "setL2 (\<lambda>x. y) A = sqrt (of_nat (card A)) * \<bar>y\<bar>"

   unfolding setL2_def by (simp add: real_sqrt_mult)

 lemma setL2_mono:

   assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"

   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"

   shows "setL2 f K \<le> setL2 g K"

   unfolding setL2_def

   by (simp add: setsum_nonneg setsum_mono power_mono prems)

 lemma setL2_strict_mono:

   assumes "finite K" and "K \<noteq> {}"

   assumes "\<And>i. i \<in> K \<Longrightarrow> f i < g i"

   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"

   shows "setL2 f K < setL2 g K"

   unfolding setL2_def

   by (simp add: setsum_strict_mono power_strict_mono assms)

 lemma setL2_right_distrib:

   "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"

   unfolding setL2_def

   apply (simp add: power_mult_distrib)

   apply (simp add: setsum_right_distrib [symmetric])

   apply (simp add: real_sqrt_mult setsum_nonneg)

   done

 lemma setL2_left_distrib:

   "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"

   unfolding setL2_def

   apply (simp add: power_mult_distrib)

   apply (simp add: setsum_left_distrib [symmetric])

   apply (simp add: real_sqrt_mult setsum_nonneg)

   done

 lemma setsum_nonneg_eq_0_iff:

   fixes f :: "'a \<Rightarrow> 'b::ordered_ab_group_add"

   shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"

   apply (induct set: finite, simp)

   apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)

   done

 lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"

   unfolding setL2_def

   by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)

 lemma setL2_triangle_ineq:

   shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"

 proof (cases "finite A")

   case False

   thus ?thesis by simp

 next

   case True

   thus ?thesis

   proof (induct set: finite)

     case empty

     show ?case by simp

   next

     case (insert x F)

     hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>

            sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"

       by (intro real_sqrt_le_mono add_left_mono power_mono insert

                 setL2_nonneg add_increasing zero_le_power2)

     also have

       "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"

       by (rule real_sqrt_sum_squares_triangle_ineq)

     finally show ?case

       using insert by simp

   qed

 qed

 lemma sqrt_sum_squares_le_sum:

   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"

   apply (rule power2_le_imp_le)

   apply (simp add: power2_sum)

   apply (simp add: mult_nonneg_nonneg)

   apply (simp add: add_nonneg_nonneg)

   done

 lemma setL2_le_setsum [rule_format]:

   "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"

   apply (cases "finite A")

   apply (induct set: finite)

   apply simp

   apply clarsimp

   apply (erule order_trans [OF sqrt_sum_squares_le_sum])

   apply simp

   apply simp

   apply simp

   done

 lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"

   apply (rule power2_le_imp_le)

   apply (simp add: power2_sum)

   apply (simp add: mult_nonneg_nonneg)

   apply (simp add: add_nonneg_nonneg)

   done

 lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"

   apply (cases "finite A")

   apply (induct set: finite)

   apply simp

   apply simp

   apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])

   apply simp

   apply simp

   done

 lemma setL2_mult_ineq_lemma:

   fixes a b c d :: real

   shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"

 proof -

   have "0 \<le> (a * d - b * c)\<twosuperior>" by simp

   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"

     by (simp only: power2_diff power_mult_distrib)

   also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"

     by simp

   finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"

     by simp

 qed

 lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"

   apply (cases "finite A")

   apply (induct set: finite)

   apply simp

   apply (rule power2_le_imp_le, simp)

   apply (rule order_trans)

   apply (rule power_mono)

   apply (erule add_left_mono)

   apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)

   apply (simp add: power2_sum)

   apply (simp add: power_mult_distrib)

   apply (simp add: right_distrib left_distrib)

   apply (rule ord_le_eq_trans)

   apply (rule setL2_mult_ineq_lemma)

   apply simp

   apply (intro mult_nonneg_nonneg setL2_nonneg)

   apply simp

   done

 lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"

   apply (rule_tac s="insert i (A - {i})" and t="A" in subst)

   apply fast

   apply (subst setL2_insert)

   apply simp

   apply simp

   apply simp

   done

 subsection {* Metric *}

 (* TODO: move somewhere else *)

 lemma finite_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"

 apply (induct set: finite, simp_all)

 apply (clarify, rename_tac y)

 apply (rule_tac x="f(x:=y)" in exI, simp)

 done

 instantiation cart :: (metric_space, finite) metric_space

 begin

 definition dist_vector_def:

   "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"

 lemma dist_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"

 unfolding dist_vector_def

 by (rule member_le_setL2) simp_all

 instance proof

   fix x y :: "'a ^ 'b"

   show "dist x y = 0 \<longleftrightarrow> x = y"

     unfolding dist_vector_def

     by (simp add: setL2_eq_0_iff Cart_eq)

 next

   fix x y z :: "'a ^ 'b"

   show "dist x y \<le> dist x z + dist y z"

     unfolding dist_vector_def

     apply (rule order_trans [OF _ setL2_triangle_ineq])

     apply (simp add: setL2_mono dist_triangle2)

     done

 next

   (* FIXME: long proof! *)

   fix S :: "('a ^ 'b) set"

   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

     unfolding open_vector_def open_dist

     apply safe

      apply (drule (1) bspec)

      apply clarify

      apply (subgoal_tac "\<exists>e>0. \<forall>i y. dist y (x$i) < e \<longrightarrow> y \<in> A i")

       apply clarify

       apply (rule_tac x=e in exI, clarify)

       apply (drule spec, erule mp, clarify)

       apply (drule spec, drule spec, erule mp)

       apply (erule le_less_trans [OF dist_nth_le])

      apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>e>0. \<forall>y. dist y (x$i) < e \<longrightarrow> y \<in> A i")

       apply (drule finite_choice [OF finite], clarify)

       apply (rule_tac x="Min (range f)" in exI, simp)

      apply clarify

      apply (drule_tac x=i in spec, clarify)

      apply (erule (1) bspec)

     apply (drule (1) bspec, clarify)

     apply (subgoal_tac "\<exists>r. (\<forall>i::'b. 0 < r i) \<and> e = setL2 r UNIV")

      apply clarify

      apply (rule_tac x="\<lambda>i. {y. dist y (x$i) < r i}" in exI)

      apply (rule conjI)

       apply clarify

       apply (rule conjI)

        apply (clarify, rename_tac y)

        apply (rule_tac x="r i - dist y (x$i)" in exI, rule conjI, simp)

        apply clarify

        apply (simp only: less_diff_eq)

        apply (erule le_less_trans [OF dist_triangle])

       apply simp

      apply clarify

      apply (drule spec, erule mp)

      apply (simp add: dist_vector_def setL2_strict_mono)

     apply (rule_tac x="\<lambda>i. e / sqrt (of_nat CARD('b))" in exI)

     apply (simp add: divide_pos_pos setL2_constant)

     done

 qed

 end

 lemma LIMSEQ_Cart_nth:

   "(X ----> a) \<Longrightarrow> (\<lambda>n. X n $ i) ----> a $ i"

 unfolding LIMSEQ_conv_tendsto by (rule tendsto_Cart_nth)

 lemma LIM_Cart_nth:

   "(f -- x --> y) \<Longrightarrow> (\<lambda>x. f x $ i) -- x --> y $ i"

 unfolding LIM_conv_tendsto by (rule tendsto_Cart_nth)

 lemma Cauchy_Cart_nth:

   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"

 unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le])

 lemma LIMSEQ_vector:

   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"

   assumes X: "\<And>i. (\<lambda>n. X n $ i) ----> (a $ i)"

   shows "X ----> a"

 proof (rule metric_LIMSEQ_I)

   fix r :: real assume "0 < r"

   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")

     by (simp add: divide_pos_pos)

   def N \<equiv> "\<lambda>i. LEAST N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"

   def M \<equiv> "Max (range N)"

   have "\<And>i. \<exists>N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"

     using X `0 < ?s` by (rule metric_LIMSEQ_D)

   hence "\<And>i. \<forall>n\<ge>N i. dist (X n $ i) (a $ i) < ?s"

     unfolding N_def by (rule LeastI_ex)

   hence M: "\<And>i. \<forall>n\<ge>M. dist (X n $ i) (a $ i) < ?s"

     unfolding M_def by simp

   {

     fix n :: nat assume "M \<le> n"

     have "dist (X n) a = setL2 (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"

       unfolding dist_vector_def ..

     also have "\<dots> \<le> setsum (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"

       by (rule setL2_le_setsum [OF zero_le_dist])

     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"

       by (rule setsum_strict_mono, simp_all add: M `M \<le> n`)

     also have "\<dots> = r"

       by simp

     finally have "dist (X n) a < r" .

   }

   hence "\<forall>n\<ge>M. dist (X n) a < r"

     by simp

   then show "\<exists>M. \<forall>n\<ge>M. dist (X n) a < r" ..

 qed

 lemma Cauchy_vector:

   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"

   assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"

   shows "Cauchy (\<lambda>n. X n)"

 proof (rule metric_CauchyI)

   fix r :: real assume "0 < r"

   then have "0 < r / of_nat CARD('n)" (is "0 < ?s")

     by (simp add: divide_pos_pos)

   def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"

   def M \<equiv> "Max (range N)"

   have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"

     using X `0 < ?s` by (rule metric_CauchyD)

   hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"

     unfolding N_def by (rule LeastI_ex)

   hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"

     unfolding M_def by simp

   {

     fix m n :: nat

     assume "M \<le> m" "M \<le> n"

     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"

       unfolding dist_vector_def ..

     also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"

       by (rule setL2_le_setsum [OF zero_le_dist])

     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"

       by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)

     also have "\<dots> = r"

       by simp

     finally have "dist (X m) (X n) < r" .

   }

   hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"

     by simp

   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..

 qed

 instance cart :: (complete_space, finite) complete_space

 proof

   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"

   have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"

     using Cauchy_Cart_nth [OF `Cauchy X`]

     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

   hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"

     by (simp add: LIMSEQ_vector)

   then show "convergent X"

     by (rule convergentI)

 qed

 subsection {* Norms *}

 instantiation cart :: (real_normed_vector, finite) real_normed_vector

 begin

 definition norm_vector_def:

   "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"

 definition vector_sgn_def:

   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"

 instance proof

   fix a :: real and x y :: "'a ^ 'b"

   show "0 \<le> norm x"

     unfolding norm_vector_def

     by (rule setL2_nonneg)

   show "norm x = 0 \<longleftrightarrow> x = 0"

     unfolding norm_vector_def

     by (simp add: setL2_eq_0_iff Cart_eq)

   show "norm (x + y) \<le> norm x + norm y"

     unfolding norm_vector_def

     apply (rule order_trans [OF _ setL2_triangle_ineq])

     apply (simp add: setL2_mono norm_triangle_ineq)

     done

   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"

     unfolding norm_vector_def

     by (simp add: setL2_right_distrib)

   show "sgn x = scaleR (inverse (norm x)) x"

     by (rule vector_sgn_def)

   show "dist x y = norm (x - y)"

     unfolding dist_vector_def norm_vector_def

     by (simp add: dist_norm)

 qed

 end

 lemma norm_nth_le: "norm (x $ i) \<le> norm x"

 unfolding norm_vector_def

 by (rule member_le_setL2) simp_all

 interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"

 apply default

 apply (rule vector_add_component)

 apply (rule vector_scaleR_component)

 apply (rule_tac x="1" in exI, simp add: norm_nth_le)

 done

 instance cart :: (banach, finite) banach ..

 subsection {* Inner products *}

 instantiation cart :: (real_inner, finite) real_inner

 begin

 definition inner_vector_def:

   "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"

 instance proof

   fix r :: real and x y z :: "'a ^ 'b"

   show "inner x y = inner y x"

     unfolding inner_vector_def

     by (simp add: inner_commute)

   show "inner (x + y) z = inner x z + inner y z"

     unfolding inner_vector_def

     by (simp add: inner_add_left setsum_addf)

   show "inner (scaleR r x) y = r * inner x y"

     unfolding inner_vector_def

     by (simp add: setsum_right_distrib)

   show "0 \<le> inner x x"

     unfolding inner_vector_def

     by (simp add: setsum_nonneg)

   show "inner x x = 0 \<longleftrightarrow> x = 0"

     unfolding inner_vector_def

     by (simp add: Cart_eq setsum_nonneg_eq_0_iff)

   show "norm x = sqrt (inner x x)"

     unfolding inner_vector_def norm_vector_def setL2_def

     by (simp add: power2_norm_eq_inner)

 qed

 end

 subsection{* Properties of the dot product.  *}

 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"

   by (vector mult_commute)

 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"

   by (vector ring_simps)

 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"

   by (vector ring_simps)

 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"

   by (vector ring_simps)

 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"

   by (vector ring_simps)

 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)

 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)

 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector

 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector

 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector

 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector

 lemma dot_pos_le[simp]: "(0::'a\<Colon>linordered_ring_strict) <= x \<bullet> x"

   by (simp add: dot_def setsum_nonneg)

 lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::ordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"

 using fS fp setsum_nonneg[OF fp]

 proof (induct set: finite)

   case empty thus ?case by simp

 next

   case (insert x F)

   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all

   from insert.hyps Fp setsum_nonneg[OF Fp]

   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis

   from add_nonneg_eq_0_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)

   show ?case by (simp add: h)

 qed

 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{linordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"

   by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)

 lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{linordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]

   by (auto simp add: le_less)

 subsection{* The collapse of the general concepts to dimension one. *}

 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"

   by (simp add: Cart_eq forall_1)

 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"

   apply auto

   apply (erule_tac x= "x$1" in allE)

   apply (simp only: vector_one[symmetric])

   done

 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"

   by (simp add: norm_vector_def UNIV_1)

 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"

   by (simp add: norm_vector_1)

 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"

   by (auto simp add: norm_real dist_norm)

 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 th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo

     {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

       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)

         apply ferrack by arith

       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 "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo

       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 @{typ "real^1"} *}

 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: ring_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_nat_equiv_def2

     by (auto simp add: nat_number)

   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 real_sqrt_ge_0_iff field_simps)

   done

 text{* Hence derive more interesting properties of the norm. *}

 text {*

   This type-specific version is only here

   to make @{text normarith.ML} happy.

 *}

 lemma norm_0: "norm (0::real ^ _) = 0"

   by (rule norm_zero)

 lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"

   by (simp add: norm_vector_def vector_component setL2_right_distrib

            abs_mult cong: strong_setL2_cong)

 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"

   by (simp add: norm_vector_def dot_def setL2_def power2_eq_square)

 lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"

   by (simp add: norm_vector_def setL2_def dot_def power2_eq_square)

 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"

   by (simp add: real_vector_norm_def)

 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)

 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"

   by vector

 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)

 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)

 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"

   by (metis vector_mul_lcancel)

 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"

   by (metis vector_mul_rcancel)

 lemma norm_cauchy_schwarz:

   fixes x y :: "real ^ 'n"

   shows "x \<bullet> y <= norm x * norm y"

 proof-

   {assume "norm x = 0"

     hence ?thesis by (simp add: dot_lzero dot_rzero)}

   moreover

   {assume "norm y = 0"

     hence ?thesis by (simp add: dot_lzero dot_rzero)}

   moreover

   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"

     let ?z = "norm y *s x - norm x *s y"

     from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)

     from dot_pos_le[of ?z]

     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"

       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)

       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)

     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p

       by (simp add: field_simps)

     hence ?thesis using h by (simp add: power2_eq_square)}

   ultimately show ?thesis by metis

 qed

 lemma norm_cauchy_schwarz_abs:

   fixes x y :: "real ^ 'n"

   shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"

   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]

   by (simp add: real_abs_def dot_rneg)

 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: ring_simps)

 lemma component_le_norm: "\<bar>x$i\<bar> <= norm x"

   apply (simp add: norm_vector_def)

   apply (rule member_le_setL2, simp_all)

   done

 lemma norm_bound_component_le: "norm x <= e ==> \<bar>x$i\<bar> <= e"

   by (metis component_le_norm order_trans)

 lemma norm_bound_component_lt: "norm x < e ==> \<bar>x$i\<bar> < e"

   by (metis component_le_norm basic_trans_rules(21))

 lemma norm_le_l1: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"

   by (simp add: norm_vector_def setL2_le_setsum)

 lemma real_abs_norm: "\<bar>norm x\<bar> = norm x"

   by (rule abs_norm_cancel)

 lemma real_abs_sub_norm: "\<bar>norm (x::real ^ 'n) - norm y\<bar> <= norm(x - y)"

   by (rule norm_triangle_ineq3)

 lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"

   by (simp add: real_vector_norm_def)

 lemma norm_lt: "norm(x::real ^ 'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"

   by (simp add: real_vector_norm_def)

 lemma norm_eq: "norm(x::real ^ 'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"

   by (simp add: order_eq_iff norm_le)

 lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"

   by (simp add: real_vector_norm_def)

 text{* Squaring equations and inequalities involving norms.  *}

 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"

   by (simp add: real_vector_norm_def)

 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"

   by (auto simp add: real_vector_norm_def)

 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"

 proof-

   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)

   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith

 finally show ?thesis ..

 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. *}

 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"

   by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)

 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"

   by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)

 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}

 lemma vector_eq: "(x:: real ^ 'n) = 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: dot_rsub dot_lsub dot_sym)

   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)

   then show "x = y" by (simp add: dot_eq_0)

 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_mono1)

   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: ring_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

 lemma vector_dist_norm:

   fixes x :: "'a::real_normed_vector"

   shows "dist x y = norm (x - y)"

   by (rule dist_norm)

 use "normarith.ML"

 method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)

 *} "Proves simple linear statements about vector norms"

 text{* Hence more metric properties. *}

 lemma dist_triangle_alt:

   fixes x y z :: "'a::metric_space"

   shows "dist y z <= dist x y + dist x z"

 using dist_triangle [of y z x] by (simp add: dist_commute)

 lemma dist_pos_lt:

   fixes x y :: "'a::metric_space"

   shows "x \<noteq> y ==> 0 < dist x y"

 by (simp add: zero_less_dist_iff)

 lemma dist_nz:

   fixes x y :: "'a::metric_space"

   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"

 by (simp add: zero_less_dist_iff)

 lemma dist_triangle_le:

   fixes x y z :: "'a::metric_space"

   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"

 by (rule order_trans [OF dist_triangle2])

 lemma dist_triangle_lt:

   fixes x y z :: "'a::metric_space"

   shows "dist x z + dist y z < e ==> dist x y < e"

 by (rule le_less_trans [OF dist_triangle2])

 lemma dist_triangle_half_l:

   fixes x1 x2 y :: "'a::metric_space"

   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"

 by (rule dist_triangle_lt [where z=y], simp)

 lemma dist_triangle_half_r:

   fixes x1 x2 y :: "'a::metric_space"

   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"

 by (rule dist_triangle_half_l, simp_all add: dist_commute)

 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 vector_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 vector_dist_norm[THEN sym]]]

   unfolding vector_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_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"

   unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..

 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_component [simp]:

   fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"

   shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"

   by (cases "finite S", induct S set: finite, simp_all)

 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"

   by (simp add: Cart_eq)

 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_cmul:

   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"

   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"

   by (simp add: Cart_eq setsum_right_distrib)

 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 real_setsum_norm:

   fixes f :: "'a \<Rightarrow> real ^'n"

   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 real_setsum_norm_le:

   fixes f :: "'a \<Rightarrow> real ^ 'n"

   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 real_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 real_setsum_norm_bound:

   fixes f :: "'a \<Rightarrow> real ^ 'n"

   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 real_setsum_norm_le[OF fS K] setsum_constant[symmetric]

   by simp

 lemma setsum_vmul:

   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"

   assumes fS: "finite S"

   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"

 proof(induct rule: finite_induct[OF fS])

   case 1 then show ?case by (simp add: vector_smult_lzero)

 next

   case (2 x F)

   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"

     by simp

   also have "\<dots> = f x *s v + setsum f F *s v"

     by (simp add: vector_sadd_rdistrib)

   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp

   finally show ?case .

 qed

 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---

  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)

     (* FIXME: Here too need stupid finiteness assumption on T!!! *)

 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 vsum_norm_allsubsets_bound:

   fixes f:: "'a \<Rightarrow> real ^'n"

   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"

   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"

 proof-

   let ?d = "real CARD('n)"

   let ?nf = "\<lambda>x. norm (f x)"

   let ?U = "UNIV :: 'n set"

   have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"

     by (rule setsum_commute)

   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)

   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"

     apply (rule setsum_mono)

     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]

       by (auto intro: abs_le_D1)

     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"

       using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]

       by (auto simp add: setsum_negf intro: abs_le_D1)

     have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"

       apply (subst thp)

       apply (rule setsum_Un_zero)

       using fP thp0 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 dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "

   by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)

 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "

   by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)

 subsection{* Basis vectors in coordinate directions. *}

 definition "basis k = (\<chi> i. if i = k then 1 else 0)"

 lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"

   unfolding basis_def by simp

 lemma delta_mult_idempotent:

   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)

 lemma norm_basis:

   shows "norm (basis k :: real ^'n) = 1"

   apply (simp add: basis_def real_vector_norm_def dot_def)

   apply (vector delta_mult_idempotent)

   using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]

   apply auto

   done

 lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"

   by (rule norm_basis)

 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"

   apply (rule exI[where x="c *s basis arbitrary"])

   by (simp only: norm_mul norm_basis)

 lemma vector_choose_dist: assumes e: "0 <= e"

   shows "\<exists>(y::real^'n). dist x y = e"

 proof-

   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"

     by blast

   then have "dist x (x - c) = e" by (simp add: dist_norm)

   then show ?thesis by blast

 qed

 lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n)"

   by (simp add: inj_on_def Cart_eq)

 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"

   by auto

 lemma basis_expansion:

   "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")

   by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)

 lemma basis_expansion_unique:

   "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"

   by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)

 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"

   by auto

 lemma dot_basis:

   shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)"

   by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)

 lemma inner_basis:

   fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"

   shows "inner (basis i) x = inner 1 (x $ i)"

     and "inner x (basis i) = inner (x $ i) 1"

   unfolding inner_vector_def basis_def

   by (auto simp add: cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)

 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"

   by (auto simp add: Cart_eq)

 lemma basis_nonzero:

   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"

   by (simp add: basis_eq_0)

 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"

   apply (auto simp add: Cart_eq dot_basis)

   apply (erule_tac x="basis i" in allE)

   apply (simp add: dot_basis)

   apply (subgoal_tac "y = z")

   apply simp

   apply (simp add: Cart_eq)

   done

 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"

   apply (auto simp add: Cart_eq dot_basis)

   apply (erule_tac x="basis i" in allE)

   apply (simp add: dot_basis)

   apply (subgoal_tac "x = y")

   apply simp

   apply (simp add: Cart_eq)

   done

 subsection{* Orthogonality. *}

 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"

 lemma orthogonal_basis:

   shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"

   by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)

 lemma orthogonal_basis_basis:

   shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j"

   unfolding orthogonal_basis[of i] basis_component[of j] by simp

   (* FIXME : Maybe some of these require less than comm_ring, but not all*)

 lemma orthogonal_clauses:

   "orthogonal a (0::'a::comm_ring ^'n)"

   "orthogonal a x ==> orthogonal a (c *s x)"

   "orthogonal a x ==> orthogonal a (-x)"

   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"

   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"

   "orthogonal 0 a"

   "orthogonal x a ==> orthogonal (c *s x) a"

   "orthogonal x a ==> orthogonal (-x) a"

   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"

   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"

   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub

   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub

   by simp_all

 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"

   by (simp add: orthogonal_def dot_sym)

 subsection{* Explicit vector construction from lists. *}

 primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"

 where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"

 lemma from_nat [simp]: "from_nat = of_nat"

 by (rule ext, induct_tac x, simp_all)

 primrec

   list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"

 where

   "list_fun n [] = (\<lambda>x. 0)"

 | "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"

 definition "vector l = (\<chi> i. list_fun 1 l i)"

 (*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)

 lemma vector_1: "(vector[x]) $1 = x"

   unfolding vector_def by simp

 lemma vector_2:

  "(vector[x,y]) $1 = x"

  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"

   unfolding vector_def by simp_all

 lemma vector_3:

  "(vector [x,y,z] ::('a::zero)^3)$1 = x"

  "(vector [x,y,z] ::('a::zero)^3)$2 = y"

  "(vector [x,y,z] ::('a::zero)^3)$3 = z"

   unfolding vector_def by simp_all

 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (subgoal_tac "vector [v$1] = v")

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_1)

   done

 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (erule_tac x="v$2" in allE)

   apply (subgoal_tac "vector [v$1, v$2] = v")

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_2)

   done

 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"

   apply auto

   apply (erule_tac x="v$1" in allE)

   apply (erule_tac x="v$2" in allE)

   apply (erule_tac x="v$3" in allE)

   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_3)

   done

 subsection{* Linear functions. *}

 definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"

 lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *s x) = c *s f x"

   shows "linear f" using assms unfolding linear_def by auto

 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"

   by (vector linear_def Cart_eq ring_simps)

 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)

 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"

   by (vector linear_def Cart_eq ring_simps)

 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"

   by (vector linear_def Cart_eq ring_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::'a::semiring_1 ^ 'n)" by (simp add: linear_def)

 lemma linear_compose_setsum:

   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^'m)"

   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"

   using lS

   apply (induct rule: finite_induct[OF fS])

   by (auto simp add: linear_zero intro: linear_compose_add)

 lemma linear_vmul_component:

   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"

   assumes lf: "linear f"

   shows "linear (\<lambda>x. f x $ k *s v)"

   using lf

   apply (auto simp add: linear_def )

   by (vector ring_simps)+

 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"

   unfolding linear_def

   apply clarsimp

   apply (erule allE[where x="0::'a"])

   apply simp

   done

 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)

 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"

   unfolding vector_sneg_minus1

   using linear_cmul[of f] by auto

 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)

 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"

   by (simp add: diff_def linear_add linear_neg)

 lemma linear_setsum:

   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"

   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:

   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"

   assumes lf: "linear f" and fS: "finite S"

   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"

   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]

   linear_cmul[OF lf] by simp

 lemma linear_injective_0:

   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"

   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

 lemma linear_bounded:

   fixes f:: "real ^'m \<Rightarrow> real ^'n"

   assumes lf: "linear f"

   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"

 proof-

   let ?S = "UNIV:: 'm set"

   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"

   have fS: "finite ?S" by simp

   {fix x:: "real ^ 'm"

     let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"

     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"

       by (simp only:  basis_expansion)

     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s 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) *s f (basis i))?S)" .

     {fix i assume i: "i \<in> ?S"

       from component_le_norm[of x i]

       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"

       unfolding norm_mul

       apply (simp only: mult_commute)

       apply (rule mult_mono)

       by (auto simp add: ring_simps norm_ge_zero) }

     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis

     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (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:: "real ^'n \<Rightarrow> real ^'m"

   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 "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)

       with C have "B * norm (1:: real ^ 'n) < 0"

         by (simp add: zero_compare_simps)

       with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp

     }

     then have Bp: "B \<ge> 0" by ferrack

     {fix x::"real ^ 'n"

       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: ring_simps split add: abs_split)

       apply (erule order_trans, simp)

       done

   }

   then show ?thesis using Kp by blast

 qed

 lemma smult_conv_scaleR: "c *s x = scaleR c x"

   unfolding vector_scalar_mult_def vector_scaleR_def by simp

 lemma linear_conv_bounded_linear:

   fixes f :: "real ^ _ \<Rightarrow> real ^ _"

   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 add: smult_conv_scaleR)

   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"

     unfolding linear_def smult_conv_scaleR

     by (simp add: f.add f.scaleR)

 qed

 lemma bounded_linearI': fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *s x) = c *s f x"

   shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]

   by(rule linearI[OF assms])

 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 *s x) y = c *s (h x y)"

   by (simp add: bilinear_def linear_def)

 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"

   by (simp add: bilinear_def linear_def)

 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"

   by (simp only: vector_sneg_minus1 bilinear_lmul)

 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"

   by (simp only: vector_sneg_minus1 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:

   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"

   using bilinear_ladd[OF bh, of 0 0 x]

     by (simp add: eq_add_iff ring_simps)

 lemma bilinear_rzero:

   fixes h :: "'a::ring^_ \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"

   using bilinear_radd[OF bh, of x 0 0 ]

     by (simp add: eq_add_iff ring_simps)

 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ _)) z = h x z - h y z"

   by (simp  add: diff_def bilinear_ladd bilinear_lneg)

 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ _)) = h z x - h z y"

   by (simp  add: diff_def bilinear_radd bilinear_rneg)

 lemma bilinear_setsum:

   fixes h:: "'a ^_ \<Rightarrow> 'a::semiring_1^_\<Rightarrow> 'a ^ _"

   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

 lemma bilinear_bounded:

   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^'k"

   assumes bh: "bilinear h"

   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"

 proof-

   let ?M = "UNIV :: 'm set"

   let ?N = "UNIV :: 'n set"

   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:: "real ^ 'm" and  y :: "real^'n"

     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..

     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s 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 real_setsum_norm_le)

       using fN fM

       apply simp

       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)

       apply (rule mult_mono)

       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)

       apply (rule mult_mono)

       apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)

       done}

   then show ?thesis by metis

 qed

 lemma bilinear_bounded_pos:

   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^'k"

   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::"real ^'m" and y :: "real ^'n"

     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 simp add: norm_ge_zero)

     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 :: "real ^ _ \<Rightarrow> real ^ _ \<Rightarrow> real ^ _"

   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 add: smult_conv_scaleR)

   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 add: smult_conv_scaleR)

   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{* Adjoints. *}

 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"

 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis

 lemma adjoint_works_lemma:

   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f"

   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"

 proof-

   let ?N = "UNIV :: 'n set"

   let ?M = "UNIV :: 'm set"

   have fN: "finite ?N" by simp

   have fM: "finite ?M" by simp

   {fix y:: "'a ^ 'm"

     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"

     {fix x

       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"

         by (simp only: basis_expansion)

       also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"

         unfolding linear_setsum[OF lf fN]

         by (simp add: linear_cmul[OF lf])

       finally have "f x \<bullet> y = x \<bullet> ?w"

         apply (simp only: )

         apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)

         done}

   }

   then show ?thesis unfolding adjoint_def

     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]

     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]

     by metis

 qed

 lemma adjoint_works:

   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f"

   shows "x \<bullet> adjoint f y = f x \<bullet> y"

   using adjoint_works_lemma[OF lf] by metis

 lemma adjoint_linear:

   fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f"

   shows "linear (adjoint f)"

   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])

 lemma adjoint_clauses:

   fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f"

   shows "x \<bullet> adjoint f y = f x \<bullet> y"

   and "adjoint f y \<bullet> x = y \<bullet> f x"

   by (simp_all add: adjoint_works[OF lf] dot_sym )

 lemma adjoint_adjoint:

   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f"

   shows "adjoint (adjoint f) = f"

   apply (rule ext)

   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])

 lemma adjoint_unique:

   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^'m"

   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"

   shows "f' = adjoint f"

   apply (rule ext)

   using u

   by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])

 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}

 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)

   where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"

 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)

   where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"

 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)

   where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"

 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"

 definition transpose where 

   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"

 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"

 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"

 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"

 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"

 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)

 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"

   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)

 lemma matrix_mul_lid:

   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"

   shows "mat 1 ** A = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)

 lemma matrix_mul_rid:

   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"

   shows "A ** mat 1 = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

   by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)

 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"

   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)

   apply (subst setsum_commute)

   apply simp

   done

 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"

   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)

   apply (subst setsum_commute)

   apply simp

   done

 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"

   apply (vector matrix_vector_mult_def mat_def)

   by (simp add: cond_value_iff cond_application_beta

     setsum_delta' cong del: if_weak_cong)

 lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"

   by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)

 lemma matrix_eq:

   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"

   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")

   apply auto

   apply (subst Cart_eq)

   apply clarify

   apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)

   apply (erule_tac x="basis ia" in allE)

   apply (erule_tac x="i" in allE)

   by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)

 lemma matrix_vector_mul_component:

   shows "((A::'a::semiring_1^_^_) *v x)$k = (A$k) \<bullet> x"

   by (simp add: matrix_vector_mult_def dot_def)

 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"

   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)

   apply (subst setsum_commute)

   by simp

 lemma transpose_mat: "transpose (mat n) = mat n"

   by (vector transpose_def mat_def)

 lemma transpose_transpose: "transpose(transpose A) = A"

   by (vector transpose_def)

 lemma row_transpose:

   fixes A:: "'a::semiring_1^_^_"

   shows "row i (transpose A) = column i A"

   by (simp add: row_def column_def transpose_def Cart_eq)

 lemma column_transpose:

   fixes A:: "'a::semiring_1^_^_"

   shows "column i (transpose A) = row i A"

   by (simp add: row_def column_def transpose_def Cart_eq)

 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"

 by (auto simp add: rows_def columns_def row_transpose intro: set_ext)

 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose)

 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}

 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"

   by (simp add: matrix_vector_mult_def dot_def)

 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"

   by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)

 lemma vector_componentwise:

   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"

   apply (subst basis_expansion[symmetric])

   by (vector Cart_eq setsum_component)

 lemma linear_componentwise:

   fixes f:: "'a::ring_1 ^'m \<Rightarrow> 'a ^ _"

   assumes lf: "linear f"

   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")

 proof-

   let ?M = "(UNIV :: 'm set)"

   let ?N = "(UNIV :: 'n set)"

   have fM: "finite ?M" by simp

   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"

     unfolding vector_smult_component[symmetric]

     unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]

     ..

   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..

 qed

 text{* Inverse matrices  (not necessarily square) *}

 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"

 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =

         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"

 text{* Correspondence between matrices and linear operators. *}

 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"

 where "matrix f = (\<chi> i j. (f(basis j))$i)"

 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ _))"

   by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)

 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"

 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)

 apply clarify

 apply (rule linear_componentwise[OF lf, symmetric])

 done

 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)

 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"

   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)

 lemma matrix_compose:

   assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> 'a^'m)"

   and lg: "linear (g::'a::comm_ring_1^'m \<Rightarrow> 'a^_)"

   shows "matrix (g o f) = matrix g ** matrix f"

   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]

   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)

 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"

   by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)

 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"

   apply (rule adjoint_unique[symmetric])

   apply (rule matrix_vector_mul_linear)

   apply (simp add: transpose_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)

   apply (subst setsum_commute)

   apply (auto simp add: mult_ac)

   done

 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^'m)"

   shows "matrix(adjoint f) = transpose(matrix f)"

   apply (subst matrix_vector_mul[OF lf])

   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..

 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 prems apply auto

   apply (erule_tac x="n" in allE)

   apply (erule_tac x="n" in allE)

   apply auto

   done

 lemma real_convex_bound_lt:

   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"

   and uv: "u + v = 1"

   shows "u * x + v * y < a"

 proof-

   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith

   have "a = a * (u + v)" unfolding uv  by simp

   hence th: "u * a + v * a = a" by (simp add: ring_simps)

   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)

   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)

   from xa ya u v have "u * x + v * y < u * a + v * a"

     apply (cases "u = 0", simp_all add: uv')

     apply(rule mult_strict_left_mono)

     using uv' apply simp_all

     apply (rule add_less_le_mono)

     apply(rule mult_strict_left_mono)

     apply simp_all

     apply (rule mult_left_mono)

     apply simp_all

     done

   thus ?thesis unfolding th .

 qed

 lemma real_convex_bound_le:

   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"

   and uv: "u + v = 1"

   shows "u * x + v * y \<le> a"

 proof-

   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)

   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)

   finally show ?thesis unfolding uv by simp

 qed

 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: zero_compare_simps)

   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)

   from y z have yz: "y + z \<ge> 0" by arith

   from power2_le_imp_le[OF th yz] show ?thesis .

 qed

 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>

    (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")

 proof-

   let ?S = "(UNIV :: 'n set)"

   {assume H: "?rhs"

     then have ?lhs by auto}

   moreover

   {assume H: "?lhs"

     then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis

     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"

     {fix i

       from f have "P i (f i)" by metis

       then have "P i (?x$i)" by auto

     }

     hence "\<forall>i. P i (?x$i)" by metis

     hence ?rhs by metis }

   ultimately show ?thesis by metis

 qed

 subsection{* Operator norm. *}

 definition "onorm f = Sup {norm (f x)| x. norm x = 1}"

 lemma norm_bound_generalize:

   fixes f:: "real ^'n \<Rightarrow> real^'m"

   assumes lf: "linear f"

   shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")

 proof-

   {assume H: ?rhs

     {fix x :: "real^'n" assume x: "norm x = 1"

       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}

     then have ?lhs by blast }

   moreover

   {assume H: ?lhs

     from H[rule_format, of "basis arbitrary"]

     have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]

       by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])

     {fix x :: "real ^'n"

       {assume "x = 0"

         then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}

       moreover

       {assume x0: "x \<noteq> 0"

         hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)

         let ?c = "1/ norm x"

         have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)

         with H have "norm (f(?c*s x)) \<le> b" by blast

         hence "?c * norm (f x) \<le> b"

           by (simp add: linear_cmul[OF lf] norm_mul)

         hence "norm (f x) \<le> b * norm x"

           using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}

       ultimately have "norm (f x) \<le> b * norm x" by blast}

     then have ?rhs by blast}

   ultimately show ?thesis by blast

 qed

 lemma onorm:

   fixes f:: "real ^'n \<Rightarrow> real ^'m"

   assumes lf: "linear f"

   shows "norm (f x) <= onorm f * norm x"

   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"

 proof-

   {

     let ?S = "{norm (f x) |x. norm x = 1}"

     have Se: "?S \<noteq> {}" using  norm_basis by auto

     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"

       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)

     {from Sup[OF Se b, unfolded onorm_def[symmetric]]

       show "norm (f x) <= onorm f * norm x"

         apply -

         apply (rule spec[where x = x])

         unfolding norm_bound_generalize[OF lf, symmetric]

         by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}

     {

       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"

         using Sup[OF Se b, unfolded onorm_def[symmetric]]

         unfolding norm_bound_generalize[OF lf, symmetric]

         by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}

   }

 qed

 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"

   using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp

 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"

   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"

   using onorm[OF lf]

   apply (auto simp add: onorm_pos_le)

   apply atomize

   apply (erule allE[where x="0::real"])

   using onorm_pos_le[OF lf]

   apply arith

   done

 lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^'m)) = norm y"

 proof-

   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"

   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"

     by(auto intro: vector_choose_size set_ext)

   show ?thesis

     unfolding onorm_def th

     apply (rule Sup_unique) by (simp_all  add: setle_def)

 qed

 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"

   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"

   unfolding onorm_eq_0[OF lf, symmetric]

   using onorm_pos_le[OF lf] by arith

 lemma onorm_compose:

   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"

   and lg: "linear (g::real^'k \<Rightarrow> real^'n)"

   shows "onorm (f o g) <= onorm f * onorm g"

   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])

   unfolding o_def

   apply (subst mult_assoc)

   apply (rule order_trans)

   apply (rule onorm(1)[OF lf])

   apply (rule mult_mono1)

   apply (rule onorm(1)[OF lg])

   apply (rule onorm_pos_le[OF lf])

   done

 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"

   shows "onorm (\<lambda>x. - f x) \<le> onorm f"

   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]

   unfolding norm_minus_cancel by metis

 lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"

   shows "onorm (\<lambda>x. - f x) = onorm f"

   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]

   by simp

 lemma onorm_triangle:

   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"

   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"

   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])

   apply (rule order_trans)

   apply (rule norm_triangle_ineq)

   apply (simp add: distrib)

   apply (rule add_mono)

   apply (rule onorm(1)[OF lf])

   apply (rule onorm(1)[OF lg])

   done

 lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e

   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"

   apply (rule order_trans)

   apply (rule onorm_triangle)

   apply assumption+

   done

 lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e

   ==> onorm(\<lambda>x. f x + g x) < e"

   apply (rule order_le_less_trans)

   apply (rule onorm_triangle)

   by assumption+

 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)

 abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"

 abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a"

   where "dest_vec1 x \<equiv> (x$1)"

 lemma vec1_component[simp]: "(vec1 x)$1 = x"

   by (simp add: )

 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"

   by (simp_all add:  Cart_eq forall_1)

 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)

 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)

 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)

 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)

 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto

 lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)

 lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)

 lemma vec_cmul: "vec(c* x) = c *s vec x " by (vector vec_def)

 lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)

 lemma range_vec1[simp]:"range vec1 = UNIV" apply(rule set_ext,rule) unfolding image_iff defer

   apply(rule_tac x="dest_vec1 x" in bexI) by auto

 lemma vec_setsum: assumes fS: "finite S"

   shows "vec(setsum f S) = setsum (vec o f) S"

   apply (induct rule: finite_induct[OF fS])

   apply (simp)

   apply (auto simp add: vec_add)

   done

 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"

   by (simp)

 lemma dest_vec1_vec: "dest_vec1(vec x) = x"

   by (simp)

 lemma dest_vec1_sum: assumes fS: "finite S"

   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"

   apply (induct rule: finite_induct[OF fS])

   apply (simp add: dest_vec1_vec)

   apply (auto simp add:vector_minus_component)

   done

 lemma norm_vec1: "norm(vec1 x) = abs(x)"

   by (simp add: vec_def norm_real)

 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"

   by (simp only: dist_real vec1_component)

 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"

   by (metis vec1_dest_vec1 norm_vec1)

 lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component

    vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def norm_vec1 real_norm_def

 lemma bounded_linear_vec1:"bounded_linear (vec1::real\<Rightarrow>real^1)"

   unfolding bounded_linear_def additive_def bounded_linear_axioms_def 

   unfolding smult_conv_scaleR[THEN sym] unfolding vec1_dest_vec1_simps

   apply(rule conjI) defer apply(rule conjI) defer apply(rule_tac x=1 in exI) by auto

 lemma linear_vmul_dest_vec1:

   fixes f:: "'a::semiring_1^_ \<Rightarrow> 'a^1"

   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"

   apply (rule linear_vmul_component)

   by auto

 lemma linear_from_scalars:

   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^_)"

   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"

   apply (rule ext)

   apply (subst matrix_works[OF lf, symmetric])

   apply (auto simp add: Cart_eq matrix_vector_mult_def column_def  mult_commute UNIV_1)

   done

 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \<Rightarrow> 'a^1)"

   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"

   apply (rule ext)

   apply (subst matrix_works[OF lf, symmetric])

   apply (simp add: Cart_eq matrix_vector_mult_def row_def dot_def mult_commute forall_1)

   done

 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"

   by (simp add: dest_vec1_eq[symmetric])

 lemma setsum_scalars: assumes fS: "finite S"

   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"

   unfolding vec_setsum[OF fS] by simp

 lemma dest_vec1_wlog_le: "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)  \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"

   apply (cases "dest_vec1 x \<le> dest_vec1 y")

   apply simp

   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")

   apply (auto)

   done

 text{* Pasting vectors. *}

 lemma linear_fstcart[intro]: "linear fstcart"

   by (auto simp add: linear_def Cart_eq)

 lemma linear_sndcart[intro]: "linear sndcart"

   by (auto simp add: linear_def Cart_eq)

 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"

   by (simp add: Cart_eq)

 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b::finite + 'c::finite)) + fstcart y"

   by (simp add: Cart_eq)

 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b::finite + 'c::finite))"

   by (simp add: Cart_eq)

 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^(_ + _))"

   by (simp add: Cart_eq)

 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^(_ + _)) - fstcart y"

   by (simp add: Cart_eq)

 lemma fstcart_setsum:

   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"

   assumes fS: "finite S"

   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"

   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)

 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"

   by (simp add: Cart_eq)

 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^(_ + _)) + sndcart y"

   by (simp add: Cart_eq)

 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^(_ + _))"

   by (simp add: Cart_eq)

 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^(_ + _))"

   by (simp add: Cart_eq)

 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^(_ + _)) - sndcart y"

   by (simp add: Cart_eq)

 lemma sndcart_setsum:

   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"

   assumes fS: "finite S"

   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"

   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)

 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"

   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)

 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"

   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)

 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"

   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)

 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"

   unfolding vector_sneg_minus1 pastecart_cmul ..

 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"

   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)

 lemma pastecart_setsum:

   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"

   assumes fS: "finite S"

   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"

   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)

 lemma setsum_Plus:

   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>

     (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"

   unfolding Plus_def

   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)

 lemma setsum_UNIV_sum:

   fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"

   shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"

   apply (subst UNIV_Plus_UNIV [symmetric])

   apply (rule setsum_Plus [OF finite finite])

   done

 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"

 proof-

   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"

     by (simp add: pastecart_fst_snd)

   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"

     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)

   then show ?thesis

     unfolding th0

     unfolding real_vector_norm_def real_sqrt_le_iff id_def

     by (simp add: dot_def)

 qed

 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"

   unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)

 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"

 proof-

   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"

     by (simp add: pastecart_fst_snd)

   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"

     by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)

   then show ?thesis

     unfolding th0

     unfolding real_vector_norm_def real_sqrt_le_iff id_def

     by (simp add: dot_def)

 qed

 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"

   unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)

 lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n) (x2::'a::{times,comm_monoid_add}^'m)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"

   by (simp add: dot_def setsum_UNIV_sum pastecart_def)