(* Title:      Library/Euclidean_Space

    ID:         $Id: 

    Author:     Amine Chaieb, University of Cambridge

 *)

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

 theory Euclidean_Space

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

   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type

   Inner_Product

   uses ("normarith.ML")

 begin

 text{* Some common special cases.*}

 lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"

   by (metis order_eq_iff)

 lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"

   by (simp add: dimindex_def)

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

 proof-

   have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith

   thus ?thesis by metis

 qed

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

 proof-

   have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith

   thus ?thesis by metis

 qed

 lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp

 lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" 

   by (simp add: atLeastAtMost_singleton)

 lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2" 

   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)

 lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" 

   by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)

 subsection{* Basic componentwise operations on vectors. *}

 instantiation "^" :: (plus,type) plus

 begin

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

 instance ..

 end

 instantiation "^" :: (times,type) times

 begin

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

   instance ..

 end

 instantiation "^" :: (minus,type) minus begin

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

 instance ..

 end

 instantiation "^" :: (uminus,type) uminus begin

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

 instance ..

 end

 instantiation "^" :: (zero,type) zero begin

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

 instance ..

 end

 instantiation "^" :: (one,type) one begin

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

 instance ..

 end

 instantiation "^" :: (ord,type) ord

  begin

 definition vector_less_eq_def:

   "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}.

   x$i <= y$i)"

 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..

   dimindex (UNIV :: 'b set)}. x$i < y$i)"

 instance by (intro_classes)

 end

 instantiation "^" :: (scaleR, type) 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" (infixr "*s" 75)

   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) {1 .. dimindex (UNIV:: 'n set)}"

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

   by (simp add: dot_def dimindex_def)

 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 dimindex_def nat_number)

 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 dimindex_def nat_number)

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

 lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]

 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

   Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac)

 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: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x" 

   by (vector vec_def) 

 lemma vector_add_component: 

   fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

   shows "(x + y)$i = x$i + y$i"

   using i by vector

 lemma vector_minus_component: 

   fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

   shows "(x - y)$i = x$i - y$i"

   using i  by vector

 lemma vector_mult_component: 

   fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

   shows "(x * y)$i = x$i * y$i"

   using i by vector

 lemma vector_smult_component: 

   fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

   shows "(c *s y)$i = c * (y$i)"

   using i by vector

 lemma vector_uminus_component: 

   fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

   shows "(- x)$i = - (x$i)"

   using i by vector

 lemma vector_scaleR_component:

   fixes x :: "'a::scaleR ^ 'n"

   assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"

   shows "(scaleR r x)$i = scaleR r (x$i)"

   using 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 "^" :: (semigroup_add,type) semigroup_add 

   apply (intro_classes) by (vector add_assoc)

 instance "^" :: (monoid_add,type) monoid_add 

   apply (intro_classes) by vector+ 

 instance "^" :: (group_add,type) group_add 

   apply (intro_classes) by (vector algebra_simps)+ 

 instance "^" :: (ab_semigroup_add,type) ab_semigroup_add 

   apply (intro_classes) by (vector add_commute)

 instance "^" :: (comm_monoid_add,type) comm_monoid_add

   apply (intro_classes) by vector

 instance "^" :: (ab_group_add,type) ab_group_add 

   apply (intro_classes) by vector+

 instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add 

   apply (intro_classes)

   by (vector Cart_eq)+

 instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add

   apply (intro_classes)

   by (vector Cart_eq)

 instance "^" :: (real_vector, type) real_vector

   by default (vector scaleR_left_distrib scaleR_right_distrib)+

 instance "^" :: (semigroup_mult,type) semigroup_mult 

   apply (intro_classes) by (vector mult_assoc)

 instance "^" :: (monoid_mult,type) monoid_mult 

   apply (intro_classes) by vector+

 instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult 

   apply (intro_classes) by (vector mult_commute)

 instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult 

   apply (intro_classes) by (vector mult_idem)

 instance "^" :: (comm_monoid_mult,type) comm_monoid_mult 

   apply (intro_classes) by vector

 fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where

   "vector_power x 0 = 1"

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

 instantiation "^" :: (recpower,type) recpower 

 begin

   definition vec_power_def: "op ^ \<equiv> vector_power"

   instance 

   apply (intro_classes) by (simp_all add: vec_power_def) 

 end

 instance "^" :: (semiring,type) semiring

   apply (intro_classes) by (vector ring_simps)+

 instance "^" :: (semiring_0,type) semiring_0

   apply (intro_classes) by (vector ring_simps)+

 instance "^" :: (semiring_1,type) semiring_1

   apply (intro_classes) apply vector using dimindex_ge_1 by auto 

 instance "^" :: (comm_semiring,type) comm_semiring

   apply (intro_classes) by (vector ring_simps)+

 instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) 

 instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..

 instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes) 

 instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes) 

 instance "^" :: (ring,type) ring by (intro_classes) 

 instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes) 

 instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)

 instance "^" :: (ring_1,type) ring_1 ..

 instance "^" :: (real_algebra,type) real_algebra

   apply intro_classes

   apply (simp_all add: vector_scaleR_def ring_simps)

   apply vector

   apply vector

   done

 instance "^" :: (real_algebra_1,type) real_algebra_1 ..

 lemma of_nat_index: 

   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"

   apply (induct n)

   apply vector

   apply vector

   done

 lemma zero_index[simp]: 

   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector

 lemma one_index[simp]: 

   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (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 "^" :: (semiring_char_0,type) semiring_char_0 

 proof (intro_classes) 

   fix m n ::nat

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

   proof(induct m arbitrary: n)

     case 0 thus ?case apply vector 

       apply (induct n,auto simp add: ring_simps)

       using dimindex_ge_1 apply auto

       apply vector

       by (auto simp add: of_nat_index one_plus_of_nat_neq_0)

   next

     case (Suc n m)

     thus ?case  apply vector

       apply (induct m, auto simp add: ring_simps of_nat_index zero_index)

       using dimindex_ge_1 apply simp apply blast

       apply (simp add: one_plus_of_nat_neq_0)

       using dimindex_ge_1 apply simp apply blast

       apply (simp add: vector_component one_index of_nat_index)

       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)

       using  dimindex_ge_1 apply simp apply blast

       apply (simp add: vector_component one_index of_nat_index)

       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)

       using dimindex_ge_1 apply simp apply blast

       apply (simp add: vector_component one_index of_nat_index)

       done

   qed

 qed

 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes

 instance "^" :: (ring_char_0,type) 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)" 

   apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )

   using dimindex_ge_1 apply auto done

 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_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_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::pordered_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 {* Norms *}

 instantiation "^" :: (real_normed_vector, type) real_normed_vector

 begin

 definition vector_norm_def:

   "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"

 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 vector_norm_def

     by (rule setL2_nonneg)

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

     unfolding vector_norm_def

     by (simp add: setL2_eq_0_iff Cart_eq)

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

     unfolding vector_norm_def

     apply (rule order_trans [OF _ setL2_triangle_ineq])

     apply (rule setL2_mono)

     apply (simp add: vector_component norm_triangle_ineq)

     apply simp

     done

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

     unfolding vector_norm_def

     by (simp add: vector_component norm_scaleR setL2_right_distrib

              cong: strong_setL2_cong)

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

     by (rule vector_sgn_def)

 qed

 end

 subsection {* Inner products *}

 instantiation "^" :: (real_inner, type) real_inner

 begin

 definition vector_inner_def:

   "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) {1 .. dimindex(UNIV::'b set)}"

 instance proof

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

   show "inner x y = inner y x"

     unfolding vector_inner_def

     by (simp add: inner_commute)

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

     unfolding vector_inner_def

     by (vector inner_left_distrib)

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

     unfolding vector_inner_def

     by (vector inner_scaleR_left)

   show "0 \<le> inner x x"

     unfolding vector_inner_def

     by (simp add: setsum_nonneg)

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

     unfolding vector_inner_def

     by (simp add: Cart_eq setsum_nonneg_eq_0_iff)

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

     unfolding vector_inner_def vector_norm_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>ordered_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::pordered_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 sum_nonneg_eq_zero_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::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"

 proof-

   {assume f: "finite (UNIV :: 'n set)"

     let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"

     have fS: "finite ?S" using f by simp

     have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp

     have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}

   moreover

   {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}

   ultimately show ?thesis by metis

 qed

 lemma dot_pos_lt: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_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 (vector dimindex_def)

 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: vector_norm_def dimindex_def)

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

   by (simp add: norm_vector_1)

 text{* Metric *}

 text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}

 definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 

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

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

   using dimindex_ge_1[of "UNIV :: 1 set"]

   by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )

 subsection {* A connectedness or intermediate value lemma with several applications. *}

 lemma connected_real_lemma:

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

   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_def, 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. *}

 lemma norm_0: "norm (0::real ^ 'n) = 0"

   by (rule norm_zero)

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

   by (simp add: vector_norm_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: vector_norm_def dot_def setL2_def power2_eq_square)

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

   by (simp add: vector_norm_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: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"

   by vector

 lemma vector_mul_lcancel: "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: "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: "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: "\<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: "norm (x::real ^'n) <= norm(y) + norm(x - y)"

   using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)

 lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"

   by (metis order_trans norm_triangle_ineq)

 lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"

   by (metis basic_trans_rules(21) norm_triangle_ineq)

 lemma setsum_delta: 

   assumes fS: "finite S"

   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"

 proof-

   let ?f = "(\<lambda>k. if k=a then b k else 0)"

   {assume a: "a \<notin> S"

     hence "\<forall> k\<in> S. ?f k = 0" by simp

     hence ?thesis  using a by simp}

   moreover 

   {assume a: "a \<in> S"

     let ?A = "S - {a}"

     let ?B = "{a}"

     have eq: "S = ?A \<union> ?B" using a by blast 

     have dj: "?A \<inter> ?B = {}" by simp

     from fS have fAB: "finite ?A" "finite ?B" by auto  

     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"

       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]

       by simp

     then have ?thesis  using a by simp}

   ultimately show ?thesis by blast

 qed

 lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"

   apply (simp add: vector_norm_def)

   apply (rule member_le_setL2, simp_all)

   done

 lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e

                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"

   by (metis component_le_norm order_trans)

 lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e

                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"

   by (metis component_le_norm basic_trans_rules(21))

 lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"

   by (simp add: vector_norm_def setL2_le_setsum)

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

   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  dot_pos_le )

 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: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"

   apply (clarsimp simp add: norm_mul)

   apply (rule mult_mono1)

   apply simp_all

   done

 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"

   apply (rule norm_triangle_le) by simp

 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"

   by (simp add: ring_simps)

 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)

 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp

 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector

 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+

 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector

 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)

 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all 

 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) 

 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"

   "c *s x + (d *s x + z) == (c + d) *s x + z"

   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+

 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector

 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" 

   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"

   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"

   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"

   by ((atomize (full)), vector)+

 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"

   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"

   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"

   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+

 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector

 lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"

   by (atomize) (auto simp add: norm_ge_zero)

 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith

 lemma norm_pths: 

   "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"

   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"

   using norm_ge_zero[of "x - y"] by auto

 use "normarith.ML"

 method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)

 *} "Proves simple linear statements about vector norms"

 text{* Hence more metric properties. *}

 lemma dist_refl: "dist x x = 0" by norm

 lemma dist_sym: "dist x y = dist y x"by norm

 lemma dist_pos_le: "0 <= dist x y" by norm

 lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm

 lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm

 lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm

 lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm 

 lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm 

 lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm 

 lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm 

 lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm 

 lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm 

 lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"

   by norm 

 lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 

   unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. 

 lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm 

 lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm 

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

   apply vector

   apply auto

   apply (cases "finite S")

   apply (rule finite_induct[of S])

   apply (auto simp add: vector_component zero_index)

   done

 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: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)

 lemma setsum_component: 

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

   assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"

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

   using i by (simp add: setsum_eq Cart_lambda_beta)

   (* This needs finiteness assumption due to the definition of fold!!! *)

 lemma setsum_superset:

   assumes fb: "finite B" and ab: "A \<subseteq> B" 

   and f0: "\<forall>x \<in> B - A. f x = 0"

   shows "setsum f B = setsum f A"

 proof-

   from ab fb have fa: "finite A" by (metis finite_subset)

   from fb have fba: "finite (B - A)" by (metis finite_Diff)

   have d: "A \<inter> (B - A) = {}" by blast

   from ab have b: "B = A \<union> (B - A)" by blast

   from setsum_Un_disjoint[OF fa fba d, of f] b

     setsum_0'[OF f0]

   show "setsum f B = setsum f A" by simp

 qed

 lemma setsum_restrict_set:

   assumes fA: "finite A"

   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"

 proof-

   from fA have fab: "finite (A \<inter> B)" by auto

   have aba: "A \<inter> B \<subseteq> A" by blast

   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"

   from setsum_superset[OF fA aba, of ?g]

   show ?thesis by simp

 qed

 lemma setsum_cases:

   assumes fA: "finite A"

   shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =

          setsum f (A \<inter> B) + setsum g (A \<inter> - B)"

 proof-

   have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" 

     by blast+

   from fA 

   have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto

   let ?g = "\<lambda>x. if x \<in> B then f x else g x"

   from setsum_Un_disjoint[OF f a(2), of ?g] a(1)

   show ?thesis by simp

 qed

 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 !! *)

 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"

   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

 proof-

   let ?A = "{m .. n}"

   let ?B = "{n + 1 .. n + p}"

   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto 

   have d: "?A \<inter> ?B = {}" by auto

   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto

 qed

 lemma setsum_reindex_nonzero: 

   assumes fS: "finite S"

   and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"

   shows "setsum h (f ` S) = setsum (h o f) S"

 using nz

 proof(induct rule: finite_induct[OF fS])

   case 1 thus ?case by simp

 next

   case (2 x F) 

   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto

     then obtain y where y: "y \<in> F" "f x = f y" by auto 

     from "2.hyps" y have xy: "x \<noteq> y" by auto

     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp

     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto

     also have "\<dots> = setsum (h o f) (insert x F)" 

       using "2.hyps" "2.prems" h0  by auto 

     finally have ?case .}

   moreover

   {assume fxF: "f x \<notin> f ` F"

     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 

       using fxF "2.hyps" by simp 

     also have "\<dots> = setsum (h o f) (insert x F)"  

       using "2.hyps" "2.prems" fxF

       apply auto apply metis done

     finally have ?case .}

   ultimately show ?case by blast

 qed

 lemma setsum_Un_nonzero:

   assumes fS: "finite S" and fF: "finite F"

   and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"

   shows "setsum f (S \<union> F) = setsum f S + setsum f F"

   using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp

 lemma setsum_natinterval_left:

   assumes mn: "(m::nat) <= n" 

   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"

 proof-

   from mn have "{m .. n} = insert m {m+1 .. n}" by auto

   then show ?thesis by auto

 qed

 lemma setsum_natinterval_difff: 

   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

           (if m <= n then f m - f(n + 1) else 0)"

 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)

 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]

 lemma setsum_setsum_restrict:

   "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"

   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)

   by (rule setsum_commute)

 lemma setsum_image_gen: assumes fS: "finite S"

   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

 proof-

   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}

   note th0 = this

   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" 

     apply (rule setsum_cong2) 

     by (simp add: th0)

   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

     apply (rule setsum_setsum_restrict[OF fS])

     by (rule finite_imageI[OF fS])

   finally show ?thesis .

 qed

     (* 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_superset[OF fT fST])

 by (auto intro: setsum_0')

 (* FIXME: Change the name to fold_image\<dots> *)

 lemma (in comm_monoid_mult) fold_1': "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"

   apply (induct set: finite)

   apply simp by (auto simp add: fold_image_insert)

 lemma (in comm_monoid_mult) fold_union_nonzero:

   assumes fS: "finite S" and fT: "finite T"

   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"

   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"

 proof-

   have "fold_image op * f 1 (S \<inter> T) = 1" 

     apply (rule fold_1')

     using fS fT I0 by auto 

   with fold_image_Un_Int[OF fS fT] show ?thesis by simp

 qed

 lemma setsum_union_nonzero:  

   assumes fS: "finite S" and fT: "finite T"

   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"

   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"

   using fS fT

   apply (simp add: setsum_def)

   apply (rule comm_monoid_add.fold_union_nonzero)

   using I0 by auto

 lemma setprod_union_nonzero:  

   assumes fS: "finite S" and fT: "finite T"

   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"

   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"

   using fS fT

   apply (simp add: setprod_def)

   apply (rule fold_union_nonzero)

   using I0 by auto

 lemma setsum_unions_nonzero: 

   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"

   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"

   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"

   using fSS f0

 proof(induct rule: finite_induct[OF fS])

   case 1 thus ?case by simp

 next

   case (2 T F)

   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 

     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)

   from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)

   from "2.prems" TF fTF

   show ?case 

     by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])

 qed

   (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)

 lemma (in comm_monoid_mult) fold_related: 

   assumes Re: "R e e" 

   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 

   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"

   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"

   using fS by (rule finite_subset_induct) (insert assms, auto)

   (* FIXME: I think we can get rid of the finite assumption!! *)	

 lemma (in comm_monoid_mult) 

   fold_eq_general:

   assumes fS: "finite S"

   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 

   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"

   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"

 proof-

   from h f12 have hS: "h ` S = S'" by auto

   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"

     from f12 h H  have "x = y" by auto }

   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast

   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 

   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp

   also have "\<dots> = fold_image (op *) (f2 o h) e S" 

     using fold_image_reindex[OF fS hinj, of f2 e] .

   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]

     by blast

   finally show ?thesis ..

 qed

 lemma (in comm_monoid_mult) fold_eq_general_inverses:

   assumes fS: "finite S" 

   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"

   shows "fold_image (op *) f e S = fold_image (op *) g e T"

   using fold_eq_general[OF fS, of T h g f e] kh hk by metis

 lemma setsum_eq_general_reverses:

   assumes fS: "finite S" and fT: "finite T"

   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"

   shows "setsum f S = setsum g T"

   apply (simp add: setsum_def fS fT)

   apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS])

   apply (erule kh)

   apply (erule hk)

   done

 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 (dimindex(UNIV :: 'n set)) *  e"

 proof-

   let ?d = "real (dimindex (UNIV ::'n set))"

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

   let ?U = "{1 .. dimindex (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 i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]

       by (auto simp add: setsum_component intro: abs_le_D1)

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

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

       by (auto simp add: setsum_negf setsum_component vector_component 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_nonzero) 

       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)

 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 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:

   assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"

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

   using k 

   apply (simp add: basis_def real_vector_norm_def dot_def)

   apply (vector delta_mult_idempotent)

   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]

   apply auto

   done

 lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"

   apply (simp add: basis_def real_vector_norm_def dot_def)

   apply (vector delta_mult_idempotent)

   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]

   apply auto

   done

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

   apply (rule exI[where x="c *s basis 1"])

   by (simp only: norm_mul norm_basis_1)

 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_def)

   then show ?thesis by blast

 qed

 lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"

   by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)

 lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)"

   by (simp add: basis_def Cart_lambda_beta)

 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) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")

   by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component 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) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)"

   by (simp add: Cart_eq setsum_component vector_component basis_component 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:

   assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"

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

   using i

   by (auto simp add: dot_def basis_def Cart_lambda_beta 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> i \<notin> {1..dimindex(UNIV ::'n set)}"

   by (auto simp add: Cart_eq basis_component zero_index)

 lemma basis_nonzero: 

   assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"

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

   using k 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 vector

   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 vector

   done

 subsection{* Orthogonality. *}

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

 lemma orthogonal_basis:

   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 

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

   using i

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

 lemma orthogonal_basis_basis:

   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}" 

   and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}" 

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

   unfolding orthogonal_basis[OF i] basis_component[OF i] 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. *}

 lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1"

   apply (rule Cart_lambda_beta[rule_format])

   using dimindex_ge_1 apply auto done

 lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1"

   by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)

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

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

   using dimindex_ge_1

   by (auto simp add: vector_def Cart_lambda_beta[rule_format])

 lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"

   by (auto simp add: dimindex_def)

 lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"

   by (auto simp add: dimindex_def)

 lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"

   by (auto simp add: dimindex_def)

 lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"

   by (auto simp add: dimindex_def)

 lemma vector_2:

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

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

   apply (simp add: vector_def)

   using Cart_lambda_beta[rule_format, OF dimindex_2, of "\<lambda>i. if i \<le> length [x,y] then [x,y] ! (i - 1) else (0::'a)"]

   apply (simp only: vector_def )

   apply auto

   done

 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"

 apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)

   using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]   using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]

   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

   by (vector vector_def dimindex_def)

 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 dimindex_def)

   apply auto

   apply (subgoal_tac "i = 1 \<or> i =2", auto)

   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 dimindex_def)

   apply auto

   apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)

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

   by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] 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" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"

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

   using lf k

   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 = "{1..dimindex(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::nat. (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 i, of x]

       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

 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^'n \<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 ^ 'n)) 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 ^ 'n)) = h z x - h z y"

   by (simp  add: diff_def bilinear_radd bilinear_rneg)

 lemma bilinear_setsum:

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

   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 = "{1 .. dimindex (UNIV :: 'm set)}"

   let ?N = "{1 .. dimindex (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

 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 = "{1 .. dimindex (UNIV :: 'n set)}"

   let ?M = "{1 .. dimindex (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_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)

 	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> _"

   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"} *}

 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)

 defs (overloaded) 

 matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"

 abbreviation 

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

   where "m ** m' == m\<star> m'"

 defs (overloaded) 

   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"

 abbreviation 

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

   where 

   "m *v v == m \<star> v"

 defs (overloaded) 

   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"

 abbreviation 

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

   where 

   "v v* m == v \<star> m"

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

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

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

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

 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"

 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"

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

 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"

   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)

 lemma setsum_delta': 

   assumes fS: "finite S" shows 

   "setsum (\<lambda>k. if a = k then b k else 0) S = 

      (if a\<in> S then b a else 0)"

   using setsum_delta[OF fS, of a b, symmetric] 

   by (auto intro: setsum_cong)

 lemma matrix_mul_lid: "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_atLeastAtMost]  mult_1_left mult_zero_left if_True)

 lemma matrix_mul_rid: "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_atLeastAtMost]  mult_1_right mult_zero_right if_True 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"

   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_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"

   by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)

 lemma matrix_eq: "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 Cart_lambda_beta cong del: if_weak_cong)

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

   apply (erule_tac x="i" in ballE)

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

 lemma matrix_vector_mul_component: 

   assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"

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

   using k

   by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)

 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) 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 Cart_lambda_beta mult_ac)

   apply (subst setsum_commute)

   by simp

 lemma transp_mat: "transp (mat n) = mat n"

   by (vector transp_def mat_def)

 lemma transp_transp: "transp(transp A) = A"

   by (vector transp_def)

 lemma row_transp: 

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

   assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"

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

   using i 

   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)

 lemma column_transp:

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

   assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"

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

   using i 

   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)

 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"

 apply (auto simp add: rows_def columns_def row_transp intro: set_ext)

 apply (rule_tac x=i in exI)

 apply (auto simp add: row_transp)

 done

 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)

 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) {1 .. dimindex(UNIV:: 'n set)}"

   by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)

 lemma vector_componentwise:

   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})"

   apply (subst basis_expansion[symmetric])

   by (vector Cart_eq Cart_lambda_beta setsum_component)

 lemma linear_componentwise:

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

   assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"

   shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")

 proof-

   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"

   let ?N = "{1 .. dimindex (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[OF j, symmetric]

     unfolding setsum_component[OF j, 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 ^ 'n))"

   by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component 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 Cart_lambda_beta mult_commute del: One_nat_def)

 apply clarify

 apply (rule linear_componentwise[OF lf, symmetric])

 apply simp

 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> _)" and lg: "linear g" 

   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^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) {1..dimindex(UNIV:: 'n set)}"

   by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)

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

   apply (rule adjoint_unique[symmetric])

   apply (rule matrix_vector_mul_linear)

   apply (simp add: transp_def dot_def Cart_lambda_beta 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) = transp(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 \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>

    (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")

 proof-

   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"

   {assume H: "?rhs"

     then have ?lhs by auto}

   moreover

   {assume H: "?lhs"

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

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

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

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

       then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto 

     }

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

     hence ?rhs by metis }

   ultimately show ?thesis by metis

 qed 

 (* Supremum and infimum of real sets *)

 definition rsup:: "real set \<Rightarrow> real" where

   "rsup S = (SOME a. isLub UNIV S a)"

 lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)

 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"

   shows "isLub UNIV S (rsup S)"

 using Se b

 unfolding rsup_def

 apply clarify

 apply (rule someI_ex)

 apply (rule reals_complete)

 by (auto simp add: isUb_def setle_def)

 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"

 proof-

   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)

   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast 

   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)

 qed

 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "rsup S = Max S"

 using fS Se

 proof-

   let ?m = "Max S"

   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis

   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)

   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S" 

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

   moreover 

   have "rsup S \<le> ?m" using Sm lub

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

   ultimately  show ?thesis by arith 

 qed

 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "rsup S \<in> S"

   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis

 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "isUb S S (rsup S)"

   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]

   unfolding isUb_def setle_def by metis

 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"

 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)

 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"

 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)

 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"

 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)

 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"

 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)

 lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"

   shows "rsup S = b"

 using b S  

 unfolding setle_def rsup_alt

 apply -

 apply (rule some_equality)

 apply (metis  linorder_not_le order_eq_iff[symmetric])+

 done

 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"

   apply (rule rsup_le)

   apply simp

   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)

 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"

   apply (rule ext)

   by (metis isUb_def)

 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)

 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"

   shows "a \<le> rsup S \<and> rsup S \<le> b"

 proof-

   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast

   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')

   from Se obtain y where y: "y \<in> S" by blast

   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')

     apply (erule ballE[where x=y])

     apply (erule ballE[where x=y])

     apply arith

     using y apply auto

     done

   with b show ?thesis by blast

 qed

 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"

   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]

   by (auto simp add: setge_def setle_def)

 lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"

 proof-

   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith

   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th 

     by  (auto simp add: setge_def setle_def)

 qed

 definition rinf:: "real set \<Rightarrow> real" where

   "rinf S = (SOME a. isGlb UNIV S a)"

 lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)

 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"

   shows "\<exists>(t::real). isGlb UNIV S t"

 proof-

   let ?M = "uminus ` S"

   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)

     by (rule_tac x="-y" in exI, auto)

   from Se have Me: "\<exists>x. x \<in> ?M" by blast

   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast

   have "isGlb UNIV S (- t)" using t

     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)

     apply (erule_tac x="-y" in allE)

     apply auto

     done

   then show ?thesis by metis

 qed

 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"

   shows "isGlb UNIV S (rinf S)"

 using Se b

 unfolding rinf_def

 apply clarify

 apply (rule someI_ex)

 apply (rule reals_complete_Glb)

 apply (auto simp add: isLb_def setle_def setge_def)

 done

 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"

 proof-

   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)

   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast 

   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)

 qed

 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "rinf S = Min S"

 using fS Se

 proof-

   let ?m = "Min S"

   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis

   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)

   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S" 

     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)

   moreover 

   have "rinf S \<ge> ?m" using Sm glb

     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)

   ultimately  show ?thesis by arith 

 qed

 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "rinf S \<in> S"

   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis

 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "isLb S S (rinf S)"

   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]

   unfolding isLb_def setge_def by metis

 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"

 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)

 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"

 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)

 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"

 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)

 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"

   shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"

 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)

 lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"

   shows "rinf S = b"

 using b S  

 unfolding setge_def rinf_alt

 apply -

 apply (rule some_equality)

 apply (metis  linorder_not_le order_eq_iff[symmetric])+

 done

 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"

   apply (rule rinf_ge)

   apply simp

   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)

 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"

   apply (rule ext)

   by (metis isLb_def)

 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"

   shows "a \<le> rinf S \<and> rinf S \<le> b"

 proof-

   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast

   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')

   from Se obtain y where y: "y \<in> S" by blast

   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')

     apply (erule ballE[where x=y])

     apply (erule ballE[where x=y])

     apply arith

     using y apply auto

     done

   with b show ?thesis by blast

 qed

 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"

   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]

   by (auto simp add: setge_def setle_def)

 lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"

 proof-

   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith

   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th 

     by  (auto simp add: setge_def setle_def)

 qed

 subsection{* Operator norm. *}

 definition "onorm f = rsup {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 1"] 

     have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]

       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_1 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 rsup[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 rsup[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 1"], unfolded norm_basis_1] 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 rsup_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"

   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 *)

 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"

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

   where "dest_vec1 x = (x$1)"

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

   by (simp add: vec1_def)

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

   by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def)

 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 forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)

 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_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 vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto

 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)

 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)

 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)

 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)

 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)

 lemma vec1_setsum: assumes fS: "finite S"

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

   apply (induct rule: finite_induct[OF fS])

   apply (simp add: vec1_vec)

   apply (auto simp add: vec1_add)

   done

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

   by (simp add: dest_vec1_def)

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

   by (simp add: vec1_vec[symmetric])

 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"

  by (metis vec1_dest_vec1 vec1_add)

 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"

  by (metis vec1_dest_vec1 vec1_sub)

 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"

  by (metis vec1_dest_vec1 vec1_cmul)

 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"

  by (metis vec1_dest_vec1 vec1_neg)

 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)

 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: dest_vec1_add)

   done

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

   by (simp add: vec1_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)

 lemma linear_vmul_dest_vec1: 

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