header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}

 theory Cartesian_Euclidean_Space

 imports Finite_Cartesian_Product Integration

 begin

 (* TODO:  real_vector^'n is instance of real_vector *)

 (* Some strange lemmas, are they really needed? *)

 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 setsum_Plus:

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

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

   unfolding Plus_def

   by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)

 lemma setsum_UNIV_sum:

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

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

   apply (subst UNIV_Plus_UNIV [symmetric])

   apply (rule setsum_Plus [OF finite finite])

   done

 lemma setsum_mult_product:

   "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"

   unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]

 proof (rule setsum_cong, simp, rule setsum_reindex_cong)

   fix i show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)

   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"

   proof safe

     fix j assume "j \<in> {i * B..<i * B + B}"

     thus "j \<in> (\<lambda>j. j + i * B) ` {..<B}"

       by (auto intro!: image_eqI[of _ _ "j - i * B"])

   qed simp

 qed simp

 subsection{* Basic componentwise operations on vectors. *}

 instantiation cart :: (times,finite) times

 begin

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

   instance ..

 end

 instantiation cart :: (one,finite) one

 begin

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

   instance ..

 end

 instantiation cart :: (ord,finite) ord

 begin

   definition vector_le_def:

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

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

   instance by (intro_classes)

 end

 text{* The ordering on one-dimensional vectors is linear. *}

 class cart_one = assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"

 begin

   subclass finite

   proof from UNIV_one show "finite (UNIV :: 'a set)"

       by (auto intro!: card_ge_0_finite) qed

 end

 instantiation cart :: (linorder,cart_one) linorder begin

 instance proof

   guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+

   hence *:"UNIV = {a}" by auto

   have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto

   fix x y z::"'a^'b::cart_one" note * = vector_le_def vector_less_def all Cart_eq

   show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps)

   { assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) }

   { assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) }

 qed end

 text{* Constant Vectors *} 

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

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

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

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

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

 method_setup vector = {*

 let

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

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

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

   val ss2 = @{simpset} addsimps

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

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

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

               @{thm vector_scaleR_def},

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

  fun vector_arith_tac ths =

    simp_tac ss1

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

          ORELSE rtac @{thm setsum_0'} i

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

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

    THEN' asm_full_simp_tac (ss2 addsimps ths)

  in

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

  end

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

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

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

 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector

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

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

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

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

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

 lemma vec_setsum: assumes fS: "finite S"

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

   apply (induct rule: finite_induct[OF fS])

   apply (simp)

   apply (auto simp add: vec_add)

   done

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

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

   by (vector vec_def)

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

   by vector

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

   by vector

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

 lemmas vector_component =

   vec_component vector_add_component vector_mult_component

   vector_smult_component vector_minus_component vector_uminus_component

   vector_scaleR_component cond_component

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

 instance cart :: (semigroup_mult,finite) semigroup_mult

   apply (intro_classes) by (vector mult_assoc)

 instance cart :: (monoid_mult,finite) monoid_mult

   apply (intro_classes) by vector+

 instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult

   apply (intro_classes) by (vector mult_commute)

 instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult

   apply (intro_classes) by (vector mult_idem)

 instance cart :: (comm_monoid_mult,finite) comm_monoid_mult

   apply (intro_classes) by vector

 instance cart :: (semiring,finite) semiring

   apply (intro_classes) by (vector field_simps)+

 instance cart :: (semiring_0,finite) semiring_0

   apply (intro_classes) by (vector field_simps)+

 instance cart :: (semiring_1,finite) semiring_1

   apply (intro_classes) by vector

 instance cart :: (comm_semiring,finite) comm_semiring

   apply (intro_classes) by (vector field_simps)+

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

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

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

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

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

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

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

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

 instance cart :: (real_algebra,finite) real_algebra

   apply intro_classes

   apply (simp_all add: vector_scaleR_def field_simps)

   apply vector

   apply vector

   done

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

 lemma of_nat_index:

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

   apply (induct n)

   apply vector

   apply vector

   done

 lemma one_index[simp]:

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

 instance cart :: (semiring_char_0,finite) semiring_char_0

 proof (intro_classes)

   fix m n ::nat

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

     by (simp add: Cart_eq of_nat_index)

 qed

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

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

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

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

   by (vector mult_assoc)

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

   by (vector field_simps)

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

   by (vector field_simps)

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

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

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

   by (vector field_simps)

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

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

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

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

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

   by (vector field_simps)

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

   by (simp add: Cart_eq)

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

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

   by vector

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

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)

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

   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)

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

   by (metis vector_mul_lcancel)

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

   by (metis vector_mul_rcancel)

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

   apply (simp add: norm_vector_def)

   apply (rule member_le_setL2, simp_all)

   done

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

   by (metis component_le_norm_cart order_trans)

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

   by (metis component_le_norm_cart basic_trans_rules(21))

 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"

   by (simp add: norm_vector_def setL2_le_setsum)

 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"

   unfolding vector_scaleR_def vector_scalar_mult_def by simp

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

   unfolding dist_norm scalar_mult_eq_scaleR

   unfolding scaleR_right_diff_distrib[symmetric] by simp

 lemma setsum_component [simp]:

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

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

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

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

   by (simp add: Cart_eq)

 lemma setsum_cmul:

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

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

   by (simp add: Cart_eq setsum_right_distrib)

 (* TODO: use setsum_norm_allsubsets_bound *)

 lemma setsum_norm_allsubsets_bound_cart:

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

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

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

 proof-

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

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

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

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

     by (rule setsum_commute)

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

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

     apply (rule setsum_mono)    by (rule norm_le_l1_cart)

   also have "\<dots> \<le> 2 * ?d * e"

     unfolding th0 th1

   proof(rule setsum_bounded)

     fix i assume i: "i \<in> ?U"

     let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"

     let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"

     have thp: "P = ?Pp \<union> ?Pn" by auto

     have thp0: "?Pp \<inter> ?Pn ={}" by auto

     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+

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

       using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]

       by (auto intro: abs_le_D1)

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

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

       by (auto simp add: setsum_negf intro: abs_le_D1)

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

       apply (subst thp)

       apply (rule setsum_Un_zero)

       using fP thp0 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

 subsection {* A bijection between 'n::finite and {..<CARD('n)} *}

 definition cart_bij_nat :: "nat \<Rightarrow> ('n::finite)" where

   "cart_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"

 abbreviation "\<pi> \<equiv> cart_bij_nat"

 definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"

 lemma bij_betw_pi:

   "bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"

   using ex_bij_betw_nat_finite[of "UNIV::'n set"]

   by (auto simp: cart_bij_nat_def atLeast0LessThan

     intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])

 lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"

   using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto

 lemma pi'_inj[intro]: "inj \<pi>'"

   using bij_betw_pi' unfolding bij_betw_def by auto

 lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"

   using bij_betw_pi' unfolding bij_betw_def by auto

 lemma \<pi>\<pi>'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"

   using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)

 lemma \<pi>'\<pi>[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"

   using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)

 lemma \<pi>\<pi>'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"

   by auto

 lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"

   using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)

 instantiation cart :: (real_basis,finite) real_basis

 begin

 definition "(basis i::'a^'b) =

   (if i < (CARD('b) * DIM('a))

   then (\<chi> j::'b. if j = \<pi>(i div DIM('a)) then basis (i mod DIM('a)) else 0)

   else 0)"

 lemma basis_eq:

   assumes "i < CARD('b)" and "j < DIM('a)"

   shows "basis (j + i * DIM('a)) = (\<chi> k. if k = \<pi> i then basis j else 0)"

 proof -

   have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)

   also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto

   finally show ?thesis

     unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps)

 qed

 lemma basis_eq_pi':

   assumes "j < DIM('a)"

   shows "basis (j + \<pi>' i * DIM('a)) $ k = (if k = i then basis j else 0)"

   apply (subst basis_eq)

   using pi'_range assms by simp_all

 lemma split_times_into_modulo[consumes 1]:

   fixes k :: nat

   assumes "k < A * B"

   obtains i j where "i < A" and "j < B" and "k = j + i * B"

 proof

   have "A * B \<noteq> 0"

   proof assume "A * B = 0" with assms show False by simp qed

   hence "0 < B" by auto

   thus "k mod B < B" using `0 < B` by auto

 next

   have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)

   also have "... < A * B" using assms by simp

   finally show "k div B < A" by auto

 qed simp

 lemma split_CARD_DIM[consumes 1]:

   fixes k :: nat

   assumes k: "k < CARD('b) * DIM('a)"

   obtains i and j::'b where "i < DIM('a)" "k = i + \<pi>' j * DIM('a)"

 proof -

   from split_times_into_modulo[OF k] guess i j . note ij = this

   show thesis

   proof

     show "j < DIM('a)" using ij by simp

     show "k = j + \<pi>' (\<pi> i :: 'b) * DIM('a)"

       using ij by simp

   qed

 qed

 lemma linear_less_than_times:

   fixes i j A B :: nat assumes "i < B" "j < A"

   shows "j + i * A < B * A"

 proof -

   have "i * A + j < (Suc i)*A" using `j < A` by simp

   also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp

   finally show ?thesis by simp

 qed

 instance

 proof

   let ?b = "basis :: nat \<Rightarrow> 'a^'b"

   let ?b' = "basis :: nat \<Rightarrow> 'a"

   { fix D :: nat and f :: "nat \<Rightarrow> 'c::real_vector"

     assume "inj_on f {..<D}"

     hence eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"

       and inj: "\<And>i. inj_on f ({..<D} - {i})"

       by (auto simp: inj_on_def)

     have *: "\<And>i. finite (f ` {..<D} - {i})" by simp

     have "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"

       unfolding dependent_def span_finite[OF *]

       by (auto simp: eq setsum_reindex[OF inj]) }

   note independentI = this

   have setsum_basis:

     "\<And>f. (\<Sum>x\<in>range basis. f (x::'a)) = f 0 + (\<Sum>i<DIM('a). f (basis i))"

     unfolding range_basis apply (subst setsum.insert)

     by (auto simp: basis_eq_0_iff setsum.insert setsum_reindex[OF basis_inj])

   have inj: "inj_on ?b {..<CARD('b)*DIM('a)}"

     by (auto intro!: inj_onI elim!: split_CARD_DIM split: split_if_asm

              simp add: Cart_eq basis_eq_pi' all_conj_distrib basis_neq_0

                        inj_on_iff[OF basis_inj])

   moreover

   hence indep: "independent (?b ` {..<CARD('b) * DIM('a)})"

   proof (rule independentI[THEN iffD2], safe elim!: split_CARD_DIM del: notI)

     fix j and i :: 'b and u :: "'a^'b \<Rightarrow> real" assume "j < DIM('a)"

     let ?x = "j + \<pi>' i * DIM('a)"

     show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) \<noteq> ?b ?x"

       unfolding Cart_eq not_all

     proof

       have "(\<lambda>j. j + \<pi>' i*DIM('a))`({..<DIM('a)}-{j}) =

         {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)} - {?x}"(is "?S = ?I - _")

       proof safe

         fix y assume "y \<in> ?I"

         moreover def k \<equiv> "y - \<pi>' i*DIM('a)"

         ultimately have "k < DIM('a)" and "y = k + \<pi>' i * DIM('a)" by auto

         moreover assume "y \<notin> ?S"

         ultimately show "y = j + \<pi>' i * DIM('a)" by auto

       qed auto

       have "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i =

           (\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k $ i)" by simp

       also have "\<dots> = (\<Sum>k\<in>?S. u(?b k) *\<^sub>R ?b k $ i)"

         unfolding `?S = ?I - {?x}`

       proof (safe intro!: setsum_mono_zero_cong_right)

         fix y assume "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"

         moreover have "Suc (\<pi>' i) * DIM('a) \<le> CARD('b) * DIM('a)"

           unfolding mult_le_cancel2 using pi'_range[of i] by simp

         ultimately show "y < CARD('b) * DIM('a)" by simp

       next

         fix y assume "y < CARD('b) * DIM('a)"

         with split_CARD_DIM guess l k . note y = this

         moreover assume "u (?b y) *\<^sub>R ?b y $ i \<noteq> 0"

         ultimately show "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"

           by (auto simp: basis_eq_pi' split: split_if_asm)

       qed simp

       also have "\<dots> = (\<Sum>k\<in>{..<DIM('a)} - {j}. u (?b (k + \<pi>' i*DIM('a))) *\<^sub>R (?b' k))"

         by (subst setsum_reindex) (auto simp: basis_eq_pi' intro!: inj_onI)

       also have "\<dots> \<noteq> ?b ?x $ i"

       proof -

         note independentI[THEN iffD1, OF basis_inj independent_basis, rule_format]

         note this[of j "\<lambda>v. u (\<chi> ka::'b. if ka = i then v else (0\<Colon>'a))"]

         thus ?thesis by (simp add: `j < DIM('a)` basis_eq pi'_range)

       qed

       finally show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i \<noteq> ?b ?x $ i" .

     qed

   qed

   ultimately

   show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"

     by (auto intro!: exI[of _ "CARD('b) * DIM('a)"] simp: basis_cart_def)

   from indep have exclude_0: "0 \<notin> ?b ` {..<CARD('b) * DIM('a)}"

     using dependent_0[of "?b ` {..<CARD('b) * DIM('a)}"] by blast

   show "span (range ?b) = UNIV"

   proof -

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

       let "?if i y" = "(\<chi> k::'b. if k = i then ?b' y else (0\<Colon>'a))"

       have The_if: "\<And>i j. j < DIM('a) \<Longrightarrow> (THE k. (?if i j) $ k \<noteq> 0) = i"

         by (rule the_equality) (simp_all split: split_if_asm add: basis_neq_0)

       { fix x :: 'a

         have "x \<in> span (range basis)"

           using span_basis by (auto simp: basis_range)

         hence "\<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = x"

           by (subst (asm) span_finite) (auto simp: setsum_basis) }

       hence "\<forall>i. \<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = i" by auto

       then obtain u where u: "\<forall>i. (\<Sum>x<DIM('a). u i (?b' x) *\<^sub>R ?b' x) = i"

         by (auto dest: choice)

       have "\<exists>u. \<forall>i. (\<Sum>j<DIM('a). u (?if i j) *\<^sub>R ?b' j) = x $ i"

         apply (rule exI[of _ "\<lambda>v. let i = (THE i. v$i \<noteq> 0) in u (x$i) (v$i)"])

         using The_if u by simp }

     moreover

     have "\<And>i::'b. {..<CARD('b)} \<inter> {x. i = \<pi> x} = {\<pi>' i}"

       using pi'_range[where 'n='b] by auto

     moreover

     have "range ?b = {0} \<union> ?b ` {..<CARD('b) * DIM('a)}"

       by (auto simp: image_def basis_cart_def)

     ultimately

     show ?thesis

       by (auto simp add: Cart_eq setsum_reindex[OF inj] basis_range

           setsum_mult_product basis_eq if_distrib setsum_cases span_finite

           setsum_reindex[OF basis_inj])

   qed

 qed

 end

 lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a::real_basis)"

 proof (safe intro!: dimension_eq elim!: split_times_into_modulo del: notI)

   fix i j assume *: "i < CARD('b)" "j < DIM('a)"

   hence A: "(i * DIM('a) + j) div DIM('a) = i"

     by (subst div_add1_eq) simp

   from * have B: "(i * DIM('a) + j) mod DIM('a) = j"

     unfolding mod_mult_self3 by simp

   show "basis (j + i * DIM('a)) \<noteq> (0::'a^'b)" unfolding basis_cart_def

     using * basis_finite[where 'a='a]

       linear_less_than_times[of i "CARD('b)" j "DIM('a)"]

     by (auto simp: A B field_simps Cart_eq basis_eq_0_iff)

 qed (auto simp: basis_cart_def)

 lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp

 lemma split_dimensions'[consumes 1]:

   assumes "k < DIM('a::real_basis^'b)"

   obtains i j where "i < CARD('b::finite)" and "j < DIM('a::real_basis)" and "k = j + i * DIM('a::real_basis)"

 using split_times_into_modulo[OF assms[simplified]] .

 lemma cart_euclidean_bound[intro]:

   assumes j:"j < DIM('a::{real_basis})"

   shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::real_basis)"

   using linear_less_than_times[OF pi'_range j, of i] .

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

 lemma (in real_basis) forall_CARD_DIM:

   "(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"

    (is "?l \<longleftrightarrow> ?r")

 proof (safe elim!: split_times_into_modulo)

   fix i :: 'b and j assume "j < DIM('a)"

   note linear_less_than_times[OF pi'_range[of i] this]

   moreover assume "?l"

   ultimately show "P (j + \<pi>' i * DIM('a))" by auto

 next

   fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r"

   from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`

   show "P (j + i * DIM('a))" by simp

 qed

 lemma (in real_basis) exists_CARD_DIM:

   "(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"

   using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast

 lemma forall_CARD:

   "(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))"

   using forall_CARD_DIM[where 'a=real, of P] by simp

 lemma exists_CARD:

   "(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))"

   using exists_CARD_DIM[where 'a=real, of P] by simp

 lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD

 lemma cart_euclidean_nth[simp]:

   fixes x :: "('a::real_basis_with_inner, 'b::finite) cart"

   assumes j:"j < DIM('a)"

   shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"

   unfolding euclidean_component_def inner_vector_def basis_eq_pi'[OF j] if_distrib cond_application_beta

   by (simp add: setsum_cases)

 lemma real_euclidean_nth:

   fixes x :: "real^'n"

   shows "x $$ \<pi>' i = (x $ i :: real)"

   using cart_euclidean_nth[where 'a=real, of 0 x i] by simp

 lemmas nth_conv_component = real_euclidean_nth[symmetric]

 lemma mult_split_eq:

   fixes A :: nat assumes "x < A" "y < A"

   shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j"

 proof

   assume *: "x + i * A = y + j * A"

   { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A"

     hence "x + i * A < Suc i * A" using `x < A` by simp

     also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp

     also have "\<dots> \<le> y + j * A" by simp

     finally have "i = j" using * by simp }

   note eq = this

   have "i = j"

   proof (cases rule: linorder_cases)

     assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp

   next

     assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp

   qed simp

   thus "x = y \<and> i = j" using * by simp

 qed simp

 instance cart :: (euclidean_space,finite) euclidean_space

 proof (default, safe elim!: split_dimensions')

   let ?b = "basis :: nat \<Rightarrow> 'a^'b"

   have if_distrib_op: "\<And>f P Q a b c d.

     f (if P then a else b) (if Q then c else d) =

       (if P then if Q then f a c else f a d else if Q then f b c else f b d)"

     by simp

   fix i j k l

   assume "i < CARD('b)" "k < CARD('b)" "j < DIM('a)" "l < DIM('a)"

   thus "?b (j + i * DIM('a)) \<bullet> ?b (l + k * DIM('a)) =

     (if j + i * DIM('a) = l + k * DIM('a) then 1 else 0)"

     using inj_on_iff[OF \<pi>_inj_on[where 'n='b], of k i]

     by (auto simp add: basis_eq inner_vector_def if_distrib_op[of inner] setsum_cases basis_orthonormal mult_split_eq)

 qed

 instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space

 proof

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

   show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i \<le> y $$ i)"

     unfolding vector_le_def apply(subst eucl_le) by (simp add: cart_simps)

   show"(x < y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i < y $$ i)"

     unfolding vector_less_def apply(subst eucl_less) by (simp add: cart_simps)

 qed

 subsection{* Basis vectors in coordinate directions. *}

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

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

   unfolding cart_basis_def by simp

 lemma norm_basis[simp]:

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

   apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vector_def

   apply (vector delta_mult_idempotent)

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

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

   by (rule norm_basis)

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

   by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp

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

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

 proof-

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

     by blast

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

   then show ?thesis by blast

 qed

 lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"

   by (simp add: inj_on_def Cart_eq)

 lemma basis_expansion:

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

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

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

   unfolding vector_scalar_mult_def vector_scaleR_def by simp

 lemma basis_expansion':

   "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"

   by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])

 lemma basis_expansion_unique:

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

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

 lemma dot_basis:

   shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"

   by (auto simp add: inner_vector_def cart_basis_def cond_application_beta if_distrib setsum_delta

            cong del: if_weak_cong)

 lemma inner_basis:

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

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

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

   unfolding inner_vector_def cart_basis_def

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

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

   by (auto simp add: Cart_eq)

 lemma basis_nonzero:

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

   by (simp add: basis_eq_0)

 text {* some lemmas to map between Eucl and Cart *}

 lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"

   unfolding basis_cart_def using pi'_range[where 'n='a]

   by (auto simp: Cart_eq Cart_lambda_beta)

 subsection {* Orthogonality on cartesian products *}

 lemma orthogonal_basis:

   shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"

   by (auto simp add: orthogonal_def inner_vector_def cart_basis_def if_distrib

                      cond_application_beta setsum_delta cong del: if_weak_cong)

 lemma orthogonal_basis_basis:

   shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"

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

 subsection {* Linearity on cartesian products *}

 lemma linear_vmul_component:

   assumes lf: "linear f"

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

   using lf

   by (auto simp add: linear_def algebra_simps)

 subsection{* Adjoints on cartesian products *}

 text {* TODO: The following lemmas about adjoints should hold for any

 Hilbert space (i.e. complete inner product space).

 (see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})

 *}

 lemma adjoint_works_lemma:

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

   assumes lf: "linear f"

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

 proof-

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

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

   have fN: "finite ?N" by simp

   have fM: "finite ?M" by simp

   {fix y:: "real ^ 'm"

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

     {fix x

       have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"

         by (simp only: basis_expansion')

       also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"

         unfolding linear_setsum[OF lf fN]

         by (simp add: linear_cmul[OF lf])

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

         apply (simp only: )

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

         done}

   }

   then show ?thesis unfolding adjoint_def

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

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

     by metis

 qed

 lemma adjoint_works:

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

   assumes lf: "linear f"

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

   using adjoint_works_lemma[OF lf] by metis

 lemma adjoint_linear:

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

   assumes lf: "linear f"

   shows "linear (adjoint f)"

   unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe

   unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto

 lemma adjoint_clauses:

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

   assumes lf: "linear f"

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

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

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

 lemma adjoint_adjoint:

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

   assumes lf: "linear f"

   shows "adjoint (adjoint f) = f"

   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])

 subsection {* Matrix operations *}

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

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

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

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

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

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

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

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

 definition transpose where 

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

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

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

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

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

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

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

   by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)

 lemma matrix_mul_lid:

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

   shows "mat 1 ** A = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

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

 lemma matrix_mul_rid:

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

   shows "A ** mat 1 = A"

   apply (simp add: matrix_matrix_mult_def mat_def)

   apply vector

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

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

   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)

   apply (subst setsum_commute)

   apply simp

   done

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

   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)

   apply (subst setsum_commute)

   apply simp

   done

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

   apply (vector matrix_vector_mult_def mat_def)

   by (simp add: if_distrib cond_application_beta

     setsum_delta' cong del: if_weak_cong)

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

   by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)

 lemma matrix_eq:

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

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

   apply auto

   apply (subst Cart_eq)

   apply clarify

   apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta Cart_eq cong del: if_weak_cong)

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

   apply (erule_tac x="i" in allE)

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

 lemma matrix_vector_mul_component:

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

   by (simp add: matrix_vector_mult_def inner_vector_def)

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

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

   apply (subst setsum_commute)

   by simp

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

   by (vector transpose_def mat_def)

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

   by (vector transpose_def)

 lemma row_transpose:

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

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

   by (simp add: row_def column_def transpose_def Cart_eq)

 lemma column_transpose:

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

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

   by (simp add: row_def column_def transpose_def Cart_eq)

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

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

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

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

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

   by (simp add: matrix_vector_mult_def inner_vector_def)

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

   by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)

 lemma vector_componentwise:

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

   apply (subst basis_expansion[symmetric])

   by (vector Cart_eq setsum_component)

 lemma linear_componentwise:

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

   assumes lf: "linear f"

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

 proof-

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

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

   have fM: "finite ?M" by simp

   have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j"

     unfolding vector_smult_component[symmetric] smult_conv_scaleR

     unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]

     ..

   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(cart_basis j))$i)"

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

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

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

 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)

 apply clarify

 apply (rule linear_componentwise[OF lf, symmetric])

 done

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

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

   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)

 lemma matrix_compose:

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

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

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

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

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

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

   by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)

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

   apply (rule adjoint_unique)

   apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)

   apply (subst setsum_commute)

   apply (auto simp add: mult_ac)

   done

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

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

   apply (subst matrix_vector_mul[OF lf])

   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..

 section {* lambda_skolem on cartesian products *}

 (* FIXME: rename do choice_cart *)

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

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

 proof-

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

   {assume H: "?rhs"

     then have ?lhs by auto}

   moreover

   {assume H: "?lhs"

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

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

     {fix i

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

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

     }

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

     hence ?rhs by metis }

   ultimately show ?thesis by metis

 qed

 subsection {* Standard bases are a spanning set, and obviously finite. *}

 lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"

 apply (rule set_ext)

 apply auto

 apply (subst basis_expansion'[symmetric])

 apply (rule span_setsum)

 apply simp

 apply auto

 apply (rule span_mul)

 apply (rule span_superset)

 apply (auto simp add: Collect_def mem_def)

 done

 lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")

 proof-

   have eq: "?S = cart_basis ` UNIV" by blast

   show ?thesis unfolding eq by auto

 qed

 lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")

 proof-

   have eq: "?S = cart_basis ` UNIV" by blast

   show ?thesis unfolding eq using card_image[OF basis_inj] by simp

 qed

 lemma independent_stdbasis_lemma:

   assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"

   and iS: "i \<notin> S"

   shows "(x$i) = 0"

 proof-

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

   let ?B = "cart_basis ` S"

   let ?P = "\<lambda>(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"

  {fix x::"real^_" assume xS: "x\<in> ?B"

    from xS have "?P x" by auto}

  moreover

  have "subspace ?P"

    by (auto simp add: subspace_def Collect_def mem_def)

  ultimately show ?thesis

    using x span_induct[of ?B ?P x] iS by blast

 qed

 lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"

 proof-

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

   let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"

   let ?B = "?b ` ?I"

   have eq: "{?b i|i. i \<in> ?I} = ?B"

     by auto

   {assume d: "dependent ?B"

     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"

       unfolding dependent_def by auto

     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp

     have eq2: "?B - {?b k} = ?b ` (?I - {k})"

       unfolding eq1

       apply (rule inj_on_image_set_diff[symmetric])

       apply (rule basis_inj) using k(1) by auto

     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .

     from independent_stdbasis_lemma[OF th0, of k, simplified]

     have False by simp}

   then show ?thesis unfolding eq dependent_def ..

 qed

 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"

   unfolding inner_simps smult_conv_scaleR by auto

 lemma linear_eq_stdbasis_cart:

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

   and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"

   shows "f = g"

 proof-

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

   let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"

   {fix x assume x: "x \<in> (UNIV :: (real^'m) set)"

     from equalityD2[OF span_stdbasis]

     have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast

     from linear_eq[OF lf lg IU] fg x

     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}

   then show ?thesis by (auto intro: ext)

 qed

 lemma bilinear_eq_stdbasis_cart:

   assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"

   and bg: "bilinear g"

   and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"

   shows "f = g"

 proof-

   from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in>  {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast

   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)

 qed

 lemma left_invertible_transpose:

   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"

   by (metis matrix_transpose_mul transpose_mat transpose_transpose)

 lemma right_invertible_transpose:

   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"

   by (metis matrix_transpose_mul transpose_mat transpose_transpose)

 lemma matrix_left_invertible_injective:

 "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"

 proof-

   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"

     from xy have "B*v (A *v x) = B *v (A*v y)" by simp

     hence "x = y"

       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}

   moreover

   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"

     hence i: "inj (op *v A)" unfolding inj_on_def by auto

     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]

     obtain g where g: "linear g" "g o op *v A = id" by blast

     have "matrix g ** A = mat 1"

       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]

       using g(2) by (simp add: o_def id_def stupid_ext)

     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}

   ultimately show ?thesis by blast

 qed

 lemma matrix_left_invertible_ker:

   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"

   unfolding matrix_left_invertible_injective

   using linear_injective_0[OF matrix_vector_mul_linear, of A]

   by (simp add: inj_on_def)

 lemma matrix_right_invertible_surjective:

 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"

 proof-

   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"

     {fix x :: "real ^ 'm"

       have "A *v (B *v x) = x"

         by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}

     hence "surj (op *v A)" unfolding surj_def by metis }

   moreover

   {assume sf: "surj (op *v A)"

     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]

     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"

       by blast

     have "A ** (matrix g) = mat 1"

       unfolding matrix_eq  matrix_vector_mul_lid

         matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]

       using g(2) unfolding o_def stupid_ext[symmetric] id_def

       .

     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast

   }

   ultimately show ?thesis unfolding surj_def by blast

 qed

 lemma matrix_left_invertible_independent_columns:

   fixes A :: "real^'n^'m"

   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"

    (is "?lhs \<longleftrightarrow> ?rhs")

 proof-

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

   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"

     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"

       and i: "i \<in> ?U"

       let ?x = "\<chi> i. c i"

       have th0:"A *v ?x = 0"

         using c

         unfolding matrix_mult_vsum Cart_eq

         by auto

       from k[rule_format, OF th0] i

       have "c i = 0" by (vector Cart_eq)}

     hence ?rhs by blast}

   moreover

   {assume H: ?rhs

     {fix x assume x: "A *v x = 0"

       let ?c = "\<lambda>i. ((x$i ):: real)"

       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]

       have "x = 0" by vector}}

   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast

 qed

 lemma matrix_right_invertible_independent_rows:

   fixes A :: "real^'n^'m"

   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"

   unfolding left_invertible_transpose[symmetric]

     matrix_left_invertible_independent_columns

   by (simp add: column_transpose)

 lemma matrix_right_invertible_span_columns:

   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")

 proof-

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

   have fU: "finite ?U" by simp

   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"

     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def

     apply (subst eq_commute) ..

   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast

   {assume h: ?lhs

     {fix x:: "real ^'n"

         from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"

           where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast

         have "x \<in> span (columns A)"

           unfolding y[symmetric]

           apply (rule span_setsum[OF fU])

           apply clarify

           unfolding smult_conv_scaleR

           apply (rule span_mul)

           apply (rule span_superset)

           unfolding columns_def

           by blast}

     then have ?rhs unfolding rhseq by blast}

   moreover

   {assume h:?rhs

     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"

     {fix y have "?P y"

       proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])

         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"

           by (rule exI[where x=0], simp)

       next

         fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"

         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"

           unfolding columns_def by blast

         from y2 obtain x:: "real ^'m" where

           x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast

         let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"

         show "?P (c*s y1 + y2)"

           proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong)

             fix j

             have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)

            else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)

               by (simp add: field_simps)

             have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)

            else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"

               apply (rule setsum_cong[OF refl])

               using th by blast

             also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"

               by (simp add: setsum_addf)

             also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"

               unfolding setsum_delta[OF fU]

               using i(1) by simp

             finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)

            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .

           qed

         next

           show "y \<in> span (columns A)" unfolding h by blast

         qed}

     then have ?lhs unfolding lhseq ..}

   ultimately show ?thesis by blast

 qed

 lemma matrix_left_invertible_span_rows:

   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"

   unfolding right_invertible_transpose[symmetric]

   unfolding columns_transpose[symmetric]

   unfolding matrix_right_invertible_span_columns

  ..

 text {* The same result in terms of square matrices. *}

 lemma matrix_left_right_inverse:

   fixes A A' :: "real ^'n^'n"

   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"

 proof-

   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"

     have sA: "surj (op *v A)"

       unfolding surj_def

       apply clarify

       apply (rule_tac x="(A' *v y)" in exI)

       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)

     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]

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

       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast

     have th: "matrix f' ** A = mat 1"

       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])

     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp

     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)

     hence "matrix f' ** A = A' ** A" by simp

     hence "A' ** A = mat 1" by (simp add: th)}

   then show ?thesis by blast

 qed

 text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}

 definition "rowvector v = (\<chi> i j. (v$j))"

 definition "columnvector v = (\<chi> i j. (v$i))"

 lemma transpose_columnvector:

  "transpose(columnvector v) = rowvector v"

   by (simp add: transpose_def rowvector_def columnvector_def Cart_eq)

 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"

   by (simp add: transpose_def columnvector_def rowvector_def Cart_eq)

 lemma dot_rowvector_columnvector:

   "columnvector (A *v v) = A ** columnvector v"

   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)

 lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"

   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def)

 lemma dot_matrix_vector_mul:

   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"

   shows "(A *v x) \<bullet> (B *v y) =

       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"

 unfolding dot_matrix_product transpose_columnvector[symmetric]

   dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..

 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"

   unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe

   apply(rule_tac x="\<pi> i" in exI) defer

   apply(rule_tac x="\<pi>' i" in exI) unfolding nth_conv_component using pi'_range by auto

 lemma infnorm_set_image_cart:

   "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =

   (\<lambda>i. abs(x$i)) ` (UNIV)" by blast

 lemma infnorm_set_lemma_cart:

   shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"

   and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"

   unfolding  infnorm_set_image_cart

   by (auto intro: finite_imageI)

 lemma component_le_infnorm_cart:

   shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"

   unfolding nth_conv_component

   using component_le_infnorm[of x] .

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

   unfolding dist_vector_def

   by (rule member_le_setL2) simp_all

 instance cart :: (perfect_space, finite) perfect_space

 proof

   fix x :: "'a ^ 'b"

   {

     fix e :: real assume "0 < e"

     def a \<equiv> "x $ undefined"

     have "a islimpt UNIV" by (rule islimpt_UNIV)

     with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"

       unfolding islimpt_approachable by auto

     def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"

     from `b \<noteq> a` have "y \<noteq> x"

       unfolding a_def y_def by (simp add: Cart_eq)

     from `dist b a < e` have "dist y x < e"

       unfolding dist_vector_def a_def y_def

       apply simp

       apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])

       apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)

       done

     from `y \<noteq> x` and `dist y x < e`

     have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto

   }

   then show "x islimpt UNIV" unfolding islimpt_approachable by blast

 qed

 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"

 proof-

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

   let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"

   {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"

     and xi: "x$i < 0"

     from xi have th0: "-x$i > 0" by arith

     from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast

       have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith

       have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith

       have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi

         apply (simp only: vector_component)

         by (rule th') auto

       have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm_cart[of "x'-x" i]

         apply (simp add: dist_norm) by norm

       from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }

   then show ?thesis unfolding closed_limpt islimpt_approachable

     unfolding not_le[symmetric] by blast

 qed

 lemma Lim_component_cart:

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

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

   unfolding tendsto_iff

   apply (clarify)

   apply (drule spec, drule (1) mp)

   apply (erule eventually_elim1)

   apply (erule le_less_trans [OF dist_nth_le_cart])

   done

 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"

 unfolding bounded_def

 apply clarify

 apply (rule_tac x="x $ i" in exI)

 apply (rule_tac x="e" in exI)

 apply clarify

 apply (rule order_trans [OF dist_nth_le_cart], simp)

 done

 lemma compact_lemma_cart:

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

   assumes "bounded s" and "\<forall>n. f n \<in> s"

   shows "\<forall>d.

         \<exists>l r. subseq r \<and>

         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"

 proof

   fix d::"'n set" have "finite d" by simp

   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>

       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"

   proof(induct d) case empty thus ?case unfolding subseq_def by auto

   next case (insert k d)

     have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component_cart)

     obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"

       using insert(3) by auto

     have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp

     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"

       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

       using r1 and r2 unfolding r_def o_def subseq_def by auto

     moreover

     def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"

     { fix e::real assume "e>0"

       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast

       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)

       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"

         by (rule eventually_subseq)

       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"

         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)

     }

     ultimately show ?case by auto

   qed

 qed

 instance cart :: (heine_borel, finite) heine_borel

 proof

   fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"

   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

   then obtain l r where r: "subseq r"

     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"

     using compact_lemma_cart [OF s f] by blast

   let ?d = "UNIV::'b set"

   { fix e::real assume "e>0"

     hence "0 < e / (real_of_nat (card ?d))"

       using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto

     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"

       by simp

     moreover

     { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"

       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"

         unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)

       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"

         by (rule setsum_strict_mono) (simp_all add: n)

       finally have "dist (f (r n)) l < e" by simp

     }

     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

       by (rule eventually_elim1)

   }

   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

 qed

 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"

 unfolding continuous_at by (intro tendsto_intros)

 lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"

 unfolding continuous_on_def by (intro ballI tendsto_intros)

 lemma interval_cart: fixes a :: "'a::ord^'n" shows

   "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and

   "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"

   by (auto simp add: expand_set_eq vector_less_def vector_le_def)

 lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows

   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"

   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"

   using interval_cart[of a b] by(auto simp add: expand_set_eq vector_less_def vector_le_def)

 lemma interval_eq_empty_cart: fixes a :: "real^'n" shows

  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and

  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)

 proof-

   { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"

     hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto

     hence "a$i < b$i" by auto

     hence False using as by auto  }

   moreover

   { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"

     let ?x = "(1/2) *\<^sub>R (a + b)"

     { fix i

       have "a$i < b$i" using as[THEN spec[where x=i]] by auto

       hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"

         unfolding vector_smult_component and vector_add_component

         by auto  }

     hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto  }

   ultimately show ?th1 by blast

   { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"

     hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto

     hence "a$i \<le> b$i" by auto

     hence False using as by auto  }

   moreover

   { assume as:"\<forall>i. \<not> (b$i < a$i)"

     let ?x = "(1/2) *\<^sub>R (a + b)"

     { fix i

       have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto

       hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"

         unfolding vector_smult_component and vector_add_component

         by auto  }

     hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }

   ultimately show ?th2 by blast

 qed

 lemma interval_ne_empty_cart: fixes a :: "real^'n" shows

   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and

   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"

   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)

     (* BH: Why doesn't just "auto" work here? *)

 lemma subset_interval_imp_cart: fixes a :: "real^'n" shows

  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart

   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)

 lemma interval_sing: fixes a :: "'a::linorder^'n" shows

  "{a .. a} = {a} \<and> {a<..<a} = {}"

 apply(auto simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)

 apply (simp add: order_eq_iff)

 apply (auto simp add: not_less less_imp_le)

 done

 lemma interval_open_subset_closed_cart:  fixes a :: "'a::preorder^'n" shows

  "{a<..<b} \<subseteq> {a .. b}"

 proof(simp add: subset_eq, rule)

   fix x

   assume x:"x \<in>{a<..<b}"

   { fix i

     have "a $ i \<le> x $ i"

       using x order_less_imp_le[of "a$i" "x$i"]

       by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)

   }

   moreover

   { fix i

     have "x $ i \<le> b $ i"

       using x order_less_imp_le[of "x$i" "b$i"]

       by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)

   }

   ultimately

   show "a \<le> x \<and> x \<le> b"

     by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)

 qed

 lemma subset_interval_cart: fixes a :: "real^'n" shows

  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and

  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and

  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and

  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)

   using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)

 lemma disjoint_interval_cart: fixes a::"real^'n" shows

   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and

   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) and

   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) and

   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)

   using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)

 lemma inter_interval_cart: fixes a :: "'a::linorder^'n" shows

  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"

   unfolding expand_set_eq and Int_iff and mem_interval_cart

   by auto

 lemma closed_interval_left_cart: fixes b::"real^'n"

   shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"

 proof-

   { fix i

     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"

     { assume "x$i > b$i"

       then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto

       hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }

     hence "x$i \<le> b$i" by(rule ccontr)auto  }

   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

 qed

 lemma closed_interval_right_cart: fixes a::"real^'n"

   shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"

 proof-

   { fix i

     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"

     { assume "a$i > x$i"

       then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto

       hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }

     hence "a$i \<le> x$i" by(rule ccontr)auto  }

   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

 qed

 lemma is_interval_cart:"is_interval (s::(real^'n) set) \<longleftrightarrow>

   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"

   unfolding is_interval_def Ball_def by (simp add: cart_simps real_euclidean_nth)

 lemma closed_halfspace_component_le_cart:

   shows "closed {x::real^'n. x$i \<le> a}"

   using closed_halfspace_le[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto

 lemma closed_halfspace_component_ge_cart:

   shows "closed {x::real^'n. x$i \<ge> a}"

   using closed_halfspace_ge[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto

 lemma open_halfspace_component_lt_cart:

   shows "open {x::real^'n. x$i < a}"

   using open_halfspace_lt[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto

 lemma open_halfspace_component_gt_cart:

   shows "open {x::real^'n. x$i  > a}"

   using open_halfspace_gt[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto

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

   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"

   shows "l$i \<le> b"

 proof-

   { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this

   show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l] unfolding *

     using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto

 qed

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

   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"

   shows "b \<le> l$i"

 proof-

   { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this

   show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l] unfolding *

     using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto

 qed

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

   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"

   shows "l$i = b"

   using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge_cart[OF net, of b i] and

     Lim_component_le_cart[OF net, of i b] by auto

 lemma connected_ivt_component_cart: fixes x::"real^'n" shows

  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"

   using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] by (auto simp add: inner_basis)

 lemma subspace_substandard_cart:

  "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"

   unfolding subspace_def by auto

 lemma closed_substandard_cart:

  "closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")

 proof-

   let ?D = "{i. P i}"

   let ?Bs = "{{x::real^'n. inner (cart_basis i) x = 0}| i. i \<in> ?D}"

   { fix x

     { assume "x\<in>?A"

       hence x:"\<forall>i\<in>?D. x $ i = 0" by auto

       hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }

     moreover

     { assume x:"x\<in>\<Inter>?Bs"

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

         then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (cart_basis i) x = 0}" by auto

         hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto  }

       hence "x\<in>?A" by auto }

     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }

   hence "?A = \<Inter> ?Bs" by auto

   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)

 qed

 lemma dim_substandard_cart:

   shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")

 proof- have *:"{x. \<forall>i<DIM((real, 'n) cart). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} = 

     {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe

     apply(erule_tac x="\<pi>' i" in allE) defer

     apply(erule_tac x="\<pi> i" in allE)

     unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])

   have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) cart)}" using pi'_range[where 'n='n] by auto

   thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 

     unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto

 qed

 lemma affinity_inverses:

   assumes m0: "m \<noteq> (0::'a::field)"

   shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"

   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"

   using m0

 apply (auto simp add: expand_fun_eq vector_add_ldistrib)

 by (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])

 lemma vector_affinity_eq:

   assumes m0: "(m::'a::field) \<noteq> 0"

   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"

 proof

   assume h: "m *s x + c = y"

   hence "m *s x = y - c" by (simp add: field_simps)

   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp

   then show "x = inverse m *s y + - (inverse m *s c)"

     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)

 next

   assume h: "x = inverse m *s y + - (inverse m *s c)"

   show "m *s x + c = y" unfolding h diff_minus[symmetric]

     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)

 qed

 lemma vector_eq_affinity:

  "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"

   using vector_affinity_eq[where m=m and x=x and y=y and c=c]

   by metis

 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"

   apply(subst euclidean_eq)

 proof safe case goal1

   hence *:"(basis i::real^'n) = cart_basis (\<pi> i)"

     unfolding basis_real_n[THEN sym] by auto

   have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *

     unfolding dot_basis by auto

   thus ?case using goal1 by auto

 qed

 section "Convex Euclidean Space"

 lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"

   apply(subst euclidean_eq)

 proof safe case goal1 thus ?case using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto

 qed

 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]

 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]

 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component

 lemma convex_box_cart:

   assumes "\<And>i. convex {x. P i x}"

   shows "convex {x. \<forall>i. P i (x$i)}"

   using assms unfolding convex_def by auto

 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"

   by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)

 lemma unit_interval_convex_hull_cart:

   "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")

   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]

   apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_ext) unfolding mem_Collect_eq

   apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer

   apply(erule_tac x="\<pi> i" in allE) by auto

 lemma cube_convex_hull_cart:

   assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" 

 proof- from cube_convex_hull[OF assms, where 'a="real^'n" and x=x] guess s . note s=this

   show thesis apply(rule that[OF s(1)]) unfolding s(2)[THEN sym] const_vector_cart ..

 qed

 lemma std_simplex_cart:

   "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =

   (insert 0 { basis i | i. i<DIM((real,'n) cart)})"

   apply(rule arg_cong[where f="\<lambda>s. (insert 0 s)"])

   unfolding basis_real_n[THEN sym] apply safe

   apply(rule_tac x="\<pi>' i" in exI) defer

   apply(rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n] by auto

 subsection "Brouwer Fixpoint"

 lemma kuhn_labelling_lemma_cart:

   assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"

   shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>

              (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>

              (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>

              (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>

              (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-

   have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto

   have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto

   show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1

     let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>

         (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"

     { assume "P x" "Q xa" hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1" using assms(2)[rule_format,of "f x" xa]

         apply(drule_tac assms(1)[rule_format]) by auto }

     hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 

 lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).

     (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"

   unfolding interval_bij_def apply(rule ext)+ apply safe

   unfolding Cart_eq Cart_lambda_beta unfolding nth_conv_component

   apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto

 lemma interval_bij_affine_cart:

  "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +

             (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"

   apply rule unfolding Cart_eq interval_bij_cart vector_component_simps

   by(auto simp add: field_simps add_divide_distrib[THEN sym]) 

 subsection "Derivative"

 lemma has_derivative_vmul_component_cart: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"

   assumes "(c has_derivative c') net"

   shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" 

   using has_derivative_vmul_component[OF assms] 

   unfolding nth_conv_component .

 lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"

   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)

 definition "jacobian f net = matrix(frechet_derivative f net)"

 lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"

   apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer

   apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption

 subsection {* Component of the differential must be zero if it exists at a local        *)

 (* maximum or minimum for that corresponding component. *}

 lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"

   assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"

   "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"

 (* FIXME: reuse proof of generic differential_zero_maxmin_component*)

 proof(rule ccontr)

   def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"

   then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto

   hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto

   note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]

   guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this

   guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this

   { fix c assume "abs c \<le> d" 

     hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" using norm_basis[of j] d by auto

     have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" 

       by(rule component_le_norm_cart)

     also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto

     finally have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp

     hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"

       unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] 

       unfolding inner_simps dot_basis smult_conv_scaleR by simp  } note * = this

   have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"

     unfolding mem_ball dist_norm using norm_basis[of j] d by auto

   hence **:"((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>

          ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)" using assms(2) by auto

   have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith

   show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"]) 

     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left

     unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)

 qed

 subsection {* Lemmas for working on real^1 *}

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

   by (metis num1_eq_iff)

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

   by auto (metis num1_eq_iff)

 lemma exhaust_2:

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

 proof (induct x)

   case (of_int z)

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

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

   then show ?case by auto

 qed

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

   by (metis exhaust_2)

 lemma exhaust_3:

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

 proof (induct x)

   case (of_int z)

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

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

   then show ?case by auto

 qed

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

   by (metis exhaust_3)

 lemma UNIV_1 [simp]: "UNIV = {1::1}"

   by (auto simp add: num1_eq_iff)

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

   using exhaust_2 by auto

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

   using exhaust_3 by auto

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

   unfolding UNIV_1 by simp

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

   unfolding UNIV_2 by simp

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

   unfolding UNIV_3 by (simp add: add_ac)

 instantiation num1 :: cart_one begin

 instance proof

   show "CARD(1) = Suc 0" by auto

 qed end

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

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

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

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

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

   by (simp_all add:  Cart_eq)

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

   by (simp_all add:  Cart_eq)

 declare vec1_dest_vec1(1) [simp]

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

   by (metis vec1_dest_vec1(1))

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

   by (metis vec1_dest_vec1(1))

 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y"

   by (metis vec1_dest_vec1(2))

 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"

   by (metis vec1_dest_vec1(1))

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

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

   by (simp add: Cart_eq)

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

   apply auto

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

   apply (simp only: vector_one[symmetric])

   done

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

   by (simp add: norm_vector_def)

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

   by (simp add: norm_vector_1)

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

   by (auto simp add: norm_real dist_norm)

 subsection{* Explicit vector construction from lists. *}

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

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

 lemma from_nat [simp]: "from_nat = of_nat"

 by (rule ext, induct_tac x, simp_all)

 primrec

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

 where

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

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

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

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

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

   unfolding vector_def by simp

 lemma vector_2:

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

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

   unfolding vector_def by simp_all

 lemma vector_3:

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

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

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

   unfolding vector_def by simp_all

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

   apply auto

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

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

   apply simp

   apply (vector vector_def)

   apply simp

   done

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

   apply auto

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

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

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

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_2)

   done

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

   apply auto

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

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

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

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

   apply simp

   apply (vector vector_def)

   apply (simp add: forall_3)

   done

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

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

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

   by (simp)

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

   by (simp)

 lemma dest_vec1_sum: assumes fS: "finite S"

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

   apply (induct rule: finite_induct[OF fS])

   apply simp

   apply auto

   done

 lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)"

   by (simp add: vec_def norm_real)

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

   by (simp only: dist_real vec1_component)

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

   by (metis vec1_dest_vec1(1) norm_vec1)

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

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

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

   unfolding bounded_linear_def additive_def bounded_linear_axioms_def 

   unfolding smult_conv_scaleR[THEN sym] unfolding vec1_dest_vec1_simps

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

 lemma linear_vmul_dest_vec1:

   fixes f:: "real^_ \<Rightarrow> real^1"

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

   unfolding smult_conv_scaleR

   by (rule linear_vmul_component)

 lemma linear_from_scalars:

   assumes lf: "linear (f::real^1 \<Rightarrow> real^_)"

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

   unfolding smult_conv_scaleR

   apply (rule ext)

   apply (subst matrix_works[OF lf, symmetric])

   apply (auto simp add: Cart_eq matrix_vector_mult_def column_def mult_commute)

   done

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

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

   apply (rule ext)

   apply (subst matrix_works[OF lf, symmetric])

   apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute)

   done

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

   by (simp add: dest_vec1_eq[symmetric])

 lemma setsum_scalars: assumes fS: "finite S"

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

   unfolding vec_setsum[OF fS] by simp

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

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

   apply simp

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

   apply (auto)

   done

 text{* Lifting and dropping *}

 lemma continuous_on_o_dest_vec1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"

   assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"

   using assms unfolding continuous_on_iff apply safe

   apply(erule_tac x="x$1" in ballE,erule_tac x=e in allE) apply safe

   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 

   apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:vector_le_def)

 lemma continuous_on_o_vec1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"

   assumes "continuous_on {a..b} f" shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"

   using assms unfolding continuous_on_iff apply safe

   apply(erule_tac x="vec x" in ballE,erule_tac x=e in allE) apply safe

   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 

   apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:vector_le_def)

 lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"

   by(rule linear_continuous_on[OF bounded_linear_vec1])

 lemma mem_interval_1: fixes x :: "real^1" shows

  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"

  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"

 by(simp_all add: Cart_eq vector_less_def vector_le_def)

 lemma vec1_interval:fixes a::"real" shows

   "vec1 ` {a .. b} = {vec1 a .. vec1 b}"

   "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"

   apply(rule_tac[!] set_ext) unfolding image_iff vector_less_def unfolding mem_interval_cart

   unfolding forall_1 unfolding vec1_dest_vec1_simps

   apply rule defer apply(rule_tac x="dest_vec1 x" in bexI) prefer 3 apply rule defer

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

 (* Some special cases for intervals in R^1.                                  *)

 lemma interval_cases_1: fixes x :: "real^1" shows

  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"

   unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)

 lemma in_interval_1: fixes x :: "real^1" shows

  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>

   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"

   unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)

 lemma interval_eq_empty_1: fixes a :: "real^1" shows

   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"

   "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"

   unfolding interval_eq_empty_cart and ex_1 by auto

 lemma subset_interval_1: fixes a :: "real^1" shows

  "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>

                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"

  "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>

                 dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"

  "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>

                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"

  "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>

                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"

   unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto

 lemma eq_interval_1: fixes a :: "real^1" shows

  "{a .. b} = {c .. d} \<longleftrightarrow>

           dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>

           dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"

 unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]

 unfolding subset_interval_1(1)[of a b c d]

 unfolding subset_interval_1(1)[of c d a b]

 by auto

 lemma disjoint_interval_1: fixes a :: "real^1" shows

   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"

   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"

   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"

   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"

   unfolding disjoint_interval_cart and ex_1 by auto

 lemma open_closed_interval_1: fixes a :: "real^1" shows

  "{a<..<b} = {a .. b} - {a, b}"

   unfolding expand_set_eq apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)

 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"

   unfolding expand_set_eq apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)

 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows

   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"

   using Lim_component_le_cart[of f l net 1 b] by auto

 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows

  "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"

   using Lim_component_ge_cart[of f l net b 1] by auto

 text{* Also more convenient formulations of monotone convergence.                *}

 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"

   assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"

   shows "\<exists>l. (s ---> l) sequentially"

 proof-

   obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto

   { fix m::nat

     have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"

       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq)  }

   hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto

   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto

   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)

     unfolding dist_norm unfolding abs_dest_vec1  by auto

 qed

 lemma dest_vec1_simps[simp]: fixes a::"real^1"

   shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)

   "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"

   by(auto simp add: vector_le_def Cart_eq)

 lemma dest_vec1_inverval:

   "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"

   "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"

   "dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"

   "dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"

   apply(rule_tac [!] equalityI)

   unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff

   apply(rule_tac [!] allI)apply(rule_tac [!] impI)

   apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)

   apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)

   by (auto simp add: vector_less_def vector_le_def)

 lemma dest_vec1_setsum: assumes "finite S"

   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"

   using dest_vec1_sum[OF assms] by auto

 lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"

 unfolding open_vector_def forall_1 by auto

 lemma tendsto_dest_vec1 [tendsto_intros]:

   "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"

 by(rule tendsto_Cart_nth)

 lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"

   unfolding continuous_def by (rule tendsto_dest_vec1)

 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))" 

   apply safe defer apply(erule_tac x="vec1 x" in allE) by auto

 lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"

   apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto 

 lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"

   apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule 

   apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto

 lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding dist_norm by auto

 lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"

   shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-

   { assume ?l guess K using linear_bounded[OF `?l`] ..

     hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)

       unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }

   thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def

     unfolding vec1_dest_vec1_simps by auto qed

 lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"

   unfolding vector_le_def by auto

 lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"

   unfolding vector_less_def by auto

 subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}

 lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows

   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)

   = (f has_derivative f') (at x within s)"

   unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]

   unfolding o_def Lim_within Ball_def unfolding forall_vec1 

   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto  

 lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows

   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"

   using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto

 lemma bounded_linear_vec1': fixes f::"'a::real_normed_vector\<Rightarrow>real"

   shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"

   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def

   unfolding vec1_dest_vec1_simps by auto

 lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"

   shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"

   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def

   unfolding vec1_dest_vec1_simps by auto

 lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows

   "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"

   unfolding has_derivative_at unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]

   unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto

 lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows

   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"

   unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def

   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto

 lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows

   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"

   using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)

 subsection {* In particular if we have a mapping into @{typ "real^1"}. *}

 lemma onorm_vec1: fixes f::"real \<Rightarrow> real"

   shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-

   have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)

   hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto

   have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto

   have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto

   have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto

   show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max) qed

 lemma convex_vec1:"convex (vec1 ` s) = convex (s::real set)"

   unfolding convex_def Ball_def forall_vec1 unfolding vec1_dest_vec1_simps image_iff by auto

 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"

   apply(rule bounded_linearI[where K=1]) 

   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto

 lemma bounded_vec1[intro]:  "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"

   unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI)

   by(auto simp add: dist_real dist_real_def)

 (*lemma content_closed_interval_cases_cart:

   "content {a..b::real^'n} =

   (if {a..b} = {} then 0 else setprod (\<lambda>i. b$i - a$i) UNIV)" 

 proof(cases "{a..b} = {}")

   case True thus ?thesis unfolding content_def by auto

 next case Falsethus ?thesis unfolding content_def unfolding if_not_P[OF False]

   proof(cases "\<forall>i. a $ i \<le> b $ i")

     case False thus ?thesis unfolding content_def using t by auto

   next case True note interval_eq_empty

    apply auto 

   sorry*)

 lemma integral_component_eq_cart[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real^'m"

   assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"

   using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .

 lemma interval_split_cart:

   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"

   "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"

   apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval_cart mem_Collect_eq

   unfolding Cart_lambda_beta by auto

 (*lemma content_split_cart:

   "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"

 proof- note simps = interval_split_cart content_closed_interval_cases_cart Cart_lambda_beta vector_le_def

   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }

   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto

   have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"

     "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 

     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto

   assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c

     \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"

     by  (auto simp add:field_simps)

   moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"

     unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto

   ultimately show ?thesis 

     unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)

 qed*)

 lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"

   shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"

 proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"

     unfolding vec_sub Cart_eq by(auto simp add: split_beta)

   show ?thesis using assms unfolding has_integral apply safe

     apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)

     apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed

 end