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

theory Cartesian_Euclidean_Space

imports Finite_Cartesian_Product Integration

begin

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}"

    then show "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 vec :: (times, finite) times

begin

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

instance ..

end

instantiation vec :: (one, finite) one

begin

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

instance ..

end

instantiation vec :: (ord, finite) ord

begin

definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"

definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"

instance ..

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 vec :: (linorder, cart_one) linorder

begin

instance

proof

  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"

  proof -

    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)

    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)

    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto

    then show thesis by (auto intro: that)

  qed

  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps

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

  show "x \<le> x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x \<le> y \<or> y \<le> x" by auto

  { assume "x\<le>y" "y\<le>z" then show "x\<le>z" by auto }

  { assume "x\<le>y" "y\<le>x" then show "x=y" by auto }

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 =

    simpset_of (put_simpset HOL_basic_ss @{context}

      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_of (@{context} addsimps

             [@{thm plus_vec_def}, @{thm times_vec_def},

              @{thm minus_vec_def}, @{thm uminus_vec_def},

              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},

              @{thm scaleR_vec_def},

              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])

  fun vector_arith_tac ctxt ths =

    simp_tac (put_simpset ss1 ctxt)

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

         ORELSE rtac @{thm setsum_0'} i

         ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)

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

    THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)

in

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

end

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

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

lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_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 "finite S"

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

  using assms

proof induct

  case empty

  then show ?case by simp

next

  case insert

  then show ?case by (auto simp add: vec_add)

qed

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 vec :: (semigroup_mult, finite) semigroup_mult

  by default (vector mult_assoc)

instance vec :: (monoid_mult, finite) monoid_mult

  by default vector+

instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult

  by default (vector mult_commute)

instance vec :: (comm_monoid_mult, finite) comm_monoid_mult

  by default vector

instance vec :: (semiring, finite) semiring

  by default (vector field_simps)+

instance vec :: (semiring_0, finite) semiring_0

  by default (vector field_simps)+

instance vec :: (semiring_1, finite) semiring_1

  by default vector

instance vec :: (comm_semiring, finite) comm_semiring

  by default (vector field_simps)+

instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..

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

instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..

instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..

instance vec :: (ring, finite) ring ..

instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..

instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..

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

instance vec :: (real_algebra, finite) real_algebra

  by default (simp_all add: vec_eq_iff)

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

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

proof (induct n)

  case 0

  then show ?case by vector

next

  case Suc

  then show ?case by vector

qed

lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1"

  by vector

instance vec :: (semiring_char_0, finite) semiring_char_0

proof

  fix m n :: nat

  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"

    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)

qed

instance vec :: (numeral, finite) numeral ..

instance vec :: (semiring_numeral, finite) semiring_numeral ..

lemma numeral_index [simp]: "numeral w $ i = numeral w"

  by (induct w) (simp_all only: numeral.simps vector_add_component one_index)

lemma neg_numeral_index [simp]: "neg_numeral w $ i = neg_numeral w"

  by (simp only: neg_numeral_def vector_uminus_component numeral_index)

instance vec :: (comm_ring_1, finite) comm_ring_1 ..

instance vec :: (ring_char_0, finite) ring_char_0 ..

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: vec_eq_iff)

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_vec_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 le_less_trans)

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

  by (simp add: norm_vec_def setL2_le_setsum)

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

  unfolding scaleR_vec_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"

proof (cases "finite S")

  case True

  then show ?thesis by induct simp_all

next

  case False

  then show ?thesis by simp

qed

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

  by (simp add: vec_eq_iff)

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: vec_eq_iff setsum_right_distrib)

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"

  using setsum_norm_allsubsets_bound[OF assms]

  by (simp add: DIM_cart Basis_real_def)

instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space

proof

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

  show "(x \<le> y) = (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"

    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: Basis_vec_def inner_axis)

  show"(x < y) = (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"

    unfolding less_vec_def apply(subst eucl_less) by (simp add: Basis_vec_def inner_axis)

qed

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

  apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]

    mult_1_left mult_zero_left if_True UNIV_I)

  done

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

  apply (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)

  done

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)

  apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)

  done

lemma matrix_transpose_mul:

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

  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff 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 vec_eq_iff)

  apply clarify

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

  apply (erule_tac x="axis ia 1" in allE)

  apply (erule_tac x="i" in allE)

  apply (auto simp add: if_distrib cond_application_beta axis_def

    setsum_delta[OF finite] cong del: if_weak_cong)

  done

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

  by (simp add: matrix_vector_mult_def inner_vec_def)

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

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

  apply (subst setsum_commute)

  apply simp

  done

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 vec_eq_iff)

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 vec_eq_iff)

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

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

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_vec_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 vec_eq_iff column_def mult_commute)

lemma vector_componentwise:

  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"

  by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)

lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"

  by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)

lemma linear_componentwise:

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

  assumes lf: "linear f"

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

proof -

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

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

  have fM: "finite ?M" by simp

  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"

    unfolding setsum_component by simp

  then show ?thesis

    unfolding linear_setsum_mul[OF lf fM, symmetric]

    unfolding scalar_mult_eq_scaleR[symmetric]

    unfolding basis_expansion

    by simp

qed

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

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(axis j 1))$i)"

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

  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff

      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 vec_eq_iff 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 vec_eq_iff 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_vec_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

  apply rule

  done

subsection {* lambda skolemization 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

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

  unfolding inner_simps scalar_mult_eq_scaleR by auto

lemma left_invertible_transpose:

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

  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: fun_eq_iff)

    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 fun_eq_iff 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 vec_eq_iff

        by auto

      from k[rule_format, OF th0] i

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

    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)

    apply rule

    done

  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 scalar_mult_eq_scaleR

        apply (rule span_mul)

        apply (rule span_superset)

        unfolding columns_def

        apply blast

        done

    }

    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 scalar_mult_eq_scaleR])

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

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

      next

        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 distrib_left 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 apply blast

            done

          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)

      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)

      done

    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 vec_eq_iff)

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

  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)

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_vec_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}"

  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)

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

  using Basis_le_infnorm[of "axis i 1" x]

  by (simp add: Basis_vec_def axis_eq_axis inner_axis)

lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"

  unfolding continuous_def by (rule tendsto_vec_nth)

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

  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)

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

  by (simp add: Collect_all_eq closed_INT closed_Collect_le)

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_vec_nth_le], simp)

  done

lemma compact_lemma_cart:

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

  assumes f: "bounded (range f)"

  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)

    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 s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)`

      by (auto intro!: bounded_component_cart)

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

    have "bounded (range (\<lambda>i. f (r1 i) $ k))"

      by (metis (lifting) bounded_subset image_subsetI f' s')

    then obtain l2 r2 where r2: "subseq r2"

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

      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)

    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 vec :: (heine_borel, finite) heine_borel

proof

  fix f :: "nat \<Rightarrow> 'a ^ 'b"

  assume f: "bounded (range f)"

  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 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 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_vec_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 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: set_eq_iff less_vec_def less_eq_vec_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)"

    and "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: set_eq_iff less_vec_def less_eq_vec_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: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)

  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: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)

  }

  moreover

  { fix i

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

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

      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)

  }

  ultimately

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

    by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)