(* Title: HOL/Analysis/Cartesian_Euclidean_Space.thy
Some material by Jose Divasón, Tim Makarios and L C Paulson
*)
section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
theory Cartesian_Euclidean_Space
imports Finite_Cartesian_Product Derivative
begin
lemma norm_le_componentwise:
"(\<And>b. b \<in> Basis \<Longrightarrow> abs(x \<bullet> b) \<le> abs(y \<bullet> b)) \<Longrightarrow> norm x \<le> norm y"
by (auto simp: norm_le euclidean_inner [of x x] euclidean_inner [of y y] abs_le_square_iff power2_eq_square intro!: sum_mono)
lemma norm_le_componentwise_cart:
fixes x :: "real^'n"
shows "(\<And>i. abs(x$i) \<le> abs(y$i)) \<Longrightarrow> norm x \<le> norm y"
unfolding cart_eq_inner_axis
by (rule norm_le_componentwise) (metis axis_index)
lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
by (simp add: subspace_def)
lemma sum_mult_product:
"sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
proof (rule sum.cong, simp, rule sum.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\<open>Basic componentwise operations on vectors\<close>
instantiation vec :: (times, finite) times
begin
definition "( * ) \<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::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
instance ..
end
text\<open>The ordering on one-dimensional vectors is linear.\<close>
class cart_one =
assumes UNIV_one: "card (UNIV :: '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
instance vec:: (order, finite) order
by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
intro: order.trans order.antisym order.strict_implies_order)
instance vec :: (linorder, cart_one) linorder
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
fix x y :: "'a^'b::cart_one"
note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
show "x \<le> y \<or> y \<le> x" by auto
qed
text\<open>Constant Vectors\<close>
definition "vec x = (\<chi> i. x)"
lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
text\<open>Also the scalar-vector multiplication.\<close>
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 \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
lemma sum_cong_aux:
"(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
by (auto intro: sum.cong)
hide_fact (open) sum_cong_aux
method_setup vector = \<open>
let
val ss1 =
simpset_of (put_simpset HOL_basic_ss @{context}
addsimps [@{thm sum.distrib} RS sym,
@{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
@{thm sum_distrib_right}, @{thm sum_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 => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i
ORELSE resolve_tac ctxt @{thms sum.neutral} 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
\<close> "lift trivial vector statements to real arith statements"
lemma vec_0[simp]: "vec 0 = 0" by vector
lemma vec_1[simp]: "vec 1 = 1" by vector
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
lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
lemma vec_neg: "vec(- x) = - vec x " by vector
lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
by vector
lemma vec_sum:
assumes "finite S"
shows "vec(sum f S) = sum (vec \<circ> 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\<open>Obvious "component-pushing".\<close>
lemma vec_component [simp]: "vec x $ i = x"
by vector
lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
by vector
lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
by vector
lemma cond_component: "(if b then x else y)$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 \<open>Some frequently useful arithmetic lemmas over vectors\<close>
instance vec :: (semigroup_mult, finite) semigroup_mult
by standard (vector mult.assoc)
instance vec :: (monoid_mult, finite) monoid_mult
by standard vector+
instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
by standard (vector mult.commute)
instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
by standard vector
instance vec :: (semiring, finite) semiring
by standard (vector field_simps)+
instance vec :: (semiring_0, finite) semiring_0
by standard (vector field_simps)+
instance vec :: (semiring_1, finite) semiring_1
by standard vector
instance vec :: (comm_semiring, finite) comm_semiring
by standard (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 standard (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
lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ '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]: "- numeral w $ i = - numeral w"
by (simp only: 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 linear_vec [simp]: "linear vec"
by (simp add: linearI vec_add vec_eq_iff)
lemma differentiable_vec:
fixes S :: "'a::euclidean_space set"
shows "vec differentiable_on S"
by (simp add: linear_linear bounded_linear_imp_differentiable_on)
lemma continuous_vec [continuous_intros]:
fixes x :: "'a::euclidean_space"
shows "isCont vec x"
apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
lemma box_vec_eq_empty [simp]:
shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
"box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
by (simp add: inner_axis' norm_eq_1)
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 component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
apply (simp add: norm_vec_def)
apply (rule member_le_L2_set, simp_all)
done
lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
by (metis component_le_norm_cart order_trans)
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
by (metis component_le_norm_cart le_less_trans)
lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
by (simp add: norm_vec_def L2_set_le_sum)
lemma scalar_mult_eq_scaleR [simp]: "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 sum_component [simp]:
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
shows "(sum f S)$i = sum (\<lambda>x. (f x)$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 sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
by (simp add: vec_eq_iff)
lemma sum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
by (simp add: vec_eq_iff sum_distrib_left)
lemma sum_norm_allsubsets_bound_cart:
fixes f:: "'a \<Rightarrow> real ^'n"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e"
using sum_norm_allsubsets_bound[OF assms]
by simp
subsection\<open>Closures and interiors of halfspaces\<close>
lemma interior_halfspace_le [simp]:
assumes "a \<noteq> 0"
shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
proof -
have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x
proof -
obtain e where "e>0" and e: "cball x e \<subseteq> S"
using \<open>open S\<close> open_contains_cball x by blast
then have "x + (e / norm a) *\<^sub>R a \<in> cball x e"
by (simp add: dist_norm)
then have "x + (e / norm a) *\<^sub>R a \<in> S"
using e by blast
then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}"
using S by blast
moreover have "e * (a \<bullet> a) / norm a > 0"
by (simp add: \<open>0 < e\<close> assms)
ultimately show ?thesis
by (simp add: algebra_simps)
qed
show ?thesis
by (rule interior_unique) (auto simp: open_halfspace_lt *)
qed
lemma interior_halfspace_ge [simp]:
"a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
using interior_halfspace_le [of "-a" "-b"] by simp
lemma interior_halfspace_component_le [simp]:
"interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
and interior_halfspace_component_ge [simp]:
"interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
proof -
have "axis k (1::real) \<noteq> 0"
by (simp add: axis_def vec_eq_iff)
moreover have "axis k (1::real) \<bullet> x = x$k" for x
by (simp add: cart_eq_inner_axis inner_commute)
ultimately show ?LE ?GE
using interior_halfspace_le [of "axis k (1::real)" a]
interior_halfspace_ge [of "axis k (1::real)" a] by auto
qed
lemma closure_halfspace_lt [simp]:
assumes "a \<noteq> 0"
shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
proof -
have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}"
by (force simp:)
then show ?thesis
using interior_halfspace_ge [of a b] assms
by (force simp: closure_interior)
qed
lemma closure_halfspace_gt [simp]:
"a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
using closure_halfspace_lt [of "-a" "-b"] by simp
lemma closure_halfspace_component_lt [simp]:
"closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
and closure_halfspace_component_gt [simp]:
"closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
proof -
have "axis k (1::real) \<noteq> 0"
by (simp add: axis_def vec_eq_iff)
moreover have "axis k (1::real) \<bullet> x = x$k" for x
by (simp add: cart_eq_inner_axis inner_commute)
ultimately show ?LE ?GE
using closure_halfspace_lt [of "axis k (1::real)" a]
closure_halfspace_gt [of "axis k (1::real)" a] by auto
qed
lemma interior_hyperplane [simp]:
assumes "a \<noteq> 0"
shows "interior {x. a \<bullet> x = b} = {}"
proof -
have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
by (force simp:)
then show ?thesis
by (auto simp: assms)
qed
lemma frontier_halfspace_le:
assumes "a \<noteq> 0 \<or> b \<noteq> 0"
shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
proof (cases "a = 0")
case True with assms show ?thesis by simp
next
case False then show ?thesis
by (force simp: frontier_def closed_halfspace_le)
qed
lemma frontier_halfspace_ge:
assumes "a \<noteq> 0 \<or> b \<noteq> 0"
shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
proof (cases "a = 0")
case True with assms show ?thesis by simp
next
case False then show ?thesis
by (force simp: frontier_def closed_halfspace_ge)
qed
lemma frontier_halfspace_lt:
assumes "a \<noteq> 0 \<or> b \<noteq> 0"
shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
proof (cases "a = 0")
case True with assms show ?thesis by simp
next
case False then show ?thesis
by (force simp: frontier_def interior_open open_halfspace_lt)
qed
lemma frontier_halfspace_gt:
assumes "a \<noteq> 0 \<or> b \<noteq> 0"
shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
proof (cases "a = 0")
case True with assms show ?thesis by simp
next
case False then show ?thesis
by (force simp: frontier_def interior_open open_halfspace_gt)
qed
lemma interior_standard_hyperplane:
"interior {x :: (real^'n). x$k = a} = {}"
proof -
have "axis k (1::real) \<noteq> 0"
by (simp add: axis_def vec_eq_iff)
moreover have "axis k (1::real) \<bullet> x = x$k" for x
by (simp add: cart_eq_inner_axis inner_commute)
ultimately show ?thesis
using interior_hyperplane [of "axis k (1::real)" a]
by force
qed
subsection \<open>Matrix operations\<close>
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
"map_matrix f x = (\<chi> i j. f (x $ i $ j))"
lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
by (simp add: map_matrix_def)
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. sum (\<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. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
(infixl "v*" 70)
where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$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 times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
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 sum.distrib[symmetric] field_simps)
lemma matrix_mul_lid [simp]:
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 sum.delta'[OF finite]
mult_1_left mult_zero_left if_True UNIV_I)
done
lemma matrix_mul_rid [simp]:
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 sum.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 sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
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
sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
apply simp
done
lemma scalar_matrix_assoc:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
lemma matrix_scalar_ac:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
lemma matrix_vector_mul_lid [simp]: "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 sum.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
sum.delta[OF finite] cong del: if_weak_cong)
done
lemma matrix_vector_mul_component: "(A *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 sum_distrib_right sum_distrib_left ac_simps)
apply (subst sum.swap)
apply simp
done
lemma transpose_mat [simp]: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)
lemma transpose_transpose [simp]: "transpose(transpose A) = A"
by (vector transpose_def)
lemma row_transpose [simp]:
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 [simp]:
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 [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
by (metis transpose_transpose rows_transpose)
lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
unfolding transpose_def
by (simp add: vec_eq_iff)
lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
by (metis transpose_transpose)
lemma matrix_mult_transpose_dot_column:
shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))"
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
lemma matrix_mult_transpose_dot_row:
shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))"
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
by (simp add: matrix_vector_mult_def inner_vec_def)
lemma matrix_mult_sum:
"(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$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 sum.If_cases vec_eq_iff)
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
lemma linear_componentwise_expansion:
fixes f:: "real ^'m \<Rightarrow> real ^ _"
assumes lf: "linear f"
shows "(f x)$j = sum (\<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 "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
unfolding sum_component by simp
then show ?thesis
unfolding linear_sum_mul[OF lf, symmetric]
unfolding scalar_mult_eq_scaleR[symmetric]
unfolding basis_expansion
by simp
qed
subsection\<open>Inverse matrices (not necessarily square)\<close>
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\<open>Correspondence between matrices and linear operators.\<close>
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_id_mat_1: "matrix id = mat 1"
by (simp add: mat_def matrix_def axis_def)
lemma matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r"
by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::('a::real_algebra_1) ^ _))"
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum.distrib scaleR_right.sum)
lemma
fixes A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear)
lemma vector_matrix_left_distrib [algebra_simps]:
shows "(x + y) v* A = x v* A + y v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps sum.distrib vec_eq_iff)
lemma matrix_vector_right_distrib [algebra_simps]:
"A *v (x + y) = A *v x + A *v y"
by (vector matrix_vector_mult_def sum.distrib distrib_left)
lemma matrix_vector_mult_diff_distrib [algebra_simps]:
fixes A :: "'a::ring_1^'n^'m"
shows "A *v (x - y) = A *v x - A *v y"
by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
lemma matrix_vector_mult_scaleR[algebra_simps]:
fixes A :: "real^'n^'m"
shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
using linear_iff matrix_vector_mul_linear by blast
lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
"(A + B) *v x = (A *v x) + (B *v x)"
by (vector matrix_vector_mult_def sum.distrib distrib_right)
lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
fixes A :: "'a :: ring_1^'n^'m"
shows "(A - B) *v x = (A *v x) - (B *v x)"
by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
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)
by (simp add: linear_componentwise_expansion lf)
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
by (simp add: ext matrix_works)
declare matrix_vector_mul [symmetric, simp]
lemma matrix_of_matrix_vector_mul [simp]: "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 \<circ> 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 = sum (\<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
sum_distrib_right sum_distrib_left)
apply (subst sum.swap)
apply (auto simp add: ac_simps)
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
lemma inj_matrix_vector_mult:
fixes A::"'a::field^'n^'m"
assumes "invertible A"
shows "inj (( *v) A)"
by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
lemma scalar_invertible:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A)"
proof -
obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
using assms unfolding invertible_def by auto
with `k \<noteq> 0`
have "(k *\<^sub>R A) ** ((1/k) *\<^sub>R A') = mat 1" "((1/k) *\<^sub>R A') ** (k *\<^sub>R A) = mat 1"
by (simp_all add: assms matrix_scalar_ac)
thus "invertible (k *\<^sub>R A)"
unfolding invertible_def by auto
qed
lemma scalar_invertible_iff:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
by (simp add: assms scalar_invertible)
lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
lemma vector_scalar_commute:
fixes A :: "'a::{field}^'m^'n"
shows "A *v (c *s x) = c *s (A *v x)"
by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
lemma scalar_vector_matrix_assoc:
fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
shows "(k *s x) v* A = k *s (x v* A)"
by (metis transpose_matrix_vector vector_scalar_commute)
lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lemma vector_matrix_mul_rid [simp]:
fixes v :: "('a::semiring_1)^'n"
shows "v v* mat 1 = v"
by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
lemma scaleR_vector_matrix_assoc:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
lemma vector_scaleR_matrix_ac:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
proof -
have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps)
with scaleR_vector_matrix_assoc
show "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
by auto
qed
subsection\<open>Some bounds on components etc. relative to operator norm\<close>
lemma norm_column_le_onorm:
fixes A :: "real^'n^'m"
shows "norm(column i A) \<le> onorm(( *v) A)"
proof -
have bl: "bounded_linear (( *v) A)"
by (simp add: linear_linear matrix_vector_mul_linear)
have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
by (simp add: matrix_mult_dot cart_eq_inner_axis)
also have "\<dots> \<le> onorm (( *v) A)"
using onorm [OF bl, of "axis i 1"] by auto
finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" .
then show ?thesis
unfolding column_def .
qed
lemma matrix_component_le_onorm:
fixes A :: "real^'n^'m"
shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
proof -
have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
also have "\<dots> \<le> onorm (( *v) A)"
by (metis (no_types) column_def norm_column_le_onorm)
finally show ?thesis .
qed
lemma component_le_onorm:
fixes f :: "real^'m \<Rightarrow> real^'n"
shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
by (metis matrix_component_le_onorm matrix_vector_mul)
lemma onorm_le_matrix_component_sum:
fixes A :: "real^'n^'m"
shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
proof (rule onorm_le)
fix x
have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
by (rule norm_le_l1_cart)
also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
proof (rule sum_mono)
fix i
have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>"
by (simp add: matrix_vector_mult_def)
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)"
by (rule sum_abs)
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" .
qed
finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
by (simp add: sum_distrib_right)
qed
lemma onorm_le_matrix_component:
fixes A :: "real^'n^'m"
assumes "\<And>i j. abs(A$i$j) \<le> B"
shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
proof (rule onorm_le)
fix x :: "real^'n::_"
have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
by (rule norm_le_l1_cart)
also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
proof (rule sum_mono)
fix i
have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x"
by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
by (simp add: mult_right_mono norm_le_l1_cart)
also have "\<dots> \<le> real (CARD('n)) * B * norm x"
by (simp add: assms sum_bounded_above mult_right_mono)
finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" .
qed
also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
by simp
finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
qed
subsection \<open>lambda skolemization on cartesian products\<close>
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$i)" by metis
hence ?rhs by metis }
ultimately show ?thesis by metis
qed
lemma rational_approximation:
assumes "e > 0"
obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
lemma matrix_rational_approximation:
fixes A :: "real^'n^'m"
assumes "e > 0"
obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
proof -
have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
by (auto simp: lambda_skolem Bex_def)
show ?thesis
proof
have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
(e / (2 * real CARD('m) * real CARD('n)))"
apply (rule onorm_le_matrix_component)
using Bclo by (simp add: abs_minus_commute less_imp_le)
also have "\<dots> < e"
using \<open>0 < e\<close> by (simp add: divide_simps)
finally show "onorm (( *v) (A - B)) < e" .
qed (use B in auto)
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:
fixes A :: "real^'n^'m"
shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)"
proof safe
fix B
assume B: "B ** A = mat 1"
show "inj (( *v) A)"
unfolding inj_on_def
by (metis B matrix_vector_mul_assoc matrix_vector_mul_lid)
next
assume "inj (( *v) A)"
with linear_injective_left_inverse[OF matrix_vector_mul_linear]
obtain g where "linear g" and g: "g \<circ> ( *v) A = id"
by blast
have "matrix g ** A = mat 1"
by (metis \<open>linear g\<close> g matrix_compose matrix_id_mat_1 matrix_of_matrix_vector_mul matrix_vector_mul_linear)
then show "\<exists>B. B ** A = mat 1"
by metis
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 (( *v) A)" unfolding surj_def by metis }
moreover
{ assume sf: "surj (( *v) A)"
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> 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. sum (\<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: "sum (\<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_sum 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$i ):: real)"
from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x]
have "x = 0" by vector }
}
ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
qed
lemma matrix_right_invertible_independent_rows:
fixes A :: "real^'n^'m"
shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
(\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
unfolding left_invertible_transpose[symmetric]
matrix_left_invertible_independent_columns
by (simp add: column_transpose)
lemma matrix_right_invertible_span_columns:
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
span (columns A) = UNIV" (is "?lhs = ?rhs")
proof -
let ?U = "UNIV :: 'm set"
have fU: "finite ?U" by simp
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def
by (simp add: 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: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
have "x \<in> span (columns A)"
unfolding y[symmetric] scalar_mult_eq_scaleR
proof (rule span_sum [OF span_mul])
show "column i A \<in> span (columns A)" for i
using columns_def span_inc by auto
qed
}
then have ?rhs unfolding rhseq by blast }
moreover
{ assume h:?rhs
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$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::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0"
by (rule exI[where x=0], simp)
next
fix c y1 y2
assume y1: "y1 \<in> columns A" and y2: "?P y2"
from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
unfolding columns_def by blast
from y2 obtain x:: "real ^'m" where
x: "sum (\<lambda>i. (x$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)$j))"
using i(1) by (simp add: field_simps)
have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
by (rule sum.cong[OF refl]) (use th in blast)
also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
by (simp add: sum.distrib)
also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
unfolding sum.delta[OF fU]
using i(1) by simp
finally show "sum (\<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)$j) + sum (\<lambda>xa. ((x$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 \<open>The same result in terms of square matrices.\<close>
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 (( *v) A)"
using AA' matrix_right_invertible_surjective by auto
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] 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')
hence "matrix f' ** A = A' ** A" by simp
hence "A' ** A = mat 1" by (simp add: th)
}
then show ?thesis by blast
qed
lemma invertible_mult:
assumes inv_A: "invertible A"
and inv_B: "invertible B"
shows "invertible (A**B)"
proof -
obtain A' where AA': "A ** A' = mat 1" and A'A: "A' ** A = mat 1"
using inv_A unfolding invertible_def by blast
obtain B' where BB': "B ** B' = mat 1" and B'B: "B' ** B = mat 1"
using inv_B unfolding invertible_def by blast
show ?thesis
proof (unfold invertible_def, rule exI[of _ "B'**A'"], rule conjI)
have "A ** B ** (B' ** A') = A ** (B ** (B' ** A'))"
using matrix_mul_assoc[of A B "(B' ** A')", symmetric] .
also have "... = A ** (B ** B' ** A')" unfolding matrix_mul_assoc[of B "B'" "A'"] ..
also have "... = A ** (mat 1 ** A')" unfolding BB' ..
also have "... = A ** A'" unfolding matrix_mul_lid ..
also have "... = mat 1" unfolding AA' ..
finally show "A ** B ** (B' ** A') = mat (1::'a)" .
have "B' ** A' ** (A ** B) = B' ** (A' ** (A ** B))" using matrix_mul_assoc[of B' A' "(A ** B)", symmetric] .
also have "... = B' ** (A' ** A ** B)" unfolding matrix_mul_assoc[of A' A B] ..
also have "... = B' ** (mat 1 ** B)" unfolding A'A ..
also have "... = B' ** B" unfolding matrix_mul_lid ..
also have "... = mat 1" unfolding B'B ..
finally show "B' ** A' ** (A ** B) = mat 1" .
qed
qed
lemma transpose_invertible:
fixes A :: "real^'n^'n"
assumes "invertible A"
shows "invertible (transpose A)"
by (meson assms invertible_def matrix_left_right_inverse right_invertible_transpose)
lemma vector_matrix_mul_assoc:
fixes v :: "('a::comm_semiring_1)^'n"
shows "(v v* M) v* N = v v* (M ** N)"
proof -
from matrix_vector_mul_assoc
have "transpose N *v (transpose M *v v) = (transpose N ** transpose M) *v v" by fast
thus "(v v* M) v* N = v v* (M ** N)"
by (simp add: matrix_transpose_mul [symmetric])
qed
lemma matrix_scaleR_vector_ac:
fixes A :: "real^('m::finite)^'n"
shows "A *v (k *\<^sub>R v) = k *\<^sub>R A *v v"
by (metis matrix_vector_mult_scaleR transpose_scalar vector_scaleR_matrix_ac vector_transpose_matrix)
lemma scaleR_matrix_vector_assoc:
fixes A :: "real^('m::finite)^'n"
shows "k *\<^sub>R (A *v v) = k *\<^sub>R A *v v"
by (metis matrix_scaleR_vector_ac matrix_vector_mult_scaleR)
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
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)$1"
unfolding dot_matrix_product transpose_columnvector[symmetric]
dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |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_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
unfolding continuous_def by (rule tendsto_vec_nth)
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
lemma continuous_on_vec_lambda[continuous_intros]:
"(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
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 continuous_on_const continuous_on_id continuous_on_component)
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 "\<exists>l r. strict_mono r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
(is "?th d")
proof -
have "\<forall>d' \<subseteq> d. ?th d'"
by (rule compact_lemma_general[where unproj=vec_lambda])
(auto intro!: f bounded_component_cart simp: vec_lambda_eta)
then show "?th d" by simp
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: "strict_mono 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 L2_set_le_sum)
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
by (rule sum_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_mono)
}
hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
qed
lemma interval_cart:
fixes a :: "real^'n"
shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
and "cbox a b = {x::real^'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 mem_box Basis_vec_def inner_axis)
lemma mem_box_cart:
fixes a :: "real^'n"
shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
and "x \<in> cbox 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 "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
and "(cbox 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>box a b"
hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_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$i"
unfolding vector_smult_component and vector_add_component
by auto }
hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto }
ultimately show ?th1 by blast
{ fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b"
hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_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$i"
unfolding vector_smult_component and vector_add_component
by auto }
hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto }
ultimately show ?th2 by blast
qed
lemma interval_ne_empty_cart:
fixes a :: "real^'n"
shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$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$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b"
and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
lemma interval_sing:
fixes a :: "'a::linorder^'n"
shows "{a .. a} = {a} \<and> {a<..<a} = {}"
apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
done
lemma subset_interval_cart:
fixes a :: "real^'n"
shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
lemma disjoint_interval_cart:
fixes a::"real^'n"
shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
lemma Int_interval_cart:
fixes a :: "real^'n"
shows "cbox a b \<inter> cbox c d = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
unfolding Int_interval
by (auto simp: mem_box less_eq_vec_def)
(auto simp: Basis_vec_def inner_axis)
lemma closed_interval_left_cart:
fixes b :: "real^'n"
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
lemma closed_interval_right_cart:
fixes a::"real^'n"
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
lemma is_interval_cart:
"is_interval (s::(real^'n) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}"
by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
lemma Lim_component_le_cart:
fixes f :: "'a \<Rightarrow> real^'n"
assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f x $i \<le> b) net"
shows "l$i \<le> b"
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
lemma Lim_component_ge_cart:
fixes f :: "'a \<Rightarrow> real^'n"
assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
shows "b \<le> l$i"
by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
lemma Lim_component_eq_cart:
fixes f :: "'a \<Rightarrow> real^'n"
assumes net: "(f \<longlongrightarrow> 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_conj_iff] and
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$k = a)"
using connected_ivt_hyperplane[of s x y "axis k 1" a]
by (auto simp add: inner_axis inner_commute)
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
unfolding subspace_def by auto
lemma closed_substandard_cart:
"closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
proof -
{ fix i::'n
have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
thus ?thesis
unfolding Collect_all_eq by (simp add: closed_INT)
qed
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
(is "dim ?A = _")
proof -
let ?a = "\<lambda>x. axis x 1 :: real^'n"
have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
have "?a ` d \<subseteq> Basis"
by (auto simp: Basis_vec_def)
thus ?thesis
using dim_substandard[of "?a ` d"] card_image[of ?a d]
by (auto simp: axis_eq_axis inj_on_def *)
qed
lemma dim_subset_UNIV_cart:
fixes S :: "(real^'n) set"
shows "dim S \<le> CARD('n)"
by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
lemma affinity_inverses:
assumes m0: "m \<noteq> (0::'a::field)"
shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
"(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
using m0
apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
done
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
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 vector_cart:
fixes f :: "real^'n \<Rightarrow> real"
shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
unfolding euclidean_eq_iff[where 'a="real^'n"]
by simp (simp add: Basis_vec_def inner_axis)
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
by (rule vector_cart)
subsection "Convex Euclidean Space"
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
using const_vector_cart[of 1] by (simp add: one_vec_def)
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 less_eq_vec_def vec_lambda_beta 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$i)}"
by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
lemma unit_interval_convex_hull_cart:
"cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
lemma cube_convex_hull_cart:
assumes "0 < d"
obtains s::"(real^'n) set"
where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
proof -
from assms obtain s where "finite s"
and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
by (rule cube_convex_hull)
with that[of s] show thesis
by (simp add: const_vector_cart)
qed
subsection "Derivative"
definition "jacobian f net = matrix(frechet_derivative f net)"
lemma jacobian_works:
"(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
(f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
next
assume ?rhs then show ?lhs
by (rule differentiableI)
qed
subsection \<open>Component of the differential must be zero if it exists at a local
maximum or minimum for that corresponding component\<close>
lemma differential_zero_maxmin_cart:
fixes f::"real^'a \<Rightarrow> real^'b"
assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$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"
using differential_zero_maxmin_component[of "axis k 1" e x f] assms
vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
by (metis (full_types) num1_eq_iff)
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
by auto (metis (full_types) 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 \<le> 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 \<le> 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 sum_1: "sum f (UNIV::1 set) = f 1"
unfolding UNIV_1 by simp
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
unfolding UNIV_2 by simp
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
unfolding UNIV_3 by (simp add: ac_simps)
lemma num1_eqI:
fixes a::num1 shows "a = b"
by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
lemma num1_eq1 [simp]:
fixes a::num1 shows "a = 1"
by (rule num1_eqI)
instantiation num1 :: cart_one
begin
instance
proof
show "CARD(1) = Suc 0" by auto
qed
end
instantiation num1 :: linorder begin
definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
instance
by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
end
instance num1 :: wellorder
by intro_classes (auto simp: less_eq_num1_def less_num1_def)
subsection\<open>The collapse of the general concepts to dimension one\<close>
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
by (simp add: vec_eq_iff)
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_vec_def)
lemma dist_vector_1:
fixes x :: "'a::real_normed_vector^1"
shows "dist x y = dist (x$1) (y$1)"
by (simp add: dist_norm norm_vector_1)
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
by (simp add: norm_vector_1)
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
by (auto simp add: norm_real dist_norm)
subsection\<open> Rank of a matrix\<close>
text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
lemma matrix_vector_mult_in_columnspace:
fixes A :: "real^'n^'m"
shows "(A *v x) \<in> span(columns A)"
apply (simp add: matrix_vector_column columns_def transpose_def column_def)
apply (intro span_sum span_mul)
apply (force intro: span_superset)
done
lemma orthogonal_nullspace_rowspace:
fixes A :: "real^'n^'m"
assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
shows "orthogonal x y"
proof (rule span_induct [OF y])
show "subspace {a. orthogonal x a}"
by (simp add: subspace_orthogonal_to_vector)
next
fix v
assume "v \<in> rows A"
then obtain i where "v = row i A"
by (auto simp: rows_def)
with 0 show "orthogonal x v"
unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
qed
lemma nullspace_inter_rowspace:
fixes A :: "real^'n^'m"
shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
using orthogonal_nullspace_rowspace orthogonal_self by auto
lemma matrix_vector_mul_injective_on_rowspace:
fixes A :: "real^'n^'m"
shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
using nullspace_inter_rowspace [of A "x-y"]
by (metis eq_iff_diff_eq_0 matrix_vector_mult_diff_distrib span_diff)
definition rank :: "real^'n^'m=>nat"
where "rank A \<equiv> dim(columns A)"
lemma dim_rows_le_dim_columns:
fixes A :: "real^'n^'m"
shows "dim(rows A) \<le> dim(columns A)"
proof -
have "dim (span (rows A)) \<le> dim (span (columns A))"
proof -
obtain B where "independent B" "span(rows A) \<subseteq> span B"
and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
using basis_exists [of "span(rows A)"] by blast
with span_subspace have eq: "span B = span(rows A)"
by auto
then have inj: "inj_on (( *v) A) (span B)"
using inj_on_def matrix_vector_mul_injective_on_rowspace by blast
then have ind: "independent (( *v) A ` B)"
by (rule independent_inj_on_image [OF \<open>independent B\<close> matrix_vector_mul_linear])
then have "finite (( *v) A ` B) \<and> card (( *v) A ` B) \<le> dim (( *v) A ` B)"
by (rule independent_bound_general)
then show ?thesis
by (metis (no_types, lifting) B ind inj eq card_image image_subset_iff independent_card_le_dim inj_on_subset matrix_vector_mult_in_columnspace)
qed
then show ?thesis
by simp
qed
lemma rank_row:
fixes A :: "real^'n^'m"
shows "rank A = dim(rows A)"
unfolding rank_def
by (metis dim_rows_le_dim_columns columns_transpose dual_order.antisym rows_transpose)
lemma rank_transpose:
fixes A :: "real^'n^'m"
shows "rank(transpose A) = rank A"
by (metis rank_def rank_row rows_transpose)
lemma matrix_vector_mult_basis:
fixes A :: "real^'n^'m"
shows "A *v (axis k 1) = column k A"
by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
lemma columns_image_basis:
fixes A :: "real^'n^'m"
shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
by (force simp: columns_def matrix_vector_mult_basis [symmetric])
lemma rank_dim_range:
fixes A :: "real^'n^'m"
shows "rank A = dim(range (\<lambda>x. A *v x))"
unfolding rank_def
proof (rule span_eq_dim)
show "span (columns A) = span (range (( *v) A))"
apply (auto simp: columns_image_basis span_linear_image matrix_vector_mul_linear)
by (metis columns_image_basis matrix_vector_mul_linear matrix_vector_mult_in_columnspace span_linear_image)
qed
lemma rank_bound:
fixes A :: "real^'n^'m"
shows "rank A \<le> min CARD('m) (CARD('n))"
by (metis (mono_tags, hide_lams) min.bounded_iff DIM_cart DIM_real dim_subset_UNIV mult.right_neutral rank_def rank_transpose)
lemma full_rank_injective:
fixes A :: "real^'n^'m"
shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows rank_row dim_eq_full [symmetric])
lemma full_rank_surjective:
fixes A :: "real^'n^'m"
shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
by (simp add: full_rank_injective inj_on_def)
lemma less_rank_noninjective:
fixes A :: "real^'n^'m"
shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
lemma matrix_nonfull_linear_equations_eq:
fixes A :: "real^'n^'m"
shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank 0 = 0"
by (auto simp: rank_dim_range matrix_eq)
lemma rank_mul_le_right:
fixes A :: "real^'n^'m" and B :: "real^'p^'n"
shows "rank(A ** B) \<le> rank B"
proof -
have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
also have "\<dots> \<le> rank B"
by (simp add: rank_dim_range matrix_vector_mul_linear dim_image_le)
finally show ?thesis .
qed
lemma rank_mul_le_left:
fixes A :: "real^'n^'m" and B :: "real^'p^'n"
shows "rank(A ** B) \<le> rank A"
by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
lemma vector_one_nth [simp]:
fixes x :: "'a^1" shows "vec (x $ 1) = x"
by (metis vec_def vector_one)
lemma vec_cbox_1_eq [simp]:
shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)"
by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
lemma vec_nth_cbox_1_eq [simp]:
fixes u v :: "'a::euclidean_space^1"
shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u$1) (v$1)"
by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component)
lemma vec_nth_1_iff_cbox [simp]:
fixes a b :: "'a::euclidean_space"
shows "(\<lambda>x::'a^1. x $ 1) ` S = cbox a b \<longleftrightarrow> S = cbox (vec a) (vec b)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs show ?rhs
proof (intro equalityI subsetI)
fix x
assume "x \<in> S"
then have "x $ 1 \<in> (\<lambda>v. v $ (1::1)) ` cbox (vec a) (vec b)"
using L by auto
then show "x \<in> cbox (vec a) (vec b)"
by (metis (no_types, lifting) imageE vector_one_nth)
next
fix x :: "'a^1"
assume "x \<in> cbox (vec a) (vec b)"
then show "x \<in> S"
by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth)
qed
qed simp
lemma tendsto_at_within_vector_1:
fixes S :: "'a :: metric_space set"
assumes "(f \<longlongrightarrow> fx) (at x within S)"
shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
proof (rule topological_tendstoI)
fix T :: "('a^1) set"
assume "open T" "vec fx \<in> T"
have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
unfolding eventually_at dist_norm [symmetric]
by (rule ex_forward)
(use \<open>open T\<close> in
\<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
qed
lemma has_derivative_vector_1:
assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
shows "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' a))
(at ((vec a)::real^1) within vec ` S)"
using der_g
apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
apply (drule tendsto_at_within_vector_1, vector)
apply (auto simp: algebra_simps eventually_at tendsto_def)
done
subsection\<open>Explicit vector construction from lists\<close>
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
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]))"
by (metis vector_1 vector_one)
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 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 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$i)}"
"cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
apply (rule_tac[!] set_eqI)
unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart
unfolding vec_lambda_beta
by auto
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
bounded_linear.uniform_limit[OF bounded_linear_component_cart]
end