(* Title: HOL/Analysis/Cartesian_Euclidean_Space.thy
Some material by Jose Divasón, Tim Makarios and L C Paulson
*)
section \<open>Finite Cartesian Products of Euclidean Spaces\<close>
theory Cartesian_Euclidean_Space
imports Derivative
begin
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
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)
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)
subsection\<open>Closures and interiors of halfspaces\<close>
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$k" for x
by (simp add: cart_eq_inner_axis inner_commute)
ultimately show ?LE ?GE
using interior_halfspace_le [of "axis k (1::real)" a]
interior_halfspace_ge [of "axis k (1::real)" a] by auto
qed
lemma closure_halfspace_component_lt [simp]:
"closure {x. x$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_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
lemma matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear ((*v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
using matrix_vector_mul_linear[of A]
by (simp add: linear_conv_bounded_linear linear_matrix_vector_mul_eq)
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_continuous_at linear_continuous_on)
subsection\<open>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 "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 matrix_vector_mul_bounded_linear, of A "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(2))
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
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
proposition 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: field_split_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 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_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)
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$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_component)
lemma closed_interval_right_cart:
fixes a::"real^'n"
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_component)
lemma is_interval_cart:
"is_interval (s::(real^'n) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$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_component)
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
by (simp add: closed_Collect_le continuous_on_component)
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
by (simp add: open_Collect_less continuous_on_component)
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}"
by (simp add: open_Collect_less 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" "\<not> 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$k = a)"
using connected_ivt_hyperplane[of s x y "axis k 1" a]
by (auto simp add: inner_axis inner_commute)
lemma subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
unfolding vec.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_component) }
thus ?thesis
unfolding Collect_all_eq by (simp add: closed_INT)
qed
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$i)}"
using assms unfolding convex_def by auto
(* Unused
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$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)
proposition 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\<^marker>\<open>tag important\<close> "jacobian f net = matrix(frechet_derivative f net)"
proposition 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
text \<open>Component of the differential must be zero if it exists at a local
maximum or minimum for that corresponding component\<close>
proposition 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\<^marker>\<open>tag unimportant\<close>\<open>Routine results connecting the types \<^typ>\<open>real^1\<close> and \<^typ>\<open>real\<close>\<close>
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 vec_nth_real_1_iff_cbox [simp]:
fixes a b :: real
shows "(\<lambda>x::real^1. x $ 1) ` S = {a..b} \<longleftrightarrow> S = cbox (vec a) (vec b)"
using vec_nth_1_iff_cbox[of S a b]
by simp
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]
end