Theory Cartesian_Euclidean_Space

```(* Title:      HOL/Analysis/Cartesian_Euclidean_Space.thy
Some material by Jose Divasón, Tim Makarios and L C Paulson
*)

section ‹Finite Cartesian Products of Euclidean Spaces›

theory Cartesian_Euclidean_Space
imports Derivative
begin

lemma subspace_special_hyperplane: "subspace {x. x \$ k = 0}"

lemma sum_mult_product:
"sum h {..<A * B :: nat} = (∑i∈{..<A}. ∑j∈{..<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 (λj. j + i * B) {..<B}" by (auto intro!: inj_onI)
show "{i * B..<i * B + B} = (λj. j + i * B) ` {..<B}"
proof safe
fix j assume "j ∈ {i * B..<i * B + B}"
then show "j ∈ (λj. 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"

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) = {} ⟷ cbox a b = {}"
"box (vec a) (vec b) = {} ⟷ box a b = {}"
by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)

subsection‹Closures and interiors of halfspaces›

lemma interior_halfspace_component_le [simp]:
"interior {x. x\$k ≤ a} = {x :: (real^'n). x\$k < a}" (is "?LE")
and interior_halfspace_component_ge [simp]:
"interior {x. x\$k ≥ a} = {x :: (real^'n). x\$k > a}" (is "?GE")
proof -
have "axis k (1::real) ≠ 0"
moreover have "axis k (1::real) ∙ x = x\$k" for x
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 ≤ a}" (is "?LE")
and closure_halfspace_component_gt [simp]:
"closure {x. x\$k > a} = {x :: (real^'n). x\$k ≥ a}" (is "?GE")
proof -
have "axis k (1::real) ≠ 0"
moreover have "axis k (1::real) ∙ x = x\$k" for x
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) ≠ 0"
moreover have "axis k (1::real) ∙ x = x\$k" for x
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]

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)"

subsection‹Bounds on components etc.\ relative to operator norm›

lemma norm_column_le_onorm:
fixes A :: "real^'n^'m"
shows "norm(column i A) ≤ onorm((*v) A)"
proof -
have "norm (χ j. A \$ j \$ i) ≤ norm (A *v axis i 1)"
also have "… ≤ onorm ((*v) A)"
using onorm [OF matrix_vector_mul_bounded_linear, of A "axis i 1"] by auto
finally have "norm (χ j. A \$ j \$ i) ≤ onorm ((*v) A)" .
then show ?thesis
unfolding column_def .
qed

lemma matrix_component_le_onorm:
fixes A :: "real^'n^'m"
shows "¦A \$ i \$ j¦ ≤ onorm((*v) A)"
proof -
have "¦A \$ i \$ j¦ ≤ norm (χ n. (A \$ n \$ j))"
by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
also have "… ≤ 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 ⇒ real^'n"
shows "linear f ⟹ ¦matrix f \$ i \$ j¦ ≤ 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) ≤ (∑i∈UNIV. ∑j∈UNIV. ¦A \$ i \$ j¦)"
proof (rule onorm_le)
fix x
have "norm (A *v x) ≤ (∑i∈UNIV. ¦(A *v x) \$ i¦)"
by (rule norm_le_l1_cart)
also have "… ≤ (∑i∈UNIV. ∑j∈UNIV. ¦A \$ i \$ j¦ * norm x)"
proof (rule sum_mono)
fix i
have "¦(A *v x) \$ i¦ ≤ ¦∑j∈UNIV. A \$ i \$ j * x \$ j¦"
also have "… ≤ (∑j∈UNIV. ¦A \$ i \$ j * x \$ j¦)"
by (rule sum_abs)
also have "… ≤ (∑j∈UNIV. ¦A \$ i \$ j¦ * norm x)"
by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
finally show "¦(A *v x) \$ i¦ ≤ (∑j∈UNIV. ¦A \$ i \$ j¦ * norm x)" .
qed
finally show "norm (A *v x) ≤ (∑i∈UNIV. ∑j∈UNIV. ¦A \$ i \$ j¦) * norm x"
qed

lemma onorm_le_matrix_component:
fixes A :: "real^'n^'m"
assumes "⋀i j. abs(A\$i\$j) ≤ B"
shows "onorm((*v) A) ≤ real (CARD('m)) * real (CARD('n)) * B"
proof (rule onorm_le)
fix x :: "real^'n::_"
have "norm (A *v x) ≤ (∑i∈UNIV. ¦(A *v x) \$ i¦)"
by (rule norm_le_l1_cart)
also have "… ≤ (∑i::'m ∈UNIV. real (CARD('n)) * B * norm x)"
proof (rule sum_mono)
fix i
have "¦(A *v x) \$ i¦ ≤ norm(A \$ i) * norm x"
also have "… ≤ (∑j∈UNIV. ¦A \$ i \$ j¦) * norm x"
also have "… ≤ real (CARD('n)) * B * norm x"
by (simp add: assms sum_bounded_above mult_right_mono)
finally show "¦(A *v x) \$ i¦ ≤ real (CARD('n)) * B * norm x" .
qed
also have "… ≤ CARD('m) * real (CARD('n)) * B * norm x"
by simp
finally show "norm (A *v x) ≤ CARD('m) * real (CARD('n)) * B * norm x" .
qed

lemma vector_sub_project_orthogonal_cart: "(b::real^'n) ∙ (x - ((b ∙ x) / (b ∙ b)) *s b) = 0"
unfolding inner_simps scalar_mult_eq_scaleR by auto

lemma infnorm_cart:"infnorm (x::real^'n) = Sup {¦x\$i¦ |i. i∈UNIV}"
by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)

lemma component_le_infnorm_cart: "¦x\$i¦ ≤ 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 ⟹ continuous F (λx. f x \$ i)"
unfolding continuous_def by (rule tendsto_vec_nth)

lemma continuous_on_component[continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. f x \$ i)"
unfolding continuous_on_def by (fast intro: tendsto_vec_nth)

lemma continuous_on_vec_lambda[continuous_intros]:
"(⋀i. continuous_on S (f i)) ⟹ continuous_on S (λx. χ i. f i x)"
unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)

lemma closed_positive_orthant: "closed {x::real^'n. ∀i. 0 ≤x\$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_component)

lemma bounded_component_cart: "bounded s ⟹ bounded ((λx. 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 ⇒ 'a::heine_borel ^ 'n"
assumes f: "bounded (range f)"
shows "∃l r. strict_mono r ∧
(∀e>0. eventually (λn. ∀i∈d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
(is "?th d")
proof -
have "∀d' ⊆ 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 ⇒ 'a ^ 'b"
assume f: "bounded (range f)"
then obtain l r where r: "strict_mono r"
and l: "∀e>0. eventually (λn. ∀i∈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 (λn. ∀i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
by simp
moreover
{ fix n
assume n: "∀i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
have "dist (f (r n)) l ≤ (∑i∈?d. dist (f (r n) \$ i) (l \$ i))"
unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum)
also have "… < (∑i∈?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 (λn. dist (f (r n)) l < e) sequentially"
by (rule eventually_mono)
}
hence "((f ∘ r) ⤏ l) sequentially" unfolding o_def tendsto_iff by simp
with r show "∃l r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially" by auto
qed

lemma interval_cart:
fixes a :: "real^'n"
shows "box a b = {x::real^'n. ∀i. a\$i < x\$i ∧ x\$i < b\$i}"
and "cbox a b = {x::real^'n. ∀i. a\$i ≤ x\$i ∧ x\$i ≤ 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 ∈ box a b ⟷ (∀i. a\$i < x\$i ∧ x\$i < b\$i)"
and "x ∈ cbox a b ⟷ (∀i. a\$i ≤ x\$i ∧ x\$i ≤ 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 = {} ⟷ (∃i. b\$i ≤ a\$i))" (is ?th1)
and "(cbox a b = {} ⟷ (∃i. b\$i < a\$i))" (is ?th2)
proof -
{ fix i x assume as:"b\$i ≤ a\$i" and x:"x∈box a b"
hence "a \$ i < x \$ i ∧ 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:"∀i. ¬ (b\$i ≤ a\$i)"
let ?x = "(1/2) *⇩R (a + b)"
{ fix i
have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
hence "a\$i < ((1/2) *⇩R (a+b)) \$ i" "((1/2) *⇩R (a+b)) \$ i < b\$i"
by auto }
hence "box a b ≠ {}" 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∈cbox a b"
hence "a \$ i ≤ x \$ i ∧ 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:"∀i. ¬ (b\$i < a\$i)"
let ?x = "(1/2) *⇩R (a + b)"
{ fix i
have "a\$i ≤ b\$i" using as[THEN spec[where x=i]] by auto
hence "a\$i ≤ ((1/2) *⇩R (a+b)) \$ i" "((1/2) *⇩R (a+b)) \$ i ≤ b\$i"
by auto }
hence "cbox a b ≠ {}" 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 ≠ {} ⟷ (∀i. a\$i ≤ b\$i)"
and "box a b ≠ {} ⟷ (∀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 "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ cbox c d ⊆ cbox a b"
and "(∀i. a\$i < c\$i ∧ d\$i < b\$i) ⟹ cbox c d ⊆ box a b"
and "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ box c d ⊆ cbox a b"
and "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ box c d ⊆ 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} ∧ {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 ⊆ cbox a b ⟷ (∀i. c\$i ≤ d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i)" (is ?th1)
and "cbox c d ⊆ box a b ⟷ (∀i. c\$i ≤ d\$i) --> (∀i. a\$i < c\$i ∧ d\$i < b\$i)" (is ?th2)
and "box c d ⊆ cbox a b ⟷ (∀i. c\$i < d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i)" (is ?th3)
and "box c d ⊆ box a b ⟷ (∀i. c\$i < d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ 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 ∩ cbox c d = {} ⟷ (∃i. (b\$i < a\$i ∨ d\$i < c\$i ∨ b\$i < c\$i ∨ d\$i < a\$i))" (is ?th1)
and "cbox a b ∩ box c d = {} ⟷ (∃i. (b\$i < a\$i ∨ d\$i ≤ c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ a\$i))" (is ?th2)
and "box a b ∩ cbox c d = {} ⟷ (∃i. (b\$i ≤ a\$i ∨ d\$i < c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ a\$i))" (is ?th3)
and "box a b ∩ box c d = {} ⟷ (∃i. (b\$i ≤ a\$i ∨ d\$i ≤ c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ 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 ∩ cbox c d =  {(χ i. max (a\$i) (c\$i)) .. (χ 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. ∀i. x\$i ≤ 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. ∀i. a\$i ≤ 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) ⟷
(∀a∈s. ∀b∈s. ∀x. (∀i. ((a\$i ≤ x\$i ∧ x\$i ≤ b\$i) ∨ (b\$i ≤ x\$i ∧ x\$i ≤ a\$i))) ⟶ x ∈ 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 ≤ a}"

lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i ≥ a}"

lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"

lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"

lemma Lim_component_le_cart:
fixes f :: "'a ⇒ real^'n"
assumes "(f ⤏ l) net" "¬ (trivial_limit net)"  "eventually (λx. f x \$i ≤ b) net"
shows "l\$i ≤ 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 ⇒ real^'n"
assumes "(f ⤏ l) net"  "¬ (trivial_limit net)"  "eventually (λx. b ≤ (f x)\$i) net"
shows "b ≤ 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 ⇒ real^'n"
assumes net: "(f ⤏ l) net" "¬ trivial_limit net" and ev:"eventually (λx. 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 ⟹ x ∈ s ⟹ y ∈ s ⟹ x\$k ≤ a ⟹ a ≤ y\$k ⟹ (∃z∈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. (∀i. P i ⟶ x\$i = 0)}"
unfolding vec.subspace_def by auto

lemma closed_substandard_cart:
"closed {x::'a::real_normed_vector ^ 'n. ∀i. P i ⟶ x\$i = 0}"
proof -
{ fix i::'n
have "closed {x::'a ^ 'n. P i ⟶ 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) = ∑Basis"
using const_vector_cart[of 1] by (simp add: one_vec_def)

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 "⋀i. convex {x. P i x}"
shows "convex {x. ∀i. P i (x\$i)}"
using assms unfolding convex_def by auto

(* Unused
lemma convex_positive_orthant_cart: "convex {x::real^'n. (∀i. 0 ≤ 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. ∀i. (x\$i = 0) ∨ (x\$i = 1)}"
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
by (rule arg_cong[where f="λx. 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 - (χ i. d)) (x + (χ i. d)) = convex hull s"
proof -
from assms obtain s where "finite s"
and "cbox (x - sum (( *⇩R) d) Basis) (x + sum (( *⇩R) d) Basis) = convex hull s"
by (rule cube_convex_hull)
with that[of s] show thesis
qed
*)

subsection‹Arbitrarily good rational approximations›

lemma rational_approximation:
assumes "e > 0"
obtains r::real where "r ∈ ℚ" "¦r - x¦ < e"
using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto

lemma Rats_closure_real: "closure ℚ = (UNIV::real set)"
proof -
have "⋀x::real. x ∈ closure ℚ"
by (metis closure_approachable dist_real_def rational_approximation)
then show ?thesis by auto
qed

proposition matrix_rational_approximation:
fixes A :: "real^'n^'m"
assumes "e > 0"
obtains B where "⋀i j. B\$i\$j ∈ ℚ" "onorm(λx. (A - B) *v x) < e"
proof -
have "∀i j. ∃q ∈ ℚ. ¦q - A \$ i \$ j¦ < 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: "⋀i j. B\$i\$j ∈ ℚ" and Bclo: "⋀i j. ¦B\$i\$j - A \$ i \$ j¦ < e / (2 * CARD('m) * CARD('n))"
by (auto simp: lambda_skolem Bex_def)
show ?thesis
proof
have "onorm ((*v) (A - B)) ≤ 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 "… < e"
using ‹0 < e› by (simp add: field_split_simps)
finally show "onorm ((*v) (A - B)) < e" .
qed (use B in auto)
qed

subsection "Derivative"

definition✐‹tag important› "jacobian f net = matrix(frechet_derivative f net)"

proposition jacobian_works:
"(f::(real^'a) ⇒ (real^'b)) differentiable net ⟷
(f has_derivative (λh. (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 ‹Component of the differential must be zero if it exists at a local
maximum or minimum for that corresponding component›

proposition differential_zero_maxmin_cart:
fixes f::"real^'a ⇒ real^'b"
assumes "0 < e" "((∀y ∈ ball x e. (f y)\$k ≤ (f x)\$k) ∨ (∀y∈ball x e. (f x)\$k ≤ (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 "λj. frechet_derivative f (at x) j \$ k"]
by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)

subsection✐‹tag unimportant›‹Routine results connecting the types \<^typ>‹real^1› and \<^typ>‹real››

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 "(λx. 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 "(λx::'a^1. x \$ 1) ` S = cbox a b ⟷ S = cbox (vec a) (vec b)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs show ?rhs
proof (intro equalityI subsetI)
fix x
assume "x ∈ S"
then have "x \$ 1 ∈ (λv. v \$ (1::1)) ` cbox (vec a) (vec b)"
using L by auto
then show "x ∈ cbox (vec a) (vec b)"
by (metis (no_types, lifting) imageE vector_one_nth)
next
fix x :: "'a^1"
assume "x ∈ cbox (vec a) (vec b)"
then show "x ∈ 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 "(λx::real^1. x \$ 1) ` S = {a..b} ⟷ 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} ∩ {x. x\$k ≤ c} = {a .. (χ i. if i = k then min (b\$k) c else b\$i)}"
"cbox a b ∩ {x. x\$k ≥ c} = {(χ i. if i = k then max (a\$k) c else a\$i) .. b}"
unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart set_eq_iff
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
```