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Quellcode-Bibliothek
© Kompilation durch diese Firma
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Datei:
options5.html
Sprache: Isabelle
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(* Author: L C Paulson, University of Cambridge
Author: Amine Chaieb, University of Cambridge
Author: Robert Himmelmann, TU Muenchen
Author: Brian Huffman, Portland State University
*)
chapter \<open>Vector Analysis\<close>
theory Topology_Euclidean_Space
imports
Elementary_Normed_Spaces
Linear_Algebra
Norm_Arith
begin
section \<open>Elementary Topology in Euclidean Space\<close>
lemma euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = L2_set (\i. dist (x \ i) (y \ i)) Basis"
unfolding dist_norm norm_eq_sqrt_inner L2_set_def
by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
lemma norm_nth_le: "norm (x \ i) \ norm x" if "i \ Basis"
proof -
have "(x \ i)\<^sup>2 = (\i\{i}. (x \ i)\<^sup>2)"
by simp
also have "\ \ (\i\Basis. (x \ i)\<^sup>2)"
by (intro sum_mono2) (auto simp: that)
finally show ?thesis
unfolding norm_conv_dist euclidean_dist_l2[of x] L2_set_def
by (auto intro!: real_le_rsqrt)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Continuity of the representation WRT an orthogonal basis\<close>
lemma orthogonal_Basis: "pairwise orthogonal Basis"
by (simp add: inner_not_same_Basis orthogonal_def pairwise_def)
lemma representation_bound:
fixes B :: "'N::real_inner set"
assumes "finite B" "independent B" "b \ B" and orth: "pairwise orthogonal B"
obtains m where "m > 0" "\x. x \ span B \ \representation B x b\ \ m * norm x"
proof
fix x
assume x: "x \ span B"
have "b \ 0"
using \<open>independent B\<close> \<open>b \<in> B\<close> dependent_zero by blast
have [simp]: "b \ b' = (if b' = b then (norm b)\<^sup>2 else 0)"
if "b \ B" "b' \ B" for b b'
using orth by (simp add: orthogonal_def pairwise_def norm_eq_sqrt_inner that)
have "norm x = norm (\b\B. representation B x b *\<^sub>R b)"
using real_vector.sum_representation_eq [OF \<open>independent B\<close> x \<open>finite B\<close> order_refl]
by simp
also have "\ = sqrt ((\b\B. representation B x b *\<^sub>R b) \ (\b\B. representation B x b *\<^sub>R b))"
by (simp add: norm_eq_sqrt_inner)
also have "\ = sqrt (\b\B. (representation B x b *\<^sub>R b) \ (representation B x b *\<^sub>R b))"
using \<open>finite B\<close>
by (simp add: inner_sum_left inner_sum_right if_distrib [of "\x. _ * x"] cong: if_cong sum.cong_simp)
also have "\ = sqrt (\b\B. (norm (representation B x b *\<^sub>R b))\<^sup>2)"
by (simp add: mult.commute mult.left_commute power2_eq_square)
also have "\ = sqrt (\b\B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)"
by (simp add: norm_mult power_mult_distrib)
finally have "norm x = sqrt (\b\B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)" .
moreover
have "sqrt ((representation B x b)\<^sup>2 * (norm b)\<^sup>2) \ sqrt (\b\B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)"
using \<open>b \<in> B\<close> \<open>finite B\<close> by (auto intro: member_le_sum)
then have "\representation B x b\ \ (1 / norm b) * sqrt (\b\B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)"
using \<open>b \<noteq> 0\<close> by (simp add: field_split_simps real_sqrt_mult del: real_sqrt_le_iff)
ultimately show "\representation B x b\ \ (1 / norm b) * norm x"
by simp
next
show "0 < 1 / norm b"
using \<open>independent B\<close> \<open>b \<in> B\<close> dependent_zero by auto
qed
lemma continuous_on_representation:
fixes B :: "'N::euclidean_space set"
assumes "finite B" "independent B" "b \ B" "pairwise orthogonal B"
shows "continuous_on (span B) (\x. representation B x b)"
proof
show "\d>0. \x'\span B. dist x' x < d \ dist (representation B x' b) (representation B x b) \ e"
if "e > 0" "x \ span B" for x e
proof -
obtain m where "m > 0" and m: "\x. x \ span B \ \representation B x b\ \ m * norm x"
using assms representation_bound by blast
show ?thesis
unfolding dist_norm
proof (intro exI conjI ballI impI)
show "e/m > 0"
by (simp add: \<open>e > 0\<close> \<open>m > 0\<close>)
show "norm (representation B x' b - representation B x b) \ e"
if x': "x' \<in> span B" and less: "norm (x'-x) < e/m" for x'
proof -
have "\representation B (x'-x) b\ \ m * norm (x'-x)"
using m [of "x'-x"] \<open>x \<in> span B\<close> span_diff x' by blast
also have "\ < e"
by (metis \<open>m > 0\<close> less mult.commute pos_less_divide_eq)
finally have "\representation B (x'-x) b\ \ e" by simp
then show ?thesis
by (simp add: \<open>x \<in> span B\<close> \<open>independent B\<close> representation_diff x')
qed
qed
qed
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Balls in Euclidean Space\<close>
lemma cball_subset_cball_iff:
fixes a :: "'a :: euclidean_space"
shows "cball a r \ cball a' r' \ dist a a' + r \ r' \ r < 0"
(is "?lhs \ ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "r < 0")
case True
then show ?rhs by simp
next
case False
then have [simp]: "r \ 0" by simp
have "norm (a - a') + r \ r'"
proof (cases "a = a'")
case True
then show ?thesis
using subsetD [where c = "a + r *\<^sub>R (SOME i. i \ Basis)", OF \?lhs\] subsetD [where c = a, OF \?lhs\]
by (force simp: SOME_Basis dist_norm)
next
case False
have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
by (simp add: algebra_simps)
also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
by (simp add: algebra_simps)
also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"
by simp (simp add: field_simps)
finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \norm (a - a') + r\"
by linarith
from \<open>a \<noteq> a'\<close> show ?thesis
using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \?lhs\]
by (simp add: dist_norm scaleR_add_left)
qed
then show ?rhs
by (simp add: dist_norm)
qed
qed metric
lemma cball_subset_ball_iff: "cball a r \ ball a' r' \ dist a a' + r < r' \ r < 0"
(is "?lhs \ ?rhs")
for a :: "'a::euclidean_space"
proof
assume ?lhs
then show ?rhs
proof (cases "r < 0")
case True then
show ?rhs by simp
next
case False
then have [simp]: "r \ 0" by simp
have "norm (a - a') + r < r'"
proof (cases "a = a'")
case True
then show ?thesis
using subsetD [where c = "a + r *\<^sub>R (SOME i. i \ Basis)", OF \?lhs\] subsetD [where c = a, OF \?lhs\]
by (force simp: SOME_Basis dist_norm)
next
case False
have False if "norm (a - a') + r \ r'"
proof -
from that have "\r' - norm (a - a')\ \ r"
by (simp split: abs_split)
(metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
then show ?thesis
using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \?lhs\] \a \ a'\
apply (simp add: dist_norm)
apply (simp add: scaleR_left_diff_distrib)
apply (simp add: field_simps)
done
qed
then show ?thesis by force
qed
then show ?rhs by (simp add: dist_norm)
qed
next
assume ?rhs
then show ?lhs
by metric
qed
lemma ball_subset_cball_iff: "ball a r \ cball a' r' \ dist a a' + r \ r' \ r \ 0"
(is "?lhs = ?rhs")
for a :: "'a::euclidean_space"
proof (cases "r \ 0")
case True
then show ?thesis
by metric
next
case False
show ?thesis
proof
assume ?lhs
then have "(cball a r \ cball a' r')"
by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
with False show ?rhs
by (fastforce iff: cball_subset_cball_iff)
next
assume ?rhs
with False show ?lhs
by metric
qed
qed
lemma ball_subset_ball_iff:
fixes a :: "'a :: euclidean_space"
shows "ball a r \ ball a' r' \ dist a a' + r \ r' \ r \ 0"
(is "?lhs = ?rhs")
proof (cases "r \ 0")
case True then show ?thesis
by metric
next
case False show ?thesis
proof
assume ?lhs
then have "0 < r'"
using False by metric
then have "(cball a r \ cball a' r')"
by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
then show ?rhs
using False cball_subset_cball_iff by fastforce
qed metric
qed
lemma ball_eq_ball_iff:
fixes x :: "'a :: euclidean_space"
shows "ball x d = ball y e \ d \ 0 \ e \ 0 \ x=y \ d=e"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "d \ 0 \ e \ 0")
case True
with \<open>?lhs\<close> show ?rhs
by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
next
case False
with \<open>?lhs\<close> show ?rhs
apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
done
qed
next
assume ?rhs then show ?lhs
by (auto simp: set_eq_subset ball_subset_ball_iff)
qed
lemma cball_eq_cball_iff:
fixes x :: "'a :: euclidean_space"
shows "cball x d = cball y e \ d < 0 \ e < 0 \ x=y \ d=e"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "d < 0 \ e < 0")
case True
with \<open>?lhs\<close> show ?rhs
by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
next
case False
with \<open>?lhs\<close> show ?rhs
apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
done
qed
next
assume ?rhs then show ?lhs
by (auto simp: set_eq_subset cball_subset_cball_iff)
qed
lemma ball_eq_cball_iff:
fixes x :: "'a :: euclidean_space"
shows "ball x d = cball y e \ d \ 0 \ e < 0" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
done
next
assume ?rhs then show ?lhs by auto
qed
lemma cball_eq_ball_iff:
fixes x :: "'a :: euclidean_space"
shows "cball x d = ball y e \ d < 0 \ e \ 0"
using ball_eq_cball_iff by blast
lemma finite_ball_avoid:
fixes S :: "'a :: euclidean_space set"
assumes "open S" "finite X" "p \ S"
shows "\e>0. \w\ball p e. w\S \ (w\p \ w\X)"
proof -
obtain e1 where "0 < e1" and e1_b:"ball p e1 \ S"
using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto
obtain e2 where "0 < e2" and "\x\X. x \ p \ e2 \ dist p x"
using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto
hence "\w\ball p (min e1 e2). w\S \ (w\p \ w\X)" using e1_b by auto
thus "\e>0. \w\ball p e. w \ S \ (w \ p \ w \ X)" using \e2>0\ \e1>0\
apply (rule_tac x="min e1 e2" in exI)
by auto
qed
lemma finite_cball_avoid:
fixes S :: "'a :: euclidean_space set"
assumes "open S" "finite X" "p \ S"
shows "\e>0. \w\cball p e. w\S \ (w\p \ w\X)"
proof -
obtain e1 where "e1>0" and e1: "\w\ball p e1. w\S \ (w\p \ w\X)"
using finite_ball_avoid[OF assms] by auto
define e2 where "e2 \ e1/2"
have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto
then have "cball p e2 \ ball p e1" by (subst cball_subset_ball_iff,auto)
then show "\e>0. \w\cball p e. w \ S \ (w \ p \ w \ X)" using \e2>0\ e1 by auto
qed
lemma dim_cball:
assumes "e > 0"
shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
proof -
{
fix x :: "'n::euclidean_space"
define y where "y = (e / norm x) *\<^sub>R x"
then have "y \ cball 0 e"
using assms by auto
moreover have *: "x = (norm x / e) *\<^sub>R y"
using y_def assms by simp
moreover from * have "x = (norm x/e) *\<^sub>R y"
by auto
ultimately have "x \ span (cball 0 e)"
using span_scale[of y "cball 0 e" "norm x/e"]
span_superset[of "cball 0 e"]
by (simp add: span_base)
}
then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
by auto
then show ?thesis
using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto)
qed
subsection \<open>Boxes\<close>
abbreviation\<^marker>\<open>tag important\<close> One :: "'a::euclidean_space" where
"One \ \Basis"
lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
proof -
have "dependent (Basis :: 'a set)"
apply (simp add: dependent_finite)
apply (rule_tac x="\i. 1" in exI)
using SOME_Basis apply (auto simp: assms)
done
with independent_Basis show False by force
qed
corollary\<^marker>\<open>tag unimportant\<close> One_neq_0[iff]: "One \<noteq> 0"
by (metis One_non_0)
corollary\<^marker>\<open>tag unimportant\<close> Zero_neq_One[iff]: "0 \<noteq> One"
by (metis One_non_0)
definition\<^marker>\<open>tag important\<close> (in euclidean_space) eucl_less (infix "<e" 50) where
"eucl_less a b \ (\i\Basis. a \ i < b \ i)"
definition\<^marker>\<open>tag important\<close> box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
definition\<^marker>\<open>tag important\<close> "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
lemma box_def: "box a b = {x. \i\Basis. a \ i < x \ i \ x \ i < b \ i}"
and in_box_eucl_less: "x \ box a b \ a x
and mem_box: "x \ box a b \ (\i\Basis. a \ i < x \ i \ x \ i < b \ i)"
"x \ cbox a b \ (\i\Basis. a \ i \ x \ i \ x \ i \ b \ i)"
by (auto simp: box_eucl_less eucl_less_def cbox_def)
lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b \ cbox c d"
by (force simp: cbox_def Basis_prod_def)
lemma cbox_Pair_iff [iff]: "(x, y) \ cbox (a, c) (b, d) \ x \ cbox a b \ y \ cbox c d"
by (force simp: cbox_Pair_eq)
lemma cbox_Complex_eq: "cbox (Complex a c) (Complex b d) = (\(x,y). Complex x y) ` (cbox a b \ cbox c d)"
apply (auto simp: cbox_def Basis_complex_def)
apply (rule_tac x = "(Re x, Im x)" in image_eqI)
using complex_eq by auto
lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} \ cbox a b = {} \ cbox c d = {}"
by (force simp: cbox_Pair_eq)
lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
by auto
lemma mem_box_real[simp]:
"(x::real) \ box a b \ a < x \ x < b"
"(x::real) \ cbox a b \ a \ x \ x \ b"
by (auto simp: mem_box)
lemma box_real[simp]:
fixes a b:: real
shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
by auto
lemma box_Int_box:
fixes a :: "'a::euclidean_space"
shows "box a b \ box c d =
box (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
unfolding set_eq_iff and Int_iff and mem_box by auto
lemma rational_boxes:
fixes x :: "'a::euclidean_space"
assumes "e > 0"
shows "\a b. (\i\Basis. a \ i \ \ \ b \ i \ \) \ x \ box a b \ box a b \ ball x e"
proof -
define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
then have e: "e' > 0"
using assms by (auto)
have "\i. \y. y \ \ \ y < x \ i \ x \ i - y < e'" (is "\i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x \ i - e'" "x \ i"] e
show "?th i" by auto
qed
from choice[OF this] obtain a where
a: "\xa. a xa \ \ \ a xa < x \ xa \ x \ xa - a xa < e'" ..
have "\i. \y. y \ \ \ x \ i < y \ y - x \ i < e'" (is "\i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x \ i" "x \ i + e'"] e
show "?th i" by auto
qed
from choice[OF this] obtain b where
b: "\xa. b xa \ \ \ x \ xa < b xa \ b xa - x \ xa < e'" ..
let ?a = "\i\Basis. a i *\<^sub>R i" and ?b = "\i\Basis. b i *\<^sub>R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a
assume *: "y \ box ?a ?b"
have "dist x y = sqrt (\i\Basis. (dist (x \ i) (y \ i))\<^sup>2)"
unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
also have "\ < sqrt (\(i::'a)\Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule sum_strict_mono)
fix i :: "'a"
assume i: "i \ Basis"
have "a i < y\i \ y\i < b i"
using * i by (auto simp: box_def)
moreover have "a i < x\i" "x\i - a i < e'"
using a by auto
moreover have "x\i < b i" "b i - x\i < e'"
using b by auto
ultimately have "\x\i - y\i\ < 2 * e'"
by auto
then have "dist (x \ i) (y \ i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x \ i) (y \ i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
by (rule power_strict_mono) auto
then show "(dist (x \ i) (y \ i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
by (simp add: power_divide)
qed auto
also have "\ = e"
using \<open>0 < e\<close> by simp
finally show "y \ ball x e"
by (auto simp: ball_def)
qed (insert a b, auto simp: box_def)
qed
lemma open_UNION_box:
fixes M :: "'a::euclidean_space set"
assumes "open M"
defines "a' \ \f :: 'a \ real \ real. (\(i::'a)\Basis. fst (f i) *\<^sub>R i)"
defines "b' \ \f :: 'a \ real \ real. (\(i::'a)\Basis. snd (f i) *\<^sub>R i)"
defines "I \ {f\Basis \\<^sub>E \ \ \. box (a' f) (b' f) \ M}"
shows "M = (\f\I. box (a' f) (b' f))"
proof -
have "x \ (\f\I. box (a' f) (b' f))" if "x \ M" for x
proof -
obtain e where e: "e > 0" "ball x e \ M"
using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
moreover obtain a b where ab:
"x \ box a b"
"\i \ Basis. a \ i \ \"
"\i\Basis. b \ i \ \"
"box a b \ ball x e"
using rational_boxes[OF e(1)] by metis
ultimately show ?thesis
by (intro UN_I[of "\i\Basis. (a \ i, b \ i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
qed
then show ?thesis by (auto simp: I_def)
qed
corollary open_countable_Union_open_box:
fixes S :: "'a :: euclidean_space set"
assumes "open S"
obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = S"
proof -
let ?a = "\f. (\(i::'a)\Basis. fst (f i) *\<^sub>R i)"
let ?b = "\f. (\(i::'a)\Basis. snd (f i) *\<^sub>R i)"
let ?I = "{f\Basis \\<^sub>E \ \ \. box (?a f) (?b f) \ S}"
let ?\<D> = "(\<lambda>f. box (?a f) (?b f)) ` ?I"
show ?thesis
proof
have "countable ?I"
by (simp add: countable_PiE countable_rat)
then show "countable ?\"
by blast
show "\?\ = S"
using open_UNION_box [OF assms] by metis
qed auto
qed
lemma rational_cboxes:
fixes x :: "'a::euclidean_space"
assumes "e > 0"
shows "\a b. (\i\Basis. a \ i \ \ \ b \ i \ \) \ x \ cbox a b \ cbox a b \ ball x e"
proof -
define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
then have e: "e' > 0"
using assms by auto
have "\i. \y. y \ \ \ y < x \ i \ x \ i - y < e'" (is "\i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x \ i - e'" "x \ i"] e
show "?th i" by auto
qed
from choice[OF this] obtain a where
a: "\u. a u \ \ \ a u < x \ u \ x \ u - a u < e'" ..
have "\i. \y. y \ \ \ x \ i < y \ y - x \ i < e'" (is "\i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x \ i" "x \ i + e'"] e
show "?th i" by auto
qed
from choice[OF this] obtain b where
b: "\u. b u \ \ \ x \ u < b u \ b u - x \ u < e'" ..
let ?a = "\i\Basis. a i *\<^sub>R i" and ?b = "\i\Basis. b i *\<^sub>R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a
assume *: "y \ cbox ?a ?b"
have "dist x y = sqrt (\i\Basis. (dist (x \ i) (y \ i))\<^sup>2)"
unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
also have "\ < sqrt (\(i::'a)\Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule sum_strict_mono)
fix i :: "'a"
assume i: "i \ Basis"
have "a i \ y\i \ y\i \ b i"
using * i by (auto simp: cbox_def)
moreover have "a i < x\i" "x\i - a i < e'"
using a by auto
moreover have "x\i < b i" "b i - x\i < e'"
using b by auto
ultimately have "\x\i - y\i\ < 2 * e'"
by auto
then have "dist (x \ i) (y \ i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x \ i) (y \ i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
by (rule power_strict_mono) auto
then show "(dist (x \ i) (y \ i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
by (simp add: power_divide)
qed auto
also have "\ = e"
using \<open>0 < e\<close> by simp
finally show "y \ ball x e"
by (auto simp: ball_def)
next
show "x \ cbox (\i\Basis. a i *\<^sub>R i) (\i\Basis. b i *\<^sub>R i)"
using a b less_imp_le by (auto simp: cbox_def)
qed (use a b cbox_def in auto)
qed
lemma open_UNION_cbox:
fixes M :: "'a::euclidean_space set"
assumes "open M"
defines "a' \ \f. (\(i::'a)\Basis. fst (f i) *\<^sub>R i)"
defines "b' \ \f. (\(i::'a)\Basis. snd (f i) *\<^sub>R i)"
defines "I \ {f\Basis \\<^sub>E \ \ \. cbox (a' f) (b' f) \ M}"
shows "M = (\f\I. cbox (a' f) (b' f))"
proof -
have "x \ (\f\I. cbox (a' f) (b' f))" if "x \ M" for x
proof -
obtain e where e: "e > 0" "ball x e \ M"
using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
moreover obtain a b where ab: "x \ cbox a b" "\i \ Basis. a \ i \ \"
"\i \ Basis. b \ i \ \" "cbox a b \ ball x e"
using rational_cboxes[OF e(1)] by metis
ultimately show ?thesis
by (intro UN_I[of "\i\Basis. (a \ i, b \ i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
qed
then show ?thesis by (auto simp: I_def)
qed
corollary open_countable_Union_open_cbox:
fixes S :: "'a :: euclidean_space set"
assumes "open S"
obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = S"
proof -
let ?a = "\f. (\(i::'a)\Basis. fst (f i) *\<^sub>R i)"
let ?b = "\f. (\(i::'a)\Basis. snd (f i) *\<^sub>R i)"
let ?I = "{f\Basis \\<^sub>E \ \ \. cbox (?a f) (?b f) \ S}"
let ?\<D> = "(\<lambda>f. cbox (?a f) (?b f)) ` ?I"
show ?thesis
proof
have "countable ?I"
by (simp add: countable_PiE countable_rat)
then show "countable ?\"
by blast
show "\?\ = S"
using open_UNION_cbox [OF assms] by metis
qed auto
qed
lemma box_eq_empty:
fixes a :: "'a::euclidean_space"
shows "(box a b = {} \ (\i\Basis. b\i \ a\i))" (is ?th1)
and "(cbox a b = {} \ (\i\Basis. b\i < a\i))" (is ?th2)
proof -
{
fix i x
assume i: "i\Basis" and as:"b\i \ a\i" and x:"x\box a b"
then have "a \ i < x \ i \ x \ i < b \ i"
unfolding mem_box by (auto simp: box_def)
then have "a\i < b\i" by auto
then have False using as by auto
}
moreover
{
assume as: "\i\Basis. \ (b\i \ a\i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{
fix i :: 'a
assume i: "i \ Basis"
have "a\i < b\i"
using as[THEN bspec[where x=i]] i by auto
then have "a\i < ((1/2) *\<^sub>R (a+b)) \ i" "((1/2) *\<^sub>R (a+b)) \ i < b\i"
by (auto simp: inner_add_left)
}
then have "box a b \ {}"
using mem_box(1)[of "?x" a b] by auto
}
ultimately show ?th1 by blast
{
fix i x
assume i: "i \ Basis" and as:"b\i < a\i" and x:"x\cbox a b"
then have "a \ i \ x \ i \ x \ i \ b \ i"
unfolding mem_box by auto
then have "a\i \ b\i" by auto
then have False using as by auto
}
moreover
{
assume as:"\i\Basis. \ (b\i < a\i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{
fix i :: 'a
assume i:"i \ Basis"
have "a\i \ b\i"
using as[THEN bspec[where x=i]] i by auto
then have "a\i \ ((1/2) *\<^sub>R (a+b)) \ i" "((1/2) *\<^sub>R (a+b)) \ i \ b\i"
by (auto simp: inner_add_left)
}
then have "cbox a b \ {}"
using mem_box(2)[of "?x" a b] by auto
}
ultimately show ?th2 by blast
qed
lemma box_ne_empty:
fixes a :: "'a::euclidean_space"
shows "cbox a b \ {} \ (\i\Basis. a\i \ b\i)"
and "box a b \ {} \ (\i\Basis. a\i < b\i)"
unfolding box_eq_empty[of a b] by fastforce+
lemma
fixes a :: "'a::euclidean_space"
shows cbox_sing [simp]: "cbox a a = {a}"
and box_sing [simp]: "box a a = {}"
unfolding set_eq_iff mem_box eq_iff [symmetric]
by (auto intro!: euclidean_eqI[where 'a='a])
(metis all_not_in_conv nonempty_Basis)
lemma subset_box_imp:
fixes a :: "'a::euclidean_space"
shows "(\i\Basis. a\i \ c\i \ d\i \ b\i) \ cbox c d \ cbox a b"
and "(\i\Basis. a\i < c\i \ d\i < b\i) \ cbox c d \ box a b"
and "(\i\Basis. a\i \ c\i \ d\i \ b\i) \ box c d \ cbox a b"
and "(\i\Basis. a\i \ c\i \ d\i \ b\i) \ box c d \ box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_box
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
lemma box_subset_cbox:
fixes a :: "'a::euclidean_space"
shows "box a b \ cbox a b"
unfolding subset_eq [unfolded Ball_def] mem_box
by (fast intro: less_imp_le)
lemma subset_box:
fixes a :: "'a::euclidean_space"
shows "cbox c d \ cbox a b \ (\i\Basis. c\i \ d\i) \ (\i\Basis. a\i \ c\i \ d\i \ b\i)" (is ?th1)
and "cbox c d \ box a b \ (\i\Basis. c\i \ d\i) \ (\i\Basis. a\i < c\i \ d\i < b\i)" (is ?th2)
and "box c d \ cbox a b \ (\i\Basis. c\i < d\i) \ (\i\Basis. a\i \ c\i \ d\i \ b\i)" (is ?th3)
and "box c d \ box a b \ (\i\Basis. c\i < d\i) \ (\i\Basis. a\i \ c\i \ d\i \ b\i)" (is ?th4)
proof -
let ?lesscd = "\i\Basis. c\i < d\i"
let ?lerhs = "\i\Basis. a\i \ c\i \ d\i \ b\i"
show ?th1 ?th2
by (fastforce simp: mem_box)+
have acdb: "a\i \ c\i \ d\i \ b\i"
if i: "i \ Basis" and box: "box c d \ cbox a b" and cd: "\i. i \ Basis \ c\i < d\i" for i
proof -
have "box c d \ {}"
using that
unfolding box_eq_empty by force
{ let ?x = "(\j\Basis. (if j=i then ((min (a\j) (d\j))+c\j)/2 else (c\j+d\j)/2) *\<^sub>R j)::'a"
assume *: "a\i > c\i"
then have "c \ j < ?x \ j \ ?x \ j < d \ j" if "j \ Basis" for j
using cd that by (fastforce simp add: i *)
then have "?x \ box c d"
unfolding mem_box by auto
moreover have "?x \ cbox a b"
using i cd * by (force simp: mem_box)
ultimately have False using box by auto
}
then have "a\i \ c\i" by force
moreover
{ let ?x = "(\j\Basis. (if j=i then ((max (b\j) (c\j))+d\j)/2 else (c\j+d\j)/2) *\<^sub>R j)::'a"
assume *: "b\i < d\i"
then have "d \ j > ?x \ j \ ?x \ j > c \ j" if "j \ Basis" for j
using cd that by (fastforce simp add: i *)
then have "?x \ box c d"
unfolding mem_box by auto
moreover have "?x \ cbox a b"
using i cd * by (force simp: mem_box)
ultimately have False using box by auto
}
then have "b\i \ d\i" by (rule ccontr) auto
ultimately show ?thesis by auto
qed
show ?th3
using acdb by (fastforce simp add: mem_box)
have acdb': "a\i \ c\i \ d\i \ b\i"
if "i \ Basis" "box c d \ box a b" "\i. i \ Basis \ c\i < d\i" for i
using box_subset_cbox[of a b] that acdb by auto
show ?th4
using acdb' by (fastforce simp add: mem_box)
qed
lemma eq_cbox: "cbox a b = cbox c d \ cbox a b = {} \ cbox c d = {} \ a = c \ b = d"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "cbox a b \ cbox c d" "cbox c d \ cbox a b"
by auto
then show ?rhs
by (force simp: subset_box box_eq_empty intro: antisym euclidean_eqI)
next
assume ?rhs
then show ?lhs
by force
qed
lemma eq_cbox_box [simp]: "cbox a b = box c d \ cbox a b = {} \ box c d = {}"
(is "?lhs \ ?rhs")
proof
assume L: ?lhs
then have "cbox a b \ box c d" "box c d \ cbox a b"
by auto
then show ?rhs
apply (simp add: subset_box)
using L box_ne_empty box_sing apply (fastforce simp add:)
done
qed force
lemma eq_box_cbox [simp]: "box a b = cbox c d \ box a b = {} \ cbox c d = {}"
by (metis eq_cbox_box)
lemma eq_box: "box a b = box c d \ box a b = {} \ box c d = {} \ a = c \ b = d"
(is "?lhs \ ?rhs")
proof
assume L: ?lhs
then have "box a b \ box c d" "box c d \ box a b"
by auto
then show ?rhs
apply (simp add: subset_box)
using box_ne_empty(2) L
apply auto
apply (meson euclidean_eqI less_eq_real_def not_less)+
done
qed force
lemma subset_box_complex:
"cbox a b \ cbox c d \
(Re a \<le> Re b \<and> Im a \<le> Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
"cbox a b \ box c d \
(Re a \<le> Re b \<and> Im a \<le> Im b) \<longrightarrow> Re a > Re c \<and> Im a > Im c \<and> Re b < Re d \<and> Im b < Im d"
"box a b \ cbox c d \
(Re a < Re b \<and> Im a < Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
"box a b \ box c d \
(Re a < Re b \<and> Im a < Im b) \<longrightarrow> Re a \<ge> Re c \<and> Im a \<ge> Im c \<and> Re b \<le> Re d \<and> Im b \<le> Im d"
by (subst subset_box; force simp: Basis_complex_def)+
lemma in_cbox_complex_iff:
"x \ cbox a b \ Re x \ {Re a..Re b} \ Im x \ {Im a..Im b}"
by (cases x; cases a; cases b) (auto simp: cbox_Complex_eq)
lemma box_Complex_eq:
"box (Complex a c) (Complex b d) = (\(x,y). Complex x y) ` (box a b \ box c d)"
by (auto simp: box_def Basis_complex_def image_iff complex_eq_iff)
lemma in_box_complex_iff:
"x \ box a b \ Re x \ {Re a<.. Im x \ {Im a<..
by (cases x; cases a; cases b) (auto simp: box_Complex_eq)
lemma Int_interval:
fixes a :: "'a::euclidean_space"
shows "cbox a b \ cbox c d =
cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
unfolding set_eq_iff and Int_iff and mem_box
by auto
lemma disjoint_interval:
fixes a::"'a::euclidean_space"
shows "cbox a b \ cbox c d = {} \ (\i\Basis. (b\i < a\i \ d\i < c\i \ b\i < c\i \ d\i < a\i))" (is ?th1)
and "cbox a b \ box c d = {} \ (\i\Basis. (b\i < a\i \ d\i \ c\i \ b\i \ c\i \ d\i \ a\i))" (is ?th2)
and "box a b \ cbox c d = {} \ (\i\Basis. (b\i \ a\i \ d\i < c\i \ b\i \ c\i \ d\i \ a\i))" (is ?th3)
and "box a b \ box c d = {} \ (\i\Basis. (b\i \ a\i \ d\i \ c\i \ b\i \ c\i \ d\i \ a\i))" (is ?th4)
proof -
let ?z = "(\i\Basis. (((max (a\i) (c\i)) + (min (b\i) (d\i))) / 2) *\<^sub>R i)::'a"
have **: "\P Q. (\i :: 'a. i \ Basis \ Q ?z i \ P i) \
(\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
show ?th1 unfolding * by (intro **) auto
show ?th2 unfolding * by (intro **) auto
show ?th3 unfolding * by (intro **) auto
show ?th4 unfolding * by (intro **) auto
qed
lemma UN_box_eq_UNIV: "(\i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
proof -
have "\x \ b\ < real_of_int (\Max ((\b. \x \ b\)`Basis)\ + 1)"
if [simp]: "b \ Basis" for x b :: 'a
proof -
have "\x \ b\ \ real_of_int \\x \ b\\"
by (rule le_of_int_ceiling)
also have "\ \ real_of_int \Max ((\b. \x \ b\)`Basis)\"
by (auto intro!: ceiling_mono)
also have "\ < real_of_int (\Max ((\b. \x \ b\)`Basis)\ + 1)"
by simp
finally show ?thesis .
qed
then have "\n::nat. \b\Basis. \x \ b\ < real n" for x :: 'a
by (metis order.strict_trans reals_Archimedean2)
moreover have "\x b::'a. \n::nat. \x \ b\ < real n \ - real n < x \ b \ x \ b < real n"
by auto
ultimately show ?thesis
by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
qed
lemma image_affinity_cbox: fixes m::real
fixes a b c :: "'a::euclidean_space"
shows "(\x. m *\<^sub>R x + c) ` cbox a b =
(if cbox a b = {} then {}
else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
proof (cases "m = 0")
case True
{
fix x
assume "\i\Basis. x \ i \ c \ i" "\i\Basis. c \ i \ x \ i"
then have "x = c"
by (simp add: dual_order.antisym euclidean_eqI)
}
moreover have "c \ cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
unfolding True by (auto)
ultimately show ?thesis using True by (auto simp: cbox_def)
next
case False
{
fix y
assume "\i\Basis. a \ i \ y \ i" "\i\Basis. y \ i \ b \ i" "m > 0"
then have "\i\Basis. (m *\<^sub>R a + c) \ i \ (m *\<^sub>R y + c) \ i" and "\i\Basis. (m *\<^sub>R y + c) \ i \ (m *\<^sub>R b + c) \ i"
by (auto simp: inner_distrib)
}
moreover
{
fix y
assume "\i\Basis. a \ i \ y \ i" "\i\Basis. y \ i \ b \ i" "m < 0"
then have "\i\Basis. (m *\<^sub>R b + c) \ i \ (m *\<^sub>R y + c) \ i" and "\i\Basis. (m *\<^sub>R y + c) \ i \ (m *\<^sub>R a + c) \ i"
by (auto simp: mult_left_mono_neg inner_distrib)
}
moreover
{
fix y
assume "m > 0" and "\i\Basis. (m *\<^sub>R a + c) \ i \ y \ i" and "\i\Basis. y \ i \ (m *\<^sub>R b + c) \ i"
then have "y \ (\x. m *\<^sub>R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
done
}
moreover
{
fix y
assume "\i\Basis. (m *\<^sub>R b + c) \ i \ y \ i" "\i\Basis. y \ i \ (m *\<^sub>R a + c) \ i" "m < 0"
then have "y \ (\x. m *\<^sub>R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
done
}
ultimately show ?thesis using False by (auto simp: cbox_def)
qed
lemma image_smult_cbox:"(\x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
(if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
using image_affinity_cbox[of m 0 a b] by auto
lemma swap_continuous:
assumes "continuous_on (cbox (a,c) (b,d)) (\(x,y). f x y)"
shows "continuous_on (cbox (c,a) (d,b)) (\(x, y). f y x)"
proof -
have "(\(x, y). f y x) = (\(x, y). f x y) \ prod.swap"
by auto
then show ?thesis
apply (rule ssubst)
apply (rule continuous_on_compose)
apply (simp add: split_def)
apply (rule continuous_intros | simp add: assms)+
done
qed
subsection \<open>General Intervals\<close>
definition\<^marker>\<open>tag important\<close> "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"
lemma is_interval_1:
"is_interval (s::real set) \ (\a\s. \b\s. \ x. a \ x \ x \ b \ x \ s)"
unfolding is_interval_def by auto
lemma is_interval_Int: "is_interval X \ is_interval Y \ is_interval (X \ Y)"
unfolding is_interval_def
by blast
lemma is_interval_cbox [simp]: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
and is_interval_box [simp]: "is_interval (box a b)" (is ?th2)
unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
by (meson order_trans le_less_trans less_le_trans less_trans)+
lemma is_interval_empty [iff]: "is_interval {}"
unfolding is_interval_def by simp
lemma is_interval_univ [iff]: "is_interval UNIV"
unfolding is_interval_def by simp
lemma mem_is_intervalI:
assumes "is_interval s"
and "a \ s" "b \ s"
and "\i. i \ Basis \ a \ i \ x \ i \ x \ i \ b \ i \ b \ i \ x \ i \ x \ i \ a \ i"
shows "x \ s"
by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
lemma interval_subst:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
and "x \ S" "y j \ S"
and "j \ Basis"
shows "(\i\Basis. (if i = j then y i \ i else x \ i) *\<^sub>R i) \ S"
by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
lemma mem_box_componentwiseI:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
assumes "\i. i \ Basis \ x \ i \ ((\x. x \ i) ` S)"
shows "x \ S"
proof -
from assms have "\i \ Basis. \s \ S. x \ i = s \ i"
by auto
with finite_Basis obtain s and bs::"'a list"
where s: "\i. i \ Basis \ x \ i = s i \ i" "\i. i \ Basis \ s i \ S"
and bs: "set bs = Basis" "distinct bs"
by (metis finite_distinct_list)
from nonempty_Basis s obtain j where j: "j \ Basis" "s j \ S"
by blast
define y where
"y = rec_list (s j) (\j _ Y. (\i\Basis. (if i = j then s i \ i else Y \ i) *\<^sub>R i))"
have "x = (\i\Basis. (if i \ set bs then s i \ i else s j \ i) *\<^sub>R i)"
using bs by (auto simp: s(1)[symmetric] euclidean_representation)
also have [symmetric]: "y bs = \"
using bs(2) bs(1)[THEN equalityD1]
by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
also have "y bs \ S"
using bs(1)[THEN equalityD1]
apply (induct bs)
apply (auto simp: y_def j)
apply (rule interval_subst[OF assms(1)])
apply (auto simp: s)
done
finally show ?thesis .
qed
lemma cbox01_nonempty [simp]: "cbox 0 One \ {}"
by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)
lemma box01_nonempty [simp]: "box 0 One \ {}"
by (simp add: box_ne_empty inner_Basis inner_sum_left)
lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast
lemma interval_subset_is_interval:
assumes "is_interval S"
shows "cbox a b \ S \ cbox a b = {} \ a \ S \ b \ S" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs using box_ne_empty(1) mem_box(2) by fastforce
next
assume ?rhs
have "cbox a b \ S" if "a \ S" "b \ S"
using assms unfolding is_interval_def
apply (clarsimp simp add: mem_box)
using that by blast
with \<open>?rhs\<close> show ?lhs
by blast
qed
lemma is_real_interval_union:
"is_interval (X \ Y)"
if X: "is_interval X" and Y: "is_interval Y" and I: "(X \ {} \ Y \ {} \ X \ Y \ {})"
for X Y::"real set"
proof -
consider "X \ {}" "Y \ {}" | "X = {}" | "Y = {}" by blast
then show ?thesis
proof cases
case 1
then obtain r where "r \ X \ X \ Y = {}" "r \ Y \ X \ Y = {}"
by blast
then show ?thesis
using I 1 X Y unfolding is_interval_1
by (metis (full_types) Un_iff le_cases)
qed (use that in auto)
qed
lemma is_interval_translationI:
assumes "is_interval X"
shows "is_interval ((+) x ` X)"
unfolding is_interval_def
proof safe
fix b d e
assume "b \ X" "d \ X"
"\i\Basis. (x + b) \ i \ e \ i \ e \ i \ (x + d) \ i \
(x + d) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (x + b) \<bullet> i"
hence "e - x \ X"
by (intro mem_is_intervalI[OF assms \<open>b \<in> X\<close> \<open>d \<in> X\<close>, of "e - x"])
(auto simp: algebra_simps)
thus "e \ (+) x ` X" by force
qed
lemma is_interval_uminusI:
assumes "is_interval X"
shows "is_interval (uminus ` X)"
unfolding is_interval_def
proof safe
fix b d e
assume "b \ X" "d \ X"
"\i\Basis. (- b) \ i \ e \ i \ e \ i \ (- d) \ i \
(- d) \<bullet> i \<le> e \<bullet> i \<and> e \<bullet> i \<le> (- b) \<bullet> i"
hence "- e \ X"
by (intro mem_is_intervalI[OF assms \<open>b \<in> X\<close> \<open>d \<in> X\<close>, of "- e"])
(auto simp: algebra_simps)
thus "e \ uminus ` X" by force
qed
lemma is_interval_uminus[simp]: "is_interval (uminus ` x) = is_interval x"
using is_interval_uminusI[of x] is_interval_uminusI[of "uminus ` x"]
by (auto simp: image_image)
lemma is_interval_neg_translationI:
assumes "is_interval X"
shows "is_interval ((-) x ` X)"
proof -
have "(-) x ` X = (+) x ` uminus ` X"
by (force simp: algebra_simps)
also have "is_interval \"
by (metis is_interval_uminusI is_interval_translationI assms)
finally show ?thesis .
qed
lemma is_interval_translation[simp]:
"is_interval ((+) x ` X) = is_interval X"
using is_interval_neg_translationI[of "(+) x ` X" x]
by (auto intro!: is_interval_translationI simp: image_image)
lemma is_interval_minus_translation[simp]:
shows "is_interval ((-) x ` X) = is_interval X"
proof -
have "(-) x ` X = (+) x ` uminus ` X"
by (force simp: algebra_simps)
also have "is_interval \ = is_interval X"
by simp
finally show ?thesis .
qed
lemma is_interval_minus_translation'[simp]:
shows "is_interval ((\x. x - c) ` X) = is_interval X"
using is_interval_translation[of "-c" X]
by (metis image_cong uminus_add_conv_diff)
lemma is_interval_cball_1[intro, simp]: "is_interval (cball a b)" for a b::real
by (simp add: cball_eq_atLeastAtMost is_interval_def)
lemma is_interval_ball_real: "is_interval (ball a b)" for a b::real
by (simp add: ball_eq_greaterThanLessThan is_interval_def)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounded Projections\<close>
lemma bounded_inner_imp_bdd_above:
assumes "bounded s"
shows "bdd_above ((\x. x \ a) ` s)"
by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)
lemma bounded_inner_imp_bdd_below:
assumes "bounded s"
shows "bdd_below ((\x. x \ a) ` s)"
by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for pointwise continuity\<close>
lemma continuous_infnorm[continuous_intros]:
"continuous F f \ continuous F (\x. infnorm (f x))"
unfolding continuous_def by (rule tendsto_infnorm)
lemma continuous_inner[continuous_intros]:
assumes "continuous F f"
and "continuous F g"
shows "continuous F (\x. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for setwise continuity\<close>
lemma continuous_on_infnorm[continuous_intros]:
"continuous_on s f \ continuous_on s (\x. infnorm (f x))"
unfolding continuous_on by (fast intro: tendsto_infnorm)
lemma continuous_on_inner[continuous_intros]:
fixes g :: "'a::topological_space \ 'b::real_inner"
assumes "continuous_on s f"
and "continuous_on s g"
shows "continuous_on s (\x. inner (f x) (g x))"
using bounded_bilinear_inner assms
by (rule bounded_bilinear.continuous_on)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Openness of halfspaces.\<close>
lemma open_halfspace_lt: "open {x. inner a x < b}"
by (simp add: open_Collect_less continuous_on_inner)
lemma open_halfspace_gt: "open {x. inner a x > b}"
by (simp add: open_Collect_less continuous_on_inner)
lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x\i < a}"
by (simp add: open_Collect_less continuous_on_inner)
lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x\i > a}"
by (simp add: open_Collect_less continuous_on_inner)
lemma eucl_less_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. x i\Basis. {x. x \ i < a \ i})"
"{x. a i\Basis. {x. a \ i < x \ i})"
by (auto simp: eucl_less_def)
lemma open_Collect_eucl_less[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "open {x. x "open {x. a
by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Closure and Interior of halfspaces and hyperplanes\<close>
lemma continuous_at_inner: "continuous (at x) (inner a)"
unfolding continuous_at by (intro tendsto_intros)
lemma closed_halfspace_le: "closed {x. inner a x \ b}"
by (simp add: closed_Collect_le continuous_on_inner)
lemma closed_halfspace_ge: "closed {x. inner a x \ b}"
by (simp add: closed_Collect_le continuous_on_inner)
lemma closed_hyperplane: "closed {x. inner a x = b}"
by (simp add: closed_Collect_eq continuous_on_inner)
lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\i \ a}"
by (simp add: closed_Collect_le continuous_on_inner)
lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\i \ a}"
by (simp add: closed_Collect_le continuous_on_inner)
lemma closed_interval_left:
fixes b :: "'a::euclidean_space"
shows "closed {x::'a. \i\Basis. x\i \ b\i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner)
lemma closed_interval_right:
fixes a :: "'a::euclidean_space"
shows "closed {x::'a. \i\Basis. a\i \ x\i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner)
lemma interior_halfspace_le [simp]:
assumes "a \ 0"
shows "interior {x. a \ x \ b} = {x. a \ x < b}"
proof -
have *: "a \ x < b" if x: "x \ S" and S: "S \ {x. a \ x \ b}" and "open S" for S x
proof -
obtain e where "e>0" and e: "cball x e \ S"
using \<open>open S\<close> open_contains_cball x by blast
then have "x + (e / norm a) *\<^sub>R a \ cball x e"
by (simp add: dist_norm)
then have "x + (e / norm a) *\<^sub>R a \ S"
using e by blast
then have "x + (e / norm a) *\<^sub>R a \ {x. a \ x \ b}"
using S by blast
moreover have "e * (a \ 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 \ 0 \ interior {x. a \ x \ b} = {x. a \ x > b}"
using interior_halfspace_le [of "-a" "-b"] by simp
lemma closure_halfspace_lt [simp]:
assumes "a \ 0"
shows "closure {x. a \ x < b} = {x. a \ x \ b}"
proof -
have [simp]: "-{x. a \ x < b} = {x. a \ x \ 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 \ 0 \ closure {x. a \ x > b} = {x. a \ x \ b}"
using closure_halfspace_lt [of "-a" "-b"] by simp
lemma interior_hyperplane [simp]:
assumes "a \ 0"
shows "interior {x. a \ x = b} = {}"
proof -
have [simp]: "{x. a \ x = b} = {x. a \ x \ b} \ {x. a \ x \ b}"
by (force simp:)
then show ?thesis
by (auto simp: assms)
qed
lemma frontier_halfspace_le:
assumes "a \ 0 \ b \ 0"
shows "frontier {x. a \ x \ b} = {x. a \ 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 \ 0 \ b \ 0"
shows "frontier {x. a \ x \ b} = {x. a \ 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 \ 0 \ b \ 0"
shows "frontier {x. a \ x < b} = {x. a \ 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 \ 0 \ b \ 0"
shows "frontier {x. a \ x > b} = {x. a \ 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
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some more convenient intermediate-value theorem formulations\<close>
lemma connected_ivt_hyperplane:
assumes "connected S" and xy: "x \ S" "y \ S" and b: "inner a x \ b" "b \ inner a y"
shows "\z \ S. inner a z = b"
proof (rule ccontr)
assume as:"\ (\z\S. inner a z = b)"
let ?A = "{x. inner a x < b}"
let ?B = "{x. inner a x > b}"
have "open ?A" "open ?B"
using open_halfspace_lt and open_halfspace_gt by auto
moreover have "?A \ ?B = {}" by auto
moreover have "S \ ?A \ ?B" using as by auto
ultimately show False
using \<open>connected S\<close>[unfolded connected_def not_ex,
THEN spec[where x="?A"], THEN spec[where x="?B"]]
using xy b by auto
qed
lemma connected_ivt_component:
fixes x::"'a::euclidean_space"
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 "k::'a" a]
by (auto simp: inner_commute)
subsection \<open>Limit Component Bounds\<close>
lemma Lim_component_le:
fixes f :: "'a \ 'b::euclidean_space"
assumes "(f \ l) net"
and "\ (trivial_limit net)"
and "eventually (\x. f(x)\i \ b) net"
shows "l\i \ b"
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
lemma Lim_component_ge:
fixes f :: "'a \ 'b::euclidean_space"
assumes "(f \ l) net"
and "\ (trivial_limit net)"
and "eventually (\x. b \ (f x)\i) net"
shows "b \ l\i"
by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
lemma Lim_component_eq:
fixes f :: "'a \ 'b::euclidean_space"
assumes net: "(f \ l) net" "\ trivial_limit net"
and ev:"eventually (\x. f(x)\i = b) net"
shows "l\i = b"
using ev[unfolded order_eq_iff eventually_conj_iff]
using Lim_component_ge[OF net, of b i]
using Lim_component_le[OF net, of i b]
by auto
lemma open_box[intro]: "open (box a b)"
proof -
have "open (\i\Basis. ((\) i) -` {a \ i <..< b \ i})"
by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
also have "(\i\Basis. ((\) i) -` {a \ i <..< b \ i}) = box a b"
by (auto simp: box_def inner_commute)
finally show ?thesis .
qed
lemma closed_cbox[intro]:
fixes a b :: "'a::euclidean_space"
shows "closed (cbox a b)"
proof -
have "closed (\i\Basis. (\x. x\i) -` {a\i .. b\i})"
by (intro closed_INT ballI continuous_closed_vimage allI
linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
also have "(\i\Basis. (\x. x\i) -` {a\i .. b\i}) = cbox a b"
by (auto simp: cbox_def)
finally show "closed (cbox a b)" .
qed
lemma interior_cbox [simp]:
fixes a b :: "'a::euclidean_space"
shows "interior (cbox a b) = box a b" (is "?L = ?R")
proof(rule subset_antisym)
show "?R \ ?L"
using box_subset_cbox open_box
by (rule interior_maximal)
{
fix x
assume "x \ interior (cbox a b)"
then obtain s where s: "open s" "x \ s" "s \ cbox a b" ..
then obtain e where "e>0" and e:"\x'. dist x' x < e \ x' \ cbox a b"
unfolding open_dist and subset_eq by auto
{
fix i :: 'a
assume i: "i \ Basis"
have "dist (x - (e / 2) *\<^sub>R i) x < e"
and "dist (x + (e / 2) *\<^sub>R i) x < e"
unfolding dist_norm
apply auto
unfolding norm_minus_cancel
using norm_Basis[OF i] \<open>e>0\<close>
apply auto
done
then have "a \ i \ (x - (e / 2) *\<^sub>R i) \ i" and "(x + (e / 2) *\<^sub>R i) \ i \ b \ i"
using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
unfolding mem_box
using i
by blast+
then have "a \ i < x \ i" and "x \ i < b \ i"
using \<open>e>0\<close> i
by (auto simp: inner_diff_left inner_Basis inner_add_left)
}
then have "x \ box a b"
unfolding mem_box by auto
}
then show "?L \ ?R" ..
qed
lemma bounded_cbox [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (cbox a b)"
proof -
let ?b = "\i\Basis. \a\i\ + \b\i\"
{
fix x :: "'a"
assume "\i. i\Basis \ a \ i \ x \ i \ x \ i \ b \ i"
then have "(\i\Basis. \x \ i\) \ ?b"
by (force simp: intro!: sum_mono)
then have "norm x \ ?b"
using norm_le_l1[of x] by auto
}
then show ?thesis
unfolding cbox_def bounded_iff by force
qed
lemma bounded_box [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (box a b)"
using bounded_cbox[of a b] box_subset_cbox[of a b] bounded_subset[of "cbox a b" "box a b"]
by simp
lemma not_interval_UNIV [simp]:
fixes a :: "'a::euclidean_space"
shows "cbox a b \ UNIV" "box a b \ UNIV"
using bounded_box[of a b] bounded_cbox[of a b] by force+
lemma not_interval_UNIV2 [simp]:
fixes a :: "'a::euclidean_space"
shows "UNIV \ cbox a b" "UNIV \ box a b"
using bounded_box[of a b] bounded_cbox[of a b] by force+
lemma box_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b \ {}"
shows "((1/2) *\<^sub>R (a + b)) \ box a b"
proof -
have "a \ i < ((1 / 2) *\<^sub>R (a + b)) \ i \ ((1 / 2) *\<^sub>R (a + b)) \ i < b \ i" if "i \ Basis" for i
using assms that by (auto simp: inner_add_left box_ne_empty)
then show ?thesis unfolding mem_box by auto
qed
lemma open_cbox_convex:
fixes x :: "'a::euclidean_space"
assumes x: "x \ box a b"
and y: "y \ cbox a b"
and e: "0 < e" "e \ 1"
shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \ box a b"
proof -
{
fix i :: 'a
assume i: "i \ Basis"
have "a \ i = e * (a \ i) + (1 - e) * (a \ i)"
unfolding left_diff_distrib by simp
also have "\ < e * (x \ i) + (1 - e) * (y \ i)"
proof (rule add_less_le_mono)
show "e * (a \ i) < e * (x \ i)"
using \<open>0 < e\<close> i mem_box(1) x by auto
show "(1 - e) * (a \ i) \ (1 - e) * (y \ i)"
by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
qed
finally have "a \ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i"
unfolding inner_simps by auto
moreover
{
have "b \ i = e * (b\i) + (1 - e) * (b\i)"
unfolding left_diff_distrib by simp
also have "\ > e * (x \ i) + (1 - e) * (y \ i)"
proof (rule add_less_le_mono)
show "e * (x \ i) < e * (b \ i)"
using \<open>0 < e\<close> i mem_box(1) x by auto
show "(1 - e) * (y \ i) \ (1 - e) * (b \ i)"
by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
qed
finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i < b \ i"
unfolding inner_simps by auto
}
ultimately have "a \ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i \ (e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i < b \ i"
by auto
}
then show ?thesis
unfolding mem_box by auto
qed
--> --------------------
--> maximum size reached
--> --------------------
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