(* Title: HOL/Analysis/Path_Connected.thy
Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
*)
section \<open>Path-Connectedness\<close>
theory Path_Connected
imports
Starlike
T1_Spaces
begin
subsection \<open>Paths and Arcs\<close>
definition\<^marker>\<open>tag important\<close> path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "path g \<equiv> continuous_on {0..1} g"
definition\<^marker>\<open>tag important\<close> pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathstart g \<equiv> g 0"
definition\<^marker>\<open>tag important\<close> pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathfinish g \<equiv> g 1"
definition\<^marker>\<open>tag important\<close> path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
where "path_image g \<equiv> g ` {0 .. 1}"
definition\<^marker>\<open>tag important\<close> reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
where "reversepath g \<equiv> (\<lambda>x. g(1 - x))"
definition\<^marker>\<open>tag important\<close> joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
(infixr \<open>+++\<close> 75)
where "g1 +++ g2 \<equiv> (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
definition\<^marker>\<open>tag important\<close> loop_free :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "loop_free g \<equiv> \<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
definition\<^marker>\<open>tag important\<close> simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "simple_path g \<equiv> path g \<and> loop_free g"
definition\<^marker>\<open>tag important\<close> arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool"
where "arc g \<equiv> path g \<and> inj_on g {0..1}"
subsection\<^marker>\<open>tag unimportant\<close>\<open>Invariance theorems\<close>
lemma path_eq: "path p \<Longrightarrow> (\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t) \<Longrightarrow> path q"
using continuous_on_eq path_def by blast
lemma path_continuous_image: "path g \<Longrightarrow> continuous_on (path_image g) f \<Longrightarrow> path(f \<circ> g)"
unfolding path_def path_image_def
using continuous_on_compose by blast
lemma path_translation_eq:
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
shows "path((\<lambda>x. a + x) \<circ> g) = path g"
using continuous_on_translation_eq path_def by blast
lemma path_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "path(f \<circ> g) = path g"
proof -
from linear_injective_left_inverse [OF assms]
obtain h where h: "linear h" "h \<circ> f = id"
by blast
with assms show ?thesis
by (metis comp_assoc id_comp linear_continuous_on linear_linear path_continuous_image)
qed
lemma pathstart_translation: "pathstart((\<lambda>x. a + x) \<circ> g) = a + pathstart g"
by (simp add: pathstart_def)
lemma pathstart_linear_image_eq: "linear f \<Longrightarrow> pathstart(f \<circ> g) = f(pathstart g)"
by (simp add: pathstart_def)
lemma pathfinish_translation: "pathfinish((\<lambda>x. a + x) \<circ> g) = a + pathfinish g"
by (simp add: pathfinish_def)
lemma pathfinish_linear_image: "linear f \<Longrightarrow> pathfinish(f \<circ> g) = f(pathfinish g)"
by (simp add: pathfinish_def)
lemma path_image_translation: "path_image((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f \<circ> g) = f ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma reversepath_translation: "reversepath((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f \<circ> g) = f \<circ> reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_translation:
"((\<lambda>x. a + x) \<circ> g1) +++ ((\<lambda>x. a + x) \<circ> g2) = (\<lambda>x. a + x) \<circ> (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f \<circ> g1) +++ (f \<circ> g2) = f \<circ> (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma loop_free_translation_eq:
fixes g :: "real \<Rightarrow> 'a::euclidean_space"
shows "loop_free((\<lambda>x. a + x) \<circ> g) = loop_free g"
by (simp add: loop_free_def)
lemma simple_path_translation_eq:
fixes g :: "real \<Rightarrow> 'a::euclidean_space"
shows "simple_path((\<lambda>x. a + x) \<circ> g) = simple_path g"
by (simp add: simple_path_def loop_free_translation_eq path_translation_eq)
lemma loop_free_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "loop_free(f \<circ> g) = loop_free g"
using assms inj_on_eq_iff [of f] by (auto simp: loop_free_def)
lemma simple_path_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "simple_path(f \<circ> g) = simple_path g"
using assms
by (simp add: loop_free_linear_image_eq path_linear_image_eq simple_path_def)
lemma simple_pathI [intro?]:
assumes "path p"
assumes "\<And>x y. 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> p x = p y \<Longrightarrow> x = 0 \<and> y = 1"
shows "simple_path p"
unfolding simple_path_def loop_free_def
proof (intro ballI conjI impI)
fix x y assume "x \<in> {0..1}" "y \<in> {0..1}" "p x = p y"
thus "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
by (metis assms(2) atLeastAtMost_iff linorder_less_linear)
qed fact+
lemma arcD: "arc p \<Longrightarrow> p x = p y \<Longrightarrow> x \<in> {0..1} \<Longrightarrow> y \<in> {0..1} \<Longrightarrow> x = y"
by (auto simp: arc_def inj_on_def)
lemma arc_translation_eq:
fixes g :: "real \<Rightarrow> 'a::euclidean_space"
shows "arc((\<lambda>x. a + x) \<circ> g) \<longleftrightarrow> arc g"
by (auto simp: arc_def inj_on_def path_translation_eq)
lemma arc_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "arc(f \<circ> g) = arc g"
using assms inj_on_eq_iff [of f]
by (auto simp: arc_def inj_on_def path_linear_image_eq)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about paths\<close>
lemma path_of_real: "path complex_of_real"
unfolding path_def by (intro continuous_intros)
lemma path_const: "path (\<lambda>t. a)" for a::"'a::real_normed_vector"
unfolding path_def by (intro continuous_intros)
lemma path_minus: "path g \<Longrightarrow> path (\<lambda>t. - g t)" for g::"real\<Rightarrow>'a::real_normed_vector"
unfolding path_def by (intro continuous_intros)
lemma path_add: "\<lbrakk>path f; path g\<rbrakk> \<Longrightarrow> path (\<lambda>t. f t + g t)" for f::"real\<Rightarrow>'a::real_normed_vector"
unfolding path_def by (intro continuous_intros)
lemma path_diff: "\<lbrakk>path f; path g\<rbrakk> \<Longrightarrow> path (\<lambda>t. f t - g t)" for f::"real\<Rightarrow>'a::real_normed_vector"
unfolding path_def by (intro continuous_intros)
lemma path_mult: "\<lbrakk>path f; path g\<rbrakk> \<Longrightarrow> path (\<lambda>t. f t * g t)" for f::"real\<Rightarrow>'a::real_normed_field"
unfolding path_def by (intro continuous_intros)
lemma pathin_iff_path_real [simp]: "pathin euclideanreal g \<longleftrightarrow> path g"
by (simp add: pathin_def path_def)
lemma continuous_on_path: "path f \<Longrightarrow> t \<subseteq> {0..1} \<Longrightarrow> continuous_on t f"
using continuous_on_subset path_def by blast
lemma inj_on_imp_loop_free: "inj_on g {0..1} \<Longrightarrow> loop_free g"
by (simp add: inj_onD loop_free_def)
lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
by (simp add: arc_def inj_on_imp_loop_free simple_path_def)
lemma arc_imp_path: "arc g \<Longrightarrow> path g"
using arc_def by blast
lemma arc_imp_inj_on: "arc g \<Longrightarrow> inj_on g {0..1}"
by (auto simp: arc_def)
lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
using simple_path_def by blast
lemma loop_free_cases: "loop_free g \<Longrightarrow> inj_on g {0..1} \<or> pathfinish g = pathstart g"
by (force simp: inj_on_def loop_free_def pathfinish_def pathstart_def)
lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
using arc_def loop_free_cases simple_path_def by blast
lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g"
using simple_path_cases by auto
lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g"
unfolding arc_def inj_on_def pathfinish_def pathstart_def
by fastforce
lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g"
using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast
lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)"
by (simp add: arc_simple_path)
lemma path_image_const [simp]: "path_image (\<lambda>t. a) = {a}"
by (force simp: path_image_def)
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
unfolding path_image_def image_is_empty box_eq_empty
by auto
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
unfolding pathstart_def path_image_def
by auto
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
unfolding pathfinish_def path_image_def
by auto
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)"
unfolding path_def path_image_def
using connected_continuous_image connected_Icc by blast
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)"
unfolding path_def path_image_def
using compact_continuous_image connected_Icc by blast
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
unfolding reversepath_def
by auto
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
unfolding pathstart_def reversepath_def pathfinish_def
by auto
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
unfolding pathstart_def reversepath_def pathfinish_def
by auto
lemma reversepath_o: "reversepath g = g \<circ> (-)1"
by (auto simp: reversepath_def)
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
unfolding pathstart_def joinpaths_def pathfinish_def
by auto
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
unfolding pathstart_def joinpaths_def pathfinish_def
by auto
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
by (metis cancel_comm_monoid_add_class.diff_cancel diff_zero image_comp
image_diff_atLeastAtMost path_image_def reversepath_o)
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
by (metis continuous_on_compose continuous_on_op_minus image_comp image_ident path_def path_image_def path_image_reversepath reversepath_o reversepath_reversepath)
lemma arc_reversepath:
assumes "arc g" shows "arc(reversepath g)"
proof -
have injg: "inj_on g {0..1}"
using assms
by (simp add: arc_def)
have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y"
by simp
show ?thesis
using assms by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
qed
lemma loop_free_reversepath:
assumes "loop_free g" shows "loop_free(reversepath g)"
using assms by (simp add: reversepath_def loop_free_def Ball_def) (smt (verit))
lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)"
by (simp add: loop_free_reversepath simple_path_def)
lemmas reversepath_simps =
path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
lemma path_join[simp]:
assumes "pathfinish g1 = pathstart g2"
shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
unfolding path_def pathfinish_def pathstart_def
proof safe
assume cont: "continuous_on {0..1} (g1 +++ g2)"
have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
using assms
by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
unfolding g1 g2
by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
next
assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
by auto
{
fix x :: real
assume "0 \<le> x" and "x \<le> 1"
then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
by (intro image_eqI[where x="x/2"]) auto
}
note 1 = this
{
fix x :: real
assume "0 \<le> x" and "x \<le> 1"
then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
by (intro image_eqI[where x="x/2 + 1/2"]) auto
}
note 2 = this
show "continuous_on {0..1} (g1 +++ g2)"
using assms
unfolding joinpaths_def 01
apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
done
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Path Images\<close>
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
by (simp add: compact_imp_bounded compact_path_image)
lemma closed_path_image:
fixes g :: "real \<Rightarrow> 'a::t2_space"
shows "path g \<Longrightarrow> closed(path_image g)"
by (metis compact_path_image compact_imp_closed)
lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)"
by (metis connected_path_image simple_path_imp_path)
lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)"
by (metis compact_path_image simple_path_imp_path)
lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)"
by (metis bounded_path_image simple_path_imp_path)
lemma closed_simple_path_image:
fixes g :: "real \<Rightarrow> 'a::t2_space"
shows "simple_path g \<Longrightarrow> closed(path_image g)"
by (metis closed_path_image simple_path_imp_path)
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
by (metis connected_path_image arc_imp_path)
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
by (metis compact_path_image arc_imp_path)
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
by (metis bounded_path_image arc_imp_path)
lemma closed_arc_image:
fixes g :: "real \<Rightarrow> 'a::t2_space"
shows "arc g \<Longrightarrow> closed(path_image g)"
by (metis closed_path_image arc_imp_path)
lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
unfolding path_image_def joinpaths_def
by auto
lemma subset_path_image_join:
assumes "path_image g1 \<subseteq> S" and "path_image g2 \<subseteq> S"
shows "path_image (g1 +++ g2) \<subseteq> S"
using path_image_join_subset[of g1 g2] and assms
by auto
lemma path_image_join:
assumes "pathfinish g1 = pathstart g2"
shows "path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"
proof -
have "path_image g1 \<subseteq> path_image (g1 +++ g2)"
proof (clarsimp simp: path_image_def joinpaths_def)
fix u::real
assume "0 \<le> u" "u \<le> 1"
then show "g1 u \<in> (\<lambda>x. g1 (2 * x)) ` ({0..1} \<inter> {x. x * 2 \<le> 1})"
by (rule_tac x="u/2" in image_eqI) auto
qed
moreover
have \<section>: "g2 u \<in> (\<lambda>x. g2 (2 * x - 1)) ` ({0..1} \<inter> {x. \<not> x * 2 \<le> 1})"
if "0 < u" "u \<le> 1" for u
using that assms
by (rule_tac x="(u+1)/2" in image_eqI) (auto simp: field_simps pathfinish_def pathstart_def)
have "g2 0 \<in> (\<lambda>x. g1 (2 * x)) ` ({0..1} \<inter> {x. x * 2 \<le> 1})"
using assms
by (rule_tac x="1/2" in image_eqI) (auto simp: pathfinish_def pathstart_def)
then have "path_image g2 \<subseteq> path_image (g1 +++ g2)"
by (auto simp: path_image_def joinpaths_def intro!: \<section>)
ultimately show ?thesis
using path_image_join_subset by blast
qed
lemma not_in_path_image_join:
assumes "x \<notin> path_image g1" and "x \<notin> path_image g2"
shows "x \<notin> path_image (g1 +++ g2)"
using assms and path_image_join_subset[of g1 g2]
by auto
lemma pathstart_compose: "pathstart(f \<circ> p) = f(pathstart p)"
by (simp add: pathstart_def)
lemma pathfinish_compose: "pathfinish(f \<circ> p) = f(pathfinish p)"
by (simp add: pathfinish_def)
lemma path_image_compose: "path_image (f \<circ> p) = f ` (path_image p)"
by (simp add: image_comp path_image_def)
lemma path_compose_join: "f \<circ> (p +++ q) = (f \<circ> p) +++ (f \<circ> q)"
by (rule ext) (simp add: joinpaths_def)
lemma path_compose_reversepath: "f \<circ> reversepath p = reversepath(f \<circ> p)"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_eq:
"(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow>
(\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
\<Longrightarrow> t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
by (auto simp: joinpaths_def)
lemma loop_free_inj_on: "loop_free g \<Longrightarrow> inj_on g {0<..<1}"
by (force simp: inj_on_def loop_free_def)
lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}"
using loop_free_inj_on simple_path_def by auto
subsection\<^marker>\<open>tag unimportant\<close>\<open>Simple paths with the endpoints removed\<close>
lemma simple_path_endless:
assumes "simple_path c"
shows "path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using less_eq_real_def by (auto simp: path_image_def pathstart_def pathfinish_def)
show "?rhs \<subseteq> ?lhs"
using assms
apply (simp add: image_subset_iff path_image_def pathstart_def pathfinish_def simple_path_def loop_free_def Ball_def)
by (smt (verit))
qed
lemma connected_simple_path_endless:
assumes "simple_path c"
shows "connected(path_image c - {pathstart c,pathfinish c})"
proof -
have "continuous_on {0<..<1} c"
using assms by (simp add: simple_path_def continuous_on_path path_def subset_iff)
then have "connected (c ` {0<..<1})"
using connected_Ioo connected_continuous_image by blast
then show ?thesis
using assms by (simp add: simple_path_endless)
qed
lemma nonempty_simple_path_endless:
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}"
by (simp add: simple_path_endless)
lemma simple_path_continuous_image:
assumes "simple_path f" "continuous_on (path_image f) g" "inj_on g (path_image f)"
shows "simple_path (g \<circ> f)"
unfolding simple_path_def
proof
show "path (g \<circ> f)"
using assms unfolding simple_path_def by (intro path_continuous_image) auto
from assms have [simp]: "g (f x) = g (f y) \<longleftrightarrow> f x = f y" if "x \<in> {0..1}" "y \<in> {0..1}" for x y
unfolding inj_on_def path_image_def using that by fastforce
show "loop_free (g \<circ> f)"
using assms(1) by (auto simp: loop_free_def simple_path_def)
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>The operations on paths\<close>
lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g"
by simp
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
by simp
lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)"
by simp
lemma continuous_on_joinpaths:
assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
shows "continuous_on {0..1} (g1 +++ g2)"
using assms path_def path_join by blast
lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)"
by simp
lemma arc_join:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2"
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
shows "arc(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}" and injg2: "inj_on g2 {0..1}"
and g11: "g1 1 = g2 0" and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}"
using assms
by (auto simp: arc_def pathfinish_def pathstart_def path_image_def)
{ fix x and y::real
assume xy: "g2 (2 * x - 1) = g1 (2 * y)" "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1"
then have "g1 (2 * y) = g2 0"
using sb by force
then have False
using xy inj_onD injg2 by fastforce
} note * = this
have "inj_on (g1 +++ g2) {0..1}"
using inj_onD [OF injg1] inj_onD [OF injg2] *
by (simp add: inj_on_def joinpaths_def Ball_def) (smt (verit))
then show ?thesis
using arc_def assms path_join_imp by blast
qed
lemma simple_path_join_loop:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1"
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
shows "simple_path(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}" and injg2: "inj_on g2 {0..1}"
using assms by (auto simp add: arc_def)
have g12: "g1 1 = g2 0"
and g21: "g2 1 = g1 0"
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}"
using assms
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
{ fix x and y::real
assume g2_eq: "g2 (2 * x - 1) = g1 (2 * y)"
and xyI: "x \<noteq> 1 \<or> y \<noteq> 0"
and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1"
then consider "g1 (2 * y) = g1 0" | "g1 (2 * y) = g2 0"
using sb by force
then have False
proof cases
case 1
then have "y = 0"
using xy g2_eq by (auto dest!: inj_onD [OF injg1])
then show ?thesis
using xy g2_eq xyI by (auto dest: inj_onD [OF injg2] simp flip: g21)
next
case 2
then have "2*x = 1"
using g2_eq g12 inj_onD [OF injg2] atLeastAtMost_iff xy(1) xy(4) by fastforce
with xy show False by auto
qed
} note * = this
have "loop_free(g1 +++ g2)"
using inj_onD [OF injg1] inj_onD [OF injg2] *
by (simp add: loop_free_def joinpaths_def Ball_def) (smt (verit))
then show ?thesis
by (simp add: arc_imp_path assms simple_path_def)
qed
lemma reversepath_joinpaths:
"pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
by (rule ext) (auto simp: mult.commute)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some reversed and "if and only if" versions of joining theorems\<close>
lemma path_join_path_ends:
fixes g1 :: "real \<Rightarrow> 'a::metric_space"
assumes "path(g1 +++ g2)" "path g2"
shows "pathfinish g1 = pathstart g2"
proof (rule ccontr)
define e where "e = dist (g1 1) (g2 0)"
assume Neg: "pathfinish g1 \<noteq> pathstart g2"
then have "0 < dist (pathfinish g1) (pathstart g2)"
by auto
then have "e > 0"
by (metis e_def pathfinish_def pathstart_def)
then have "\<forall>e>0. \<exists>d>0. \<forall>x'\<in>{0..1}. dist x' 0 < d \<longrightarrow> dist (g2 x') (g2 0) < e"
using \<open>path g2\<close> atLeastAtMost_iff zero_le_one unfolding path_def continuous_on_iff
by blast
then obtain d1 where "d1 > 0"
and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; norm x' < d1\<rbrakk> \<Longrightarrow> dist (g2 x') (g2 0) < e/2"
by (metis \<open>0 < e\<close> half_gt_zero_iff norm_conv_dist)
obtain d2 where "d2 > 0"
and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk>
\<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2"
using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff
apply (drule_tac x="1/2" in bspec, simp)
apply (drule_tac x="e/2" in spec, force simp: joinpaths_def)
done
have int01_1: "min (1/2) (min d1 d2) / 2 \<in> {0..1}"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \<in> {0..1}"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
have [simp]: "\<not> min (1 / 2) (min d1 d2) \<le> 0"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
"dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
then have "dist (g1 1) (g2 0) < e/2 + e/2"
using dist_triangle_half_r e_def by blast
then show False
by (simp add: e_def [symmetric])
qed
lemma path_join_eq [simp]:
fixes g1 :: "real \<Rightarrow> 'a::metric_space"
assumes "path g1" "path g2"
shows "path(g1 +++ g2) \<longleftrightarrow> pathfinish g1 = pathstart g2"
using assms by (metis path_join_path_ends path_join_imp)
lemma simple_path_joinE:
assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
obtains "arc g1" "arc g2"
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
proof -
have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk>
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
using assms by (simp add: simple_path_def loop_free_def)
have "path g1"
using assms path_join simple_path_imp_path by blast
moreover have "inj_on g1 {0..1}"
proof (clarsimp simp: inj_on_def)
fix x y
assume "g1 x = g1 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
then show "x = y"
using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
qed
ultimately have "arc g1"
using assms by (simp add: arc_def)
have [simp]: "g2 0 = g1 1"
using assms by (metis pathfinish_def pathstart_def)
have "path g2"
using assms path_join simple_path_imp_path by blast
moreover have "inj_on g2 {0..1}"
proof (clarsimp simp: inj_on_def)
fix x y
assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
then show "x = y"
using * [of "(x+1) / 2" "(y+1) / 2"]
by (force simp: joinpaths_def split_ifs field_split_simps)
qed
ultimately have "arc g2"
using assms by (simp add: arc_def)
have "g2 y = g1 0 \<or> g2 y = g1 1"
if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y
using * [of "x / 2" "(y + 1) / 2"] that
by (auto simp: joinpaths_def split_ifs field_split_simps)
then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
by (fastforce simp: pathstart_def pathfinish_def path_image_def)
with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
qed
lemma simple_path_join_loop_eq:
assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2"
shows "simple_path(g1 +++ g2) \<longleftrightarrow>
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
by (metis assms simple_path_joinE simple_path_join_loop)
lemma arc_join_eq:
assumes "pathfinish g1 = pathstart g2"
shows "arc(g1 +++ g2) \<longleftrightarrow>
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using reversepath_simps assms
by (smt (verit, best) Int_commute arc_reversepath arc_simple_path in_mono insertE pathstart_join
reversepath_joinpaths simple_path_joinE subsetI)
next
assume ?rhs then show ?lhs
using assms
by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
qed
lemma arc_join_eq_alt:
"pathfinish g1 = pathstart g2
\<Longrightarrow> arc(g1 +++ g2) \<longleftrightarrow> arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 = {pathstart g2}"
using pathfinish_in_path_image by (fastforce simp: arc_join_eq)
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Symmetry and loops\<close>
lemma path_sym:
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> path(p +++ q) \<longleftrightarrow> path(q +++ p)"
by auto
lemma simple_path_sym:
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
\<Longrightarrow> simple_path(p +++ q) \<longleftrightarrow> simple_path(q +++ p)"
by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)
lemma path_image_sym:
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
\<Longrightarrow> path_image(p +++ q) = path_image(q +++ p)"
by (simp add: path_image_join sup_commute)
lemma simple_path_joinI:
assumes "arc p1" "arc p2" "pathfinish p1 = pathstart p2"
assumes "path_image p1 \<inter> path_image p2
\<subseteq> insert (pathstart p2) (if pathstart p1 = pathfinish p2 then {pathstart p1} else {})"
shows "simple_path (p1 +++ p2)"
by (smt (verit, best) Int_commute arc_imp_simple_path arc_join assms insert_commute simple_path_join_loop simple_path_sym)
lemma simple_path_join3I:
assumes "arc p1" "arc p2" "arc p3"
assumes "path_image p1 \<inter> path_image p2 \<subseteq> {pathstart p2}"
assumes "path_image p2 \<inter> path_image p3 \<subseteq> {pathstart p3}"
assumes "path_image p1 \<inter> path_image p3 \<subseteq> {pathstart p1} \<inter> {pathfinish p3}"
assumes "pathfinish p1 = pathstart p2" "pathfinish p2 = pathstart p3"
shows "simple_path (p1 +++ p2 +++ p3)"
using assms by (intro simple_path_joinI arc_join) (auto simp: path_image_join)
subsection\<^marker>\<open>tag unimportant\<close>\<open>The joining of paths is associative\<close>
lemma path_assoc:
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
\<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)"
by simp
lemma simple_path_assoc:
assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r"
shows "simple_path (p +++ (q +++ r)) \<longleftrightarrow> simple_path ((p +++ q) +++ r)"
proof (cases "pathstart p = pathfinish r")
case True show ?thesis
proof
assume "simple_path (p +++ q +++ r)"
with assms True show "simple_path ((p +++ q) +++ r)"
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join
dest: arc_distinct_ends [of r])
next
assume 0: "simple_path ((p +++ q) +++ r)"
with assms True have q: "pathfinish r \<notin> path_image q"
using arc_distinct_ends
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
have "pathstart r \<notin> path_image p"
using assms
by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff
pathfinish_in_path_image pathfinish_join simple_path_joinE)
with assms 0 q True show "simple_path (p +++ q +++ r)"
by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join
dest!: subsetD [OF _ IntI])
qed
next
case False
{ fix x :: 'a
assume a: "path_image p \<inter> path_image q \<subseteq> {pathstart q}"
"(path_image p \<union> path_image q) \<inter> path_image r \<subseteq> {pathstart r}"
"x \<in> path_image p" "x \<in> path_image r"
have "pathstart r \<in> path_image q"
by (metis assms(2) pathfinish_in_path_image)
with a have "x = pathstart q"
by blast
}
with False assms show ?thesis
by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
qed
lemma arc_assoc:
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
\<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)"
by (simp add: arc_simple_path simple_path_assoc)
subsection\<open>Subpath\<close>
definition\<^marker>\<open>tag important\<close> subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)"
lemma path_image_subpath_gen:
fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
shows "path_image(subpath u v g) = g ` (closed_segment u v)"
by (auto simp add: closed_segment_real_eq path_image_def subpath_def)
lemma path_image_subpath:
fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
lemma path_image_subpath_commute:
fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
shows "path_image(subpath u v g) = path_image(subpath v u g)"
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
lemma path_subpath [simp]:
fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
shows "path(subpath u v g)"
proof -
have "continuous_on {u..v} g" "continuous_on {v..u} g"
using assms continuous_on_path by fastforce+
then have "continuous_on {0..1} (g \<circ> (\<lambda>x. ((v-u) * x+ u)))"
by (intro continuous_intros; simp add: image_affinity_atLeastAtMost [where c=u])
then show ?thesis
by (simp add: path_def subpath_def)
qed
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
by (simp add: pathstart_def subpath_def)
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
by (simp add: pathfinish_def subpath_def)
lemma subpath_trivial [simp]: "subpath 0 1 g = g"
by (simp add: subpath_def)
lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
by (simp add: reversepath_def subpath_def)
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
by (simp add: reversepath_def subpath_def algebra_simps)
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma subpath_image: "subpath u v (f \<circ> g) = f \<circ> subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma affine_ineq:
fixes x :: "'a::linordered_idom"
assumes "x \<le> 1" "v \<le> u"
shows "v + x * u \<le> u + x * v"
proof -
have "(1-x)*(u-v) \<ge> 0"
using assms by auto
then show ?thesis
by (simp add: algebra_simps)
qed
lemma sum_le_prod1:
fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b"
by (metis add.commute affine_ineq mult.right_neutral)
lemma simple_path_subpath_eq:
"simple_path(subpath u v g) \<longleftrightarrow>
path(subpath u v g) \<and> u\<noteq>v \<and>
(\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y
\<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have p: "path (\<lambda>x. g ((v - u) * x + u))"
and sim: "(\<And>x y. \<lbrakk>x\<in>{0..1}; y\<in>{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\<rbrakk>
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
by (auto simp: simple_path_def loop_free_def subpath_def)
{ fix x y
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
(simp_all add: field_split_simps)
} moreover
have "path(subpath u v g) \<and> u\<noteq>v"
using sim [of "1/3" "2/3"] p
by (auto simp: subpath_def)
ultimately show ?rhs
by metis
next
assume ?rhs
then
have d1: "\<And>x y. \<lbrakk>g x = g y; u \<le> x; x \<le> v; u \<le> y; y \<le> v\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
and d2: "\<And>x y. \<lbrakk>g x = g y; v \<le> x; x \<le> u; v \<le> y; y \<le> u\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
and ne: "u < v \<or> v < u"
and psp: "path (subpath u v g)"
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1"
by algebra
show ?lhs using psp ne
unfolding simple_path_def loop_free_def subpath_def
by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed
lemma arc_subpath_eq:
"arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)"
by (smt (verit, best) arc_simple_path closed_segment_commute ends_in_segment(2) inj_on_def pathfinish_subpath pathstart_subpath simple_path_subpath_eq)
lemma simple_path_subpath:
assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v"
shows "simple_path(subpath u v g)"
using assms
unfolding simple_path_subpath_eq
by (force simp: simple_path_def loop_free_def closed_segment_real_eq image_affinity_atLeastAtMost)
lemma arc_simple_path_subpath:
"\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
by (force intro: simple_path_subpath simple_path_imp_arc)
lemma arc_subpath_arc:
"\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)
lemma arc_simple_path_subpath_interior:
"\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v; \<bar>u-v\<bar> < 1\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
by (force simp: simple_path_def loop_free_def intro: arc_simple_path_subpath)
lemma path_image_subpath_subset:
"\<lbrakk>u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
by (metis atLeastAtMost_iff atLeastatMost_subset_iff path_image_def path_image_subpath subset_image_iff)
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps)
subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close>
lemma subpath_to_frontier_explicit:
fixes S :: "'a::metric_space set"
assumes g: "path g" and "pathfinish g \<notin> S"
obtains u where "0 \<le> u" "u \<le> 1"
"\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
"(g u \<notin> interior S)" "(u = 0 \<or> g u \<in> closure S)"
proof -
have gcon: "continuous_on {0..1} g"
using g by (simp add: path_def)
moreover have "bounded ({u. g u \<in> closure (- S)} \<inter> {0..1})"
using compact_eq_bounded_closed by fastforce
ultimately have com: "compact ({0..1} \<inter> {u. g u \<in> closure (- S)})"
using closed_vimage_Int
by (metis (full_types) Int_commute closed_atLeastAtMost closed_closure compact_eq_bounded_closed vimage_def)
have "1 \<in> {u. g u \<in> closure (- S)}"
using assms by (simp add: pathfinish_def closure_def)
then have dis: "{0..1} \<inter> {u. g u \<in> closure (- S)} \<noteq> {}"
using atLeastAtMost_iff zero_le_one by blast
then obtain u where "0 \<le> u" "u \<le> 1" and gu: "g u \<in> closure (- S)"
and umin: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; g t \<in> closure (- S)\<rbrakk> \<Longrightarrow> u \<le> t"
using compact_attains_inf [OF com dis] by fastforce
then have umin': "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; t < u\<rbrakk> \<Longrightarrow> g t \<in> S"
using closure_def by fastforce
have \<section>: "g u \<in> closure S" if "u \<noteq> 0"
proof -
have "u > 0" using that \<open>0 \<le> u\<close> by auto
{ fix e::real assume "e > 0"
obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u \<le> d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e"
using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
have *: "dist (max 0 (u - d / 2)) u \<le> d"
using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
have "\<exists>y\<in>S. dist y (g u) < e"
using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
by (force intro: d [OF _ *] umin')
}
then show ?thesis
by (simp add: frontier_def closure_approachable)
qed
show ?thesis
proof
show "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
using \<open>u \<le> 1\<close> interior_closure umin by fastforce
show "g u \<notin> interior S"
by (simp add: gu interior_closure)
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<section> in auto)
qed
lemma subpath_to_frontier_strong:
assumes g: "path g" and "pathfinish g \<notin> S"
obtains u where "0 \<le> u" "u \<le> 1" "g u \<notin> interior S"
"u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
proof -
obtain u where "0 \<le> u" "u \<le> 1"
and gxin: "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
and gunot: "(g u \<notin> interior S)" and u0: "(u = 0 \<or> g u \<in> closure S)"
using subpath_to_frontier_explicit [OF assms] by blast
show ?thesis
proof
show "g u \<notin> interior S"
using gunot by blast
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> u0 in \<open>(force simp: subpath_def gxin)+\<close>)
qed
lemma subpath_to_frontier:
assumes g: "path g" and g0: "pathstart g \<in> closure S" and g1: "pathfinish g \<notin> S"
obtains u where "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "path_image(subpath 0 u g) - {g u} \<subseteq> interior S"
proof -
obtain u where "0 \<le> u" "u \<le> 1"
and notin: "g u \<notin> interior S"
and disj: "u = 0 \<or>
(\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
(is "_ \<or> ?P")
using subpath_to_frontier_strong [OF g g1] by blast
show ?thesis
proof
show "g u \<in> frontier S"
by (metis DiffI disj frontier_def g0 notin pathstart_def)
show "path_image (subpath 0 u g) - {g u} \<subseteq> interior S"
using disj
proof
assume "u = 0"
then show ?thesis
by (simp add: path_image_subpath)
next
assume P: ?P
show ?thesis
proof (clarsimp simp add: path_image_subpath_gen)
fix y
assume y: "y \<in> closed_segment 0 u" "g y \<notin> interior S"
with \<open>0 \<le> u\<close> have "0 \<le> y" "y \<le> u"
by (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
then have "y=u \<or> subpath 0 u g (y/u) \<in> interior S"
using P less_eq_real_def by force
then show "g y = g u"
using y by (auto simp: subpath_def split: if_split_asm)
qed
qed
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> in auto)
qed
lemma exists_path_subpath_to_frontier:
fixes S :: "'a::real_normed_vector set"
assumes "path g" "pathstart g \<in> closure S" "pathfinish g \<notin> S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
"path_image h - {pathfinish h} \<subseteq> interior S"
"pathfinish h \<in> frontier S"
proof -
obtain u where u: "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "(path_image(subpath 0 u g) - {g u}) \<subseteq> interior S"
using subpath_to_frontier [OF assms] by blast
show ?thesis
proof
show "path_image (subpath 0 u g) \<subseteq> path_image g"
by (simp add: path_image_subpath_subset u)
show "pathstart (subpath 0 u g) = pathstart g"
by (metis pathstart_def pathstart_subpath)
qed (use assms u in \<open>auto simp: path_image_subpath\<close>)
qed
lemma exists_path_subpath_to_frontier_closed:
fixes S :: "'a::real_normed_vector set"
assumes S: "closed S" and g: "path g" and g0: "pathstart g \<in> S" and g1: "pathfinish g \<notin> S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g \<inter> S"
"pathfinish h \<in> frontier S"
by (smt (verit, del_insts) Diff_iff Int_iff S closure_closed exists_path_subpath_to_frontier
frontier_def g g0 g1 interior_subset singletonD subset_eq)
subsection \<open>Shift Path to Start at Some Given Point\<close>
definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))"
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
by (auto simp: shiftpath_def)
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
lemma pathfinish_shiftpath:
assumes "0 \<le> a"
and "pathfinish g = pathstart g"
shows "pathfinish (shiftpath a g) = g a"
using assms
unfolding pathstart_def pathfinish_def shiftpath_def
by auto
lemma endpoints_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \<in> {0 .. 1}"
shows "pathfinish (shiftpath a g) = g a"
and "pathstart (shiftpath a g) = g a"
using assms
by (simp_all add: pathstart_shiftpath pathfinish_shiftpath)
lemma closed_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \<in> {0..1}"
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
using endpoints_shiftpath[OF assms]
by auto
lemma path_shiftpath:
assumes "path g"
and "pathfinish g = pathstart g"
and "a \<in> {0..1}"
shows "path (shiftpath a g)"
proof -
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
using assms(3) by auto
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
by (smt (verit, best) assms(2) pathfinish_def pathstart_def)
show ?thesis
unfolding path_def shiftpath_def *
proof (rule continuous_on_closed_Un)
have contg: "continuous_on {0..1} g"
using \<open>path g\<close> path_def by blast
show "continuous_on {0..1-a} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {0..1-a} (g \<circ> (+) a)"
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
qed auto
show "continuous_on {1-a..1} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {1-a..1} (g \<circ> (+) (a - 1))"
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
qed (auto simp: "**" add.commute add_diff_eq)
qed auto
qed
lemma shiftpath_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \<in> {0..1}"
and "x \<in> {0..1}"
shows "shiftpath (1 - a) (shiftpath a g) x = g x"
using assms
unfolding pathfinish_def pathstart_def shiftpath_def
by auto
lemma path_image_shiftpath:
assumes a: "a \<in> {0..1}"
and "pathfinish g = pathstart g"
shows "path_image (shiftpath a g) = path_image g"
proof -
{ fix x
assume g: "g 1 = g 0" "x \<in> {0..1::real}" and gne: "\<And>y. y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1} \<Longrightarrow> g x \<noteq> g (a + y - 1)"
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
proof (cases "a \<le> x")
case False
then show ?thesis
apply (rule_tac x="1 + x - a" in bexI)
using g gne[of "1 + x - a"] a by (force simp: field_simps)+
next
case True
then show ?thesis
using g a by (rule_tac x="x - a" in bexI) (auto simp: field_simps)
qed
}
then show ?thesis
using assms
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
by (auto simp: image_iff)
qed
lemma loop_free_shiftpath:
assumes "loop_free g" "pathfinish g = pathstart g" and a: "0 \<le> a" "a \<le> 1"
shows "loop_free (shiftpath a g)"
unfolding loop_free_def
proof (intro conjI impI ballI)
show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
if "x \<in> {0..1}" "y \<in> {0..1}" "shiftpath a g x = shiftpath a g y" for x y
using that a assms unfolding shiftpath_def loop_free_def
by (smt (verit, ccfv_threshold) atLeastAtMost_iff)
qed
lemma simple_path_shiftpath:
assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \<le> a" "a \<le> 1"
shows "simple_path (shiftpath a g)"
using assms loop_free_shiftpath path_shiftpath simple_path_def by fastforce
subsection \<open>Straight-Line Paths\<close>
definition\<^marker>\<open>tag important\<close> linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
unfolding pathstart_def linepath_def
by auto
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
unfolding pathfinish_def linepath_def
by auto
lemma linepath_inner: "linepath a b x \<bullet> v = linepath (a \<bullet> v) (b \<bullet> v) x"
by (simp add: linepath_def algebra_simps)
lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x"
by (simp add: linepath_def)
lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x"
by (simp add: linepath_def)
lemma linepath_0': "linepath a b 0 = a"
by (simp add: linepath_def)
lemma linepath_1': "linepath a b 1 = b"
by (simp add: linepath_def)
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
unfolding linepath_def
by (intro continuous_intros)
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
using continuous_linepath_at
by (auto intro!: continuous_at_imp_continuous_on)
lemma path_linepath[iff]: "path (linepath a b)"
unfolding path_def
by (rule continuous_on_linepath)
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
unfolding path_image_def segment linepath_def
by auto
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
unfolding reversepath_def linepath_def
by auto
lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b"
by (simp add: linepath_def)
lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
by (simp add: linepath_def)
lemma arc_linepath:
assumes "a \<noteq> b" shows [simp]: "arc (linepath a b)"
proof -
{
fix x y :: "real"
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
by (simp add: algebra_simps)
with assms have "x = y"
by simp
}
then show ?thesis
unfolding arc_def inj_on_def
by (fastforce simp: algebra_simps linepath_def)
qed
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
by (simp add: arc_imp_simple_path)
lemma linepath_trivial [simp]: "linepath a a x = a"
by (simp add: linepath_def real_vector.scale_left_diff_distrib)
lemma linepath_refl: "linepath a a = (\<lambda>x. a)"
by auto
lemma subpath_refl: "subpath a a g = linepath (g a) (g a)"
by (simp add: subpath_def linepath_def algebra_simps)
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
by (simp add: scaleR_conv_of_real linepath_def)
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
lemma inj_on_linepath:
assumes "a \<noteq> b" shows "inj_on (linepath a b) {0..1}"
using arc_imp_inj_on arc_linepath assms by blast
lemma linepath_le_1:
fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"
using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto
lemma linepath_in_path:
shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_in_convex_hull:
fixes x::real
assumes "a \<in> convex hull S"
and "b \<in> convex hull S"
and "0\<le>x" "x\<le>1"
shows "linepath a b x \<in> convex hull S"
by (meson assms atLeastAtMost_iff convex_contains_segment convex_convex_hull linepath_in_path subset_eq)
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
by (simp add: linepath_def)
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
by (simp add: linepath_def)
lemma
assumes "x \<in> closed_segment y z"
shows in_closed_segment_imp_Re_in_closed_segment: "Re x \<in> closed_segment (Re y) (Re z)" (is ?th1)
and in_closed_segment_imp_Im_in_closed_segment: "Im x \<in> closed_segment (Im y) (Im z)" (is ?th2)
proof -
from assms obtain t where t: "t \<in> {0..1}" "x = linepath y z t"
by (metis imageE linepath_image_01)
have "Re x = linepath (Re y) (Re z) t" "Im x = linepath (Im y) (Im z) t"
by (simp_all add: t Re_linepath' Im_linepath')
with t(1) show ?th1 ?th2
using linepath_in_path[of t "Re y" "Re z"] linepath_in_path[of t "Im y" "Im z"] by simp_all
qed
lemma linepath_in_open_segment: "t \<in> {0<..<1} \<Longrightarrow> x \<noteq> y \<Longrightarrow> linepath x y t \<in> open_segment x y"
unfolding greaterThanLessThan_iff by (metis in_segment(2) linepath_def)
lemma in_open_segment_imp_Re_in_open_segment:
assumes "x \<in> open_segment y z" "Re y \<noteq> Re z"
shows "Re x \<in> open_segment (Re y) (Re z)"
proof -
from assms obtain t where t: "t \<in> {0<..<1}" "x = linepath y z t"
by (metis greaterThanLessThan_iff in_segment(2) linepath_def)
have "Re x = linepath (Re y) (Re z) t"
by (simp_all add: t Re_linepath')
with t(1) show ?thesis
using linepath_in_open_segment[of t "Re y" "Re z"] assms by auto
qed
lemma in_open_segment_imp_Im_in_open_segment:
assumes "x \<in> open_segment y z" "Im y \<noteq> Im z"
shows "Im x \<in> open_segment (Im y) (Im z)"
proof -
from assms obtain t where t: "t \<in> {0<..<1}" "x = linepath y z t"
by (metis greaterThanLessThan_iff in_segment(2) linepath_def)
have "Im x = linepath (Im y) (Im z) t"
by (simp_all add: t Im_linepath')
with t(1) show ?thesis
using linepath_in_open_segment[of t "Im y" "Im z"] assms by auto
qed
lemma bounded_linear_linepath:
assumes "bounded_linear f"
shows "f (linepath a b x) = linepath (f a) (f b) x"
proof -
interpret f: bounded_linear f by fact
show ?thesis by (simp add: linepath_def f.add f.scale)
qed
lemma bounded_linear_linepath':
assumes "bounded_linear f"
shows "f \<circ> linepath a b = linepath (f a) (f b)"
using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)
lemma linepath_cnj': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)"
by (simp add: linepath_def fun_eq_iff)
lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
by (auto simp: linepath_def)
lemma has_vector_derivative_linepath_within:
"(linepath a b has_vector_derivative (b - a)) (at x within S)"
by (force intro: derivative_eq_intros simp add: linepath_def has_vector_derivative_def algebra_simps)
lemma linepath_real_ge_left:
fixes x y :: real
assumes "x \<le> y" "t \<ge> 0"
shows "linepath x y t \<ge> x"
proof -
have "x + 0 \<le> x + t *\<^sub>R (y - x)"
using assms by (intro add_left_mono) auto
also have "\<dots> = linepath x y t"
by (simp add: linepath_def algebra_simps)
finally show ?thesis by simp
qed
lemma linepath_real_le_right:
fixes x y :: real
assumes "x \<le> y" "t \<le> 1"
shows "linepath x y t \<le> y"
proof -
have "y + 0 \<ge> y + (1 - t) *\<^sub>R (x - y)"
using assms by (intro add_left_mono) (auto intro: mult_nonneg_nonpos)
also have "y + (1 - t) *\<^sub>R (x - y) = linepath x y t"
by (simp add: linepath_def algebra_simps)
finally show ?thesis by simp
qed
lemma linepath_translate: "(+) c \<circ> linepath a b = linepath (a + c) (b + c)"
by (auto simp: linepath_def algebra_simps)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close>
lemma segments_subset_convex_hull:
"closed_segment a b \<subseteq> (convex hull {a,b,c})"
"closed_segment a c \<subseteq> (convex hull {a,b,c})"
"closed_segment b c \<subseteq> (convex hull {a,b,c})"
"closed_segment b a \<subseteq> (convex hull {a,b,c})"
"closed_segment c a \<subseteq> (convex hull {a,b,c})"
"closed_segment c b \<subseteq> (convex hull {a,b,c})"
by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono])
lemma midpoints_in_convex_hull:
assumes "x \<in> convex hull s" "y \<in> convex hull s"
shows "midpoint x y \<in> convex hull s"
using assms closed_segment_subset_convex_hull csegment_midpoint_subset by blast
lemma not_in_interior_convex_hull_3:
fixes a :: "complex"
shows "a \<notin> interior(convex hull {a,b,c})"
"b \<notin> interior(convex hull {a,b,c})"
"c \<notin> interior(convex hull {a,b,c})"
by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
lemma midpoint_in_closed_segment [simp]: "midpoint a b \<in> closed_segment a b"
using midpoints_in_convex_hull segment_convex_hull by blast
lemma midpoint_in_open_segment [simp]: "midpoint a b \<in> open_segment a b \<longleftrightarrow> a \<noteq> b"
by (simp add: open_segment_def)
lemma continuous_IVT_local_extremum:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes contf: "continuous_on (closed_segment a b) f"
and ab: "a \<noteq> b" "f a = f b"
obtains z where "z \<in> open_segment a b"
"(\<forall>w \<in> closed_segment a b. (f w) \<le> (f z)) \<or>
(\<forall>w \<in> closed_segment a b. (f z) \<le> (f w))"
proof -
obtain c where "c \<in> closed_segment a b" and c: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f y \<le> f c"
using continuous_attains_sup [of "closed_segment a b" f] contf by auto
moreover
obtain d where "d \<in> closed_segment a b" and d: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f d \<le> f y"
using continuous_attains_inf [of "closed_segment a b" f] contf by auto
ultimately show ?thesis
by (smt (verit) UnE ab closed_segment_eq_open empty_iff insert_iff midpoint_in_open_segment that)
qed
text\<open>An injective map into R is also an open map w.r.T. the universe, and conversely. \<close>
proposition injective_eq_1d_open_map_UNIV:
fixes f :: "real \<Rightarrow> real"
assumes contf: "continuous_on S f" and S: "is_interval S"
shows "inj_on f S \<longleftrightarrow> (\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open(f ` T))"
(is "?lhs = ?rhs")
proof safe
fix T
assume injf: ?lhs and "open T" and "T \<subseteq> S"
have "\<exists>U. open U \<and> f x \<in> U \<and> U \<subseteq> f ` T" if "x \<in> T" for x
proof -
obtain \<delta> where "\<delta> > 0" and \<delta>: "cball x \<delta> \<subseteq> T"
using \<open>open T\<close> \<open>x \<in> T\<close> open_contains_cball_eq by blast
show ?thesis
proof (intro exI conjI)
have "closed_segment (x-\<delta>) (x+\<delta>) = {x-\<delta>..x+\<delta>}"
using \<open>0 < \<delta>\<close> by (auto simp: closed_segment_eq_real_ivl)
also have "\<dots> \<subseteq> S"
using \<delta> \<open>T \<subseteq> S\<close> by (auto simp: dist_norm subset_eq)
finally have "f ` (open_segment (x-\<delta>) (x+\<delta>)) = open_segment (f (x-\<delta>)) (f (x+\<delta>))"
using continuous_injective_image_open_segment_1
by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
then show "open (f ` {x-\<delta><..<x+\<delta>})"
using \<open>0 < \<delta>\<close> by (simp add: open_segment_eq_real_ivl)
show "f x \<in> f ` {x - \<delta><..<x + \<delta>}"
by (auto simp: \<open>\<delta> > 0\<close>)
show "f ` {x - \<delta><..<x + \<delta>} \<subseteq> f ` T"
using \<delta> by (auto simp: dist_norm subset_iff)
qed
qed
with open_subopen show "open (f ` T)"
by blast
next
assume R: ?rhs
have False if xy: "x \<in> S" "y \<in> S" and "f x = f y" "x \<noteq> y" for x y
proof -
have "open (f ` open_segment x y)"
using R
by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy)
moreover
have "continuous_on (closed_segment x y) f"
by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that)
then obtain \<xi> where "\<xi> \<in> open_segment x y"
and \<xi>: "(\<forall>w \<in> closed_segment x y. (f w) \<le> (f \<xi>)) \<or>
(\<forall>w \<in> closed_segment x y. (f \<xi>) \<le> (f w))"
using continuous_IVT_local_extremum [of x y f] \<open>f x = f y\<close> \<open>x \<noteq> y\<close> by blast
ultimately obtain e where "e>0" and e: "\<And>u. dist u (f \<xi>) < e \<Longrightarrow> u \<in> f ` open_segment x y"
using open_dist by (metis image_eqI)
have fin: "f \<xi> + (e/2) \<in> f ` open_segment x y" "f \<xi> - (e/2) \<in> f ` open_segment x y"
using e [of "f \<xi> + (e/2)"] e [of "f \<xi> - (e/2)"] \<open>e > 0\<close> by (auto simp: dist_norm)
show ?thesis
using \<xi> \<open>0 < e\<close> fin open_closed_segment by fastforce
qed
then show ?lhs
by (force simp: inj_on_def)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounding a point away from a path\<close>
lemma not_on_path_ball:
fixes g :: "real \<Rightarrow> 'a::heine_borel"
assumes "path g"
and z: "z \<notin> path_image g"
shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
proof -
have "closed (path_image g)"
by (simp add: \<open>path g\<close> closed_path_image)
then obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z])
then show ?thesis
by (rule_tac x="dist z a" in exI) (use dist_commute z in auto)
qed
lemma not_on_path_cball:
fixes g :: "real \<Rightarrow> 'a::heine_borel"
assumes "path g"
and "z \<notin> path_image g"
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
by (smt (verit, ccfv_threshold) open_ball assms centre_in_ball inf.orderE inf_assoc
inf_bot_right not_on_path_ball open_contains_cball_eq)
subsection \<open>Path component\<close>
text \<open>Original formalization by Tom Hales\<close>
definition\<^marker>\<open>tag important\<close> "path_component S x y \<equiv>
(\<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
abbreviation\<^marker>\<open>tag important\<close>
"path_component_set S x \<equiv> Collect (path_component S x)"
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
lemma path_component_mem:
assumes "path_component S x y"
shows "x \<in> S" and "y \<in> S"
using assms
unfolding path_defs
by auto
lemma path_component_refl:
assumes "x \<in> S"
shows "path_component S x x"
using assms
unfolding path_defs
by (metis (full_types) assms continuous_on_const image_subset_iff path_image_def)
lemma path_component_refl_eq: "path_component S x x \<longleftrightarrow> x \<in> S"
by (auto intro!: path_component_mem path_component_refl)
lemma path_component_sym: "path_component S x y \<Longrightarrow> path_component S y x"
unfolding path_component_def
by (metis (no_types) path_image_reversepath path_reversepath pathfinish_reversepath pathstart_reversepath)
lemma path_component_trans:
assumes "path_component S x y" and "path_component S y z"
shows "path_component S x z"
using assms
unfolding path_component_def
by (metis path_join pathfinish_join pathstart_join subset_path_image_join)
lemma path_component_of_subset: "S \<subseteq> T \<Longrightarrow> path_component S x y \<Longrightarrow> path_component T x y"
unfolding path_component_def by auto
lemma path_component_linepath:
fixes S :: "'a::real_normed_vector set"
shows "closed_segment a b \<subseteq> S \<Longrightarrow> path_component S a b"
unfolding path_component_def by fastforce
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Path components as sets\<close>
lemma path_component_set:
"path_component_set S x =
{y. (\<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)}"
by (auto simp: path_component_def)
lemma path_component_subset: "path_component_set S x \<subseteq> S"
by (auto simp: path_component_mem(2))
lemma path_component_eq_empty: "path_component_set S x = {} \<longleftrightarrow> x \<notin> S"
using path_component_mem path_component_refl_eq
by fastforce
lemma path_component_mono:
"S \<subseteq> T \<Longrightarrow> (path_component_set S x) \<subseteq> (path_component_set T x)"
by (simp add: Collect_mono path_component_of_subset)
lemma path_component_eq:
"y \<in> path_component_set S x \<Longrightarrow> path_component_set S y = path_component_set S x"
by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)
subsection \<open>Path connectedness of a space\<close>
definition\<^marker>\<open>tag important\<close> "path_connected S \<longleftrightarrow>
(\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
lemma path_connectedin_iff_path_connected_real [simp]:
"path_connectedin euclideanreal S \<longleftrightarrow> path_connected S"
by (simp add: path_connectedin path_connected_def path_defs image_subset_iff_funcset)
lemma path_connected_component: "path_connected S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. path_component S x y)"
unfolding path_connected_def path_component_def by auto
lemma path_connected_component_set: "path_connected S \<longleftrightarrow> (\<forall>x\<in>S. path_component_set S x = S)"
unfolding path_connected_component path_component_subset
using path_component_mem by blast
lemma path_component_maximal:
"\<lbrakk>x \<in> T; path_connected T; T \<subseteq> S\<rbrakk> \<Longrightarrow> T \<subseteq> (path_component_set S x)"
by (metis path_component_mono path_connected_component_set)
lemma convex_imp_path_connected:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "path_connected S"
unfolding path_connected_def
using assms convex_contains_segment by fastforce
lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)"
by (simp add: convex_imp_path_connected)
lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)"
using path_connected_component_set by auto
lemma path_connected_imp_connected:
assumes "path_connected S"
shows "connected S"
proof (rule connectedI)
fix e1 e2
assume as: "open e1" "open e2" "S \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> S = {}" "e1 \<inter> S \<noteq> {}" "e2 \<inter> S \<noteq> {}"
then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> S" "x2 \<in> e2 \<inter> S"
by auto
then obtain g where g: "path g" "path_image g \<subseteq> S" and pg: "pathstart g = x1" "pathfinish g = x2"
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
have *: "connected {0..1::real}"
by (auto intro!: convex_connected)
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}"
using as(3) g(2)[unfolded path_defs] by blast
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}"
using as(4) g(2)[unfolded path_defs]
unfolding subset_eq
by auto
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}"
by (smt (verit, ccfv_threshold) IntE atLeastAtMost_iff empty_iff pg mem_Collect_eq obt pathfinish_def pathstart_def)
ultimately show False
using *[unfolded connected_local not_ex, rule_format,
of "{0..1} \<inter> g -` e1" "{0..1} \<inter> g -` e2"]
using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)]
using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)]
by auto
qed
lemma open_path_component:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open (path_component_set S x)"
unfolding open_contains_ball
by (metis assms centre_in_ball convex_ball convex_imp_path_connected equals0D openE
path_component_eq path_component_eq_empty path_component_maximal)
lemma open_non_path_component:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open (S - path_component_set S x)"
unfolding open_contains_ball
proof
fix y
assume y: "y \<in> S - path_component_set S x"
then obtain e where e: "e > 0" "ball y e \<subseteq> S"
using assms openE by auto
show "\<exists>e>0. ball y e \<subseteq> S - path_component_set S x"
proof (intro exI conjI subsetI DiffI notI)
show "\<And>x. x \<in> ball y e \<Longrightarrow> x \<in> S"
using e by blast
show False if "z \<in> ball y e" "z \<in> path_component_set S x" for z
by (metis (no_types, lifting) Diff_iff centre_in_ball convex_ball convex_imp_path_connected
path_component_eq path_component_maximal subsetD that y e)
qed (use e in auto)
qed
lemma connected_open_path_connected:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
and "connected S"
shows "path_connected S"
unfolding path_connected_component_set
proof (rule, rule, rule path_component_subset, rule)
fix x y
assume "x \<in> S" and "y \<in> S"
show "y \<in> path_component_set S x"
proof (rule ccontr)
assume "\<not> ?thesis"
moreover have "path_component_set S x \<inter> S \<noteq> {}"
using \<open>x \<in> S\<close> path_component_eq_empty path_component_subset[of S x]
by auto
ultimately
show False
using \<open>y \<in> S\<close> open_non_path_component[OF \<open>open S\<close>] open_path_component[OF \<open>open S\<close>]
using \<open>connected S\<close>[unfolded connected_def not_ex, rule_format,
of "path_component_set S x" "S - path_component_set S x"]
by auto
qed
qed
lemma path_connected_continuous_image:
assumes contf: "continuous_on S f"
and "path_connected S"
shows "path_connected (f ` S)"
unfolding path_connected_def
proof clarsimp
fix x y
assume x: "x \<in> S" and y: "y \<in> S"
with \<open>path_connected S\<close>
show "\<exists>g. path g \<and> path_image g \<subseteq> f ` S \<and> pathstart g = f x \<and> pathfinish g = f y"
unfolding path_defs path_connected_def
using continuous_on_subset[OF contf]
by (smt (verit, ccfv_threshold) continuous_on_compose2 image_eqI image_subset_iff)
qed
lemma path_connected_translationI:
fixes a :: "'a :: topological_group_add"
assumes "path_connected S" shows "path_connected ((\<lambda>x. a + x) ` S)"
by (intro path_connected_continuous_image assms continuous_intros)
lemma path_connected_translation:
fixes a :: "'a :: topological_group_add"
shows "path_connected ((\<lambda>x. a + x) ` S) = path_connected S"
proof -
have "\<forall>x y. (+) (x::'a) ` (+) (0 - x) ` y = y"
by (simp add: image_image)
then show ?thesis
by (metis (no_types) path_connected_translationI)
qed
lemma path_connected_segment [simp]:
fixes a :: "'a::real_normed_vector"
shows "path_connected (closed_segment a b)"
by (simp add: convex_imp_path_connected)
lemma path_connected_open_segment [simp]:
fixes a :: "'a::real_normed_vector"
shows "path_connected (open_segment a b)"
by (simp add: convex_imp_path_connected)
lemma homeomorphic_path_connectedness:
"S homeomorphic T \<Longrightarrow> path_connected S \<longleftrightarrow> path_connected T"
unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image)
lemma path_connected_empty [simp]: "path_connected {}"
unfolding path_connected_def by auto
lemma path_connected_singleton [simp]: "path_connected {a}"
unfolding path_connected_def pathstart_def pathfinish_def path_image_def
using path_def by fastforce
lemma path_connected_Un:
assumes "path_connected S"
and "path_connected T"
and "S \<inter> T \<noteq> {}"
shows "path_connected (S \<union> T)"
unfolding path_connected_component
proof (intro ballI)
fix x y
assume x: "x \<in> S \<union> T" and y: "y \<in> S \<union> T"
from assms obtain z where z: "z \<in> S" "z \<in> T"
by auto
with x y show "path_component (S \<union> T) x y"
by (smt (verit) assms(1,2) in_mono mem_Collect_eq path_component_eq path_component_maximal
sup.bounded_iff sup.cobounded2 sup_ge1)
qed
lemma path_connected_UNION:
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
shows "path_connected (\<Union>i\<in>A. S i)"
unfolding path_connected_component
proof clarify
fix x i y j
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
then have "path_component (S i) x z" and "path_component (S j) z y"
using assms by (simp_all add: path_connected_component)
then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y"
using *(1,3) by (meson SUP_upper path_component_of_subset)+
then show "path_component (\<Union>i\<in>A. S i) x y"
by (rule path_component_trans)
qed
lemma path_component_path_image_pathstart:
assumes p: "path p" and x: "x \<in> path_image p"
shows "path_component (path_image p) (pathstart p) x"
proof -
obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1"
using x by (auto simp: path_image_def)
show ?thesis
unfolding path_component_def
proof (intro exI conjI)
have "continuous_on ((*) y ` {0..1}) p"
by (simp add: continuous_on_path image_mult_atLeastAtMost_if p y)
then have "continuous_on {0..1} (p \<circ> ((*) y))"
using continuous_on_compose continuous_on_mult_const by blast
then show "path (\<lambda>u. p (y * u))"
by (simp add: path_def)
show "path_image (\<lambda>u. p (y * u)) \<subseteq> path_image p"
using y mult_le_one by (fastforce simp: path_image_def image_iff)
qed (auto simp: pathstart_def pathfinish_def x)
qed
lemma path_connected_path_image: "path p \<Longrightarrow> path_connected(path_image p)"
unfolding path_connected_component
by (meson path_component_path_image_pathstart path_component_sym path_component_trans)
lemma path_connected_path_component [simp]:
"path_connected (path_component_set S x)"
by (smt (verit) mem_Collect_eq path_component_def path_component_eq path_component_maximal
path_connected_component path_connected_path_image pathstart_in_path_image)
lemma path_component:
"path_component S x y \<longleftrightarrow> (\<exists>t. path_connected t \<and> t \<subseteq> S \<and> x \<in> t \<and> y \<in> t)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis path_component_def path_connected_path_image pathfinish_in_path_image pathstart_in_path_image)
next
assume ?rhs then show ?lhs
by (meson path_component_of_subset path_connected_component)
qed
lemma path_component_path_component [simp]:
"path_component_set (path_component_set S x) x = path_component_set S x"
by (metis (full_types) mem_Collect_eq path_component_eq_empty path_component_refl path_connected_component_set path_connected_path_component)
lemma path_component_subset_connected_component:
"(path_component_set S x) \<subseteq> (connected_component_set S x)"
proof (cases "x \<in> S")
case True show ?thesis
by (simp add: True connected_component_maximal path_component_refl path_component_subset path_connected_imp_connected)
next
case False then show ?thesis
using path_component_eq_empty by auto
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Lemmas about path-connectedness\<close>
lemma path_connected_linear_image:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "path_connected S" "bounded_linear f"
shows "path_connected(f ` S)"
by (auto simp: linear_continuous_on assms path_connected_continuous_image)
lemma is_interval_path_connected: "is_interval S \<Longrightarrow> path_connected S"
by (simp add: convex_imp_path_connected is_interval_convex)
lemma path_connected_Ioi[simp]: "path_connected {a<..}" for a :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Ici[simp]: "path_connected {a..}" for a :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Iio[simp]: "path_connected {..<a}" for a :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Iic[simp]: "path_connected {..a}" for a :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Ioo[simp]: "path_connected {a<..<b}" for a b :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Ioc[simp]: "path_connected {a<..b}" for a b :: real
by (simp add: convex_imp_path_connected)
lemma path_connected_Ico[simp]: "path_connected {a..<b}" for a b :: real
by (simp add: convex_imp_path_connected)
lemma path_connectedin_path_image:
assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))"
unfolding pathin_def
proof (rule path_connectedin_continuous_map_image)
show "continuous_map (subtopology euclideanreal {0..1}) X g"
using assms pathin_def by blast
qed (auto simp: is_interval_1 is_interval_path_connected)
lemma path_connected_space_subconnected:
"path_connected_space X \<longleftrightarrow>
(\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. \<exists>S. path_connectedin X S \<and> x \<in> S \<and> y \<in> S)"
by (metis path_connectedin path_connectedin_topspace path_connected_space_def)
lemma connectedin_path_image: "pathin X g \<Longrightarrow> connectedin X (g ` ({0..1}))"
by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image)
lemma compactin_path_image: "pathin X g \<Longrightarrow> compactin X (g ` ({0..1}))"
unfolding pathin_def
by (rule image_compactin [of "top_of_set {0..1}"]) auto
lemma linear_homeomorphism_image:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
obtains g where "homeomorphism (f ` S) S g f"
proof -
obtain g where "linear g" "g \<circ> f = id"
using assms linear_injective_left_inverse by blast
then have "homeomorphism (f ` S) S g f"
using assms unfolding homeomorphism_def
by (auto simp: eq_id_iff [symmetric] image_comp linear_conv_bounded_linear linear_continuous_on)
then show thesis ..
qed
lemma linear_homeomorphic_image:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
shows "S homeomorphic f ` S"
by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms])
lemma path_connected_Times:
assumes "path_connected s" "path_connected t"
shows "path_connected (s \<times> t)"
proof (simp add: path_connected_def Sigma_def, clarify)
fix x1 y1 x2 y2
assume "x1 \<in> s" "y1 \<in> t" "x2 \<in> s" "y2 \<in> t"
obtain g where "path g" and g: "path_image g \<subseteq> s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2"
using \<open>x1 \<in> s\<close> \<open>x2 \<in> s\<close> assms by (force simp: path_connected_def)
obtain h where "path h" and h: "path_image h \<subseteq> t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2"
using \<open>y1 \<in> t\<close> \<open>y2 \<in> t\<close> assms by (force simp: path_connected_def)
have "path (\<lambda>z. (x1, h z))"
using \<open>path h\<close>
unfolding path_def
by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force)
moreover have "path (\<lambda>z. (g z, y2))"
using \<open>path g\<close>
unfolding path_def
by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force)
ultimately have 1: "path ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2)))"
by (metis hf gs path_join_imp pathstart_def pathfinish_def)
have "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> path_image (\<lambda>z. (x1, h z)) \<union> path_image (\<lambda>z. (g z, y2))"
by (rule Path_Connected.path_image_join_subset)
also have "\<dots> \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})"
using g h \<open>x1 \<in> s\<close> \<open>y2 \<in> t\<close> by (force simp: path_image_def)
finally have 2: "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" .
show "\<exists>g. path g \<and> path_image g \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)}) \<and>
pathstart g = (x1, y1) \<and> pathfinish g = (x2, y2)"
using 1 2 gf hs
by (metis (no_types, lifting) pathfinish_def pathfinish_join pathstart_def pathstart_join)
qed
lemma is_interval_path_connected_1:
fixes s :: "real set"
shows "is_interval s \<longleftrightarrow> path_connected s"
using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast
subsection\<^marker>\<open>tag unimportant\<close>\<open>Path components\<close>
lemma Union_path_component [simp]:
"Union {path_component_set S x |x. x \<in> S} = S"
using path_component_subset path_component_refl by blast
lemma path_component_disjoint:
"disjnt (path_component_set S a) (path_component_set S b) \<longleftrightarrow>
(a \<notin> path_component_set S b)"
unfolding disjnt_iff
using path_component_sym path_component_trans by blast
lemma path_component_eq_eq:
"path_component S x = path_component S y \<longleftrightarrow>
(x \<notin> S) \<and> (y \<notin> S) \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis (no_types) path_component_mem(1) path_component_refl)
next
assume ?rhs then show ?lhs
proof
assume "x \<notin> S \<and> y \<notin> S" then show ?lhs
by (metis Collect_empty_eq_bot path_component_eq_empty)
next
assume S: "x \<in> S \<and> y \<in> S \<and> path_component S x y" show ?lhs
by (rule ext) (metis S path_component_trans path_component_sym)
qed
qed
lemma path_component_unique:
assumes "x \<in> C" "C \<subseteq> S" "path_connected C"
"\<And>C'. \<lbrakk>x \<in> C'; C' \<subseteq> S; path_connected C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C"
shows "path_component_set S x = C"
by (smt (verit, best) Collect_cong assms path_component path_component_of_subset path_connected_component_set)
lemma path_component_intermediate_subset:
"path_component_set U a \<subseteq> T \<and> T \<subseteq> U
\<Longrightarrow> path_component_set T a = path_component_set U a"
by (metis (no_types) path_component_mono path_component_path_component subset_antisym)
lemma complement_path_component_Union:
fixes x :: "'a :: topological_space"
shows "S - path_component_set S x =
\<Union>({path_component_set S y| y. y \<in> S} - {path_component_set S x})"
proof -
have *: "(\<And>x. x \<in> S - {a} \<Longrightarrow> disjnt a x) \<Longrightarrow> \<Union>S - a = \<Union>(S - {a})"
for a::"'a set" and S
by (auto simp: disjnt_def)
have "\<And>y. y \<in> {path_component_set S x |x. x \<in> S} - {path_component_set S x}
\<Longrightarrow> disjnt (path_component_set S x) y"
using path_component_disjoint path_component_eq by fastforce
then have "\<Union>{path_component_set S x |x. x \<in> S} - path_component_set S x =
\<Union>({path_component_set S y |y. y \<in> S} - {path_component_set S x})"
by (meson *)
then show ?thesis by simp
qed
subsection\<open>Path components\<close>
definition path_component_of
where "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
abbreviation path_component_of_set
where "path_component_of_set X x \<equiv> Collect (path_component_of X x)"
definition path_components_of :: "'a topology \<Rightarrow> 'a set set"
where "path_components_of X \<equiv> path_component_of_set X ` topspace X"
lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g \<in> {0..1} \<rightarrow> T"
by (simp add: path_def pathin_def image_subset_iff_funcset)
lemma path_component_of_canon_iff [simp]:
"path_component_of (top_of_set T) a b \<longleftrightarrow> path_component T a b"
by (simp add: path_component_of_def pathin_canon_iff path_defs image_subset_iff_funcset)
lemma path_component_in_topspace:
"path_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
by (auto simp: path_component_of_def pathin_def continuous_map_def)
lemma path_component_of_refl:
"path_component_of X x x \<longleftrightarrow> x \<in> topspace X"
by (metis path_component_in_topspace path_component_of_def pathin_const)
lemma path_component_of_sym:
assumes "path_component_of X x y"
shows "path_component_of X y x"
using assms
apply (clarsimp simp: path_component_of_def pathin_def)
apply (rule_tac x="g \<circ> (\<lambda>t. 1 - t)" in exI)
apply (auto intro!: continuous_map_compose simp: continuous_map_in_subtopology continuous_on_op_minus)
done
lemma path_component_of_sym_iff:
"path_component_of X x y \<longleftrightarrow> path_component_of X y x"
by (metis path_component_of_sym)
lemma continuous_map_cases_le:
assumes contp: "continuous_map X euclideanreal p"
and contq: "continuous_map X euclideanreal q"
and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
proof -
have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0..} then f x else g x)"
proof (rule continuous_map_cases_function)
show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
by (intro contp contq continuous_intros)
show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0..}}) Y f"
by (simp add: contf)
show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g"
by (simp add: contg flip: Compl_eq_Diff_UNIV)
qed (auto simp: fg)
then show ?thesis
by simp
qed
lemma continuous_map_cases_lt:
assumes contp: "continuous_map X euclideanreal p"
and contq: "continuous_map X euclideanreal q"
and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
shows "continuous_map X Y (\<lambda>x. if p x < q x then f x else g x)"
proof -
have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0<..} then f x else g x)"
proof (rule continuous_map_cases_function)
show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
by (intro contp contq continuous_intros)
show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0<..}}) Y f"
by (simp add: contf)
show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g"
by (simp add: contg flip: Compl_eq_Diff_UNIV)
qed (auto simp: fg)
then show ?thesis
by simp
qed
lemma path_component_of_trans:
assumes "path_component_of X x y" and "path_component_of X y z"
shows "path_component_of X x z"
unfolding path_component_of_def pathin_def
proof -
let ?T01 = "top_of_set {0..1::real}"
obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1"
and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1"
using assms unfolding path_component_of_def pathin_def by blast
let ?g = "\<lambda>x. if x \<le> 1/2 then (g1 \<circ> (\<lambda>t. 2 * t)) x else (g2 \<circ> (\<lambda>t. 2 * t -1)) x"
show "\<exists>g. continuous_map ?T01 X g \<and> g 0 = x \<and> g 1 = z"
proof (intro exI conjI)
show "continuous_map (subtopology euclideanreal {0..1}) X ?g"
proof (intro continuous_map_cases_le continuous_map_compose, force, force)
show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. x \<le> 1/2}) ?T01 ((*) 2)"
by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology)
have "continuous_map
(subtopology (top_of_set {0..1}) {x. 0 \<le> x \<and> x \<le> 1 \<and> 1 \<le> x * 2})
euclideanreal (\<lambda>t. 2 * t - 1)"
by (intro continuous_intros) (force intro: continuous_map_from_subtopology)
then show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. 1/2 \<le> x}) ?T01 (\<lambda>t. 2 * t - 1)"
by (force simp: continuous_map_in_subtopology)
show "(g1 \<circ> (*) 2) x = (g2 \<circ> (\<lambda>t. 2 * t - 1)) x" if "x \<in> topspace ?T01" "x = 1/2" for x
using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology)
qed (auto simp: g1 g2)
qed (auto simp: g1 g2)
qed
lemma path_component_of_mono:
"\<lbrakk>path_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk> \<Longrightarrow> path_component_of (subtopology X T) x y"
unfolding path_component_of_def
by (metis subsetD pathin_subtopology)
lemma path_component_of:
"path_component_of X x y \<longleftrightarrow> (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis atLeastAtMost_iff image_eqI order_refl path_component_of_def path_connectedin_path_image zero_le_one)
next
assume ?rhs then show ?lhs
by (metis path_component_of_def path_connectedin)
qed
lemma path_component_of_set:
"path_component_of X x y \<longleftrightarrow> (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)"
by (auto simp: path_component_of_def)
lemma path_component_of_subset_topspace:
"Collect(path_component_of X x) \<subseteq> topspace X"
using path_component_in_topspace by fastforce
lemma path_component_of_eq_empty:
"Collect(path_component_of X x) = {} \<longleftrightarrow> (x \<notin> topspace X)"
using path_component_in_topspace path_component_of_refl by fastforce
lemma path_connected_space_iff_path_component:
"path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. path_component_of X x y)"
by (simp add: path_component_of path_connected_space_subconnected)
lemma path_connected_space_imp_path_component_of:
"\<lbrakk>path_connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk>
\<Longrightarrow> path_component_of X a b"
by (simp add: path_connected_space_iff_path_component)
lemma path_connected_space_path_component_set:
"path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. Collect(path_component_of X x) = topspace X)"
using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce
lemma path_component_of_maximal:
"\<lbrakk>path_connectedin X s; x \<in> s\<rbrakk> \<Longrightarrow> s \<subseteq> Collect(path_component_of X x)"
using path_component_of by fastforce
lemma path_component_of_equiv:
"path_component_of X x y \<longleftrightarrow> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding fun_eq_iff path_component_in_topspace
by (metis path_component_in_topspace path_component_of_sym path_component_of_trans)
qed (simp add: path_component_of_refl)
lemma path_component_of_disjoint:
"disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) \<longleftrightarrow>
~(path_component_of X x y)"
by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv)
lemma path_component_of_eq:
"path_component_of X x = path_component_of X y \<longleftrightarrow>
(x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y"
by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv)
lemma path_component_of_aux:
"path_component_of X x y
\<Longrightarrow> path_component_of (subtopology X (Collect (path_component_of X x))) x y"
by (meson path_component_of path_component_of_maximal path_connectedin_subtopology)
lemma path_connectedin_path_component_of:
"path_connectedin X (Collect (path_component_of X x))"
proof -
have "topspace (subtopology X (path_component_of_set X x)) = path_component_of_set X x"
by (meson path_component_of_subset_topspace topspace_subtopology_subset)
then have "path_connected_space (subtopology X (path_component_of_set X x))"
by (metis mem_Collect_eq path_component_of_aux path_component_of_equiv path_connected_space_iff_path_component)
then show ?thesis
by (simp add: path_component_of_subset_topspace path_connectedin_def)
qed
lemma path_connectedin_euclidean [simp]:
"path_connectedin euclidean S \<longleftrightarrow> path_connected S"
by (auto simp: path_connectedin_def path_connected_space_iff_path_component path_connected_component)
lemma path_connected_space_euclidean_subtopology [simp]:
"path_connected_space(subtopology euclidean S) \<longleftrightarrow> path_connected S"
using path_connectedin_topspace by force
lemma Union_path_components_of:
"\<Union>(path_components_of X) = topspace X"
by (auto simp: path_components_of_def path_component_of_equiv)
lemma path_components_of_maximal:
"\<lbrakk>C \<in> path_components_of X; path_connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C"
by (smt (verit, ccfv_SIG) disjnt_iff imageE mem_Collect_eq path_component_of_equiv
path_component_of_maximal path_components_of_def)
lemma pairwise_disjoint_path_components_of:
"pairwise disjnt (path_components_of X)"
by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv)
lemma complement_path_components_of_Union:
"C \<in> path_components_of X \<Longrightarrow> topspace X - C = \<Union>(path_components_of X - {C})"
by (metis Union_path_components_of bot.extremum ccpo_Sup_singleton diff_Union_pairwise_disjoint
insert_subsetI pairwise_disjoint_path_components_of)
lemma nonempty_path_components_of:
assumes "C \<in> path_components_of X" shows "C \<noteq> {}"
by (metis assms imageE path_component_of_eq_empty path_components_of_def)
lemma path_components_of_subset: "C \<in> path_components_of X \<Longrightarrow> C \<subseteq> topspace X"
by (auto simp: path_components_of_def path_component_of_equiv)
lemma path_connectedin_path_components_of:
"C \<in> path_components_of X \<Longrightarrow> path_connectedin X C"
by (auto simp: path_components_of_def path_connectedin_path_component_of)
lemma path_component_in_path_components_of:
"Collect (path_component_of X a) \<in> path_components_of X \<longleftrightarrow> a \<in> topspace X"
by (metis imageI nonempty_path_components_of path_component_of_eq_empty path_components_of_def)
lemma path_connectedin_Union:
assumes \<A>: "\<And>S. S \<in> \<A> \<Longrightarrow> path_connectedin X S" and "\<Inter>\<A> \<noteq> {}"
shows "path_connectedin X (\<Union>\<A>)"
proof -
obtain a where "\<And>S. S \<in> \<A> \<Longrightarrow> a \<in> S"
using assms by blast
then have "\<And>x. x \<in> topspace (subtopology X (\<Union>\<A>)) \<Longrightarrow> path_component_of (subtopology X (\<Union>\<A>)) a x"
unfolding topspace_subtopology path_component_of
by (metis (full_types) IntD2 Union_iff Union_upper \<A> path_connectedin_subtopology)
then show ?thesis
using \<A> unfolding path_connectedin_def
by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component)
qed
lemma path_connectedin_Un:
"\<lbrakk>path_connectedin X S; path_connectedin X T; S \<inter> T \<noteq> {}\<rbrakk>
\<Longrightarrow> path_connectedin X (S \<union> T)"
by (blast intro: path_connectedin_Union [of "{S,T}", simplified])
lemma path_connected_space_iff_components_eq:
"path_connected_space X \<longleftrightarrow>
(\<forall>C \<in> path_components_of X. \<forall>C' \<in> path_components_of X. C = C')"
unfolding path_components_of_def
proof (intro iffI ballI)
assume "\<forall>C \<in> path_component_of_set X ` topspace X.
\<forall>C' \<in> path_component_of_set X ` topspace X. C = C'"
then show "path_connected_space X"
using path_component_of_refl path_connected_space_iff_path_component by fastforce
qed (auto simp: path_connected_space_path_component_set)
lemma path_components_of_eq_empty:
"path_components_of X = {} \<longleftrightarrow> X = trivial_topology"
by (metis image_is_empty path_components_of_def subtopology_eq_discrete_topology_empty)
lemma path_components_of_empty_space:
"path_components_of trivial_topology = {}"
by (simp add: path_components_of_eq_empty)
lemma path_components_of_subset_singleton:
"path_components_of X \<subseteq> {S} \<longleftrightarrow>
path_connected_space X \<and> (topspace X = {} \<or> topspace X = S)"
proof (cases "topspace X = {}")
case True
then show ?thesis
by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty)
next
case False
have "(path_components_of X = {S}) \<longleftrightarrow> (path_connected_space X \<and> topspace X = S)"
by (metis False Set.set_insert ex_in_conv insert_iff path_component_in_path_components_of
path_connected_space_iff_components_eq path_connected_space_path_component_set)
with False show ?thesis
by (simp add: path_components_of_eq_empty subset_singleton_iff)
qed
lemma path_connected_space_iff_components_subset_singleton:
"path_connected_space X \<longleftrightarrow> (\<exists>a. path_components_of X \<subseteq> {a})"
by (simp add: path_components_of_subset_singleton)
lemma path_components_of_eq_singleton:
"path_components_of X = {S} \<longleftrightarrow> path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X"
by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff)
lemma path_components_of_path_connected_space:
"path_connected_space X \<Longrightarrow> path_components_of X = (if topspace X = {} then {} else {topspace X})"
by (simp add: path_components_of_eq_empty path_components_of_eq_singleton)
lemma path_component_subset_connected_component_of:
"path_component_of_set X x \<subseteq> connected_component_of_set X x"
proof (cases "x \<in> topspace X")
case True
then show ?thesis
by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of)
next
case False
then show ?thesis
using path_component_of_eq_empty by fastforce
qed
lemma exists_path_component_of_superset:
assumes S: "path_connectedin X S" and ne: "topspace X \<noteq> {}"
obtains C where "C \<in> path_components_of X" "S \<subseteq> C"
by (metis S ne ex_in_conv path_component_in_path_components_of path_component_of_maximal path_component_of_subset_topspace subset_eq that)
lemma path_component_of_eq_overlap:
"path_component_of X x = path_component_of X y \<longleftrightarrow>
(x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {}"
by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty)
lemma path_component_of_nonoverlap:
"Collect (path_component_of X x) \<inter> Collect (path_component_of X y) = {} \<longleftrightarrow>
(x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
path_component_of X x \<noteq> path_component_of X y"
by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap)
lemma path_component_of_overlap:
"Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {} \<longleftrightarrow>
x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
by (meson path_component_of_nonoverlap)
lemma path_components_of_disjoint:
"\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> disjnt C C' \<longleftrightarrow> C \<noteq> C'"
by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv)
lemma path_components_of_overlap:
"\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> C \<inter> C' \<noteq> {} \<longleftrightarrow> C = C'"
by (auto simp: path_components_of_def path_component_of_equiv)
lemma path_component_of_unique:
"\<lbrakk>x \<in> C; path_connectedin X C; \<And>C'. \<lbrakk>x \<in> C'; path_connectedin X C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C\<rbrakk>
\<Longrightarrow> Collect (path_component_of X x) = C"
by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of)
lemma path_component_of_discrete_topology [simp]:
"Collect (path_component_of (discrete_topology U) x) = (if x \<in> U then {x} else {})"
proof -
have "\<And>C'. \<lbrakk>x \<in> C'; path_connectedin (discrete_topology U) C'\<rbrakk> \<Longrightarrow> C' \<subseteq> {x}"
by (metis path_connectedin_discrete_topology subsetD singletonD)
then have "x \<in> U \<Longrightarrow> Collect (path_component_of (discrete_topology U) x) = {x}"
by (simp add: path_component_of_unique)
then show ?thesis
using path_component_in_topspace by fastforce
qed
lemma path_component_of_discrete_topology_iff [simp]:
"path_component_of (discrete_topology U) x y \<longleftrightarrow> x \<in> U \<and> y=x"
by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD)
lemma path_components_of_discrete_topology [simp]:
"path_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U"
by (auto simp: path_components_of_def image_def fun_eq_iff)
lemma homeomorphic_map_path_component_of:
assumes f: "homeomorphic_map X Y f" and x: "x \<in> topspace X"
shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)"
proof -
obtain g where g: "homeomorphic_maps X Y f g"
using f homeomorphic_map_maps by blast
show ?thesis
proof
have "Collect (path_component_of Y (f x)) \<subseteq> topspace Y"
by (simp add: path_component_of_subset_topspace)
moreover have "g ` Collect(path_component_of Y (f x)) \<subseteq> Collect (path_component_of X (g (f x)))"
using f g x unfolding homeomorphic_maps_def
by (metis image_Collect_subsetI image_eqI mem_Collect_eq path_component_of_equiv path_component_of_maximal
path_connectedin_continuous_map_image path_connectedin_path_component_of)
ultimately show "Collect (path_component_of Y (f x)) \<subseteq> f ` Collect (path_component_of X x)"
using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff
by metis
show "f ` Collect (path_component_of X x) \<subseteq> Collect (path_component_of Y (f x))"
proof (rule path_component_of_maximal)
show "path_connectedin Y (f ` Collect (path_component_of X x))"
by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of)
qed (simp add: path_component_of_refl x)
qed
qed
lemma homeomorphic_map_path_components_of:
assumes "homeomorphic_map X Y f"
shows "path_components_of Y = (image f) ` (path_components_of X)"
unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric]
using assms homeomorphic_map_path_component_of by fastforce
subsection\<open>Paths and path-connectedness\<close>
lemma path_connected_space_quotient_map_image:
"\<lbrakk>quotient_map X Y q; path_connected_space X\<rbrakk> \<Longrightarrow> path_connected_space Y"
by (metis path_connectedin_continuous_map_image path_connectedin_topspace quotient_imp_continuous_map quotient_imp_surjective_map)
lemma path_connected_space_retraction_map_image:
"\<lbrakk>retraction_map X Y r; path_connected_space X\<rbrakk> \<Longrightarrow> path_connected_space Y"
using path_connected_space_quotient_map_image retraction_imp_quotient_map by blast
lemma path_connected_space_prod_topology:
"path_connected_space(prod_topology X Y) \<longleftrightarrow>
topspace(prod_topology X Y) = {} \<or> path_connected_space X \<and> path_connected_space Y"
proof (cases "topspace(prod_topology X Y) = {}")
case True
then show ?thesis
using path_connected_space_topspace_empty by force
next
case False
have "path_connected_space (prod_topology X Y)"
if X: "path_connected_space X" and Y: "path_connected_space Y"
proof (clarsimp simp: path_connected_space_def)
fix x y x' y'
assume "x \<in> topspace X" and "y \<in> topspace Y" and "x' \<in> topspace X" and "y' \<in> topspace Y"
obtain f where "pathin X f" "f 0 = x" "f 1 = x'"
by (meson X \<open>x \<in> topspace X\<close> \<open>x' \<in> topspace X\<close> path_connected_space_def)
obtain g where "pathin Y g" "g 0 = y" "g 1 = y'"
by (meson Y \<open>y \<in> topspace Y\<close> \<open>y' \<in> topspace Y\<close> path_connected_space_def)
show "\<exists>h. pathin (prod_topology X Y) h \<and> h 0 = (x,y) \<and> h 1 = (x',y')"
proof (intro exI conjI)
show "pathin (prod_topology X Y) (\<lambda>t. (f t, g t))"
using \<open>pathin X f\<close> \<open>pathin Y g\<close> by (simp add: continuous_map_paired pathin_def)
show "(\<lambda>t. (f t, g t)) 0 = (x, y)"
using \<open>f 0 = x\<close> \<open>g 0 = y\<close> by blast
show "(\<lambda>t. (f t, g t)) 1 = (x', y')"
using \<open>f 1 = x'\<close> \<open>g 1 = y'\<close> by blast
qed
qed
then show ?thesis
by (metis False path_connected_space_quotient_map_image prod_topology_trivial1 prod_topology_trivial2
quotient_map_fst quotient_map_snd topspace_discrete_topology)
qed
lemma path_connectedin_Times:
"path_connectedin (prod_topology X Y) (S \<times> T) \<longleftrightarrow>
S = {} \<or> T = {} \<or> path_connectedin X S \<and> path_connectedin Y T"
by (auto simp add: path_connectedin_def subtopology_Times path_connected_space_prod_topology)
subsection\<open>Path components\<close>
lemma path_component_of_subtopology_eq:
"path_component_of (subtopology X U) x = path_component_of X x \<longleftrightarrow> path_component_of_set X x \<subseteq> U"
(is "?lhs = ?rhs")
proof
show "?lhs \<Longrightarrow> ?rhs"
by (metis path_connectedin_path_component_of path_connectedin_subtopology)
next
show "?rhs \<Longrightarrow> ?lhs"
unfolding fun_eq_iff
by (metis path_connectedin_subtopology path_component_of path_component_of_aux path_component_of_mono)
qed
lemma path_components_of_subtopology:
assumes "C \<in> path_components_of X" "C \<subseteq> U"
shows "C \<in> path_components_of (subtopology X U)"
using assms path_component_of_refl path_component_of_subtopology_eq topspace_subtopology
by (smt (verit) imageE path_component_in_path_components_of path_components_of_def)
lemma path_imp_connected_component_of:
"path_component_of X x y \<Longrightarrow> connected_component_of X x y"
by (metis in_mono mem_Collect_eq path_component_subset_connected_component_of)
lemma finite_path_components_of_finite:
"finite(topspace X) \<Longrightarrow> finite(path_components_of X)"
by (simp add: Union_path_components_of finite_UnionD)
lemma path_component_of_continuous_image:
"\<lbrakk>continuous_map X X' f; path_component_of X x y\<rbrakk> \<Longrightarrow> path_component_of X' (f x) (f y)"
by (meson image_eqI path_component_of path_connectedin_continuous_map_image)
lemma path_component_of_pair [simp]:
"path_component_of_set (prod_topology X Y) (x,y) =
path_component_of_set X x \<times> path_component_of_set Y y" (is "?lhs = ?rhs")
proof (cases "?lhs = {}")
case True
then show ?thesis
by (metis Sigma_empty1 Sigma_empty2 mem_Sigma_iff path_component_of_eq_empty topspace_prod_topology)
next
case False
then have "path_component_of X x x" "path_component_of Y y y"
using path_component_of_eq_empty path_component_of_refl by fastforce+
moreover
have "path_connectedin (prod_topology X Y) (path_component_of_set X x \<times> path_component_of_set Y y)"
by (metis path_connectedin_Times path_connectedin_path_component_of)
moreover have "path_component_of X x a" "path_component_of Y y b"
if "(x, y) \<in> C'" "(a,b) \<in> C'" and "path_connectedin (prod_topology X Y) C'" for C' a b
by (smt (verit, best) that continuous_map_fst continuous_map_snd fst_conv snd_conv path_component_of path_component_of_continuous_image)+
ultimately show ?thesis
by (intro path_component_of_unique) auto
qed
lemma path_components_of_prod_topology:
"path_components_of (prod_topology X Y) =
(\<lambda>(C,D). C \<times> D) ` (path_components_of X \<times> path_components_of Y)"
by (force simp add: image_iff path_components_of_def)
lemma path_components_of_prod_topology':
"path_components_of (prod_topology X Y) =
{C \<times> D |C D. C \<in> path_components_of X \<and> D \<in> path_components_of Y}"
by (auto simp: path_components_of_prod_topology)
lemma path_component_of_product_topology:
"path_component_of_set (product_topology X I) f =
(if f \<in> extensional I then PiE I (\<lambda>i. path_component_of_set (X i) (f i)) else {})"
(is "?lhs = ?rhs")
proof (cases "path_component_of_set (product_topology X I) f = {}")
case True
then show ?thesis
by (smt (verit) PiE_eq_empty_iff PiE_iff path_component_of_eq_empty topspace_product_topology)
next
case False
then have [simp]: "f \<in> extensional I"
by (auto simp: path_component_of_eq_empty PiE_iff path_component_of_equiv)
show ?thesis
proof (intro path_component_of_unique)
show "f \<in> ?rhs"
using False path_component_of_eq_empty path_component_of_refl by force
show "path_connectedin (product_topology X I) (if f \<in> extensional I then \<Pi>\<^sub>E i\<in>I. path_component_of_set (X i) (f i) else {})"
by (simp add: path_connectedin_PiE path_connectedin_path_component_of)
fix C'
assume "f \<in> C'" and C': "path_connectedin (product_topology X I) C'"
show "C' \<subseteq> ?rhs"
proof -
have "C' \<subseteq> extensional I"
using PiE_def C' path_connectedin_subset_topspace by fastforce
with \<open>f \<in> C'\<close> C' show ?thesis
apply (clarsimp simp: PiE_iff subset_iff)
by (smt (verit, ccfv_threshold) continuous_map_product_projection path_component_of path_component_of_continuous_image)
qed
qed
qed
lemma path_components_of_product_topology:
"path_components_of (product_topology X I) =
{PiE I B |B. \<forall>i \<in> I. B i \<in> path_components_of(X i)}" (is "?lhs=?rhs")
proof
show "?lhs \<subseteq> ?rhs"
unfolding path_components_of_def image_subset_iff
by (smt (verit) image_iff mem_Collect_eq path_component_of_product_topology topspace_product_topology_alt)
next
show "?rhs \<subseteq> ?lhs"
proof
fix F
assume "F \<in> ?rhs"
then obtain B where B: "F = Pi\<^sub>E I B"
and "\<forall>i\<in>I. \<exists>x\<in>topspace (X i). B i = path_component_of_set (X i) x"
by (force simp add: path_components_of_def image_iff)
then obtain f where ftop: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> topspace (X i)"
and BF: "\<And>i. i \<in> I \<Longrightarrow> B i = path_component_of_set (X i) (f i)"
by metis
then have "F = path_component_of_set (product_topology X I) (restrict f I)"
by (metis (mono_tags, lifting) B PiE_cong path_component_of_product_topology restrict_apply' restrict_extensional)
then show "F \<in> ?lhs"
by (simp add: ftop path_component_in_path_components_of)
qed
qed
subsection \<open>Sphere is path-connected\<close>
lemma path_connected_punctured_universe:
assumes "2 \<le> DIM('a::euclidean_space)"
shows "path_connected (- {a::'a})"
proof -
let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
have A: "path_connected ?A"
unfolding Collect_bex_eq
proof (rule path_connected_UNION)
fix i :: 'a
assume "i \<in> Basis"
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}"
by simp
show "path_connected {x. x \<bullet> i < a \<bullet> i}"
using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
by (simp add: inner_commute)
qed
have B: "path_connected ?B"
unfolding Collect_bex_eq
proof (rule path_connected_UNION)
fix i :: 'a
assume "i \<in> Basis"
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}"
by simp
show "path_connected {x. a \<bullet> i < x \<bullet> i}"
using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
by (simp add: inner_commute)
qed
obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)"
using obtain_subset_with_card_n[OF assms] by (force simp add: eval_nat_numeral)
then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1"
unfolding card_Suc_eq by auto
then have "a + b0 - b1 \<in> ?A \<inter> ?B"
by (auto simp: inner_simps inner_Basis)
then have "?A \<inter> ?B \<noteq> {}"
by fast
with A B have "path_connected (?A \<union> ?B)"
by (rule path_connected_Un)
also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
also have "\<dots> = {x. x \<noteq> a}"
unfolding euclidean_eq_iff [where 'a='a]
by (simp add: Bex_def)
also have "\<dots> = - {a}"
by auto
finally show ?thesis .
qed
corollary connected_punctured_universe:
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})"
by (simp add: path_connected_punctured_universe path_connected_imp_connected)
proposition path_connected_sphere:
fixes a :: "'a :: euclidean_space"
assumes "2 \<le> DIM('a)"
shows "path_connected(sphere a r)"
proof (cases r "0::real" rule: linorder_cases)
case greater
then have eq: "(sphere (0::'a) r) = (\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a})"
by (force simp: image_iff split: if_split_asm)
have "continuous_on (- {0::'a}) (\<lambda>x. (r / norm x) *\<^sub>R x)"
by (intro continuous_intros) auto
then have "path_connected ((\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a}))"
by (intro path_connected_continuous_image path_connected_punctured_universe assms)
with eq have "path_connected((+) a ` (sphere (0::'a) r))"
by (simp add: path_connected_translation)
then show ?thesis
by (metis add.right_neutral sphere_translation)
qed auto
lemma connected_sphere:
fixes a :: "'a :: euclidean_space"
assumes "2 \<le> DIM('a)"
shows "connected(sphere a r)"
using path_connected_sphere [OF assms]
by (simp add: path_connected_imp_connected)
corollary path_connected_complement_bounded_convex:
fixes S :: "'a :: euclidean_space set"
assumes "bounded S" "convex S" and 2: "2 \<le> DIM('a)"
shows "path_connected (- S)"
proof (cases "S = {}")
case True then show ?thesis
using convex_imp_path_connected by auto
next
case False
then obtain a where "a \<in> S" by auto
have \<section> [rule_format]: "\<forall>y\<in>S. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> (1 - u) *\<^sub>R a + u *\<^sub>R y \<in> S"
using \<open>convex S\<close> \<open>a \<in> S\<close> by (simp add: convex_alt)
{ fix x y assume "x \<notin> S" "y \<notin> S"
then have "x \<noteq> a" "y \<noteq> a" using \<open>a \<in> S\<close> by auto
then have bxy: "bounded(insert x (insert y S))"
by (simp add: \<open>bounded S\<close>)
then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
and "S \<subseteq> ball a B"
using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
define C where "C = B / norm(x - a)"
let ?Cxa = "a + C *\<^sub>R (x - a)"
{ fix u
assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R ?Cxa \<in> S" and "0 \<le> u" "u \<le> 1"
have CC: "1 \<le> 1 + (C - 1) * u"
using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close> Bx
by (auto simp add: C_def norm_minus_commute)
have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) =
(1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x"
by (simp add: algebra_simps)
also have "\<dots> = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x"
using CC by (simp add: field_simps)
also have "\<dots> = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x"
by (simp add: algebra_simps)
also have "\<dots> = x + ((1 / (1 + C * u - u)) *\<^sub>R a +
((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))"
using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x"
by (simp add: algebra_simps)
have False
using \<section> [of "a + (1 + (C - 1) * u) *\<^sub>R (x - a)" "1 / (1 + (C - 1) * u)"]
using u \<open>x \<noteq> a\<close> \<open>x \<notin> S\<close> \<open>0 \<le> u\<close> CC
by (auto simp: xeq *)
}
then have pcx: "path_component (- S) x ?Cxa"
by (force simp: closed_segment_def intro!: path_component_linepath)
define D where "D = B / norm(y - a)" \<comment> \<open>massive duplication with the proof above\<close>
let ?Dya = "a + D *\<^sub>R (y - a)"
{ fix u
assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R ?Dya \<in> S" and "0 \<le> u" "u \<le> 1"
have DD: "1 \<le> 1 + (D - 1) * u"
using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close> By
by (auto simp add: D_def norm_minus_commute)
have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) =
(1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y"
by (simp add: algebra_simps)
also have "\<dots> = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y"
using DD by (simp add: field_simps)
also have "\<dots> = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y"
by (simp add: algebra_simps)
also have "\<dots> = y + ((1 / (1 + D * u - u)) *\<^sub>R a +
((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))"
using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y"
by (simp add: algebra_simps)
have False
using \<section> [of "a + (1 + (D - 1) * u) *\<^sub>R (y - a)" "1 / (1 + (D - 1) * u)"]
using u \<open>y \<noteq> a\<close> \<open>y \<notin> S\<close> \<open>0 \<le> u\<close> DD
by (auto simp: xeq *)
}
then have pdy: "path_component (- S) y ?Dya"
by (force simp: closed_segment_def intro!: path_component_linepath)
have pyx: "path_component (- S) ?Dya ?Cxa"
proof (rule path_component_of_subset)
show "sphere a B \<subseteq> - S"
using \<open>S \<subseteq> ball a B\<close> by (force simp: ball_def dist_norm norm_minus_commute)
have aB: "?Dya \<in> sphere a B" "?Cxa \<in> sphere a B"
using \<open>x \<noteq> a\<close> using \<open>y \<noteq> a\<close> B by (auto simp: dist_norm C_def D_def)
then show "path_component (sphere a B) ?Dya ?Cxa"
using path_connected_sphere [OF 2] path_connected_component by blast
qed
have "path_component (- S) x y"
by (metis path_component_trans path_component_sym pcx pdy pyx)
}
then show ?thesis
by (auto simp: path_connected_component)
qed
lemma connected_complement_bounded_convex:
fixes S :: "'a :: euclidean_space set"
assumes "bounded S" "convex S" "2 \<le> DIM('a)"
shows "connected (- S)"
using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast
lemma connected_diff_ball:
fixes S :: "'a :: euclidean_space set"
assumes "connected S" "cball a r \<subseteq> S" "2 \<le> DIM('a)"
shows "connected (S - ball a r)"
proof (rule connected_diff_open_from_closed [OF ball_subset_cball])
show "connected (cball a r - ball a r)"
using assms connected_sphere by (auto simp: cball_diff_eq_sphere)
qed (auto simp: assms dist_norm)
proposition connected_open_delete:
assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
shows "connected(S - {a::'N})"
proof (cases "a \<in> S")
case True
with \<open>open S\<close> obtain \<epsilon> where "\<epsilon> > 0" and \<epsilon>: "cball a \<epsilon> \<subseteq> S"
using open_contains_cball_eq by blast
define b where "b \<equiv> a + \<epsilon> *\<^sub>R (SOME i. i \<in> Basis)"
have "dist a b = \<epsilon>"
by (simp add: b_def dist_norm SOME_Basis \<open>0 < \<epsilon>\<close> less_imp_le)
with \<epsilon> have "b \<in> \<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}"
by auto
then have nonemp: "(\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}) = {} \<Longrightarrow> False"
by auto
have con: "\<And>r. r < \<epsilon> \<Longrightarrow> connected (S - ball a r)"
using \<epsilon> by (force intro: connected_diff_ball [OF \<open>connected S\<close> _ 2])
have "x \<in> \<Union>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}" if "x \<in> S - {a}" for x
using that \<open>0 < \<epsilon>\<close>
by (intro UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"]) auto
then have "S - {a} = \<Union>{S - ball a r | r. 0 < r \<and> r < \<epsilon>}"
by auto
then show ?thesis
by (auto intro: connected_Union con dest!: nonemp)
next
case False then show ?thesis
by (simp add: \<open>connected S\<close>)
qed
corollary path_connected_open_delete:
assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
shows "path_connected(S - {a::'N})"
by (simp add: assms connected_open_delete connected_open_path_connected open_delete)
corollary path_connected_punctured_ball:
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> path_connected(ball a r - {a::'N})"
by (simp add: path_connected_open_delete)
corollary connected_punctured_ball:
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(ball a r - {a::'N})"
by (simp add: connected_open_delete)
corollary connected_open_delete_finite:
fixes S T::"'a::euclidean_space set"
assumes S: "open S" "connected S" and 2: "2 \<le> DIM('a)" and "finite T"
shows "connected(S - T)"
using \<open>finite T\<close> S
proof (induct T)
case empty
show ?case using \<open>connected S\<close> by simp
next
case (insert x T)
then have "connected (S-T)"
by auto
moreover have "open (S - T)"
using finite_imp_closed[OF \<open>finite T\<close>] \<open>open S\<close> by auto
ultimately have "connected (S - T - {x})"
using connected_open_delete[OF _ _ 2] by auto
thus ?case by (metis Diff_insert)
qed
lemma sphere_1D_doubleton_zero:
assumes 1: "DIM('a) = 1" and "r > 0"
obtains x y::"'a::euclidean_space"
where "sphere 0 r = {x,y} \<and> dist x y = 2*r"
proof -
obtain b::'a where b: "Basis = {b}"
using 1 card_1_singletonE by blast
show ?thesis
proof (intro that conjI)
have "x = norm x *\<^sub>R b \<or> x = - norm x *\<^sub>R b" if "r = norm x" for x
proof -
have xb: "(x \<bullet> b) *\<^sub>R b = x"
using euclidean_representation [of x, unfolded b] by force
then have "norm ((x \<bullet> b) *\<^sub>R b) = norm x"
by simp
with b have "\<bar>x \<bullet> b\<bar> = norm x"
using norm_Basis by (simp add: b)
with xb show ?thesis
by (metis (mono_tags, opaque_lifting) abs_eq_iff abs_norm_cancel)
qed
with \<open>r > 0\<close> b show "sphere 0 r = {r *\<^sub>R b, - r *\<^sub>R b}"
by (force simp: sphere_def dist_norm)
have "dist (r *\<^sub>R b) (- r *\<^sub>R b) = norm (r *\<^sub>R b + r *\<^sub>R b)"
by (simp add: dist_norm)
also have "\<dots> = norm ((2*r) *\<^sub>R b)"
by (metis mult_2 scaleR_add_left)
also have "\<dots> = 2*r"
using \<open>r > 0\<close> b norm_Basis by fastforce
finally show "dist (r *\<^sub>R b) (- r *\<^sub>R b) = 2*r" .
qed
qed
lemma sphere_1D_doubleton:
fixes a :: "'a :: euclidean_space"
assumes "DIM('a) = 1" and "r > 0"
obtains x y where "sphere a r = {x,y} \<and> dist x y = 2*r"
using sphere_1D_doubleton_zero [OF assms] dist_add_cancel image_empty image_insert
by (metis (no_types, opaque_lifting) add.right_neutral sphere_translation)
lemma psubset_sphere_Compl_connected:
fixes S :: "'a::euclidean_space set"
assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)"
shows "connected(- S)"
proof -
have "S \<subseteq> sphere a r"
using S by blast
obtain b where "dist a b = r" and "b \<notin> S"
using S mem_sphere by blast
have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}"
by auto
have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}"
using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast
moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}"
using assms
by (force intro: connected_intermediate_closure [of "ball a r"])
moreover have "connected {x. r \<le> dist a x \<and> x \<notin> S}"
proof (rule connected_intermediate_closure [of "- cball a r"])
show "{x. r \<le> dist a x \<and> x \<notin> S} \<subseteq> closure (- cball a r)"
using interior_closure by (force intro: connected_complement_bounded_convex)
qed (use assms connected_complement_bounded_convex in auto)
ultimately show ?thesis
by (simp add: CS connected_Un)
qed
subsection\<open>Every annulus is a connected set\<close>
lemma path_connected_2DIM_I:
fixes a :: "'N::euclidean_space"
assumes 2: "2 \<le> DIM('N)" and pc: "path_connected {r. 0 \<le> r \<and> P r}"
shows "path_connected {x. P(norm(x - a))}"
proof -
have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}"
by force
moreover have "path_connected {x::'N. P(norm x)}"
proof -
let ?D = "{x. 0 \<le> x \<and> P x} \<times> sphere (0::'N) 1"
have "x \<in> (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
if "P (norm x)" for x::'N
proof (cases "x=0")
case True
with that show ?thesis
apply (simp add: image_iff)
by (metis (no_types) mem_sphere_0 order_refl vector_choose_size zero_le_one)
next
case False
with that show ?thesis
by (rule_tac x="(norm x, x /\<^sub>R norm x)" in image_eqI) auto
qed
then have *: "{x::'N. P(norm x)} = (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
by auto
have "continuous_on ?D (\<lambda>z:: real\<times>'N. fst z *\<^sub>R snd z)"
by (intro continuous_intros)
moreover have "path_connected ?D"
by (metis path_connected_Times [OF pc] path_connected_sphere 2)
ultimately show ?thesis
by (simp add: "*" path_connected_continuous_image)
qed
ultimately show ?thesis
using path_connected_translation by metis
qed
proposition path_connected_annulus:
fixes a :: "'N::euclidean_space"
assumes "2 \<le> DIM('N)"
shows "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
"path_connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
"path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
"path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms])
proposition connected_annulus:
fixes a :: "'N::euclidean_space"
assumes "2 \<le> DIM('N::euclidean_space)"
shows "connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
"connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
"connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
"connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations between components and path components\<close>
lemma open_connected_component:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open (connected_component_set S x)"
proof (clarsimp simp: open_contains_ball)
fix y
assume xy: "connected_component S x y"
then obtain e where "e>0" "ball y e \<subseteq> S"
using assms connected_component_in openE by blast
then show "\<exists>e>0. ball y e \<subseteq> connected_component_set S x"
by (metis xy centre_in_ball connected_ball connected_component_eq_eq connected_component_in connected_component_maximal)
qed
corollary open_components:
fixes S :: "'a::real_normed_vector set"
shows "\<lbrakk>open u; S \<in> components u\<rbrakk> \<Longrightarrow> open S"
by (simp add: components_iff) (metis open_connected_component)
lemma in_closure_connected_component:
fixes S :: "'a::real_normed_vector set"
assumes x: "x \<in> S" and S: "open S"
shows "x \<in> closure (connected_component_set S y) \<longleftrightarrow> x \<in> connected_component_set S y"
proof -
have "x islimpt connected_component_set S y \<Longrightarrow> connected_component S y x"
by (metis (no_types, lifting) S connected_component_eq connected_component_refl islimptE mem_Collect_eq open_connected_component x)
then show ?thesis
by (auto simp: closure_def)
qed
lemma connected_disjoint_Union_open_pick:
assumes "pairwise disjnt B"
"\<And>S. S \<in> A \<Longrightarrow> connected S \<and> S \<noteq> {}"
"\<And>S. S \<in> B \<Longrightarrow> open S"
"\<Union>A \<subseteq> \<Union>B"
"S \<in> A"
obtains T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
proof -
have "S \<subseteq> \<Union>B" "connected S" "S \<noteq> {}"
using assms \<open>S \<in> A\<close> by blast+
then obtain T where "T \<in> B" "S \<inter> T \<noteq> {}"
by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute)
have 1: "open T" by (simp add: \<open>T \<in> B\<close> assms)
have 2: "open (\<Union>(B-{T}))" using assms by blast
have 3: "S \<subseteq> T \<union> \<Union>(B - {T})" using \<open>S \<subseteq> \<Union>B\<close> by blast
have "T \<inter> \<Union>(B - {T}) = {}" using \<open>T \<in> B\<close> \<open>pairwise disjnt B\<close>
by (auto simp: pairwise_def disjnt_def)
then have 4: "T \<inter> \<Union>(B - {T}) \<inter> S = {}" by auto
from connectedD [OF \<open>connected S\<close> 1 2 4 3]
have "S \<inter> \<Union>(B-{T}) = {}"
by (auto simp: Int_commute \<open>S \<inter> T \<noteq> {}\<close>)
with \<open>T \<in> B\<close> 3 that show ?thesis
by (metis IntI UnE empty_iff subsetD subsetI)
qed
lemma connected_disjoint_Union_open_subset:
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
and eq [simp]: "\<Union>A = \<Union>B"
shows "A \<subseteq> B"
proof
fix S
assume "S \<in> A"
obtain T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
using SA SB \<open>S \<in> A\<close> connected_disjoint_Union_open_pick [OF B, of A] eq order_refl by blast
moreover obtain S' where "S' \<in> A" "T \<subseteq> S'" "T \<inter> \<Union>(A - {S'}) = {}"
using SA SB \<open>T \<in> B\<close> connected_disjoint_Union_open_pick [OF A, of B] eq order_refl by blast
ultimately have "S' = S"
by (metis A Int_subset_iff SA \<open>S \<in> A\<close> disjnt_def inf.orderE pairwise_def)
with \<open>T \<subseteq> S'\<close> have "T \<subseteq> S" by simp
with \<open>S \<subseteq> T\<close> have "S = T" by blast
with \<open>T \<in> B\<close> show "S \<in> B" by simp
qed
lemma connected_disjoint_Union_open_unique:
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
and eq [simp]: "\<Union>A = \<Union>B"
shows "A = B"
by (metis subset_antisym connected_disjoint_Union_open_subset assms)
proposition components_open_unique:
fixes S :: "'a::real_normed_vector set"
assumes "pairwise disjnt A" "\<Union>A = S"
"\<And>X. X \<in> A \<Longrightarrow> open X \<and> connected X \<and> X \<noteq> {}"
shows "components S = A"
proof -
have "open S" using assms by blast
show ?thesis
proof (rule connected_disjoint_Union_open_unique)
show "disjoint (components S)"
by (simp add: components_eq disjnt_def pairwise_def)
qed (use \<open>open S\<close> in \<open>simp_all add: assms open_components in_components_connected in_components_nonempty\<close>)
qed
subsection\<^marker>\<open>tag unimportant\<close>\<open>Existence of unbounded components\<close>
lemma cobounded_unbounded_component:
fixes S :: "'a :: euclidean_space set"
assumes "bounded (-S)"
shows "\<exists>x. x \<in> S \<and> \<not> bounded (connected_component_set S x)"
proof -
obtain i::'a where i: "i \<in> Basis"
using nonempty_Basis by blast
obtain B where B: "B>0" "-S \<subseteq> ball 0 B"
using bounded_subset_ballD [OF assms, of 0] by auto
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> S"
by (force simp: ball_def dist_norm)
have unbounded_inner: "\<not> bounded {x. inner i x \<ge> B}"
proof (clarsimp simp: bounded_def dist_norm)
fix e x
show "\<exists>y. B \<le> i \<bullet> y \<and> \<not> norm (x - y) \<le> e"
using i
by (rule_tac x="x + (max B e + 1 + \<bar>i \<bullet> x\<bar>) *\<^sub>R i" in exI) (auto simp: inner_right_distrib)
qed
have \<section>: "\<And>x. B \<le> i \<bullet> x \<Longrightarrow> x \<in> S"
using * Basis_le_norm [OF i] by (metis abs_ge_self inner_commute order_trans)
have "{x. B \<le> i \<bullet> x} \<subseteq> connected_component_set S (B *\<^sub>R i)"
by (intro connected_component_maximal) (auto simp: i intro: convex_connected convex_halfspace_ge [of B] \<section>)
then have "\<not> bounded (connected_component_set S (B *\<^sub>R i))"
using bounded_subset unbounded_inner by blast
moreover have "B *\<^sub>R i \<in> S"
by (rule *) (simp add: norm_Basis [OF i])
ultimately show ?thesis
by blast
qed
lemma cobounded_unique_unbounded_component:
fixes S :: "'a :: euclidean_space set"
assumes bs: "bounded (-S)" and "2 \<le> DIM('a)"
and bo: "\<not> bounded(connected_component_set S x)"
"\<not> bounded(connected_component_set S y)"
shows "connected_component_set S x = connected_component_set S y"
proof -
obtain i::'a where i: "i \<in> Basis"
using nonempty_Basis by blast
obtain B where "B>0" and B: "-S \<subseteq> ball 0 B"
using bounded_subset_ballD [OF bs, of 0] by auto
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> S"
by (force simp: ball_def dist_norm)
obtain x' y' where x': "connected_component S x x'" "norm x' > B"
and y': "connected_component S y y'" "norm y' > B"
using \<open>B>0\<close> bo bounded_pos by (metis linorder_not_le mem_Collect_eq)
have x'y': "connected_component S x' y'"
unfolding connected_component_def
proof (intro exI conjI)
show "connected (- ball 0 B :: 'a set)"
using assms by (auto intro: connected_complement_bounded_convex)
qed (use x' y' dist_norm * in auto)
show ?thesis
using x' y' x'y'
by (metis connected_component_eq mem_Collect_eq)
qed
lemma cobounded_unbounded_components:
fixes S :: "'a :: euclidean_space set"
shows "bounded (-S) \<Longrightarrow> \<exists>c. c \<in> components S \<and> \<not>bounded c"
by (metis cobounded_unbounded_component components_def imageI)
lemma cobounded_unique_unbounded_components:
fixes S :: "'a :: euclidean_space set"
shows "\<lbrakk>bounded (- S); c \<in> components S; \<not> bounded c; c' \<in> components S; \<not> bounded c'; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> c' = c"
unfolding components_iff
by (metis cobounded_unique_unbounded_component)
lemma cobounded_has_bounded_component:
fixes S :: "'a :: euclidean_space set"
assumes "bounded (- S)" "\<not> connected S" "2 \<le> DIM('a)"
obtains C where "C \<in> components S" "bounded C"
by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
subsection\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
text\<^marker>\<open>tag important\<close>\<open>The inside comprises the points in a bounded connected component of the set's complement.
The outside comprises the points in unbounded connected component of the complement.\<close>
definition\<^marker>\<open>tag important\<close> inside where
"inside S \<equiv> {x. (x \<notin> S) \<and> bounded(connected_component_set ( - S) x)}"
definition\<^marker>\<open>tag important\<close> outside where
"outside S \<equiv> -S \<inter> {x. \<not> bounded(connected_component_set (- S) x)}"
lemma outside: "outside S = {x. \<not> bounded(connected_component_set (- S) x)}"
by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty)
lemma inside_no_overlap [simp]: "inside S \<inter> S = {}"
by (auto simp: inside_def)
lemma outside_no_overlap [simp]:
"outside S \<inter> S = {}"
by (auto simp: outside_def)
lemma inside_Int_outside [simp]: "inside S \<inter> outside S = {}"
by (auto simp: inside_def outside_def)
lemma inside_Un_outside [simp]: "inside S \<union> outside S = (- S)"
by (auto simp: inside_def outside_def)
lemma inside_eq_outside:
"inside S = outside S \<longleftrightarrow> S = UNIV"
by (auto simp: inside_def outside_def)
lemma inside_outside: "inside S = (- (S \<union> outside S))"
by (force simp: inside_def outside)
lemma outside_inside: "outside S = (- (S \<union> inside S))"
by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap)
lemma union_with_inside: "S \<union> inside S = - outside S"
by (auto simp: inside_outside) (simp add: outside_inside)
lemma union_with_outside: "S \<union> outside S = - inside S"
by (simp add: inside_outside)
lemma outside_mono: "S \<subseteq> T \<Longrightarrow> outside T \<subseteq> outside S"
by (auto simp: outside bounded_subset connected_component_mono)
lemma inside_mono: "S \<subseteq> T \<Longrightarrow> inside S - T \<subseteq> inside T"
by (auto simp: inside_def bounded_subset connected_component_mono)
lemma segment_bound_lemma:
fixes u::real
assumes "x \<ge> B" "y \<ge> B" "0 \<le> u" "u \<le> 1"
shows "(1 - u) * x + u * y \<ge> B"
by (smt (verit) assms convex_bound_le ge_iff_diff_ge_0 minus_add_distrib
mult_minus_right neg_le_iff_le)
lemma cobounded_outside:
fixes S :: "'a :: real_normed_vector set"
assumes "bounded S" shows "bounded (- outside S)"
proof -
obtain B where B: "B>0" "S \<subseteq> ball 0 B"
using bounded_subset_ballD [OF assms, of 0] by auto
{ fix x::'a and C::real
assume Bno: "B \<le> norm x" and C: "0 < C"
have "\<exists>y. connected_component (- S) x y \<and> norm y > C"
proof (cases "x = 0")
case True with B Bno show ?thesis by force
next
case False
have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \<subseteq> - ball 0 B"
proof
fix w
assume "w \<in> closed_segment x (((B + C) / norm x) *\<^sub>R x)"
then obtain u where
w: "w = (1 - u + u * (B + C) / norm x) *\<^sub>R x" "0 \<le> u" "u \<le> 1"
by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric])
with False B C have "B \<le> (1 - u) * norm x + u * (B + C)"
using segment_bound_lemma [of B "norm x" "B + C" u] Bno
by simp
with False B C show "w \<in> - ball 0 B"
using distrib_right [of _ _ "norm x"]
by (simp add: ball_def w not_less)
qed
also have "... \<subseteq> -S"
by (simp add: B)
finally have "\<exists>T. connected T \<and> T \<subseteq> - S \<and> x \<in> T \<and> ((B + C) / norm x) *\<^sub>R x \<in> T"
by (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) simp
with False B
show ?thesis
by (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) (simp add: connected_component_def)
qed
}
then show ?thesis
apply (simp add: outside_def assms)
apply (rule bounded_subset [OF bounded_ball [of 0 B]])
apply (force simp: dist_norm not_less bounded_pos)
done
qed
lemma unbounded_outside:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "bounded S \<Longrightarrow> \<not> bounded(outside S)"
using cobounded_imp_unbounded cobounded_outside by blast
lemma bounded_inside:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "bounded S \<Longrightarrow> bounded(inside S)"
by (simp add: bounded_Int cobounded_outside inside_outside)
lemma connected_outside:
fixes S :: "'a::euclidean_space set"
assumes "bounded S" "2 \<le> DIM('a)"
shows "connected(outside S)"
apply (clarsimp simp add: connected_iff_connected_component outside)
apply (rule_tac S="connected_component_set (- S) x" in connected_component_of_subset)
apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq)
by (simp add: Collect_mono connected_component_eq)
lemma outside_connected_component_lt:
"outside S = {x. \<forall>B. \<exists>y. B < norm(y) \<and> connected_component (- S) x y}"
proof -
have "\<And>x B. x \<in> outside S \<Longrightarrow> \<exists>y. B < norm y \<and> connected_component (- S) x y"
by (metis boundedI linorder_not_less mem_Collect_eq outside)
moreover
have "\<And>x. \<forall>B. \<exists>y. B < norm y \<and> connected_component (- S) x y \<Longrightarrow> x \<in> outside S"
by (metis bounded_pos linorder_not_less mem_Collect_eq outside)
ultimately show ?thesis by auto
qed
lemma outside_connected_component_le:
"outside S = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> connected_component (- S) x y}"
apply (simp add: outside_connected_component_lt Set.set_eq_iff)
by (meson gt_ex leD le_less_linear less_imp_le order.trans)
lemma not_outside_connected_component_lt:
fixes S :: "'a::euclidean_space set"
assumes S: "bounded S" and "2 \<le> DIM('a)"
shows "- (outside S) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y}"
proof -
obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
using S [simplified bounded_pos] by auto
have cyz: "connected_component (- S) y z"
if yz: "B < norm z" "B < norm y" for y::'a and z::'a
proof -
have "connected_component (- cball 0 B) y z"
using assms yz
by (force simp: dist_norm intro: connected_componentI [OF _ subset_refl] connected_complement_bounded_convex)
then show ?thesis
by (metis connected_component_of_subset Bno Compl_anti_mono mem_cball_0 subset_iff)
qed
show ?thesis
apply (auto simp: outside bounded_pos)
apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le)
by (metis B connected_component_trans cyz not_le)
qed
lemma not_outside_connected_component_le:
fixes S :: "'a::euclidean_space set"
assumes S: "bounded S" "2 \<le> DIM('a)"
shows "- (outside S) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y}"
unfolding not_outside_connected_component_lt [OF assms]
by (metis (no_types, opaque_lifting) dual_order.strict_trans1 gt_ex pinf(8))
lemma inside_connected_component_lt:
fixes S :: "'a::euclidean_space set"
assumes S: "bounded S" "2 \<le> DIM('a)"
shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y)}"
by (auto simp: inside_outside not_outside_connected_component_lt [OF assms])
lemma inside_connected_component_le:
fixes S :: "'a::euclidean_space set"
assumes S: "bounded S" "2 \<le> DIM('a)"
shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y)}"
by (auto simp: inside_outside not_outside_connected_component_le [OF assms])
lemma inside_subset:
assumes "connected U" and "\<not> bounded U" and "T \<union> U = - S"
shows "inside S \<subseteq> T"
using bounded_subset [of "connected_component_set (- S) _" U] assms
by (metis (no_types, lifting) ComplI Un_iff connected_component_maximal inside_def mem_Collect_eq subsetI)
lemma frontier_not_empty:
fixes S :: "'a :: real_normed_vector set"
shows "\<lbrakk>S \<noteq> {}; S \<noteq> UNIV\<rbrakk> \<Longrightarrow> frontier S \<noteq> {}"
using connected_Int_frontier [of UNIV S] by auto
lemma frontier_eq_empty:
fixes S :: "'a :: real_normed_vector set"
shows "frontier S = {} \<longleftrightarrow> S = {} \<or> S = UNIV"
using frontier_UNIV frontier_empty frontier_not_empty by blast
lemma frontier_of_connected_component_subset:
fixes S :: "'a::real_normed_vector set"
shows "frontier(connected_component_set S x) \<subseteq> frontier S"
proof -
{ fix y
assume y1: "y \<in> closure (connected_component_set S x)"
and y2: "y \<notin> interior (connected_component_set S x)"
have "y \<in> closure S"
using y1 closure_mono connected_component_subset by blast
moreover have "z \<in> interior (connected_component_set S x)"
if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z
proof -
have "ball y e \<subseteq> connected_component_set S y"
using connected_component_maximal that interior_subset
by (metis centre_in_ball connected_ball subset_trans)
then show ?thesis
using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2)
qed
then have "y \<notin> interior S"
using y2 by (force simp: open_contains_ball_eq [OF open_interior])
ultimately have "y \<in> frontier S"
by (auto simp: frontier_def)
}
then show ?thesis by (auto simp: frontier_def)
qed
lemma frontier_Union_subset_closure:
fixes F :: "'a::real_normed_vector set set"
shows "frontier(\<Union>F) \<subseteq> closure(\<Union>t \<in> F. frontier t)"
proof -
have "\<exists>y\<in>F. \<exists>y\<in>frontier y. dist y x < e"
if "T \<in> F" "y \<in> T" "dist y x < e"
"x \<notin> interior (\<Union>F)" "0 < e" for x y e T
proof (cases "x \<in> T")
case True with that show ?thesis
by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono)
next
case False
have \<section>: "closed_segment x y \<inter> T \<noteq> {}" "closed_segment x y - T \<noteq> {}"
using \<open>y \<in> T\<close> False by blast+
obtain c where "c \<in> closed_segment x y" "c \<in> frontier T"
using False connected_Int_frontier [OF connected_segment \<section>] by auto
with that show ?thesis
by (smt (verit) dist_norm segment_bound1)
qed
then show ?thesis
by (fastforce simp add: frontier_def closure_approachable)
qed
lemma frontier_Union_subset:
fixes F :: "'a::real_normed_vector set set"
shows "finite F \<Longrightarrow> frontier(\<Union>F) \<subseteq> (\<Union>t \<in> F. frontier t)"
by (metis closed_UN closure_closed frontier_Union_subset_closure frontier_closed)
lemma frontier_of_components_subset:
fixes S :: "'a::real_normed_vector set"
shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S"
by (metis Path_Connected.frontier_of_connected_component_subset components_iff)
lemma frontier_of_components_closed_complement:
fixes S :: "'a::real_normed_vector set"
shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S"
using frontier_complement frontier_of_components_subset frontier_subset_eq by blast
lemma frontier_minimal_separating_closed:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
and nconn: "\<not> connected(- S)"
and C: "C \<in> components (- S)"
and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)"
shows "frontier C = S"
proof (rule ccontr)
assume "frontier C \<noteq> S"
then have "frontier C \<subset> S"
using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast
then have "connected(- (frontier C))"
by (simp add: conn)
have "\<not> connected(- (frontier C))"
unfolding connected_def not_not
proof (intro exI conjI)
show "open C"
using C \<open>closed S\<close> open_components by blast
show "open (- closure C)"
by blast
show "C \<inter> - closure C \<inter> - frontier C = {}"
using closure_subset by blast
show "C \<inter> - frontier C \<noteq> {}"
using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce
show "- frontier C \<subseteq> C \<union> - closure C"
by (simp add: \<open>open C\<close> closed_Compl frontier_closures)
then show "- closure C \<inter> - frontier C \<noteq> {}"
by (metis C Compl_Diff_eq Un_Int_eq(4) Un_commute \<open>frontier C \<subset> S\<close> \<open>open C\<close> compl_le_compl_iff frontier_def in_components_subset interior_eq leD sup_bot.right_neutral)
qed
then show False
using \<open>connected (- frontier C)\<close> by blast
qed
lemma connected_component_UNIV [simp]:
fixes x :: "'a::real_normed_vector"
shows "connected_component_set UNIV x = UNIV"
using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV
by auto
lemma connected_component_eq_UNIV:
fixes x :: "'a::real_normed_vector"
shows "connected_component_set s x = UNIV \<longleftrightarrow> s = UNIV"
using connected_component_in connected_component_UNIV by blast
lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}"
by (auto simp: components_eq_sing_iff)
lemma interior_inside_frontier:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "interior S \<subseteq> inside (frontier S)"
proof -
{ fix x y
assume x: "x \<in> interior S" and y: "y \<notin> S"
and cc: "connected_component (- frontier S) x y"
have "connected_component_set (- frontier S) x \<inter> frontier S \<noteq> {}"
proof (rule connected_Int_frontier; simp add: set_eq_iff)
show "\<exists>u. connected_component (- frontier S) x u \<and> u \<in> S"
by (meson cc connected_component_in connected_component_refl_eq interior_subset subsetD x)
show "\<exists>u. connected_component (- frontier S) x u \<and> u \<notin> S"
using y cc by blast
qed
then have "bounded (connected_component_set (- frontier S) x)"
using connected_component_in by auto
}
then show ?thesis
using bounded_subset [OF assms]
by (metis (no_types, lifting) Diff_iff frontier_def inside_def mem_Collect_eq subsetI)
qed
lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)"
by (simp add: inside_def)
lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)"
using inside_empty inside_Un_outside by blast
lemma inside_same_component:
"\<lbrakk>connected_component (- S) x y; x \<in> inside S\<rbrakk> \<Longrightarrow> y \<in> inside S"
using connected_component_eq connected_component_in
by (fastforce simp add: inside_def)
lemma outside_same_component:
"\<lbrakk>connected_component (- S) x y; x \<in> outside S\<rbrakk> \<Longrightarrow> y \<in> outside S"
using connected_component_eq connected_component_in
by (fastforce simp add: outside_def)
lemma convex_in_outside:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes S: "convex S" and z: "z \<notin> S"
shows "z \<in> outside S"
proof (cases "S={}")
case True then show ?thesis by simp
next
case False then obtain a where "a \<in> S" by blast
with z have zna: "z \<noteq> a" by auto
{ assume "bounded (connected_component_set (- S) z)"
with bounded_pos_less obtain B where "B>0" and B: "\<And>x. connected_component (- S) z x \<Longrightarrow> norm x < B"
by (metis mem_Collect_eq)
define C where "C = (B + 1 + norm z) / norm (z-a)"
have "C > 0"
using \<open>0 < B\<close> zna by (simp add: C_def field_split_simps add_strict_increasing)
have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z"
by (metis add_diff_cancel norm_triangle_ineq3)
moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj)
ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith
{ fix u::real
assume u: "0\<le>u" "u\<le>1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \<in> S"
then have Cpos: "1 + u * C > 0"
by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z"
by (simp add: scaleR_add_left [symmetric] field_split_simps)
then have False
using convexD_alt [OF S \<open>a \<in> S\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> S\<close> Cpos u
by (simp add: * field_split_simps)
} note contra = this
have "connected_component (- S) z (z + C *\<^sub>R (z-a))"
proof (rule connected_componentI [OF connected_segment])
show "closed_segment z (z + C *\<^sub>R (z - a)) \<subseteq> - S"
using contra by (force simp add: closed_segment_def)
qed auto
then have False
using zna B [of "z + C *\<^sub>R (z-a)"] C
by (auto simp: field_split_simps max_mult_distrib_right)
}
then show ?thesis
by (auto simp: outside_def z)
qed
lemma outside_convex:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes "convex S"
shows "outside S = - S"
by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2)
lemma outside_singleton [simp]:
fixes x :: "'a :: {real_normed_vector, perfect_space}"
shows "outside {x} = -{x}"
by (auto simp: outside_convex)
lemma inside_convex:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
shows "convex S \<Longrightarrow> inside S = {}"
by (simp add: inside_outside outside_convex)
lemma inside_singleton [simp]:
fixes x :: "'a :: {real_normed_vector, perfect_space}"
shows "inside {x} = {}"
by (auto simp: inside_convex)
lemma outside_subset_convex:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
shows "\<lbrakk>convex T; S \<subseteq> T\<rbrakk> \<Longrightarrow> - T \<subseteq> outside S"
using outside_convex outside_mono by blast
lemma outside_Un_outside_Un:
fixes S :: "'a::real_normed_vector set"
assumes "S \<inter> outside(T \<union> U) = {}"
shows "outside(T \<union> U) \<subseteq> outside(T \<union> S)"
proof
fix x
assume x: "x \<in> outside (T \<union> U)"
have "Y \<subseteq> - S" if "connected Y" "Y \<subseteq> - T" "Y \<subseteq> - U" "x \<in> Y" "u \<in> Y" for u Y
proof -
have "Y \<subseteq> connected_component_set (- (T \<union> U)) x"
by (simp add: connected_component_maximal that)
also have "\<dots> \<subseteq> outside(T \<union> U)"
by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x)
finally have "Y \<subseteq> outside(T \<union> U)" .
with assms show ?thesis by auto
qed
with x show "x \<in> outside (T \<union> S)"
by (simp add: outside_connected_component_lt connected_component_def) meson
qed
lemma outside_frontier_misses_closure:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "outside(frontier S) \<subseteq> - closure S"
using assms frontier_def interior_inside_frontier outside_inside by fastforce
lemma outside_frontier_eq_complement_closure:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes "bounded S" "convex S"
shows "outside(frontier S) = - closure S"
by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure
outside_subset_convex subset_antisym)
lemma inside_frontier_eq_interior:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
shows "\<lbrakk>bounded S; convex S\<rbrakk> \<Longrightarrow> inside(frontier S) = interior S"
unfolding inside_outside outside_frontier_eq_complement_closure
using closure_subset interior_subset by (auto simp: frontier_def)
lemma open_inside:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "open (inside S)"
proof -
{ fix x assume x: "x \<in> inside S"
have "open (connected_component_set (- S) x)"
using assms open_connected_component by blast
then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- S) x y"
using dist_not_less_zero
apply (simp add: open_dist)
by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x)
then have "\<exists>e>0. ball x e \<subseteq> inside S"
by (metis e dist_commute inside_same_component mem_ball subsetI x)
}
then show ?thesis
by (simp add: open_contains_ball)
qed
lemma open_outside:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "open (outside S)"
proof -
{ fix x assume x: "x \<in> outside S"
have "open (connected_component_set (- S) x)"
using assms open_connected_component by blast
then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- S) x y"
using dist_not_less_zero x
by (auto simp add: open_dist outside_def intro: connected_component_refl)
then have "\<exists>e>0. ball x e \<subseteq> outside S"
by (metis e dist_commute outside_same_component mem_ball subsetI x)
}
then show ?thesis
by (simp add: open_contains_ball)
qed
lemma closure_inside_subset:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closure(inside S) \<subseteq> S \<union> inside S"
by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside)
lemma frontier_inside_subset:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "frontier(inside S) \<subseteq> S"
using assms closure_inside_subset frontier_closures frontier_disjoint_eq open_inside by fastforce
lemma closure_outside_subset:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closure(outside S) \<subseteq> S \<union> outside S"
by (metis assms closed_open closure_minimal inside_outside open_inside sup_ge2)
lemma closed_path_image_Un_inside:
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
assumes "path g"
shows "closed (path_image g \<union> inside (path_image g))"
by (simp add: assms closed_Compl closed_path_image open_outside union_with_inside)
lemma frontier_outside_subset:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "frontier(outside S) \<subseteq> S"
unfolding frontier_def
by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup_aci(1))
lemma inside_complement_unbounded_connected_empty:
"\<lbrakk>connected (- S); \<not> bounded (- S)\<rbrakk> \<Longrightarrow> inside S = {}"
using inside_subset by blast
lemma inside_bounded_complement_connected_empty:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "\<lbrakk>connected (- S); bounded S\<rbrakk> \<Longrightarrow> inside S = {}"
by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded)
lemma inside_inside:
assumes "S \<subseteq> inside T"
shows "inside S - T \<subseteq> inside T"
unfolding inside_def
proof clarify
fix x
assume x: "x \<notin> T" "x \<notin> S" and bo: "bounded (connected_component_set (- S) x)"
show "bounded (connected_component_set (- T) x)"
proof (cases "S \<inter> connected_component_set (- T) x = {}")
case True then show ?thesis
by (metis bounded_subset [OF bo] compl_le_compl_iff connected_component_idemp connected_component_mono disjoint_eq_subset_Compl double_compl)
next
case False
then obtain y where y: "y \<in> S" "y \<in> connected_component_set (- T) x"
by (meson disjoint_iff)
then have "bounded (connected_component_set (- T) y)"
using assms [unfolded inside_def] by blast
with y show ?thesis
by (metis connected_component_eq)
qed
qed
lemma inside_inside_subset: "inside(inside S) \<subseteq> S"
using inside_inside union_with_outside by fastforce
lemma inside_outside_intersect_connected:
"\<lbrakk>connected T; inside S \<inter> T \<noteq> {}; outside S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> S \<inter> T \<noteq> {}"
apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify)
by (metis compl_le_swap1 connected_componentI connected_component_eq mem_Collect_eq)
lemma outside_bounded_nonempty:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes "bounded S" shows "outside S \<noteq> {}"
using assms unbounded_outside by force
lemma outside_compact_in_open:
fixes S :: "'a :: {real_normed_vector,perfect_space} set"
assumes S: "compact S" and T: "open T" and "S \<subseteq> T" "T \<noteq> {}"
shows "outside S \<inter> T \<noteq> {}"
proof -
have "outside S \<noteq> {}"
by (simp add: compact_imp_bounded outside_bounded_nonempty S)
with assms obtain a b where a: "a \<in> outside S" and b: "b \<in> T" by auto
show ?thesis
proof (cases "a \<in> T")
case True with a show ?thesis by blast
next
case False
have front: "frontier T \<subseteq> - S"
using \<open>S \<subseteq> T\<close> frontier_disjoint_eq T by auto
{ fix \<gamma>
assume "path \<gamma>" and pimg_sbs: "path_image \<gamma> - {pathfinish \<gamma>} \<subseteq> interior (- T)"
and pf: "pathfinish \<gamma> \<in> frontier T" and ps: "pathstart \<gamma> = a"
define c where "c = pathfinish \<gamma>"
have "c \<in> -S" unfolding c_def using front pf by blast
moreover have "open (-S)" using S compact_imp_closed by blast
ultimately obtain \<epsilon>::real where "\<epsilon> > 0" and \<epsilon>: "cball c \<epsilon> \<subseteq> -S"
using open_contains_cball[of "-S"] S by blast
then obtain d where "d \<in> T" and d: "dist d c < \<epsilon>"
using closure_approachable [of c T] pf unfolding c_def
by (metis Diff_iff frontier_def)
then have "d \<in> -S" using \<epsilon>
using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq)
have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -S"
using \<open>c \<in> - S\<close> \<open>S \<subseteq> T\<close> c_def interior_subset pimg_sbs by fastforce
have "closed_segment c d \<le> cball c \<epsilon>"
by (metis \<open>0 < \<epsilon>\<close> centre_in_cball closed_segment_subset convex_cball d dist_commute less_eq_real_def mem_cball)
with \<epsilon> have "closed_segment c d \<subseteq> -S" by blast
moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)"
by (rule connected_Un) (auto simp: c_def \<open>path \<gamma>\<close> connected_path_image)
ultimately have "connected_component (- S) a d"
unfolding connected_component_def using pimg_sbs_cos ps by blast
then have "outside S \<inter> T \<noteq> {}"
using outside_same_component [OF _ a] by (metis IntI \<open>d \<in> T\<close> empty_iff)
} note * = this
have pal: "pathstart (linepath a b) \<in> closure (- T)"
by (auto simp: False closure_def)
show ?thesis
by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b)
qed
qed
lemma inside_inside_compact_connected:
fixes S :: "'a :: euclidean_space set"
assumes S: "closed S" and T: "compact T" and "connected T" "S \<subseteq> inside T"
shows "inside S \<subseteq> inside T"
proof (cases "inside T = {}")
case True with assms show ?thesis by auto
next
case False
consider "DIM('a) = 1" | "DIM('a) \<ge> 2"
using antisym not_less_eq_eq by fastforce
then show ?thesis
proof cases
case 1 then show ?thesis
using connected_convex_1_gen assms False inside_convex by blast
next
case 2
have "bounded S"
using assms by (meson bounded_inside bounded_subset compact_imp_bounded)
then have coms: "compact S"
by (simp add: S compact_eq_bounded_closed)
then have bst: "bounded (S \<union> T)"
by (simp add: compact_imp_bounded T)
then obtain r where "0 < r" and r: "S \<union> T \<subseteq> ball 0 r"
using bounded_subset_ballD by blast
have outst: "outside S \<inter> outside T \<noteq> {}"
by (metis bounded_Un bounded_subset bst cobounded_outside disjoint_eq_subset_Compl unbounded_outside)
have "S \<inter> T = {}" using assms
by (metis disjoint_iff_not_equal inside_no_overlap subsetCE)
moreover have "outside S \<inter> inside T \<noteq> {}"
by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open T)
ultimately have "inside S \<inter> T = {}"
using inside_outside_intersect_connected [OF \<open>connected T\<close>, of S]
by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst)
then show ?thesis
using inside_inside [OF \<open>S \<subseteq> inside T\<close>] by blast
qed
qed
lemma connected_with_inside:
fixes S :: "'a :: real_normed_vector set"
assumes S: "closed S" and cons: "connected S"
shows "connected(S \<union> inside S)"
proof (cases "S \<union> inside S = UNIV")
case True with assms show ?thesis by auto
next
case False
then obtain b where b: "b \<notin> S" "b \<notin> inside S" by blast
have *: "\<exists>y T. y \<in> S \<and> connected T \<and> a \<in> T \<and> y \<in> T \<and> T \<subseteq> (S \<union> inside S)"
if "a \<in> S \<union> inside S" for a
using that
proof
assume "a \<in> S" then show ?thesis
using cons by blast
next
assume a: "a \<in> inside S"
then have ain: "a \<in> closure (inside S)"
by (simp add: closure_def)
obtain h where h: "path h" "pathstart h = a"
"path_image h - {pathfinish h} \<subseteq> interior (inside S)"
"pathfinish h \<in> frontier (inside S)"
using ain b
by (metis exists_path_subpath_to_frontier path_linepath pathfinish_linepath pathstart_linepath)
moreover
have h1S: "pathfinish h \<in> S"
using S h frontier_inside_subset by blast
moreover
have "path_image h \<subseteq> S \<union> inside S"
using IntD1 S h1S h interior_eq open_inside by fastforce
ultimately show ?thesis by blast
qed
show ?thesis
apply (simp add: connected_iff_connected_component)
apply (clarsimp simp add: connected_component_def dest!: *)
subgoal for x y u u' T t'
by (rule_tac x = "S \<union> T \<union> t'" in exI) (auto intro!: connected_Un cons)
done
qed
text\<open>The proof is virtually the same as that above.\<close>
lemma connected_with_outside:
fixes S :: "'a :: real_normed_vector set"
assumes S: "closed S" and cons: "connected S"
shows "connected(S \<union> outside S)"
proof (cases "S \<union> outside S = UNIV")
case True with assms show ?thesis by auto
next
case False
then obtain b where b: "b \<notin> S" "b \<notin> outside S" by blast
have *: "\<exists>y T. y \<in> S \<and> connected T \<and> a \<in> T \<and> y \<in> T \<and> T \<subseteq> (S \<union> outside S)" if "a \<in> (S \<union> outside S)" for a
using that proof
assume "a \<in> S" then show ?thesis
by (rule_tac x=a in exI, rule_tac x="{a}" in exI, simp)
next
assume a: "a \<in> outside S"
then have ain: "a \<in> closure (outside S)"
by (simp add: closure_def)
obtain h where h: "path h" "pathstart h = a"
"path_image h - {pathfinish h} \<subseteq> interior (outside S)"
"pathfinish h \<in> frontier (outside S)"
using ain b
by (metis exists_path_subpath_to_frontier path_linepath pathfinish_linepath pathstart_linepath)
moreover
have h1S: "pathfinish h \<in> S"
using S frontier_outside_subset h(4) by blast
moreover
have "path_image h \<subseteq> S \<union> outside S"
using IntD1 S h1S h interior_eq open_outside by fastforce
ultimately show ?thesis
by blast
qed
show ?thesis
apply (simp add: connected_iff_connected_component)
apply (clarsimp simp add: connected_component_def dest!: *)
subgoal for x y u u' T t'
by (rule_tac x="(S \<union> T \<union> t')" in exI) (auto intro!: connected_Un cons)
done
qed
lemma inside_inside_eq_empty [simp]:
fixes S :: "'a :: {real_normed_vector, perfect_space} set"
assumes S: "closed S" and cons: "connected S"
shows "inside (inside S) = {}"
proof -
have "connected (- inside S)"
by (metis S connected_with_outside cons union_with_outside)
then show ?thesis
by (metis bounded_Un inside_complement_unbounded_connected_empty unbounded_outside union_with_outside)
qed
lemma inside_in_components:
"inside S \<in> components (- S) \<longleftrightarrow> connected(inside S) \<and> inside S \<noteq> {}" (is "?lhs = ?rhs")
proof
assume R: ?rhs
then have "\<And>x. \<lbrakk>x \<in> S; x \<in> inside S\<rbrakk> \<Longrightarrow> \<not> connected (inside S)"
by (simp add: inside_outside)
with R show ?lhs
unfolding in_components_maximal
by (auto intro: inside_same_component connected_componentI)
qed (simp add: in_components_maximal)
text\<open>The proof is like that above.\<close>
lemma outside_in_components:
"outside S \<in> components (- S) \<longleftrightarrow> connected(outside S) \<and> outside S \<noteq> {}" (is "?lhs = ?rhs")
proof
assume R: ?rhs
then have "\<And>x. \<lbrakk>x \<in> S; x \<in> outside S\<rbrakk> \<Longrightarrow> \<not> connected (outside S)"
by (meson disjoint_iff outside_no_overlap)
with R show ?lhs
unfolding in_components_maximal
by (auto intro: outside_same_component connected_componentI)
qed (simp add: in_components_maximal)
lemma bounded_unique_outside:
fixes S :: "'a :: euclidean_space set"
assumes "bounded S" "DIM('a) \<ge> 2"
shows "(c \<in> components (- S) \<and> \<not> bounded c) \<longleftrightarrow> c = outside S"
using assms
by (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty
outside_in_components unbounded_outside)
subsection\<open>Condition for an open map's image to contain a ball\<close>
proposition ball_subset_open_map_image:
fixes f :: "'a::heine_borel \<Rightarrow> 'b :: {real_normed_vector,heine_borel}"
assumes contf: "continuous_on (closure S) f"
and oint: "open (f ` interior S)"
and le_no: "\<And>z. z \<in> frontier S \<Longrightarrow> r \<le> norm(f z - f a)"
and "bounded S" "a \<in> S" "0 < r"
shows "ball (f a) r \<subseteq> f ` S"
proof (cases "f ` S = UNIV")
case True then show ?thesis by simp
next
case False
then have "closed (frontier (f ` S))" "frontier (f ` S) \<noteq> {}"
using \<open>a \<in> S\<close> by (auto simp: frontier_eq_empty)
then obtain w where w: "w \<in> frontier (f ` S)"
and dw_le: "\<And>y. y \<in> frontier (f ` S) \<Longrightarrow> norm (f a - w) \<le> norm (f a - y)"
by (auto simp add: dist_norm intro: distance_attains_inf [of "frontier(f ` S)" "f a"])
then obtain \<xi> where \<xi>: "\<And>n. \<xi> n \<in> f ` S" and tendsw: "\<xi> \<longlonglongrightarrow> w"
by (metis Diff_iff frontier_def closure_sequential)
then have "\<And>n. \<exists>x \<in> S. \<xi> n = f x" by force
then obtain z where zs: "\<And>n. z n \<in> S" and fz: "\<And>n. \<xi> n = f (z n)"
by metis
then obtain y K where y: "y \<in> closure S" and "strict_mono (K :: nat \<Rightarrow> nat)"
and Klim: "(z \<circ> K) \<longlonglongrightarrow> y"
using \<open>bounded S\<close>
unfolding compact_closure [symmetric] compact_def by (meson closure_subset subset_iff)
then have ftendsw: "((\<lambda>n. f (z n)) \<circ> K) \<longlonglongrightarrow> w"
by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
have zKs: "\<And>n. (z \<circ> K) n \<in> S" by (simp add: zs)
have fz: "f \<circ> z = \<xi>" "(\<lambda>n. f (z n)) = \<xi>"
using fz by auto
then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y"
by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially)
with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto
have "r \<le> norm (f y - f a)"
proof (rule le_no)
show "y \<in> frontier S"
using w wy oint by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
qed
then have "\<And>y. \<lbrakk>norm (f a - y) < r; y \<in> frontier (f ` S)\<rbrakk> \<Longrightarrow> False"
by (metis dw_le norm_minus_commute not_less order_trans wy)
then have "ball (f a) r \<inter> frontier (f ` S) = {}"
by (metis disjoint_iff_not_equal dist_norm mem_ball)
moreover
have "ball (f a) r \<inter> f ` S \<noteq> {}"
using \<open>a \<in> S\<close> \<open>0 < r\<close> centre_in_ball by blast
ultimately show ?thesis
by (meson connected_Int_frontier connected_ball diff_shunt_var)
qed
subsubsection\<open>Special characterizations of classes of functions into and out of R.\<close>
lemma Hausdorff_space_euclidean [simp]: "Hausdorff_space (euclidean :: 'a::metric_space topology)"
proof -
have "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V"
if "x \<noteq> y" for x y :: 'a
proof (intro exI conjI)
let ?r = "dist x y / 2"
have [simp]: "?r > 0"
by (simp add: that)
show "open (ball x ?r)" "open (ball y ?r)" "x \<in> (ball x ?r)" "y \<in> (ball y ?r)"
by (auto simp add: that)
show "disjnt (ball x ?r) (ball y ?r)"
unfolding disjnt_def by (simp add: disjoint_ballI)
qed
then show ?thesis
by (simp add: Hausdorff_space_def)
qed
proposition embedding_map_into_euclideanreal:
assumes "path_connected_space X"
shows "embedding_map X euclideanreal f \<longleftrightarrow>
continuous_map X euclideanreal f \<and> inj_on f (topspace X)"
proof safe
show "continuous_map X euclideanreal f"
if "embedding_map X euclideanreal f"
using continuous_map_in_subtopology homeomorphic_imp_continuous_map that
unfolding embedding_map_def by blast
show "inj_on f (topspace X)"
if "embedding_map X euclideanreal f"
using that homeomorphic_imp_injective_map
unfolding embedding_map_def by blast
show "embedding_map X euclideanreal f"
if cont: "continuous_map X euclideanreal f" and inj: "inj_on f (topspace X)"
proof -
obtain g where gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x"
using inv_into_f_f [OF inj] by auto
show ?thesis
unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def
proof (intro exI conjI)
show "continuous_map X (top_of_set (f ` topspace X)) f"
by (simp add: cont continuous_map_in_subtopology)
let ?S = "f ` topspace X"
have eq: "{x \<in> ?S. g x \<in> U} = f ` U" if "openin X U" for U
using openin_subset [OF that] by (auto simp: gf)
have 1: "g ` ?S \<subseteq> topspace X"
using eq by blast
have "openin (top_of_set ?S) {x \<in> ?S. g x \<in> T}"
if "openin X T" for T
proof -
have "T \<subseteq> topspace X"
by (simp add: openin_subset that)
have RR: "\<forall>x \<in> ?S \<inter> g -` T. \<exists>d>0. \<forall>x' \<in> ?S \<inter> ball x d. g x' \<in> T"
proof (clarsimp simp add: gf)
have pcS: "path_connectedin euclidean ?S"
using assms cont path_connectedin_continuous_map_image path_connectedin_topspace by blast
show "\<exists>d>0. \<forall>x'\<in>f ` topspace X \<inter> ball (f x) d. g x' \<in> T"
if "x \<in> T" for x
proof -
have x: "x \<in> topspace X"
using \<open>T \<subseteq> topspace X\<close> \<open>x \<in> T\<close> by blast
obtain u v d where "0 < d" "u \<in> topspace X" "v \<in> topspace X"
and sub_fuv: "?S \<inter> {f x - d .. f x + d} \<subseteq> {f u..f v}"
proof (cases "\<exists>u \<in> topspace X. f u < f x")
case True
then obtain u where u: "u \<in> topspace X" "f u < f x" ..
show ?thesis
proof (cases "\<exists>v \<in> topspace X. f x < f v")
case True
then obtain v where v: "v \<in> topspace X" "f x < f v" ..
show ?thesis
proof
let ?d = "min (f x - f u) (f v - f x)"
show "0 < ?d"
by (simp add: \<open>f u < f x\<close> \<open>f x < f v\<close>)
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f v}"
by fastforce
qed (auto simp: u v)
next
case False
show ?thesis
proof
let ?d = "f x - f u"
show "0 < ?d"
by (simp add: u)
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f x}"
using x u False by auto
qed (auto simp: x u)
qed
next
case False
note no_u = False
show ?thesis
proof (cases "\<exists>v \<in> topspace X. f x < f v")
case True
then obtain v where v: "v \<in> topspace X" "f x < f v" ..
show ?thesis
proof
let ?d = "f v - f x"
show "0 < ?d"
by (simp add: v)
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f x..f v}"
using False by auto
qed (auto simp: x v)
next
case False
show ?thesis
proof
show "f ` topspace X \<inter> {f x - 1..f x + 1} \<subseteq> {f x..f x}"
using False no_u by fastforce
qed (auto simp: x)
qed
qed
then obtain h where "pathin X h" "h 0 = u" "h 1 = v"
using assms unfolding path_connected_space_def by blast
obtain C where "compactin X C" "connectedin X C" "u \<in> C" "v \<in> C"
proof
show "compactin X (h ` {0..1})"
using that by (simp add: \<open>pathin X h\<close> compactin_path_image)
show "connectedin X (h ` {0..1})"
using \<open>pathin X h\<close> connectedin_path_image by blast
qed (use \<open>h 0 = u\<close> \<open>h 1 = v\<close> in auto)
have "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) (subtopology X C) g"
proof (rule continuous_inverse_map)
show "compact_space (subtopology X C)"
using \<open>compactin X C\<close> compactin_subspace by blast
show "continuous_map (subtopology X C) euclideanreal f"
by (simp add: cont continuous_map_from_subtopology)
have "{f u .. f v} \<subseteq> f ` topspace (subtopology X C)"
proof (rule connected_contains_Icc)
show "connected (f ` topspace (subtopology X C))"
using connectedin_continuous_map_image [OF cont]
by (simp add: \<open>compactin X C\<close> \<open>connectedin X C\<close> compactin_subset_topspace inf_absorb2)
show "f u \<in> f ` topspace (subtopology X C)"
by (simp add: \<open>u \<in> C\<close> \<open>u \<in> topspace X\<close>)
show "f v \<in> f ` topspace (subtopology X C)"
by (simp add: \<open>v \<in> C\<close> \<open>v \<in> topspace X\<close>)
qed
then show "f ` topspace X \<inter> {f x - d..f x + d} \<subseteq> f ` topspace (subtopology X C)"
using sub_fuv by blast
qed (auto simp: gf)
then have contg: "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) X g"
using continuous_map_in_subtopology by blast
have "\<exists>e>0. \<forall>x \<in> ?S \<inter> {f x - d .. f x + d} \<inter> ball (f x) e. g x \<in> T"
using openin_continuous_map_preimage [OF contg \<open>openin X T\<close>] x \<open>x \<in> T\<close> \<open>0 < d\<close>
unfolding openin_euclidean_subtopology_iff
by (force simp: gf dist_commute)
then obtain e where "e > 0 \<and> (\<forall>x\<in>f ` topspace X \<inter> {f x - d..f x + d} \<inter> ball (f x) e. g x \<in> T)"
by metis
with \<open>0 < d\<close> have "min d e > 0" "\<forall>u. u \<in> topspace X \<longrightarrow> \<bar>f x - f u\<bar> < min d e \<longrightarrow> u \<in> T"
using dist_real_def gf by force+
then show ?thesis
by (metis (full_types) Int_iff dist_real_def image_iff mem_ball gf)
qed
qed
then obtain d where d: "\<And>r. r \<in> ?S \<inter> g -` T \<Longrightarrow>
d r > 0 \<and> (\<forall>x \<in> ?S \<inter> ball r (d r). g x \<in> T)"
by metis
show ?thesis
unfolding openin_subtopology
proof (intro exI conjI)
show "{x \<in> ?S. g x \<in> T} = (\<Union>r \<in> ?S \<inter> g -` T. ball r (d r)) \<inter> f ` topspace X"
using d by (auto simp: gf)
qed auto
qed
then show "continuous_map (top_of_set ?S) X g"
by (simp add: "1" continuous_map)
qed (auto simp: gf)
qed
qed
subsubsection \<open>An injective function into R is a homeomorphism and so an open map.\<close>
lemma injective_into_1d_eq_homeomorphism:
fixes f :: "'a::topological_space \<Rightarrow> real"
assumes f: "continuous_on S f" and S: "path_connected S"
shows "inj_on f S \<longleftrightarrow> (\<exists>g. homeomorphism S (f ` S) f g)"
proof
show "\<exists>g. homeomorphism S (f ` S) f g"
if "inj_on f S"
proof -
have "embedding_map (top_of_set S) euclideanreal f"
using that embedding_map_into_euclideanreal [of "top_of_set S" f] assms by auto
then show ?thesis
unfolding embedding_map_def topspace_euclidean_subtopology
by (metis f homeomorphic_map_closedness_eq homeomorphism_injective_closed_map that)
qed
qed (metis homeomorphism_def inj_onI)
lemma injective_into_1d_imp_open_map:
fixes f :: "'a::topological_space \<Rightarrow> real"
assumes "continuous_on S f" "path_connected S" "inj_on f S" "openin (subtopology euclidean S) T"
shows "openin (subtopology euclidean (f ` S)) (f ` T)"
using assms homeomorphism_imp_open_map injective_into_1d_eq_homeomorphism by blast
lemma homeomorphism_into_1d:
fixes f :: "'a::topological_space \<Rightarrow> real"
assumes "path_connected S" "continuous_on S f" "f ` S = T" "inj_on f S"
shows "\<exists>g. homeomorphism S T f g"
using assms injective_into_1d_eq_homeomorphism by blast
subsection\<^marker>\<open>tag unimportant\<close> \<open>Rectangular paths\<close>
definition\<^marker>\<open>tag unimportant\<close> rectpath where
"rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
lemma path_rectpath [simp, intro]: "path (rectpath a b)"
by (simp add: Let_def rectpath_def)
lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
by (simp add: rectpath_def Let_def)
lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
by (simp add: rectpath_def Let_def)
lemma simple_path_rectpath [simp, intro]:
assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3"
shows "simple_path (rectpath a1 a3)"
unfolding rectpath_def Let_def using assms
by (intro simple_path_join_loop arc_join arc_linepath)
(auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)
lemma path_image_rectpath:
assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
shows "path_image (rectpath a1 a3) =
{z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union>
{z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs")
proof -
define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union>
closed_segment a4 a3 \<union> closed_segment a1 a4"
by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
a2_def a4_def Un_assoc)
also have "\<dots> = ?rhs" using assms
by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
finally show ?thesis .
qed
lemma path_image_rectpath_subset_cbox:
assumes "Re a \<le> Re b" "Im a \<le> Im b"
shows "path_image (rectpath a b) \<subseteq> cbox a b"
using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)
lemma path_image_rectpath_inter_box:
assumes "Re a \<le> Re b" "Im a \<le> Im b"
shows "path_image (rectpath a b) \<inter> box a b = {}"
using assms by (auto simp: path_image_rectpath in_box_complex_iff)
lemma path_image_rectpath_cbox_minus_box:
assumes "Re a \<le> Re b" "Im a \<le> Im b"
shows "path_image (rectpath a b) = cbox a b - box a b"
using assms by (auto simp: path_image_rectpath in_cbox_complex_iff in_box_complex_iff)
end