(* Title: HOL/Multivariate_Analysis/Path_Connected.thy Author: Robert Himmelmann, TU Muenchen *) section {* Continuous paths and path-connected sets *} theory Path_Connected imports Convex_Euclidean_Space begin subsection {* Paths. *} definition path :: "(real => 'a::topological_space) => bool" where "path g <-> continuous_on {0..1} g" definition pathstart :: "(real => 'a::topological_space) => 'a" where "pathstart g = g 0" definition pathfinish :: "(real => 'a::topological_space) => 'a" where "pathfinish g = g 1" definition path_image :: "(real => 'a::topological_space) => 'a set" where "path_image g = g ` {0 .. 1}" definition reversepath :: "(real => 'a::topological_space) => real => 'a" where "reversepath g = (λx. g(1 - x))" definition joinpaths :: "(real => 'a::topological_space) => (real => 'a) => real => 'a" (infixr "+++" 75) where "g1 +++ g2 = (λx. if x ≤ 1/2 then g1 (2 * x) else g2 (2 * x - 1))" definition simple_path :: "(real => 'a::topological_space) => bool" where "simple_path g <-> (∀x∈{0..1}. ∀y∈{0..1}. g x = g y --> x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0)" definition injective_path :: "(real => 'a::topological_space) => bool" where "injective_path g <-> (∀x∈{0..1}. ∀y∈{0..1}. g x = g y --> x = y)" subsection {* Some lemmas about these concepts. *} lemma injective_imp_simple_path: "injective_path g ==> simple_path g" unfolding injective_path_def simple_path_def by auto lemma path_image_nonempty: "path_image g ≠ {}" unfolding path_image_def image_is_empty box_eq_empty by auto lemma pathstart_in_path_image[intro]: "pathstart g ∈ path_image g" unfolding pathstart_def path_image_def by auto lemma pathfinish_in_path_image[intro]: "pathfinish g ∈ path_image g" unfolding pathfinish_def path_image_def by auto lemma connected_path_image[intro]: "path g ==> connected (path_image g)" unfolding path_def path_image_def apply (erule connected_continuous_image) apply (rule convex_connected, rule convex_real_interval) done lemma compact_path_image[intro]: "path g ==> compact (path_image g)" unfolding path_def path_image_def apply (erule compact_continuous_image) apply (rule compact_Icc) done 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 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" proof - have *: "!!g. path_image (reversepath g) ⊆ path_image g" unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply rule apply rule apply (erule bexE) apply (rule_tac x="1 - xa" in bexI) apply auto done show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed lemma path_reversepath [simp]: "path (reversepath g) <-> path g" proof - have *: "!!g. path g ==> path (reversepath g)" unfolding path_def reversepath_def apply (rule continuous_on_compose[unfolded o_def, of _ "λx. 1 - x"]) apply (intro continuous_intros) apply (rule continuous_on_subset[of "{0..1}"]) apply assumption apply auto done show ?thesis using *[of "reversepath g"] *[of g] unfolding reversepath_reversepath by (rule iffI) qed 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) <-> path g1 ∧ 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 <-> continuous_on {0..1} ((g1 +++ g2) o (λx. x / 2))" by (intro continuous_on_cong refl) (auto simp: joinpaths_def) have g2: "continuous_on {0..1} g2 <-> continuous_on {0..1} ((g1 +++ g2) o (λx. 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} ∪ {1/2 .. 1::real}" by auto { fix x :: real assume "0 ≤ x" and "x ≤ 1" then have "x ∈ (λx. x * 2) ` {0..1 / 2}" by (intro image_eqI[where x="x/2"]) auto } note 1 = this { fix x :: real assume "0 ≤ x" and "x ≤ 1" then have "x ∈ (λx. 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 lemma path_image_join_subset: "path_image (g1 +++ g2) ⊆ path_image g1 ∪ path_image g2" unfolding path_image_def joinpaths_def by auto lemma subset_path_image_join: assumes "path_image g1 ⊆ s" and "path_image g2 ⊆ s" shows "path_image (g1 +++ g2) ⊆ 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 ∪ path_image g2" apply rule apply (rule path_image_join_subset) apply rule unfolding Un_iff proof (erule disjE) fix x assume "x ∈ path_image g1" then obtain y where y: "y ∈ {0..1}" "x = g1 y" unfolding path_image_def image_iff by auto then show "x ∈ path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff apply (rule_tac x="(1/2) *⇩_{R}y" in bexI) apply auto done next fix x assume "x ∈ path_image g2" then obtain y where y: "y ∈ {0..1}" "x = g2 y" unfolding path_image_def image_iff by auto then show "x ∈ path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff apply (rule_tac x="(1/2) *⇩_{R}(y + 1)" in bexI) using assms(1)[unfolded pathfinish_def pathstart_def] apply (auto simp add: add_divide_distrib) done qed lemma not_in_path_image_join: assumes "x ∉ path_image g1" and "x ∉ path_image g2" shows "x ∉ path_image (g1 +++ g2)" using assms and path_image_join_subset[of g1 g2] by auto lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)" using assms unfolding simple_path_def reversepath_def apply - apply (rule ballI)+ apply (erule_tac x="1-x" in ballE) apply (erule_tac x="1-y" in ballE) apply auto done lemma simple_path_join_loop: assumes "injective_path g1" and "injective_path g2" and "pathfinish g2 = pathstart g1" and "path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}" shows "simple_path (g1 +++ g2)" unfolding simple_path_def proof (intro ballI impI) let ?g = "g1 +++ g2" note inj = assms(1,2)[unfolded injective_path_def, rule_format] fix x y :: real assume xy: "x ∈ {0..1}" "y ∈ {0..1}" "?g x = ?g y" show "x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0" proof (cases "x ≤ 1/2", case_tac[!] "y ≤ 1/2", unfold not_le) assume as: "x ≤ 1 / 2" "y ≤ 1 / 2" then have "g1 (2 *⇩_{R}x) = g1 (2 *⇩_{R}y)" using xy(3) unfolding joinpaths_def by auto moreover have "2 *⇩_{R}x ∈ {0..1}" "2 *⇩_{R}y ∈ {0..1}" using xy(1,2) as by auto ultimately show ?thesis using inj(1)[of "2*⇩_{R}x" "2*⇩_{R}y"] by auto next assume as: "x > 1 / 2" "y > 1 / 2" then have "g2 (2 *⇩_{R}x - 1) = g2 (2 *⇩_{R}y - 1)" using xy(3) unfolding joinpaths_def by auto moreover have "2 *⇩_{R}x - 1 ∈ {0..1}" "2 *⇩_{R}y - 1 ∈ {0..1}" using xy(1,2) as by auto ultimately show ?thesis using inj(2)[of "2*⇩_{R}x - 1" "2*⇩_{R}y - 1"] by auto next assume as: "x ≤ 1 / 2" "y > 1 / 2" then have "?g x ∈ path_image g1" "?g y ∈ path_image g2" unfolding path_image_def joinpaths_def using xy(1,2) by auto moreover have "?g y ≠ pathstart g2" using as(2) unfolding pathstart_def joinpaths_def using inj(2)[of "2 *⇩_{R}y - 1" 0] and xy(2) by (auto simp add: field_simps) ultimately have *: "?g x = pathstart g1" using assms(4) unfolding xy(3) by auto then have "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1) using inj(1)[of "2 *⇩_{R}x" 0] by auto moreover have "y = 1" using * unfolding xy(3) assms(3)[symmetric] unfolding joinpaths_def pathfinish_def using as(2) and xy(2) using inj(2)[of "2 *⇩_{R}y - 1" 1] by auto ultimately show ?thesis by auto next assume as: "x > 1 / 2" "y ≤ 1 / 2" then have "?g x ∈ path_image g2" and "?g y ∈ path_image g1" unfolding path_image_def joinpaths_def using xy(1,2) by auto moreover have "?g x ≠ pathstart g2" using as(1) unfolding pathstart_def joinpaths_def using inj(2)[of "2 *⇩_{R}x - 1" 0] and xy(1) by (auto simp add: field_simps) ultimately have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto then have "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2) using inj(1)[of "2 *⇩_{R}y" 0] by auto moreover have "x = 1" using * unfolding xy(3)[symmetric] assms(3)[symmetric] unfolding joinpaths_def pathfinish_def using as(1) and xy(1) using inj(2)[of "2 *⇩_{R}x - 1" 1] by auto ultimately show ?thesis by auto qed qed lemma injective_path_join: assumes "injective_path g1" and "injective_path g2" and "pathfinish g1 = pathstart g2" and "path_image g1 ∩ path_image g2 ⊆ {pathstart g2}" shows "injective_path (g1 +++ g2)" unfolding injective_path_def proof (rule, rule, rule) let ?g = "g1 +++ g2" note inj = assms(1,2)[unfolded injective_path_def, rule_format] fix x y assume xy: "x ∈ {0..1}" "y ∈ {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y" show "x = y" proof (cases "x ≤ 1/2", case_tac[!] "y ≤ 1/2", unfold not_le) assume "x ≤ 1 / 2" and "y ≤ 1 / 2" then show ?thesis using inj(1)[of "2*⇩_{R}x" "2*⇩_{R}y"] and xy unfolding joinpaths_def by auto next assume "x > 1 / 2" and "y > 1 / 2" then show ?thesis using inj(2)[of "2*⇩_{R}x - 1" "2*⇩_{R}y - 1"] and xy unfolding joinpaths_def by auto next assume as: "x ≤ 1 / 2" "y > 1 / 2" then have "?g x ∈ path_image g1" and "?g y ∈ path_image g2" unfolding path_image_def joinpaths_def using xy(1,2) by auto then have "?g x = pathfinish g1" and "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto then show ?thesis using as and inj(1)[of "2 *⇩_{R}x" 1] inj(2)[of "2 *⇩_{R}y - 1" 0] and xy(1,2) unfolding pathstart_def pathfinish_def joinpaths_def by auto next assume as:"x > 1 / 2" "y ≤ 1 / 2" then have "?g x ∈ path_image g2" and "?g y ∈ path_image g1" unfolding path_image_def joinpaths_def using xy(1,2) by auto then have "?g x = pathstart g2" and "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto then show ?thesis using as and inj(2)[of "2 *⇩_{R}x - 1" 0] inj(1)[of "2 *⇩_{R}y" 1] and xy(1,2) unfolding pathstart_def pathfinish_def joinpaths_def by auto qed qed lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join subsection {* Reparametrizing a closed curve to start at some chosen point *} definition shiftpath :: "real => (real => 'a::topological_space) => real => 'a" where "shiftpath a f = (λx. if (a + x) ≤ 1 then f (a + x) else f (a + x - 1))" lemma pathstart_shiftpath: "a ≤ 1 ==> pathstart (shiftpath a g) = g a" unfolding pathstart_def shiftpath_def by auto lemma pathfinish_shiftpath: assumes "0 ≤ 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 ∈ {0 .. 1}" shows "pathfinish (shiftpath a g) = g a" and "pathstart (shiftpath a g) = g a" using assms by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) lemma closed_shiftpath: assumes "pathfinish g = pathstart g" and "a ∈ {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 ∈ {0..1}" shows "path (shiftpath a g)" proof - have *: "{0 .. 1} = {0 .. 1-a} ∪ {1-a .. 1}" using assms(3) by auto have **: "!!x. x + a = 1 ==> g (x + a - 1) = g (x + a)" using assms(2)[unfolded pathfinish_def pathstart_def] by auto show ?thesis unfolding path_def shiftpath_def * apply (rule continuous_on_union) apply (rule closed_real_atLeastAtMost)+ apply (rule continuous_on_eq[of _ "g o (λx. a + x)"]) prefer 3 apply (rule continuous_on_eq[of _ "g o (λx. a - 1 + x)"]) defer prefer 3 apply (rule continuous_intros)+ prefer 2 apply (rule continuous_intros)+ apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) using assms(3) and ** apply auto apply (auto simp add: field_simps) done qed lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" and "a ∈ {0..1}" and "x ∈ {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 ∈ {0..1}" and "pathfinish g = pathstart g" shows "path_image (shiftpath a g) = path_image g" proof - { fix x assume as: "g 1 = g 0" "x ∈ {0..1::real}" " ∀y∈{0..1} ∩ {x. ¬ a + x ≤ 1}. g x ≠ g (a + y - 1)" then have "∃y∈{0..1} ∩ {x. a + x ≤ 1}. g x = g (a + y)" proof (cases "a ≤ x") case False then show ?thesis apply (rule_tac x="1 + x - a" in bexI) using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) apply (auto simp add: field_simps atomize_not) done next case True then show ?thesis using as(1-2) and assms(1) apply (rule_tac x="x - a" in bexI) apply (auto simp add: field_simps) done qed } then show ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def by (auto simp add: image_iff) qed subsection {* Special case of straight-line paths *} definition linepath :: "'a::real_normed_vector => 'a => real => 'a" where "linepath a b = (λx. (1 - x) *⇩_{R}a + x *⇩_{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 continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" unfolding linepath_def by (intro continuous_intros) lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" using continuous_linepath_at by (auto intro!: continuous_at_imp_continuous_on) lemma path_linepath[intro]: "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 apply (rule set_eqI) apply rule defer unfolding mem_Collect_eq image_iff apply (erule exE) apply (rule_tac x="u *⇩_{R}1" in bexI) apply auto done lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" unfolding reversepath_def linepath_def by auto lemma injective_path_linepath: assumes "a ≠ b" shows "injective_path (linepath a b)" proof - { fix x y :: "real" assume "x *⇩_{R}b + y *⇩_{R}a = x *⇩_{R}a + y *⇩_{R}b" then have "(x - y) *⇩_{R}a = (x - y) *⇩_{R}b" by (simp add: algebra_simps) with assms have "x = y" by simp } then show ?thesis unfolding injective_path_def linepath_def by (auto simp add: algebra_simps) qed lemma simple_path_linepath[intro]: "a ≠ b ==> simple_path (linepath a b)" by (auto intro!: injective_imp_simple_path injective_path_linepath) subsection {* Bounding a point away from a path *} lemma not_on_path_ball: fixes g :: "real => 'a::heine_borel" assumes "path g" and "z ∉ path_image g" shows "∃e > 0. ball z e ∩ path_image g = {}" proof - obtain a where "a ∈ path_image g" "∀y ∈ path_image g. dist z a ≤ dist z y" using distance_attains_inf[OF _ path_image_nonempty, of g z] using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto then show ?thesis apply (rule_tac x="dist z a" in exI) using assms(2) apply (auto intro!: dist_pos_lt) done qed lemma not_on_path_cball: fixes g :: "real => 'a::heine_borel" assumes "path g" and "z ∉ path_image g" shows "∃e>0. cball z e ∩ (path_image g) = {}" proof - obtain e where "ball z e ∩ path_image g = {}" "e > 0" using not_on_path_ball[OF assms] by auto moreover have "cball z (e/2) ⊆ ball z e" using `e > 0` by auto ultimately show ?thesis apply (rule_tac x="e/2" in exI) apply auto done qed subsection {* Path component, considered as a "joinability" relation (from Tom Hales) *} definition "path_component s x y <-> (∃g. path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y)" 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 ∈ s" and "y ∈ s" using assms unfolding path_defs by auto lemma path_component_refl: assumes "x ∈ s" shows "path_component s x x" unfolding path_defs apply (rule_tac x="λu. x" in exI) using assms apply (auto intro!: continuous_intros) done lemma path_component_refl_eq: "path_component s x x <-> x ∈ s" by (auto intro!: path_component_mem path_component_refl) lemma path_component_sym: "path_component s x y ==> path_component s y x" using assms unfolding path_component_def apply (erule exE) apply (rule_tac x="reversepath g" in exI) apply auto done 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 apply (elim exE) apply (rule_tac x="g +++ ga" in exI) apply (auto simp add: path_image_join) done lemma path_component_of_subset: "s ⊆ t ==> path_component s x y ==> path_component t x y" unfolding path_component_def by auto text {* Can also consider it as a set, as the name suggests. *} lemma path_component_set: "{y. path_component s x y} = {y. (∃g. path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y)}" apply (rule set_eqI) unfolding mem_Collect_eq unfolding path_component_def apply auto done lemma path_component_subset: "{y. path_component s x y} ⊆ s" apply rule apply (rule path_component_mem(2)) apply auto done lemma path_component_eq_empty: "{y. path_component s x y} = {} <-> x ∉ s" apply rule apply (drule equals0D[of _ x]) defer apply (rule equals0I) unfolding mem_Collect_eq apply (drule path_component_mem(1)) using path_component_refl apply auto done subsection {* Path connectedness of a space *} definition "path_connected s <-> (∀x∈s. ∀y∈s. ∃g. path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y)" lemma path_connected_component: "path_connected s <-> (∀x∈s. ∀y∈s. path_component s x y)" unfolding path_connected_def path_component_def by auto lemma path_connected_component_set: "path_connected s <-> (∀x∈s. {y. path_component s x y} = s)" unfolding path_connected_component apply rule apply rule apply rule apply (rule path_component_subset) unfolding subset_eq mem_Collect_eq Ball_def apply auto done subsection {* Some useful lemmas about path-connectedness *} lemma convex_imp_path_connected: fixes s :: "'a::real_normed_vector set" assumes "convex s" shows "path_connected s" unfolding path_connected_def apply rule apply rule apply (rule_tac x = "linepath x y" in exI) unfolding path_image_linepath using assms [unfolded convex_contains_segment] apply auto done lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s" unfolding connected_def not_ex apply rule apply rule apply (rule ccontr) unfolding not_not apply (elim conjE) proof - fix e1 e2 assume as: "open e1" "open e2" "s ⊆ e1 ∪ e2" "e1 ∩ e2 ∩ s = {}" "e1 ∩ s ≠ {}" "e2 ∩ s ≠ {}" then obtain x1 x2 where obt:"x1 ∈ e1 ∩ s" "x2 ∈ e2 ∩ s" by auto then obtain g where g: "path g" "path_image g ⊆ s" "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 convex_real_interval) have "{0..1} ⊆ {x ∈ {0..1}. g x ∈ e1} ∪ {x ∈ {0..1}. g x ∈ e2}" using as(3) g(2)[unfolded path_defs] by blast moreover have "{x ∈ {0..1}. g x ∈ e1} ∩ {x ∈ {0..1}. g x ∈ e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto moreover have "{x ∈ {0..1}. g x ∈ e1} ≠ {} ∧ {x ∈ {0..1}. g x ∈ e2} ≠ {}" using g(3,4)[unfolded path_defs] using obt by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) ultimately show False using *[unfolded connected_local not_ex, rule_format, of "{x∈{0..1}. g x ∈ e1}" "{x∈{0..1}. g x ∈ e2}"] using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)] using continuous_open_in_preimage[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 {y. path_component s x y}" unfolding open_contains_ball proof fix y assume as: "y ∈ {y. path_component s x y}" then have "y ∈ s" apply - apply (rule path_component_mem(2)) unfolding mem_Collect_eq apply auto done then obtain e where e: "e > 0" "ball y e ⊆ s" using assms[unfolded open_contains_ball] by auto show "∃e > 0. ball y e ⊆ {y. path_component s x y}" apply (rule_tac x=e in exI) apply (rule,rule `e>0`) apply rule unfolding mem_ball mem_Collect_eq proof - fix z assume "dist y z < e" then show "path_component s x z" apply (rule_tac path_component_trans[of _ _ y]) defer apply (rule path_component_of_subset[OF e(2)]) apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using `e > 0` as apply auto done qed qed lemma open_non_path_component: fixes s :: "'a::real_normed_vector set" assumes "open s" shows "open (s - {y. path_component s x y})" unfolding open_contains_ball proof fix y assume as: "y ∈ s - {y. path_component s x y}" then obtain e where e: "e > 0" "ball y e ⊆ s" using assms [unfolded open_contains_ball] by auto show "∃e>0. ball y e ⊆ s - {y. path_component s x y}" apply (rule_tac x=e in exI) apply rule apply (rule `e>0`) apply rule apply rule defer proof (rule ccontr) fix z assume "z ∈ ball y e" "¬ z ∉ {y. path_component s x y}" then have "y ∈ {y. path_component s x y}" unfolding not_not mem_Collect_eq using `e>0` apply - apply (rule path_component_trans, assumption) apply (rule path_component_of_subset[OF e(2)]) apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) apply auto done then show False using as by auto qed (insert e(2), 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 ∈ s" and "y ∈ s" show "y ∈ {y. path_component s x y}" proof (rule ccontr) assume "¬ ?thesis" moreover have "{y. path_component s x y} ∩ s ≠ {}" using `x ∈ s` path_component_eq_empty path_component_subset[of s x] by auto ultimately show False using `y ∈ s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"] by auto qed qed lemma path_connected_continuous_image: assumes "continuous_on s f" and "path_connected s" shows "path_connected (f ` s)" unfolding path_connected_def proof (rule, rule) fix x' y' assume "x' ∈ f ` s" "y' ∈ f ` s" then obtain x y where x: "x ∈ s" and y: "y ∈ s" and x': "x' = f x" and y': "y' = f y" by auto from x y obtain g where "path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y" using assms(2)[unfolded path_connected_def] by fast then show "∃g. path g ∧ path_image g ⊆ f ` s ∧ pathstart g = x' ∧ pathfinish g = y'" unfolding x' y' apply (rule_tac x="f o g" in exI) unfolding path_defs apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)]) apply auto done qed lemma homeomorphic_path_connectedness: "s homeomorphic t ==> path_connected s <-> path_connected t" unfolding homeomorphic_def homeomorphism_def apply (erule exE|erule conjE)+ apply rule apply (drule_tac f=f in path_connected_continuous_image) prefer 3 apply (drule_tac f=g in path_connected_continuous_image) apply auto done lemma path_connected_empty: "path_connected {}" unfolding path_connected_def by auto lemma path_connected_singleton: "path_connected {a}" unfolding path_connected_def pathstart_def pathfinish_def path_image_def apply clarify apply (rule_tac x="λx. a" in exI) apply (simp add: image_constant_conv) apply (simp add: path_def continuous_on_const) done lemma path_connected_Un: assumes "path_connected s" and "path_connected t" and "s ∩ t ≠ {}" shows "path_connected (s ∪ t)" unfolding path_connected_component proof (rule, rule) fix x y assume as: "x ∈ s ∪ t" "y ∈ s ∪ t" from assms(3) obtain z where "z ∈ s ∩ t" by auto then show "path_component (s ∪ t) x y" using as and assms(1-2)[unfolded path_connected_component] apply - apply (erule_tac[!] UnE)+ apply (rule_tac[2-3] path_component_trans[of _ _ z]) apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) done qed lemma path_connected_UNION: assumes "!!i. i ∈ A ==> path_connected (S i)" and "!!i. i ∈ A ==> z ∈ S i" shows "path_connected (\<Union>i∈A. S i)" unfolding path_connected_component proof clarify fix x i y j assume *: "i ∈ A" "x ∈ S i" "j ∈ A" "y ∈ 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∈A. S i) x z" and "path_component (\<Union>i∈A. S i) z y" using *(1,3) by (auto elim!: path_component_of_subset [rotated]) then show "path_component (\<Union>i∈A. S i) x y" by (rule path_component_trans) qed subsection {* Sphere is path-connected *} lemma path_connected_punctured_universe: assumes "2 ≤ DIM('a::euclidean_space)" shows "path_connected ((UNIV::'a set) - {a})" proof - let ?A = "{x::'a. ∃i∈Basis. x • i < a • i}" let ?B = "{x::'a. ∃i∈Basis. a • i < x • i}" have A: "path_connected ?A" unfolding Collect_bex_eq proof (rule path_connected_UNION) fix i :: 'a assume "i ∈ Basis" then show "(∑i∈Basis. (a • i - 1)*⇩_{R}i) ∈ {x::'a. x • i < a • i}" by simp show "path_connected {x. x • i < a • i}" using convex_imp_path_connected [OF convex_halfspace_lt, of i "a • 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 ∈ Basis" then show "(∑i∈Basis. (a • i + 1) *⇩_{R}i) ∈ {x::'a. a • i < x • i}" by simp show "path_connected {x. a • i < x • i}" using convex_imp_path_connected [OF convex_halfspace_gt, of "a • i" i] by (simp add: inner_commute) qed obtain S :: "'a set" where "S ⊆ Basis" and "card S = Suc (Suc 0)" using ex_card[OF assms] by auto then obtain b0 b1 :: 'a where "b0 ∈ Basis" and "b1 ∈ Basis" and "b0 ≠ b1" unfolding card_Suc_eq by auto then have "a + b0 - b1 ∈ ?A ∩ ?B" by (auto simp: inner_simps inner_Basis) then have "?A ∩ ?B ≠ {}" by fast with A B have "path_connected (?A ∪ ?B)" by (rule path_connected_Un) also have "?A ∪ ?B = {x. ∃i∈Basis. x • i ≠ a • i}" unfolding neq_iff bex_disj_distrib Collect_disj_eq .. also have "… = {x. x ≠ a}" unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def) also have "… = UNIV - {a}" by auto finally show ?thesis . qed lemma path_connected_sphere: assumes "2 ≤ DIM('a::euclidean_space)" shows "path_connected {x::'a. norm (x - a) = r}" proof (rule linorder_cases [of r 0]) assume "r < 0" then have "{x::'a. norm(x - a) = r} = {}" by auto then show ?thesis using path_connected_empty by simp next assume "r = 0" then show ?thesis using path_connected_singleton by simp next assume r: "0 < r" have *: "{x::'a. norm(x - a) = r} = (λx. a + r *⇩_{R}x) ` {x. norm x = 1}" apply (rule set_eqI) apply rule unfolding image_iff apply (rule_tac x="(1/r) *⇩_{R}(x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR using r apply (auto simp add: scaleR_right_diff_distrib) done have **: "{x::'a. norm x = 1} = (λx. (1/norm x) *⇩_{R}x) ` (UNIV - {0})" apply (rule set_eqI) apply rule unfolding image_iff apply (rule_tac x=x in bexI) unfolding mem_Collect_eq apply (auto split: split_if_asm) done have "continuous_on (UNIV - {0}) (λx::'a. 1 / norm x)" by (auto intro!: continuous_intros) then show ?thesis unfolding * ** using path_connected_punctured_universe[OF assms] by (auto intro!: path_connected_continuous_image continuous_intros) qed lemma connected_sphere: "2 ≤ DIM('a::euclidean_space) ==> connected {x::'a. norm (x - a) = r}" using path_connected_sphere path_connected_imp_connected by auto end