(* Title: HOL/Multivariate_Analysis/Path_Connected.thy

Author: Robert Himmelmann, TU Muenchen

*)

header {* 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 interval_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_interval)

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_on_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_on_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_on_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_on_intros)+

prefer 2

apply (rule continuous_on_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_on_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)"

unfolding field_divide_inverse

by (simp add: continuous_on_intros)

then show ?thesis

unfolding * **

using path_connected_punctured_universe[OF assms]

by (auto intro!: path_connected_continuous_image continuous_on_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