section ‹Continuous paths and path-connected sets›
theory Path_Connected
imports Continuous_Extension Continuum_Not_Denumerable
begin
subsection ‹Paths and Arcs›
definition%important path :: "(real ⇒ 'a::topological_space) ⇒ bool"
where "path g ⟷ continuous_on {0..1} g"
definition%important pathstart :: "(real ⇒ 'a::topological_space) ⇒ 'a"
where "pathstart g = g 0"
definition%important pathfinish :: "(real ⇒ 'a::topological_space) ⇒ 'a"
where "pathfinish g = g 1"
definition%important path_image :: "(real ⇒ 'a::topological_space) ⇒ 'a set"
where "path_image g = g ` {0 .. 1}"
definition%important reversepath :: "(real ⇒ 'a::topological_space) ⇒ real ⇒ 'a"
where "reversepath g = (λx. g(1 - x))"
definition%important 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%important simple_path :: "(real ⇒ 'a::topological_space) ⇒ bool"
where "simple_path g ⟷
path g ∧ (∀x∈{0..1}. ∀y∈{0..1}. g x = g y ⟶ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0)"
definition%important arc :: "(real ⇒ 'a :: topological_space) ⇒ bool"
where "arc g ⟷ path g ∧ inj_on g {0..1}"
subsection%unimportant‹Invariance theorems›
lemma path_eq: "path p ⟹ (⋀t. t ∈ {0..1} ⟹ p t = q t) ⟹ path q"
using continuous_on_eq path_def by blast
lemma path_continuous_image: "path g ⟹ continuous_on (path_image g) f ⟹ path(f ∘ g)"
unfolding path_def path_image_def
using continuous_on_compose by blast
lemma path_translation_eq:
fixes g :: "real ⇒ 'a :: real_normed_vector"
shows "path((λx. a + x) ∘ g) = path g"
proof -
have g: "g = (λx. -a + x) ∘ ((λx. a + x) ∘ g)"
by (rule ext) simp
show ?thesis
unfolding path_def
apply safe
apply (subst g)
apply (rule continuous_on_compose)
apply (auto intro: continuous_intros)
done
qed
lemma path_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "path(f ∘ g) = path g"
proof -
from linear_injective_left_inverse [OF assms]
obtain h where h: "linear h" "h ∘ f = id"
by blast
then have g: "g = h ∘ (f ∘ g)"
by (metis comp_assoc id_comp)
show ?thesis
unfolding path_def
using h assms
by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear)
qed
lemma pathstart_translation: "pathstart((λx. a + x) ∘ g) = a + pathstart g"
by (simp add: pathstart_def)
lemma pathstart_linear_image_eq: "linear f ⟹ pathstart(f ∘ g) = f(pathstart g)"
by (simp add: pathstart_def)
lemma pathfinish_translation: "pathfinish((λx. a + x) ∘ g) = a + pathfinish g"
by (simp add: pathfinish_def)
lemma pathfinish_linear_image: "linear f ⟹ pathfinish(f ∘ g) = f(pathfinish g)"
by (simp add: pathfinish_def)
lemma path_image_translation: "path_image((λx. a + x) ∘ g) = (λx. a + x) ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma path_image_linear_image: "linear f ⟹ path_image(f ∘ g) = f ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma reversepath_translation: "reversepath((λx. a + x) ∘ g) = (λx. a + x) ∘ reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma reversepath_linear_image: "linear f ⟹ reversepath(f ∘ g) = f ∘ reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_translation:
"((λx. a + x) ∘ g1) +++ ((λx. a + x) ∘ g2) = (λx. a + x) ∘ (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma joinpaths_linear_image: "linear f ⟹ (f ∘ g1) +++ (f ∘ g2) = f ∘ (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma simple_path_translation_eq:
fixes g :: "real ⇒ 'a::euclidean_space"
shows "simple_path((λx. a + x) ∘ g) = simple_path g"
by (simp add: simple_path_def path_translation_eq)
lemma simple_path_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "simple_path(f ∘ g) = simple_path g"
using assms inj_on_eq_iff [of f]
by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
lemma arc_translation_eq:
fixes g :: "real ⇒ 'a::euclidean_space"
shows "arc((λx. a + x) ∘ g) = arc g"
by (auto simp: arc_def inj_on_def path_translation_eq)
lemma arc_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "arc(f ∘ g) = arc g"
using assms inj_on_eq_iff [of f]
by (auto simp: arc_def inj_on_def path_linear_image_eq)
subsection%unimportant‹Basic lemmas about paths›
lemma continuous_on_path: "path f ⟹ t ⊆ {0..1} ⟹ continuous_on t f"
using continuous_on_subset path_def by blast
lemma arc_imp_simple_path: "arc g ⟹ simple_path g"
by (simp add: arc_def inj_on_def simple_path_def)
lemma arc_imp_path: "arc g ⟹ path g"
using arc_def by blast
lemma arc_imp_inj_on: "arc g ⟹ inj_on g {0..1}"
by (auto simp: arc_def)
lemma simple_path_imp_path: "simple_path g ⟹ path g"
using simple_path_def by blast
lemma simple_path_cases: "simple_path g ⟹ arc g ∨ pathfinish g = pathstart g"
unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
by force
lemma simple_path_imp_arc: "simple_path g ⟹ pathfinish g ≠ pathstart g ⟹ arc g"
using simple_path_cases by auto
lemma arc_distinct_ends: "arc g ⟹ pathfinish g ≠ pathstart g"
unfolding arc_def inj_on_def pathfinish_def pathstart_def
by fastforce
lemma arc_simple_path: "arc g ⟷ simple_path g ∧ pathfinish g ≠ pathstart g"
using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast
lemma simple_path_eq_arc: "pathfinish g ≠ pathstart g ⟹ (simple_path g = arc g)"
by (simp add: arc_simple_path)
lemma path_image_const [simp]: "path_image (λt. a) = {a}"
by (force simp: path_image_def)
lemma path_image_nonempty [simp]: "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
using connected_continuous_image connected_Icc by blast
lemma compact_path_image[intro]: "path g ⟹ 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 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
by force
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 (auto intro: continuous_intros continuous_on_subset[of "{0..1}"])
done
show ?thesis
using *[of "reversepath g"] *[of g]
unfolding reversepath_reversepath
by (rule iffI)
qed
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 **: "⋀x y::real. 1-x = 1-y ⟹ x = y"
by simp
show ?thesis
using assms by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
qed
lemma simple_path_reversepath: "simple_path g ⟹ simple_path (reversepath g)"
apply (simp add: simple_path_def)
apply (force simp: reversepath_def)
done
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) ∘ (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) ∘ (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
section%unimportant ‹Path Images›
lemma bounded_path_image: "path g ⟹ bounded(path_image g)"
by (simp add: compact_imp_bounded compact_path_image)
lemma closed_path_image:
fixes g :: "real ⇒ 'a::t2_space"
shows "path g ⟹ closed(path_image g)"
by (metis compact_path_image compact_imp_closed)
lemma connected_simple_path_image: "simple_path g ⟹ connected(path_image g)"
by (metis connected_path_image simple_path_imp_path)
lemma compact_simple_path_image: "simple_path g ⟹ compact(path_image g)"
by (metis compact_path_image simple_path_imp_path)
lemma bounded_simple_path_image: "simple_path g ⟹ bounded(path_image g)"
by (metis bounded_path_image simple_path_imp_path)
lemma closed_simple_path_image:
fixes g :: "real ⇒ 'a::t2_space"
shows "simple_path g ⟹ closed(path_image g)"
by (metis closed_path_image simple_path_imp_path)
lemma connected_arc_image: "arc g ⟹ connected(path_image g)"
by (metis connected_path_image arc_imp_path)
lemma compact_arc_image: "arc g ⟹ compact(path_image g)"
by (metis compact_path_image arc_imp_path)
lemma bounded_arc_image: "arc g ⟹ bounded(path_image g)"
by (metis bounded_path_image arc_imp_path)
lemma closed_arc_image:
fixes g :: "real ⇒ 'a::t2_space"
shows "arc g ⟹ closed(path_image g)"
by (metis closed_path_image arc_imp_path)
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:
"pathfinish g1 = pathstart g2 ⟹ path_image(g1 +++ g2) = path_image g1 ∪ path_image g2"
apply (rule subset_antisym [OF path_image_join_subset])
apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def)
apply (drule sym)
apply (rule_tac x="xa/2" in bexI, auto)
apply (rule ccontr)
apply (drule_tac x="(xa+1)/2" in bspec)
apply (auto simp: field_simps)
apply (drule_tac x="1/2" in bspec, auto)
done
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 pathstart_compose: "pathstart(f ∘ p) = f(pathstart p)"
by (simp add: pathstart_def)
lemma pathfinish_compose: "pathfinish(f ∘ p) = f(pathfinish p)"
by (simp add: pathfinish_def)
lemma path_image_compose: "path_image (f ∘ p) = f ` (path_image p)"
by (simp add: image_comp path_image_def)
lemma path_compose_join: "f ∘ (p +++ q) = (f ∘ p) +++ (f ∘ q)"
by (rule ext) (simp add: joinpaths_def)
lemma path_compose_reversepath: "f ∘ reversepath p = reversepath(f ∘ p)"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_eq:
"(⋀t. t ∈ {0..1} ⟹ p t = p' t) ⟹
(⋀t. t ∈ {0..1} ⟹ q t = q' t)
⟹ t ∈ {0..1} ⟹ (p +++ q) t = (p' +++ q') t"
by (auto simp: joinpaths_def)
lemma simple_path_inj_on: "simple_path g ⟹ inj_on g {0<..<1}"
by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def)
subsection%unimportant‹Simple paths with the endpoints removed›
lemma simple_path_endless:
"simple_path c ⟹ path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}"
apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def)
apply (metis eq_iff le_less_linear)
apply (metis leD linear)
using less_eq_real_def zero_le_one apply blast
using less_eq_real_def zero_le_one apply blast
done
lemma connected_simple_path_endless:
"simple_path c ⟹ connected(path_image c - {pathstart c,pathfinish c})"
apply (simp add: simple_path_endless)
apply (rule connected_continuous_image)
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path)
by auto
lemma nonempty_simple_path_endless:
"simple_path c ⟹ path_image c - {pathstart c,pathfinish c} ≠ {}"
by (simp add: simple_path_endless)
subsection%unimportant‹The operations on paths›
lemma path_image_subset_reversepath: "path_image(reversepath g) ≤ path_image g"
by (auto simp: path_image_def reversepath_def)
lemma path_imp_reversepath: "path g ⟹ path(reversepath g)"
apply (auto simp: path_def reversepath_def)
using continuous_on_compose [of "{0..1}" "λx. 1 - x" g]
apply (auto simp: continuous_on_op_minus)
done
lemma half_bounded_equal: "1 ≤ x * 2 ⟹ x * 2 ≤ 1 ⟷ 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)"
proof -
have *: "{0..1::real} = {0..1/2} ∪ {1/2..1}"
by auto
have gg: "g2 0 = g1 1"
by (metis assms(3) pathfinish_def pathstart_def)
have 1: "continuous_on {0..1/2} (g1 +++ g2)"
apply (rule continuous_on_eq [of _ "g1 ∘ (λx. 2*x)"])
apply (rule continuous_intros | simp add: joinpaths_def assms)+
done
have "continuous_on {1/2..1} (g2 ∘ (λx. 2*x-1))"
apply (rule continuous_on_subset [of "{1/2..1}"])
apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+
done
then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
apply (rule continuous_on_eq [of "{1/2..1}" "g2 ∘ (λx. 2*x-1)"])
apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+
done
show ?thesis
apply (subst *)
apply (rule continuous_on_closed_Un)
using 1 2
apply auto
done
qed
lemma path_join_imp: "⟦path g1; path g2; pathfinish g1 = pathstart g2⟧ ⟹ path(g1 +++ g2)"
by (simp add: path_join)
lemma simple_path_join_loop:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1"
"path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
shows "simple_path(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}"
using assms
by (simp add: arc_def)
have injg2: "inj_on g2 {0..1}"
using assms
by (simp add: arc_def)
have g12: "g1 1 = g2 0"
and g21: "g2 1 = g1 0"
and sb: "g1 ` {0..1} ∩ g2 ` {0..1} ⊆ {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 xyI: "x = 1 ⟶ y ≠ 0"
and xy: "x ≤ 1" "0 ≤ y" " y * 2 ≤ 1" "¬ x * 2 ≤ 1" "g2 (2 * x - 1) = g1 (2 * y)"
have g1im: "g1 (2 * y) ∈ g1 ` {0..1} ∩ g2 ` {0..1}"
using xy
apply simp
apply (rule_tac x="2 * x - 1" in image_eqI, auto)
done
have False
using subsetD [OF sb g1im] xy
apply auto
apply (drule inj_onD [OF injg1])
using g21 [symmetric] xyI
apply (auto dest: inj_onD [OF injg2])
done
} note * = this
{ fix x and y::real
assume xy: "y ≤ 1" "0 ≤ x" "¬ y * 2 ≤ 1" "x * 2 ≤ 1" "g1 (2 * x) = g2 (2 * y - 1)"
have g1im: "g1 (2 * x) ∈ g1 ` {0..1} ∩ g2 ` {0..1}"
using xy
apply simp
apply (rule_tac x="2 * x" in image_eqI, auto)
done
have "x = 0 ∧ y = 1"
using subsetD [OF sb g1im] xy
apply auto
apply (force dest: inj_onD [OF injg1])
using g21 [symmetric]
apply (auto dest: inj_onD [OF injg2])
done
} note ** = this
show ?thesis
using assms
apply (simp add: arc_def simple_path_def path_join, clarify)
apply (simp add: joinpaths_def split: if_split_asm)
apply (force dest: inj_onD [OF injg1])
apply (metis *)
apply (metis **)
apply (force dest: inj_onD [OF injg2])
done
qed
lemma arc_join:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2"
"path_image g1 ∩ path_image g2 ⊆ {pathstart g2}"
shows "arc(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}"
using assms
by (simp add: arc_def)
have injg2: "inj_on g2 {0..1}"
using assms
by (simp add: arc_def)
have g11: "g1 1 = g2 0"
and sb: "g1 ` {0..1} ∩ g2 ` {0..1} ⊆ {g2 0}"
using assms
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
{ fix x and y::real
assume xy: "x ≤ 1" "0 ≤ y" " y * 2 ≤ 1" "¬ x * 2 ≤ 1" "g2 (2 * x - 1) = g1 (2 * y)"
have g1im: "g1 (2 * y) ∈ g1 ` {0..1} ∩ g2 ` {0..1}"
using xy
apply simp
apply (rule_tac x="2 * x - 1" in image_eqI, auto)
done
have False
using subsetD [OF sb g1im] xy
by (auto dest: inj_onD [OF injg2])
} note * = this
show ?thesis
apply (simp add: arc_def inj_on_def)
apply (clarsimp simp add: arc_imp_path assms path_join)
apply (simp add: joinpaths_def split: if_split_asm)
apply (force dest: inj_onD [OF injg1])
apply (metis *)
apply (metis *)
apply (force dest: inj_onD [OF injg2])
done
qed
lemma reversepath_joinpaths:
"pathfinish g1 = pathstart g2 ⟹ reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
by (rule ext) (auto simp: mult.commute)
subsection%unimportant‹Some reversed and "if and only if" versions of joining theorems›
lemma path_join_path_ends:
fixes g1 :: "real ⇒ '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 ≠ 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 obtain d1 where "d1 > 0"
and d1: "⋀x'. ⟦x'∈{0..1}; norm x' < d1⟧ ⟹ dist (g2 x') (g2 0) < e/2"
using assms(2) unfolding path_def continuous_on_iff
apply (drule_tac x=0 in bspec, simp)
by (metis half_gt_zero_iff norm_conv_dist)
obtain d2 where "d2 > 0"
and d2: "⋀x'. ⟦x'∈{0..1}; dist x' (1/2) < d2⟧
⟹ dist ((g1 +++ g2) x') (g1 1) < e/2"
using assms(1) ‹e > 0› unfolding path_def continuous_on_iff
apply (drule_tac x="1/2" in bspec, simp)
apply (drule_tac x="e/2" in spec)
apply (force simp: joinpaths_def)
done
have int01_1: "min (1/2) (min d1 d2) / 2 ∈ {0..1}"
using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def)
have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def dist_norm)
have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 ∈ {0..1}"
using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def)
have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def dist_norm)
have [simp]: "~ min (1 / 2) (min d1 d2) ≤ 0"
using ‹d1 > 0› ‹d2 > 0› 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 ⇒ 'a::metric_space"
assumes "path g1" "path g2"
shows "path(g1 +++ g2) ⟷ 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 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
proof -
have *: "⋀x y. ⟦0 ≤ x; x ≤ 1; 0 ≤ y; y ≤ 1; (g1 +++ g2) x = (g1 +++ g2) y⟧
⟹ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
using assms by (simp add: simple_path_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 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 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 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1"
then show "x = y"
using * [of "(x + 1) / 2" "(y + 1) / 2"]
by (force simp: joinpaths_def split_ifs divide_simps)
qed
ultimately have "arc g2"
using assms by (simp add: arc_def)
have "g2 y = g1 0 ∨ g2 y = g1 1"
if "g1 x = g2 y" "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1" for x y
using * [of "x / 2" "(y + 1) / 2"] that
by (auto simp: joinpaths_def split_ifs divide_simps)
then have "path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
by (fastforce simp: pathstart_def pathfinish_def path_image_def)
with ‹arc g1› ‹arc g2› 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) ⟷
arc g1 ∧ arc g2 ∧ path_image g1 ∩ path_image g2 ⊆ {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) ⟷
arc g1 ∧ arc g2 ∧ path_image g1 ∩ path_image g2 ⊆ {pathstart g2}"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
then have *: "⋀x y. ⟦0 ≤ x; x ≤ 1; 0 ≤ y; y ≤ 1; (g1 +++ g2) x = (g1 +++ g2) y⟧
⟹ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
using assms by (simp add: simple_path_def)
have False if "g1 0 = g2 u" "0 ≤ u" "u ≤ 1" for u
using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF ‹?lhs›]
by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs divide_simps)
then have n1: "~ (pathstart g1 ∈ path_image g2)"
unfolding pathstart_def path_image_def
using atLeastAtMost_iff by blast
show ?rhs using ‹?lhs›
apply (rule simple_path_joinE [OF arc_imp_simple_path assms])
using n1 by force
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
⟹ (arc(g1 +++ g2) ⟷
arc g1 ∧ arc g2 ∧
path_image g1 ∩ path_image g2 = {pathstart g2})"
using pathfinish_in_path_image by (fastforce simp: arc_join_eq)
subsection%unimportant‹The joining of paths is associative›
lemma path_assoc:
"⟦pathfinish p = pathstart q; pathfinish q = pathstart r⟧
⟹ path(p +++ (q +++ r)) ⟷ 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)) ⟷ 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 ∉ 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 ∉ 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 ∩ path_image q ⊆ {pathstart q}"
"(path_image p ∪ path_image q) ∩ path_image r ⊆ {pathstart r}"
"x ∈ path_image p" "x ∈ path_image r"
have "pathstart r ∈ 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:
"⟦pathfinish p = pathstart q; pathfinish q = pathstart r⟧
⟹ arc(p +++ (q +++ r)) ⟷ arc((p +++ q) +++ r)"
by (simp add: arc_simple_path simple_path_assoc)
subsubsection%unimportant‹Symmetry and loops›
lemma path_sym:
"⟦pathfinish p = pathstart q; pathfinish q = pathstart p⟧ ⟹ path(p +++ q) ⟷ path(q +++ p)"
by auto
lemma simple_path_sym:
"⟦pathfinish p = pathstart q; pathfinish q = pathstart p⟧
⟹ simple_path(p +++ q) ⟷ simple_path(q +++ p)"
by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)
lemma path_image_sym:
"⟦pathfinish p = pathstart q; pathfinish q = pathstart p⟧
⟹ path_image(p +++ q) = path_image(q +++ p)"
by (simp add: path_image_join sup_commute)
section‹Choosing a subpath of an existing path›
definition%important subpath :: "real ⇒ real ⇒ (real ⇒ 'a) ⇒ real ⇒ 'a::real_normed_vector"
where "subpath a b g ≡ λx. g((b - a) * x + a)"
lemma path_image_subpath_gen:
fixes g :: "_ ⇒ 'a::real_normed_vector"
shows "path_image(subpath u v g) = g ` (closed_segment u v)"
apply (simp add: closed_segment_real_eq path_image_def subpath_def)
apply (subst o_def [of g, symmetric])
apply (simp add: image_comp [symmetric])
done
lemma path_image_subpath:
fixes g :: "real ⇒ 'a::real_normed_vector"
shows "path_image(subpath u v g) = (if u ≤ 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 ⇒ '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 ⇒ 'a::real_normed_vector"
assumes "path g" "u ∈ {0..1}" "v ∈ {0..1}"
shows "path(subpath u v g)"
proof -
have "continuous_on {0..1} (g ∘ (λx. ((v-u) * x+ u)))"
apply (rule continuous_intros | simp)+
apply (simp add: image_affinity_atLeastAtMost [where c=u])
using assms
apply (auto simp: path_def continuous_on_subset)
done
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 ((λx. a + x) ∘ g) = (λx. a + x) ∘ subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma subpath_linear_image: "linear f ⟹ subpath u v (f ∘ g) = f ∘ subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma affine_ineq:
fixes x :: "'a::linordered_idom"
assumes "x ≤ 1" "v ≤ u"
shows "v + x * u ≤ u + x * v"
proof -
have "(1-x)*(u-v) ≥ 0"
using assms by auto
then show ?thesis
by (simp add: algebra_simps)
qed
lemma sum_le_prod1:
fixes a::real shows "⟦a ≤ 1; b ≤ 1⟧ ⟹ a + b ≤ 1 + a * b"
by (metis add.commute affine_ineq less_eq_real_def mult.right_neutral)
lemma simple_path_subpath_eq:
"simple_path(subpath u v g) ⟷
path(subpath u v g) ∧ u≠v ∧
(∀x y. x ∈ closed_segment u v ∧ y ∈ closed_segment u v ∧ g x = g y
⟶ x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u)"
(is "?lhs = ?rhs")
proof (rule iffI)
assume ?lhs
then have p: "path (λx. g ((v - u) * x + u))"
and sim: "(⋀x y. ⟦x∈{0..1}; y∈{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)⟧
⟹ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0)"
by (auto simp: simple_path_def subpath_def)
{ fix x y
assume "x ∈ closed_segment u v" "y ∈ closed_segment u v" "g x = g y"
then have "x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
split: if_split_asm)
} moreover
have "path(subpath u v g) ∧ u≠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: "⋀x y. ⟦g x = g y; u ≤ x; x ≤ v; u ≤ y; y ≤ v⟧ ⟹ x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
and d2: "⋀x y. ⟦g x = g y; v ≤ x; x ≤ u; v ≤ y; y ≤ u⟧ ⟹ x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
and ne: "u < v ∨ v < u"
and psp: "path (subpath u v g)"
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
have [simp]: "⋀x. u + x * v = v + x * u ⟷ u=v ∨ x=1"
by algebra
show ?lhs using psp ne
unfolding simple_path_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) ⟷ path(subpath u v g) ∧ u≠v ∧ inj_on g (closed_segment u v)"
(is "?lhs = ?rhs")
proof (rule iffI)
assume ?lhs
then have p: "path (λx. g ((v - u) * x + u))"
and sim: "(⋀x y. ⟦x∈{0..1}; y∈{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)⟧
⟹ x = y)"
by (auto simp: arc_def inj_on_def subpath_def)
{ fix x y
assume "x ∈ closed_segment u v" "y ∈ closed_segment u v" "g x = g y"
then have "x = y"
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
by (force simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
split: if_split_asm)
} moreover
have "path(subpath u v g) ∧ u≠v"
using sim [of "1/3" "2/3"] p
by (auto simp: subpath_def)
ultimately show ?rhs
unfolding inj_on_def
by metis
next
assume ?rhs
then
have d1: "⋀x y. ⟦g x = g y; u ≤ x; x ≤ v; u ≤ y; y ≤ v⟧ ⟹ x = y"
and d2: "⋀x y. ⟦g x = g y; v ≤ x; x ≤ u; v ≤ y; y ≤ u⟧ ⟹ x = y"
and ne: "u < v ∨ v < u"
and psp: "path (subpath u v g)"
by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost)
show ?lhs using psp ne
unfolding arc_def subpath_def inj_on_def
by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed
lemma simple_path_subpath:
assumes "simple_path g" "u ∈ {0..1}" "v ∈ {0..1}" "u ≠ v"
shows "simple_path(subpath u v g)"
using assms
apply (simp add: simple_path_subpath_eq simple_path_imp_path)
apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce)
done
lemma arc_simple_path_subpath:
"⟦simple_path g; u ∈ {0..1}; v ∈ {0..1}; g u ≠ g v⟧ ⟹ arc(subpath u v g)"
by (force intro: simple_path_subpath simple_path_imp_arc)
lemma arc_subpath_arc:
"⟦arc g; u ∈ {0..1}; v ∈ {0..1}; u ≠ v⟧ ⟹ 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:
"⟦simple_path g; u ∈ {0..1}; v ∈ {0..1}; u ≠ v; ¦u-v¦ < 1⟧ ⟹ arc(subpath u v g)"
apply (rule arc_simple_path_subpath)
apply (force simp: simple_path_def)+
done
lemma path_image_subpath_subset:
"⟦u ∈ {0..1}; v ∈ {0..1}⟧ ⟹ path_image(subpath u v g) ⊆ path_image g"
apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath)
apply (auto simp: path_image_def)
done
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)
subsection%unimportant‹There is a subpath to the frontier›
lemma subpath_to_frontier_explicit:
fixes S :: "'a::metric_space set"
assumes g: "path g" and "pathfinish g ∉ S"
obtains u where "0 ≤ u" "u ≤ 1"
"⋀x. 0 ≤ x ∧ x < u ⟹ g x ∈ interior S"
"(g u ∉ interior S)" "(u = 0 ∨ g u ∈ closure S)"
proof -
have gcon: "continuous_on {0..1} g" using g by (simp add: path_def)
then have com: "compact ({0..1} ∩ {u. g u ∈ closure (- S)})"
apply (simp add: Int_commute [of "{0..1}"] compact_eq_bounded_closed closed_vimage_Int [unfolded vimage_def])
using compact_eq_bounded_closed apply fastforce
done
have "1 ∈ {u. g u ∈ closure (- S)}"
using assms by (simp add: pathfinish_def closure_def)
then have dis: "{0..1} ∩ {u. g u ∈ closure (- S)} ≠ {}"
using atLeastAtMost_iff zero_le_one by blast
then obtain u where "0 ≤ u" "u ≤ 1" and gu: "g u ∈ closure (- S)"
and umin: "⋀t. ⟦0 ≤ t; t ≤ 1; g t ∈ closure (- S)⟧ ⟹ u ≤ t"
using compact_attains_inf [OF com dis] by fastforce
then have umin': "⋀t. ⟦0 ≤ t; t ≤ 1; t < u⟧ ⟹ g t ∈ S"
using closure_def by fastforce
{ assume "u ≠ 0"
then have "u > 0" using ‹0 ≤ u› by auto
{ fix e::real assume "e > 0"
obtain d where "d>0" and d: "⋀x'. ⟦x' ∈ {0..1}; dist x' u ≤ d⟧ ⟹ dist (g x') (g u) < e"
using continuous_onE [OF gcon _ ‹e > 0›] ‹0 ≤ _› ‹_ ≤ 1› atLeastAtMost_iff by auto
have *: "dist (max 0 (u - d / 2)) u ≤ d"
using ‹0 ≤ u› ‹u ≤ 1› ‹d > 0› by (simp add: dist_real_def)
have "∃y∈S. dist y (g u) < e"
using ‹0 < u› ‹u ≤ 1› ‹d > 0›
by (force intro: d [OF _ *] umin')
}
then have "g u ∈ closure S"
by (simp add: frontier_def closure_approachable)
}
then show ?thesis
apply (rule_tac u=u in that)
apply (auto simp: ‹0 ≤ u› ‹u ≤ 1› gu interior_closure umin)
using ‹_ ≤ 1› interior_closure umin apply fastforce
done
qed
lemma subpath_to_frontier_strong:
assumes g: "path g" and "pathfinish g ∉ S"
obtains u where "0 ≤ u" "u ≤ 1" "g u ∉ interior S"
"u = 0 ∨ (∀x. 0 ≤ x ∧ x < 1 ⟶ subpath 0 u g x ∈ interior S) ∧ g u ∈ closure S"
proof -
obtain u where "0 ≤ u" "u ≤ 1"
and gxin: "⋀x. 0 ≤ x ∧ x < u ⟹ g x ∈ interior S"
and gunot: "(g u ∉ interior S)" and u0: "(u = 0 ∨ g u ∈ closure S)"
using subpath_to_frontier_explicit [OF assms] by blast
show ?thesis
apply (rule that [OF ‹0 ≤ u› ‹u ≤ 1›])
apply (simp add: gunot)
using ‹0 ≤ u› u0 by (force simp: subpath_def gxin)
qed
lemma subpath_to_frontier:
assumes g: "path g" and g0: "pathstart g ∈ closure S" and g1: "pathfinish g ∉ S"
obtains u where "0 ≤ u" "u ≤ 1" "g u ∈ frontier S" "(path_image(subpath 0 u g) - {g u}) ⊆ interior S"
proof -
obtain u where "0 ≤ u" "u ≤ 1"
and notin: "g u ∉ interior S"
and disj: "u = 0 ∨
(∀x. 0 ≤ x ∧ x < 1 ⟶ subpath 0 u g x ∈ interior S) ∧ g u ∈ closure S"
using subpath_to_frontier_strong [OF g g1] by blast
show ?thesis
apply (rule that [OF ‹0 ≤ u› ‹u ≤ 1›])
apply (metis DiffI disj frontier_def g0 notin pathstart_def)
using ‹0 ≤ u› g0 disj
apply (simp add: path_image_subpath_gen)
apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
apply (rename_tac y)
apply (drule_tac x="y/u" in spec)
apply (auto split: if_split_asm)
done
qed
lemma exists_path_subpath_to_frontier:
fixes S :: "'a::real_normed_vector set"
assumes "path g" "pathstart g ∈ closure S" "pathfinish g ∉ S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h ⊆ path_image g"
"path_image h - {pathfinish h} ⊆ interior S"
"pathfinish h ∈ frontier S"
proof -
obtain u where u: "0 ≤ u" "u ≤ 1" "g u ∈ frontier S" "(path_image(subpath 0 u g) - {g u}) ⊆ interior S"
using subpath_to_frontier [OF assms] by blast
show ?thesis
apply (rule that [of "subpath 0 u g"])
using assms u
apply (simp_all add: path_image_subpath)
apply (simp add: pathstart_def)
apply (force simp: closed_segment_eq_real_ivl path_image_def)
done
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 ∈ S" and g1: "pathfinish g ∉ S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h ⊆ path_image g ∩ S"
"pathfinish h ∈ frontier S"
proof -
obtain h where h: "path h" "pathstart h = pathstart g" "path_image h ⊆ path_image g"
"path_image h - {pathfinish h} ⊆ interior S"
"pathfinish h ∈ frontier S"
using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
show ?thesis
apply (rule that [OF ‹path h›])
using assms h
apply auto
apply (metis Diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
done
qed
subsection ‹shiftpath: Reparametrizing a closed curve to start at some chosen point›
definition%important 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 *
proof (rule continuous_on_closed_Un)
have contg: "continuous_on {0..1} g"
using ‹path g› path_def by blast
show "continuous_on {0..1-a} (λx. if a + x ≤ 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {0..1-a} (g ∘ (+) a)"
by (intro continuous_intros continuous_on_subset [OF contg]) (use ‹a ∈ {0..1}› in auto)
qed auto
show "continuous_on {1-a..1} (λx. if a + x ≤ 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {1-a..1} (g ∘ (+) (a - 1))"
by (intro continuous_intros continuous_on_subset [OF contg]) (use ‹a ∈ {0..1}› in auto)
qed (auto simp: "**" add.commute add_diff_eq)
qed auto
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: "a ∈ {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 ∈ {0..1::real}" and gne: "⋀y. 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 g gne[of "1 + x - a"] a
apply (force simp: field_simps)+
done
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 simple_path_shiftpath:
assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 ≤ a" "a ≤ 1"
shows "simple_path (shiftpath a g)"
unfolding simple_path_def
proof (intro conjI impI ballI)
show "path (shiftpath a g)"
by (simp add: assms path_shiftpath simple_path_imp_path)
have *: "⋀x y. ⟦g x = g y; x ∈ {0..1}; y ∈ {0..1}⟧ ⟹ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
using assms by (simp add: simple_path_def)
show "x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
if "x ∈ {0..1}" "y ∈ {0..1}" "shiftpath a g x = shiftpath a g y" for x y
using that a unfolding shiftpath_def
by (force split: if_split_asm dest!: *)
qed
subsection ‹Special case of straight-line paths›
definition%important 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_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 *⇩R b"
by (simp add: linepath_def)
lemma arc_linepath:
assumes "a ≠ b" shows [simp]: "arc (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 arc_def inj_on_def
by (fastforce simp: algebra_simps linepath_def)
qed
lemma simple_path_linepath[intro]: "a ≠ b ⟹ 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 = (λx. 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 ≠ b" shows "inj_on (linepath a b) {0..1}"
proof (clarsimp simp: inj_on_def linepath_def)
fix x y
assume "(1 - x) *⇩R a + x *⇩R b = (1 - y) *⇩R a + y *⇩R b" "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1"
then have "x *⇩R (a - b) = y *⇩R (a - b)"
by (auto simp: algebra_simps)
then show "x=y"
using assms by auto
qed
subsection%unimportant‹Segments via convex hulls›
lemma segments_subset_convex_hull:
"closed_segment a b ⊆ (convex hull {a,b,c})"
"closed_segment a c ⊆ (convex hull {a,b,c})"
"closed_segment b c ⊆ (convex hull {a,b,c})"
"closed_segment b a ⊆ (convex hull {a,b,c})"
"closed_segment c a ⊆ (convex hull {a,b,c})"
"closed_segment c b ⊆ (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 ∈ convex hull s" "y ∈ convex hull s"
shows "midpoint x y ∈ convex hull s"
proof -
have "(1 - inverse(2)) *⇩R x + inverse(2) *⇩R y ∈ convex hull s"
by (rule convexD_alt) (use assms in auto)
then show ?thesis
by (simp add: midpoint_def algebra_simps)
qed
lemma not_in_interior_convex_hull_3:
fixes a :: "complex"
shows "a ∉ interior(convex hull {a,b,c})"
"b ∉ interior(convex hull {a,b,c})"
"c ∉ 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 ∈ closed_segment a b"
using midpoints_in_convex_hull segment_convex_hull by blast
lemma midpoint_in_open_segment [simp]: "midpoint a b ∈ open_segment a b ⟷ a ≠ b"
by (simp add: open_segment_def)
lemma continuous_IVT_local_extremum:
fixes f :: "'a::euclidean_space ⇒ real"
assumes contf: "continuous_on (closed_segment a b) f"
and "a ≠ b" "f a = f b"
obtains z where "z ∈ open_segment a b"
"(∀w ∈ closed_segment a b. (f w) ≤ (f z)) ∨
(∀w ∈ closed_segment a b. (f z) ≤ (f w))"
proof -
obtain c where "c ∈ closed_segment a b" and c: "⋀y. y ∈ closed_segment a b ⟹ f y ≤ f c"
using continuous_attains_sup [of "closed_segment a b" f] contf by auto
obtain d where "d ∈ closed_segment a b" and d: "⋀y. y ∈ closed_segment a b ⟹ f d ≤ f y"
using continuous_attains_inf [of "closed_segment a b" f] contf by auto
show ?thesis
proof (cases "c ∈ open_segment a b ∨ d ∈ open_segment a b")
case True
then show ?thesis
using c d that by blast
next
case False
then have "(c = a ∨ c = b) ∧ (d = a ∨ d = b)"
by (simp add: ‹c ∈ closed_segment a b› ‹d ∈ closed_segment a b› open_segment_def)
with ‹a ≠ b› ‹f a = f b› c d show ?thesis
by (rule_tac z = "midpoint a b" in that) (fastforce+)
qed
qed
text‹An injective map into R is also an open map w.r.T. the universe, and conversely. ›
proposition injective_eq_1d_open_map_UNIV:
fixes f :: "real ⇒ real"
assumes contf: "continuous_on S f" and S: "is_interval S"
shows "inj_on f S ⟷ (∀T. open T ∧ T ⊆ S ⟶ open(f ` T))"
(is "?lhs = ?rhs")
proof safe
fix T
assume injf: ?lhs and "open T" and "T ⊆ S"
have "∃U. open U ∧ f x ∈ U ∧ U ⊆ f ` T" if "x ∈ T" for x
proof -
obtain δ where "δ > 0" and δ: "cball x δ ⊆ T"
using ‹open T› ‹x ∈ T› open_contains_cball_eq by blast
show ?thesis
proof (intro exI conjI)
have "closed_segment (x-δ) (x+δ) = {x-δ..x+δ}"
using ‹0 < δ› by (auto simp: closed_segment_eq_real_ivl)
also have "… ⊆ S"
using δ ‹T ⊆ S› by (auto simp: dist_norm subset_eq)
finally have "f ` (open_segment (x-δ) (x+δ)) = open_segment (f (x-δ)) (f (x+δ))"
using continuous_injective_image_open_segment_1
by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
then show "open (f ` {x-δ<..<x+δ})"
using ‹0 < δ› by (simp add: open_segment_eq_real_ivl)
show "f x ∈ f ` {x - δ<..<x + δ}"
by (auto simp: ‹δ > 0›)
show "f ` {x - δ<..<x + δ} ⊆ f ` T"
using δ 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 ∈ S" "y ∈ S" and "f x = f y" "x ≠ 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 ξ where "ξ ∈ open_segment x y"
and ξ: "(∀w ∈ closed_segment x y. (f w) ≤ (f ξ)) ∨
(∀w ∈ closed_segment x y. (f ξ) ≤ (f w))"
using continuous_IVT_local_extremum [of x y f] ‹f x = f y› ‹x ≠ y› by blast
ultimately obtain e where "e>0" and e: "⋀u. dist u (f ξ) < e ⟹ u ∈ f ` open_segment x y"
using open_dist by (metis image_eqI)
have fin: "f ξ + (e/2) ∈ f ` open_segment x y" "f ξ - (e/2) ∈ f ` open_segment x y"
using e [of "f ξ + (e/2)"] e [of "f ξ - (e/2)"] ‹e > 0› by (auto simp: dist_norm)
show ?thesis
using ξ ‹0 < e› fin open_closed_segment by fastforce
qed
then show ?lhs
by (force simp: inj_on_def)
qed
subsection%unimportant ‹Bounding a point away from a path›
lemma not_on_path_ball:
fixes g :: "real ⇒ 'a::heine_borel"
assumes "path g"
and z: "z ∉ path_image g"
shows "∃e > 0. ball z e ∩ path_image g = {}"
proof -
have "closed (path_image g)"
by (simp add: ‹path g› closed_path_image)
then obtain a where "a ∈ path_image g" "∀y ∈ path_image g. dist z a ≤ 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 ⇒ '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
by (rule_tac x="e/2" in exI) auto
qed
section ‹Path component, considered as a "joinability" relation (from Tom Hales)›
definition%important "path_component s x y ⟷
(∃g. path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y)"
abbreviation%important
"path_component_set s x ≡ 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 ∈ 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"
unfolding path_component_def
apply (erule exE)
apply (rule_tac x="reversepath g" in exI, 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: 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
lemma path_connected_linepath:
fixes s :: "'a::real_normed_vector set"
shows "closed_segment a b ⊆ s ⟹ path_component s a b"
unfolding path_component_def
by (rule_tac x="linepath a b" in exI, auto)
subsubsection%unimportant ‹Path components as sets›
lemma path_component_set:
"path_component_set s x =
{y. (∃g. path g ∧ path_image g ⊆ s ∧ pathstart g = x ∧ pathfinish g = y)}"
by (auto simp: path_component_def)
lemma path_component_subset: "path_component_set s x ⊆ s"
by (auto simp: path_component_mem(2))
lemma path_component_eq_empty: "path_component_set s x = {} ⟷ x ∉ s"
using path_component_mem path_component_refl_eq
by fastforce
lemma path_component_mono:
"s ⊆ t ⟹ (path_component_set s x) ⊆ (path_component_set t x)"
by (simp add: Collect_mono path_component_of_subset)
lemma path_component_eq:
"y ∈ path_component_set s x ⟹ 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 ‹Path connectedness of a space›
definition%important "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. path_component_set s x = s)"
unfolding path_connected_component path_component_subset
using path_component_mem by blast
lemma path_component_maximal:
"⟦x ∈ t; path_connected t; t ⊆ s⟧ ⟹ t ⊆ (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 ⊆ 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 "{0..1} ∩ g -` e1" "{0..1} ∩ 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
proof
fix y
assume as: "y ∈ path_component_set S x"
then have "y ∈ S"
by (simp add: path_component_mem(2))
then obtain e where e: "e > 0" "ball y e ⊆ S"
using assms[unfolded open_contains_ball]
by auto
have "⋀u. dist y u < e ⟹ path_component S x u"
by (metis (full_types) as centre_in_ball convex_ball convex_imp_path_connected e mem_Collect_eq mem_ball path_component_eq path_component_of_subset path_connected_component)
then show "∃e > 0. ball y e ⊆ path_component_set S x"
using ‹e>0› by auto
qed
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 ∈ S - path_component_set S x"
then obtain e where e: "e > 0" "ball y e ⊆ S"
using assms openE by auto
show "∃e>0. ball y e ⊆ S - path_component_set S x"
proof (intro exI conjI subsetI DiffI notI)
show "⋀x. x ∈ ball y e ⟹ x ∈ S"
using e by blast
show False if "z ∈ ball y e" "z ∈ path_component_set S x" for z
proof -
have "y ∈ path_component_set S z"
by (meson assms convex_ball convex_imp_path_connected e open_contains_ball_eq open_path_component path_component_maximal that(1))
then have "y ∈ path_component_set S x"
using path_component_eq that(2) by blast
then show False
using y by blast
qed
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 ∈ S" and "y ∈ S"
show "y ∈ path_component_set S x"
proof (rule ccontr)
assume "¬ ?thesis"
moreover have "path_component_set S x ∩ 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 "path_component_set S x" "S - path_component_set S x"]
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 ∘ g" in exI)
unfolding path_defs
apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)])
apply auto
done
qed
lemma path_connected_translationI:
fixes a :: "'a :: topological_group_add"
assumes "path_connected S" shows "path_connected ((λx. 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 ((λx. a + x) ` S) = path_connected S"
proof -
have "∀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 ⟹ path_connected S ⟷ 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
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 (intro ballI)
fix x y
assume x: "x ∈ S ∪ T" and y: "y ∈ S ∪ T"
from assms obtain z where z: "z ∈ S" "z ∈ T"
by auto
show "path_component (S ∪ T) x y"
using x y
proof safe
assume "x ∈ S" "y ∈ S"
then show "path_component (S ∪ T) x y"
by (meson Un_upper1 ‹path_connected S› path_component_of_subset path_connected_component)
next
assume "x ∈ S" "y ∈ T"
then show "path_component (S ∪ T) x y"
by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
next
assume "x ∈ T" "y ∈ S"
then show "path_component (S ∪ T) x y"
by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
next
assume "x ∈ T" "y ∈ T"
then show "path_component (S ∪ T) x y"
by (metis Un_upper1 assms(2) path_component_of_subset path_connected_component sup_commute)
qed
qed
lemma path_connected_UNION:
assumes "⋀i. i ∈ A ⟹ path_connected (S i)"
and "⋀i. i ∈ A ⟹ z ∈ S i"
shows "path_connected (⋃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 (⋃i∈A. S i) x z" and "path_component (⋃i∈A. S i) z y"
using *(1,3) by (auto elim!: path_component_of_subset [rotated])
then show "path_component (⋃i∈A. S i) x y"
by (rule path_component_trans)
qed
lemma path_component_path_image_pathstart:
assumes p: "path p" and x: "x ∈ path_image p"
shows "path_component (path_image p) (pathstart p) x"
proof -
obtain y where x: "x = p y" and y: "0 ≤ y" "y ≤ 1"
using x by (auto simp: path_image_def)
show ?thesis
unfolding path_component_def
proof (intro exI conjI)
have "continuous_on {0..1} (p ∘ (( *) y))"
apply (rule continuous_intros)+
using p [unfolded path_def] y
apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
done
then show "path (λu. p (y * u))"
by (simp add: path_def)
show "path_image (λu. p (y * u)) ⊆ 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 ⟹ 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)"
proof -
{ fix y z
assume pa: "path_component s x y" "path_component s x z"
then have pae: "path_component_set s x = path_component_set s y"
using path_component_eq by auto
have yz: "path_component s y z"
using pa path_component_sym path_component_trans by blast
then have "∃g. path g ∧ path_image g ⊆ path_component_set s x ∧ pathstart g = y ∧ pathfinish g = z"
apply (simp add: path_component_def, clarify)
apply (rule_tac x=g in exI)
by (simp add: pae path_component_maximal path_connected_path_image pathstart_in_path_image)
}
then show ?thesis
by (simp add: path_connected_def)
qed
lemma path_component: "path_component S x y ⟷ (∃t. path_connected t ∧ t ⊆ S ∧ x ∈ t ∧ y ∈ t)"
apply (intro iffI)
apply (metis path_connected_path_image path_defs(5) pathfinish_in_path_image pathstart_in_path_image)
using path_component_of_subset path_connected_component by blast
lemma path_component_path_component [simp]:
"path_component_set (path_component_set S x) x = path_component_set S x"
proof (cases "x ∈ S")
case True show ?thesis
apply (rule subset_antisym)
apply (simp add: path_component_subset)
by (simp add: True path_component_maximal path_component_refl path_connected_path_component)
next
case False then show ?thesis
by (metis False empty_iff path_component_eq_empty)
qed
lemma path_component_subset_connected_component:
"(path_component_set S x) ⊆ (connected_component_set S x)"
proof (cases "x ∈ S")
case True show ?thesis
apply (rule connected_component_maximal)
apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected)
done
next
case False then show ?thesis
using path_component_eq_empty by auto
qed
subsection%unimportant‹Lemmas about path-connectedness›
lemma path_connected_linear_image:
fixes f :: "'a::real_normed_vector ⇒ '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 ⟹ path_connected S"
by (simp add: convex_imp_path_connected is_interval_convex)
lemma linear_homeomorphism_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
obtains g where "homeomorphism (f ` S) S g f"
using linear_injective_left_inverse [OF assms]
apply clarify
apply (rule_tac g=g in that)
using assms
apply (auto simp: homeomorphism_def eq_id_iff [symmetric] image_comp comp_def linear_conv_bounded_linear linear_continuous_on)
done
lemma linear_homeomorphic_image:
fixes f :: "'a::euclidean_space ⇒ '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 × t)"
proof (simp add: path_connected_def Sigma_def, clarify)
fix x1 y1 x2 y2
assume "x1 ∈ s" "y1 ∈ t" "x2 ∈ s" "y2 ∈ t"
obtain g where "path g" and g: "path_image g ⊆ s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2"
using ‹x1 ∈ s› ‹x2 ∈ s› assms by (force simp: path_connected_def)
obtain h where "path h" and h: "path_image h ⊆ t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2"
using ‹y1 ∈ t› ‹y2 ∈ t› assms by (force simp: path_connected_def)
have "path (λz. (x1, h z))"
using ‹path h›
apply (simp add: path_def)
apply (rule continuous_on_compose2 [where f = h])
apply (rule continuous_intros | force)+
done
moreover have "path (λz. (g z, y2))"
using ‹path g›
apply (simp add: path_def)
apply (rule continuous_on_compose2 [where f = g])
apply (rule continuous_intros | force)+
done
ultimately have 1: "path ((λz. (x1, h z)) +++ (λz. (g z, y2)))"
by (metis hf gs path_join_imp pathstart_def pathfinish_def)
have "path_image ((λz. (x1, h z)) +++ (λz. (g z, y2))) ⊆ path_image (λz. (x1, h z)) ∪ path_image (λz. (g z, y2))"
by (rule Path_Connected.path_image_join_subset)
also have "… ⊆ (⋃x∈s. ⋃x1∈t. {(x, x1)})"
using g h ‹x1 ∈ s› ‹y2 ∈ t› by (force simp: path_image_def)
finally have 2: "path_image ((λz. (x1, h z)) +++ (λz. (g z, y2))) ⊆ (⋃x∈s. ⋃x1∈t. {(x, x1)})" .
show "∃g. path g ∧ path_image g ⊆ (⋃x∈s. ⋃x1∈t. {(x, x1)}) ∧
pathstart g = (x1, y1) ∧ pathfinish g = (x2, y2)"
apply (intro exI conjI)
apply (rule 1)
apply (rule 2)
apply (metis hs pathstart_def pathstart_join)
by (metis gf pathfinish_def pathfinish_join)
qed
lemma is_interval_path_connected_1:
fixes s :: "real set"
shows "is_interval s ⟷ path_connected s"
using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast
subsection%unimportant‹Path components›
lemma Union_path_component [simp]:
"Union {path_component_set S x |x. x ∈ S} = S"
apply (rule subset_antisym)
using path_component_subset apply force
using path_component_refl by auto
lemma path_component_disjoint:
"disjnt (path_component_set S a) (path_component_set S b) ⟷
(a ∉ path_component_set S b)"
apply (auto simp: disjnt_def)
using path_component_eq apply fastforce
using path_component_sym path_component_trans by blast
lemma path_component_eq_eq:
"path_component S x = path_component S y ⟷
(x ∉ S) ∧ (y ∉ S) ∨ x ∈ S ∧ y ∈ S ∧ path_component S x y"
apply (rule iffI, metis (no_types) path_component_mem(1) path_component_refl)
apply (erule disjE, metis Collect_empty_eq_bot path_component_eq_empty)
apply (rule ext)
apply (metis path_component_trans path_component_sym)
done
lemma path_component_unique:
assumes "x ∈ c" "c ⊆ S" "path_connected c"
"⋀c'. ⟦x ∈ c'; c' ⊆ S; path_connected c'⟧ ⟹ c' ⊆ c"
shows "path_component_set S x = c"
apply (rule subset_antisym)
using assms
apply (metis mem_Collect_eq subsetCE path_component_eq_eq path_component_subset path_connected_path_component)
by (simp add: assms path_component_maximal)
lemma path_component_intermediate_subset:
"path_component_set u a ⊆ t ∧ t ⊆ u
⟹ 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 =
⋃({path_component_set S y| y. y ∈ S} - {path_component_set S x})"
proof -
have *: "(⋀x. x ∈ S - {a} ⟹ disjnt a x) ⟹ ⋃S - a = ⋃(S - {a})"
for a::"'a set" and S
by (auto simp: disjnt_def)
have "⋀y. y ∈ {path_component_set S x |x. x ∈ S} - {path_component_set S x}
⟹ disjnt (path_component_set S x) y"
using path_component_disjoint path_component_eq by fastforce
then have "⋃{path_component_set S x |x. x ∈ S} - path_component_set S x =
⋃({path_component_set S y |y. y ∈ S} - {path_component_set S x})"
by (meson *)
then show ?thesis by simp
qed
subsection ‹Sphere is path-connected›
lemma path_connected_punctured_universe:
assumes "2 ≤ DIM('a::euclidean_space)"
shows "path_connected (- {a::'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 "… = - {a}"
by auto
finally show ?thesis .
qed
corollary connected_punctured_universe:
"2 ≤ DIM('N::euclidean_space) ⟹ 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 ≤ DIM('a)"
shows "path_connected(sphere a r)"
proof (cases r "0::real" rule: linorder_cases)
case less
then show ?thesis
by (simp add: path_connected_empty)
next
case equal
then show ?thesis
by (simp add: path_connected_singleton)
next
case greater
then have eq: "(sphere (0::'a) r) = (λx. (r / norm x) *⇩R x) ` (- {0::'a})"
by (force simp: image_iff split: if_split_asm)
have "continuous_on (- {0::'a}) (λx. (r / norm x) *⇩R x)"
by (intro continuous_intros) auto
then have "path_connected ((λx. (r / norm x) *⇩R x) ` (- {0::'a}))"
by (intro path_connected_continuous_image path_connected_punctured_universe assms)
with eq have "path_connected (sphere (0::'a) r)"
by auto
then 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
lemma connected_sphere:
fixes a :: "'a :: euclidean_space"
assumes "2 ≤ 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 ≤ 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 ∈ s" by auto
{ fix x y assume "x ∉ s" "y ∉ s"
then have "x ≠ a" "y ≠ a" using ‹a ∈ s› by auto
then have bxy: "bounded(insert x (insert y s))"
by (simp add: ‹bounded s›)
then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
and "s ⊆ ball a B"
using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
define C where "C = B / norm(x - a)"
{ fix u
assume u: "(1 - u) *⇩R x + u *⇩R (a + C *⇩R (x - a)) ∈ s" and "0 ≤ u" "u ≤ 1"
have CC: "1 ≤ 1 + (C - 1) * u"
using ‹x ≠ a› ‹0 ≤ u›
apply (simp add: C_def divide_simps norm_minus_commute)
using Bx by auto
have *: "⋀v. (1 - u) *⇩R x + u *⇩R (a + v *⇩R (x - a)) = a + (1 + (v - 1) * u) *⇩R (x - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + C * u - u)) *⇩R x + ((u / (1 + C * u - u)) *⇩R a + (C * u / (1 + C * u - u)) *⇩R x)) =
(1 + (u / (1 + C * u - u))) *⇩R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *⇩R x"
by (simp add: algebra_simps)
also have "… = (1 + (u / (1 + C * u - u))) *⇩R a + (1 + (u / (1 + C * u - u))) *⇩R x"
using CC by (simp add: field_simps)
also have "… = x + (1 + (u / (1 + C * u - u))) *⇩R a + (u / (1 + C * u - u)) *⇩R x"
by (simp add: algebra_simps)
also have "… = x + ((1 / (1 + C * u - u)) *⇩R a +
((u / (1 + C * u - u)) *⇩R x + (C * u / (1 + C * u - u)) *⇩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)) *⇩R a + (1 / (1 + (C - 1) * u)) *⇩R (a + (1 + (C - 1) * u) *⇩R (x - a)) = x"
by (simp add: algebra_simps)
have False
using ‹convex s›
apply (simp add: convex_alt)
apply (drule_tac x=a in bspec)
apply (rule ‹a ∈ s›)
apply (drule_tac x="a + (1 + (C - 1) * u) *⇩R (x - a)" in bspec)
using u apply (simp add: *)
apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec)
using ‹x ≠ a› ‹x ∉ s› ‹0 ≤ u› CC
apply (auto simp: xeq)
done
}
then have pcx: "path_component (- s) x (a + C *⇩R (x - a))"
by (force simp: closed_segment_def intro!: path_connected_linepath)
define D where "D = B / norm(y - a)"
{ fix u
assume u: "(1 - u) *⇩R y + u *⇩R (a + D *⇩R (y - a)) ∈ s" and "0 ≤ u" "u ≤ 1"
have DD: "1 ≤ 1 + (D - 1) * u"
using ‹y ≠ a› ‹0 ≤ u›
apply (simp add: D_def divide_simps norm_minus_commute)
using By by auto
have *: "⋀v. (1 - u) *⇩R y + u *⇩R (a + v *⇩R (y - a)) = a + (1 + (v - 1) * u) *⇩R (y - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + D * u - u)) *⇩R y + ((u / (1 + D * u - u)) *⇩R a + (D * u / (1 + D * u - u)) *⇩R y)) =
(1 + (u / (1 + D * u - u))) *⇩R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *⇩R y"
by (simp add: algebra_simps)
also have "… = (1 + (u / (1 + D * u - u))) *⇩R a + (1 + (u / (1 + D * u - u))) *⇩R y"
using DD by (simp add: field_simps)
also have "… = y + (1 + (u / (1 + D * u - u))) *⇩R a + (u / (1 + D * u - u)) *⇩R y"
by (simp add: algebra_simps)
also have "… = y + ((1 / (1 + D * u - u)) *⇩R a +
((u / (1 + D * u - u)) *⇩R y + (D * u / (1 + D * u - u)) *⇩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)) *⇩R a + (1 / (1 + (D - 1) * u)) *⇩R (a + (1 + (D - 1) * u) *⇩R (y - a)) = y"
by (simp add: algebra_simps)
have False
using ‹convex s›
apply (simp add: convex_alt)
apply (drule_tac x=a in bspec)
apply (rule ‹a ∈ s›)
apply (drule_tac x="a + (1 + (D - 1) * u) *⇩R (y - a)" in bspec)
using u apply (simp add: *)
apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec)
using ‹y ≠ a› ‹y ∉ s› ‹0 ≤ u› DD
apply (auto simp: xeq)
done
}
then have pdy: "path_component (- s) y (a + D *⇩R (y - a))"
by (force simp: closed_segment_def intro!: path_connected_linepath)
have pyx: "path_component (- s) (a + D *⇩R (y - a)) (a + C *⇩R (x - a))"
apply (rule path_component_of_subset [of "sphere a B"])
using ‹s ⊆ ball a B›
apply (force simp: ball_def dist_norm norm_minus_commute)
apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format])
using ‹x ≠ a› using ‹y ≠ a› B apply (auto simp: dist_norm C_def D_def)
done
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 ≤ 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 ⊆ s" "2 ≤ DIM('a)"
shows "connected (s - ball a r)"
apply (rule connected_diff_open_from_closed [OF ball_subset_cball])
using assms connected_sphere
apply (auto simp: cball_diff_eq_sphere dist_norm)
done
proposition connected_open_delete:
assumes "open S" "connected S" and 2: "2 ≤ DIM('N::euclidean_space)"
shows "connected(S - {a::'N})"
proof (cases "a ∈ S")
case True
with ‹open S› obtain ε where "ε > 0" and ε: "cball a ε ⊆ S"
using open_contains_cball_eq by blast
have "dist a (a + ε *⇩R (SOME i. i ∈ Basis)) = ε"
by (simp add: dist_norm SOME_Basis ‹0 < ε› less_imp_le)
with ε have "⋂{S - ball a r |r. 0 < r ∧ r < ε} ⊆ {} ⟹ False"
apply (drule_tac c="a + scaleR (ε) ((SOME i. i ∈ Basis))" in subsetD)
by auto
then have nonemp: "(⋂{S - ball a r |r. 0 < r ∧ r < ε}) = {} ⟹ False"
by auto
have con: "⋀r. r < ε ⟹ connected (S - ball a r)"
using ε by (force intro: connected_diff_ball [OF ‹connected S› _ 2])
have "x ∈ ⋃{S - ball a r |r. 0 < r ∧ r < ε}" if "x ∈ S - {a}" for x
apply (rule UnionI [of "S - ball a (min ε (dist a x) / 2)"])
using that ‹0 < ε› apply simp_all
apply (rule_tac x="min ε (dist a x) / 2" in exI)
apply auto
done
then have "S - {a} = ⋃{S - ball a r | r. 0 < r ∧ r < ε}"
by auto
then show ?thesis
by (auto intro: connected_Union con dest!: nonemp)
next
case False then show ?thesis
by (simp add: ‹connected S›)
qed
corollary path_connected_open_delete:
assumes "open S" "connected S" and 2: "2 ≤ 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 ≤ DIM('N::euclidean_space) ⟹ path_connected(ball a r - {a::'N})"
by (simp add: path_connected_open_delete)
corollary connected_punctured_ball:
"2 ≤ DIM('N::euclidean_space) ⟹ 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 ≤ DIM('a)" and "finite T"
shows "connected(S - T)"
using ‹finite T› S
proof (induct T)
case empty
show ?case using ‹connected S› by simp
next
case (insert x F)
then have "connected (S-F)" by auto
moreover have "open (S - F)" using finite_imp_closed[OF ‹finite F›] ‹open S› by auto
ultimately have "connected (S - F - {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} ∧ 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 *⇩R b ∨ x = - norm x *⇩R b" if "r = norm x" for x
proof -
have xb: "(x ∙ b) *⇩R b = x"
using euclidean_representation [of x, unfolded b] by force
then have "norm ((x ∙ b) *⇩R b) = norm x"
by simp
with b have "¦x ∙ b¦ = norm x"
using norm_Basis by (simp add: b)
with xb show ?thesis
apply (simp add: abs_if split: if_split_asm)
apply (metis add.inverse_inverse real_vector.scale_minus_left xb)
done
qed
with ‹r > 0› b show "sphere 0 r = {r *⇩R b, - r *⇩R b}"
by (force simp: sphere_def dist_norm)
have "dist (r *⇩R b) (- r *⇩R b) = norm (r *⇩R b + r *⇩R b)"
by (simp add: dist_norm)
also have "… = norm ((2*r) *⇩R b)"
by (metis mult_2 scaleR_add_left)
also have "… = 2*r"
using ‹r > 0› b norm_Basis by fastforce
finally show "dist (r *⇩R b) (- r *⇩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} ∧ dist x y = 2*r"
proof -
have "sphere a r = (+) a ` sphere 0 r"
by (metis add.right_neutral sphere_translation)
then show ?thesis
using sphere_1D_doubleton_zero [OF assms]
by (metis (mono_tags, lifting) dist_add_cancel image_empty image_insert that)
qed
lemma psubset_sphere_Compl_connected:
fixes S :: "'a::euclidean_space set"
assumes S: "S ⊂ sphere a r" and "0 < r" and 2: "2 ≤ DIM('a)"
shows "connected(- S)"
proof -
have "S ⊆ sphere a r"
using S by blast
obtain b where "dist a b = r" and "b ∉ S"
using S mem_sphere by blast
have CS: "- S = {x. dist a x ≤ r ∧ (x ∉ S)} ∪ {x. r ≤ dist a x ∧ (x ∉ S)}"
by auto
have "{x. dist a x ≤ r ∧ x ∉ S} ∩ {x. r ≤ dist a x ∧ x ∉ S} ≠ {}"
using ‹b ∉ S› ‹dist a b = r› by blast
moreover have "connected {x. dist a x ≤ r ∧ x ∉ S}"
apply (rule connected_intermediate_closure [of "ball a r"])
using assms by auto
moreover
have "connected {x. r ≤ dist a x ∧ x ∉ S}"
apply (rule connected_intermediate_closure [of "- cball a r"])
using assms apply (auto intro: connected_complement_bounded_convex)
apply (metis ComplI interior_cball interior_closure mem_ball not_less)
done
ultimately show ?thesis
by (simp add: CS connected_Un)
qed
subsection‹Every annulus is a connected set›
lemma path_connected_2DIM_I:
fixes a :: "'N::euclidean_space"
assumes 2: "2 ≤ DIM('N)" and pc: "path_connected {r. 0 ≤ r ∧ 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 ≤ x ∧ P x} × sphere (0::'N) 1"
have "x ∈ (λz. fst z *⇩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)
apply (rule_tac x=0 in exI, simp)
using vector_choose_size zero_le_one by blast
next
case False
with that show ?thesis
by (rule_tac x="(norm x, x /⇩R norm x)" in image_eqI) auto
qed
then have *: "{x::'N. P(norm x)} = (λz. fst z *⇩R snd z) ` ?D"
by auto
have "continuous_on ?D (λz:: real×'N. fst z *⇩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
apply (subst *)
apply (rule path_connected_continuous_image, auto)
done
qed
ultimately show ?thesis
using path_connected_translation by metis
qed
proposition path_connected_annulus:
fixes a :: "'N::euclidean_space"
assumes "2 ≤ DIM('N)"
shows "path_connected {x. r1 < norm(x - a) ∧ norm(x - a) < r2}"
"path_connected {x. r1 < norm(x - a) ∧ norm(x - a) ≤ r2}"
"path_connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) < r2}"
"path_connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) ≤ 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 ≤ DIM('N::euclidean_space)"
shows "connected {x. r1 < norm(x - a) ∧ norm(x - a) < r2}"
"connected {x. r1 < norm(x - a) ∧ norm(x - a) ≤ r2}"
"connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) < r2}"
"connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) ≤ r2}"
by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected)
subsection%unimportant‹Relations between components and path components›
lemma open_connected_component:
fixes s :: "'a::real_normed_vector set"
shows "open s ⟹ open (connected_component_set s x)"
apply (simp add: open_contains_ball, clarify)
apply (rename_tac y)
apply (drule_tac x=y in bspec)
apply (simp add: connected_component_in, clarify)
apply (rule_tac x=e in exI)
by (metis mem_Collect_eq connected_component_eq connected_component_maximal centre_in_ball connected_ball)
corollary open_components:
fixes s :: "'a::real_normed_vector set"
shows "⟦open u; s ∈ components u⟧ ⟹ 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 ∈ s" and s: "open s"
shows "x ∈ closure (connected_component_set s y) ⟷ x ∈ connected_component_set s y"
proof -
{ assume "x ∈ closure (connected_component_set s y)"
moreover have "x ∈ connected_component_set s x"
using x by simp
ultimately have "x ∈ connected_component_set s y"
using s by (meson Compl_disjoint closure_iff_nhds_not_empty connected_component_disjoint disjoint_eq_subset_Compl open_connected_component)
}
then show ?thesis
by (auto simp: closure_def)
qed
lemma connected_disjoint_Union_open_pick:
assumes "pairwise disjnt B"
"⋀S. S ∈ A ⟹ connected S ∧ S ≠ {}"
"⋀S. S ∈ B ⟹ open S"
"⋃A ⊆ ⋃B"
"S ∈ A"
obtains T where "T ∈ B" "S ⊆ T" "S ∩ ⋃(B - {T}) = {}"
proof -
have "S ⊆ ⋃B" "connected S" "S ≠ {}"
using assms ‹S ∈ A› by blast+
then obtain T where "T ∈ B" "S ∩ T ≠ {}"
by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute)
have 1: "open T" by (simp add: ‹T ∈ B› assms)
have 2: "open (⋃(B-{T}))" using assms by blast
have 3: "S ⊆ T ∪ ⋃(B - {T})" using ‹S ⊆ ⋃B› by blast
have "T ∩ ⋃(B - {T}) = {}" using ‹T ∈ B› ‹pairwise disjnt B›
by (auto simp: pairwise_def disjnt_def)
then have 4: "T ∩ ⋃(B - {T}) ∩ S = {}" by auto
from connectedD [OF ‹connected S› 1 2 3 4]
have "S ∩ ⋃(B-{T}) = {}"
by (auto simp: Int_commute ‹S ∩ T ≠ {}›)
with ‹T ∈ B› have "S ⊆ T"
using "3" by auto
show ?thesis
using ‹S ∩ ⋃(B - {T}) = {}› ‹S ⊆ T› ‹T ∈ B› that by auto
qed
lemma connected_disjoint_Union_open_subset:
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
and SA: "⋀S. S ∈ A ⟹ open S ∧ connected S ∧ S ≠ {}"
and SB: "⋀S. S ∈ B ⟹ open S ∧ connected S ∧ S ≠ {}"
and eq [simp]: "⋃A = ⋃B"
shows "A ⊆ B"
proof
fix S
assume "S ∈ A"
obtain T where "T ∈ B" "S ⊆ T" "S ∩ ⋃(B - {T}) = {}"
apply (rule connected_disjoint_Union_open_pick [OF B, of A])
using SA SB ‹S ∈ A› by auto
moreover obtain S' where "S' ∈ A" "T ⊆ S'" "T ∩ ⋃(A - {S'}) = {}"
apply (rule connected_disjoint_Union_open_pick [OF A, of B])
using SA SB ‹T ∈ B› by auto
ultimately have "S' = S"
by (metis A Int_subset_iff SA ‹S ∈ A› disjnt_def inf.orderE pairwise_def)
with ‹T ⊆ S'› have "T ⊆ S" by simp
with ‹S ⊆ T› have "S = T" by blast
with ‹T ∈ B› show "S ∈ B" by simp
qed
lemma connected_disjoint_Union_open_unique:
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
and SA: "⋀S. S ∈ A ⟹ open S ∧ connected S ∧ S ≠ {}"
and SB: "⋀S. S ∈ B ⟹ open S ∧ connected S ∧ S ≠ {}"
and eq [simp]: "⋃A = ⋃B"
shows "A = B"
by (rule subset_antisym; metis connected_disjoint_Union_open_subset assms)
proposition components_open_unique:
fixes S :: "'a::real_normed_vector set"
assumes "pairwise disjnt A" "⋃A = S"
"⋀X. X ∈ A ⟹ open X ∧ connected X ∧ X ≠ {}"
shows "components S = A"
proof -
have "open S" using assms by blast
show ?thesis
apply (rule connected_disjoint_Union_open_unique)
apply (simp add: components_eq disjnt_def pairwise_def)
using ‹open S›
apply (simp_all add: assms open_components in_components_connected in_components_nonempty)
done
qed
subsection%unimportant‹Existence of unbounded components›
lemma cobounded_unbounded_component:
fixes s :: "'a :: euclidean_space set"
assumes "bounded (-s)"
shows "∃x. x ∈ s ∧ ~ bounded (connected_component_set s x)"
proof -
obtain i::'a where i: "i ∈ Basis"
using nonempty_Basis by blast
obtain B where B: "B>0" "-s ⊆ ball 0 B"
using bounded_subset_ballD [OF assms, of 0] by auto
then have *: "⋀x. B ≤ norm x ⟹ x ∈ s"
by (force simp: ball_def dist_norm)
have unbounded_inner: "~ bounded {x. inner i x ≥ B}"
apply (auto simp: bounded_def dist_norm)
apply (rule_tac x="x + (max B e + 1 + ¦i ∙ x¦) *⇩R i" in exI)
apply simp
using i
apply (auto simp: algebra_simps)
done
have **: "{x. B ≤ i ∙ x} ⊆ connected_component_set s (B *⇩R i)"
apply (rule connected_component_maximal)
apply (auto simp: i intro: convex_connected convex_halfspace_ge [of B])
apply (rule *)
apply (rule order_trans [OF _ Basis_le_norm [OF i]])
by (simp add: inner_commute)
have "B *⇩R i ∈ s"
by (rule *) (simp add: norm_Basis [OF i])
then show ?thesis
apply (rule_tac x="B *⇩R i" in exI, clarify)
apply (frule bounded_subset [of _ "{x. B ≤ i ∙ x}", OF _ **])
using unbounded_inner apply blast
done
qed
lemma cobounded_unique_unbounded_component:
fixes s :: "'a :: euclidean_space set"
assumes bs: "bounded (-s)" and "2 ≤ DIM('a)"
and bo: "~ bounded(connected_component_set s x)"
"~ bounded(connected_component_set s y)"
shows "connected_component_set s x = connected_component_set s y"
proof -
obtain i::'a where i: "i ∈ Basis"
using nonempty_Basis by blast
obtain B where B: "B>0" "-s ⊆ ball 0 B"
using bounded_subset_ballD [OF bs, of 0] by auto
then have *: "⋀x. B ≤ norm x ⟹ x ∈ s"
by (force simp: ball_def dist_norm)
have ccb: "connected (- ball 0 B :: 'a set)"
using assms by (auto intro: connected_complement_bounded_convex)
obtain x' where x': "connected_component s x x'" "norm x' > B"
using bo [unfolded bounded_def dist_norm, simplified, rule_format]
by (metis diff_zero norm_minus_commute not_less)
obtain y' where y': "connected_component s y y'" "norm y' > B"
using bo [unfolded bounded_def dist_norm, simplified, rule_format]
by (metis diff_zero norm_minus_commute not_less)
have x'y': "connected_component s x' y'"
apply (simp add: connected_component_def)
apply (rule_tac x="- ball 0 B" in exI)
using x' y'
apply (auto simp: ccb dist_norm *)
done
show ?thesis
apply (rule connected_component_eq)
using x' y' x'y'
by (metis (no_types, lifting) connected_component_eq_empty connected_component_eq_eq connected_component_idemp connected_component_in)
qed
lemma cobounded_unbounded_components:
fixes s :: "'a :: euclidean_space set"
shows "bounded (-s) ⟹ ∃c. c ∈ components s ∧ ~bounded c"
by (metis cobounded_unbounded_component components_def imageI)
lemma cobounded_unique_unbounded_components:
fixes s :: "'a :: euclidean_space set"
shows "⟦bounded (- s); c ∈ components s; ¬ bounded c; c' ∈ components s; ¬ bounded c'; 2 ≤ DIM('a)⟧ ⟹ 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)" "¬ connected S" "2 ≤ DIM('a)"
obtains C where "C ∈ components S" "bounded C"
by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
section‹The "inside" and "outside" of a set›
text%important‹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.›
definition%important inside where
"inside S ≡ {x. (x ∉ S) ∧ bounded(connected_component_set ( - S) x)}"
definition%important outside where
"outside S ≡ -S ∩ {x. ~ bounded(connected_component_set (- S) x)}"
lemma outside: "outside S = {x. ~ 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 ∩ S = {}"
by (auto simp: inside_def)
lemma outside_no_overlap [simp]:
"outside S ∩ S = {}"
by (auto simp: outside_def)
lemma inside_Int_outside [simp]: "inside S ∩ outside S = {}"
by (auto simp: inside_def outside_def)
lemma inside_Un_outside [simp]: "inside S ∪ outside S = (- S)"
by (auto simp: inside_def outside_def)
lemma inside_eq_outside:
"inside S = outside S ⟷ S = UNIV"
by (auto simp: inside_def outside_def)
lemma inside_outside: "inside S = (- (S ∪ outside S))"
by (force simp: inside_def outside)
lemma outside_inside: "outside S = (- (S ∪ inside S))"
by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap)
lemma union_with_inside: "S ∪ inside S = - outside S"
by (auto simp: inside_outside) (simp add: outside_inside)
lemma union_with_outside: "S ∪ outside S = - inside S"
by (simp add: inside_outside)
lemma outside_mono: "S ⊆ T ⟹ outside T ⊆ outside S"
by (auto simp: outside bounded_subset connected_component_mono)
lemma inside_mono: "S ⊆ T ⟹ inside S - T ⊆ inside T"
by (auto simp: inside_def bounded_subset connected_component_mono)
lemma segment_bound_lemma:
fixes u::real
assumes "x ≥ B" "y ≥ B" "0 ≤ u" "u ≤ 1"
shows "(1 - u) * x + u * y ≥ B"
proof -
obtain dx dy where "dx ≥ 0" "dy ≥ 0" "x = B + dx" "y = B + dy"
using assms by auto (metis add.commute diff_add_cancel)
with ‹0 ≤ u› ‹u ≤ 1› show ?thesis
by (simp add: add_increasing2 mult_left_le field_simps)
qed
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 ⊆ ball 0 B"
using bounded_subset_ballD [OF assms, of 0] by auto
{ fix x::'a and C::real
assume Bno: "B ≤ norm x" and C: "0 < C"
have "∃y. connected_component (- S) x y ∧ norm y > C"
proof (cases "x = 0")
case True with B Bno show ?thesis by force
next
case False
with B C
have "closed_segment x (((B + C) / norm x) *⇩R x) ⊆ - ball 0 B"
apply (clarsimp simp add: closed_segment_def ball_def dist_norm real_vector_class.scaleR_add_left [symmetric] divide_simps)
using segment_bound_lemma [of B "norm x" "B+C" ] Bno
by (meson le_add_same_cancel1 less_eq_real_def not_le)
also have "... ⊆ -S"
by (simp add: B)
finally have "∃T. connected T ∧ T ⊆ - S ∧ x ∈ T ∧ ((B + C) / norm x) *⇩R x ∈ T"
by (rule_tac x="closed_segment x (((B+C)/norm x) *⇩R x)" in exI) simp
with False B
show ?thesis
by (rule_tac x="((B+C)/norm x) *⇩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 ⟹ ~ 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 ⟹ 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 ≤ 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)
apply clarify
apply (metis connected_component_eq_eq connected_component_in)
done
lemma outside_connected_component_lt:
"outside S = {x. ∀B. ∃y. B < norm(y) ∧ connected_component (- S) x y}"
apply (auto simp: outside bounded_def dist_norm)
apply (metis diff_0 norm_minus_cancel not_less)
by (metis less_diff_eq norm_minus_commute norm_triangle_ineq2 order.trans pinf(6))
lemma outside_connected_component_le:
"outside S =
{x. ∀B. ∃y. B ≤ norm(y) ∧
connected_component (- S) x y}"
apply (simp add: outside_connected_component_lt)
apply (simp add: 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 ≤ DIM('a)"
shows "- (outside S) = {x. ∀B. ∃y. B < norm(y) ∧ ~ (connected_component (- S) x y)}"
proof -
obtain B::real where B: "0 < B" and Bno: "⋀x. x ∈ S ⟹ norm x ≤ B"
using S [simplified bounded_pos] by auto
{ fix y::'a and z::'a
assume yz: "B < norm z" "B < norm y"
have "connected_component (- cball 0 B) y z"
apply (rule connected_componentI [OF _ subset_refl])
apply (rule connected_complement_bounded_convex)
using assms yz
by (auto simp: dist_norm)
then have "connected_component (- S) y z"
apply (rule connected_component_of_subset)
apply (metis Bno Compl_anti_mono mem_cball_0 subset_iff)
done
} note cyz = this
show ?thesis
apply (auto simp: outside)
apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le)
apply (simp add: bounded_pos)
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 ≤ DIM('a)"
shows "- (outside S) = {x. ∀B. ∃y. B ≤ norm(y) ∧ ~ (connected_component (- S) x y)}"
apply (auto intro: less_imp_le simp: not_outside_connected_component_lt [OF assms])
by (meson gt_ex less_le_trans)
lemma inside_connected_component_lt:
fixes S :: "'a::euclidean_space set"
assumes S: "bounded S" "2 ≤ DIM('a)"
shows "inside S = {x. (x ∉ S) ∧ (∀B. ∃y. B < norm(y) ∧ ~(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 ≤ DIM('a)"
shows "inside S = {x. (x ∉ S) ∧ (∀B. ∃y. B ≤ norm(y) ∧ ~(connected_component (- S) x y))}"
by (auto simp: inside_outside not_outside_connected_component_le [OF assms])
lemma inside_subset:
assumes "connected U" and "~bounded U" and "T ∪ U = - S"
shows "inside S ⊆ T"
apply (auto simp: inside_def)
by (metis bounded_subset [of "connected_component_set (- S) _"] connected_component_maximal
Compl_iff Un_iff assms subsetI)
lemma frontier_not_empty:
fixes S :: "'a :: real_normed_vector set"
shows "⟦S ≠ {}; S ≠ UNIV⟧ ⟹ frontier S ≠ {}"
using connected_Int_frontier [of UNIV S] by auto
lemma frontier_eq_empty:
fixes S :: "'a :: real_normed_vector set"
shows "frontier S = {} ⟷ S = {} ∨ 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) ⊆ frontier S"
proof -
{ fix y
assume y1: "y ∈ closure (connected_component_set S x)"
and y2: "y ∉ interior (connected_component_set S x)"
have "y ∈ closure S"
using y1 closure_mono connected_component_subset by blast
moreover have "z ∈ interior (connected_component_set S x)"
if "0 < e" "ball y e ⊆ interior S" "dist y z < e" for e z
proof -
have "ball y e ⊆ connected_component_set S y"
apply (rule connected_component_maximal)
using that interior_subset mem_ball apply auto
done
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 ‹0 < e› y2)
qed
then have "y ∉ interior S"
using y2 by (force simp: open_contains_ball_eq [OF open_interior])
ultimately have "y ∈ 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(⋃F) ⊆ closure(⋃t ∈ F. frontier t)"
proof -
have "∃y∈F. ∃y∈frontier y. dist y x < e"
if "T ∈ F" "y ∈ T" "dist y x < e"
"x ∉ interior (⋃F)" "0 < e" for x y e T
proof (cases "x ∈ 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 1: "closed_segment x y ∩ T ≠ {}" using ‹y ∈ T› by blast
have 2: "closed_segment x y - T ≠ {}"
using False by blast
obtain c where "c ∈ closed_segment x y" "c ∈ frontier T"
using False connected_Int_frontier [OF connected_segment 1 2] by auto
then show ?thesis
proof -
have "norm (y - x) < e"
by (metis dist_norm ‹dist y x < e›)
moreover have "norm (c - x) ≤ norm (y - x)"
by (simp add: ‹c ∈ closed_segment x y› segment_bound(1))
ultimately have "norm (c - x) < e"
by linarith
then show ?thesis
by (metis (no_types) ‹c ∈ frontier T› dist_norm that(1))
qed
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 ⟹ frontier(⋃F) ⊆ (⋃t ∈ F. frontier t)"
by (rule order_trans [OF frontier_Union_subset_closure])
(auto simp: closure_subset_eq)
lemma frontier_of_components_subset:
fixes S :: "'a::real_normed_vector set"
shows "C ∈ components S ⟹ frontier C ⊆ 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 "⟦closed S; C ∈ components (- S)⟧ ⟹ frontier C ⊆ 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: "~ connected(- S)"
and C: "C ∈ components (- S)"
and conn: "⋀T. ⟦closed T; T ⊂ S⟧ ⟹ connected(- T)"
shows "frontier C = S"
proof (rule ccontr)
assume "frontier C ≠ S"
then have "frontier C ⊂ S"
using frontier_of_components_closed_complement [OF ‹closed S› C] by blast
then have "connected(- (frontier C))"
by (simp add: conn)
have "¬ connected(- (frontier C))"
unfolding connected_def not_not
proof (intro exI conjI)
show "open C"
using C ‹closed S› open_components by blast
show "open (- closure C)"
by blast
show "C ∩ - closure C ∩ - frontier C = {}"
using closure_subset by blast
show "C ∩ - frontier C ≠ {}"
using C ‹open C› components_eq frontier_disjoint_eq by fastforce
show "- frontier C ⊆ C ∪ - closure C"
by (simp add: ‹open C› closed_Compl frontier_closures)
then show "- closure C ∩ - frontier C ≠ {}"
by (metis (no_types, lifting) C Compl_subset_Compl_iff ‹frontier C ⊂ S› compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb)
qed
then show False
using ‹connected (- frontier C)› 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 ⟷ 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 ⊆ inside (frontier s)"
proof -
{ fix x y
assume x: "x ∈ interior s" and y: "y ∉ s"
and cc: "connected_component (- frontier s) x y"
have "connected_component_set (- frontier s) x ∩ frontier s ≠ {}"
apply (rule connected_Int_frontier, simp)
apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq set_rev_mp x)
using y cc
by blast
then have "bounded (connected_component_set (- frontier s) x)"
using connected_component_in by auto
}
then show ?thesis
apply (auto simp: inside_def frontier_def)
apply (rule classical)
apply (rule bounded_subset [OF assms], blast)
done
qed
lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)"
by (simp add: inside_def connected_component_UNIV)
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:
"⟦connected_component (- s) x y; x ∈ inside s⟧ ⟹ y ∈ inside s"
using connected_component_eq connected_component_in
by (fastforce simp add: inside_def)
lemma outside_same_component:
"⟦connected_component (- s) x y; x ∈ outside s⟧ ⟹ y ∈ 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 ∉ s"
shows "z ∈ outside s"
proof (cases "s={}")
case True then show ?thesis by simp
next
case False then obtain a where "a ∈ s" by blast
with z have zna: "z ≠ a" by auto
{ assume "bounded (connected_component_set (- s) z)"
with bounded_pos_less obtain B where "B>0" and B: "⋀x. connected_component (- s) z x ⟹ norm x < B"
by (metis mem_Collect_eq)
define C where "C = (B + 1 + norm z) / norm (z-a)"
have "C > 0"
using ‹0 < B› zna by (simp add: C_def divide_simps add_strict_increasing)
have "¦norm (z + C *⇩R (z-a)) - norm (C *⇩R (z-a))¦ ≤ norm z"
by (metis add_diff_cancel norm_triangle_ineq3)
moreover have "norm (C *⇩R (z-a)) > norm z + B"
using zna ‹B>0› by (simp add: C_def le_max_iff_disj field_simps)
ultimately have C: "norm (z + C *⇩R (z-a)) > B" by linarith
{ fix u::real
assume u: "0≤u" "u≤1" and ins: "(1 - u) *⇩R z + u *⇩R (z + C *⇩R (z - a)) ∈ s"
then have Cpos: "1 + u * C > 0"
by (meson ‹0 < C› add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
then have *: "(1 / (1 + u * C)) *⇩R z + (u * C / (1 + u * C)) *⇩R z = z"
by (simp add: scaleR_add_left [symmetric] divide_simps)
then have False
using convexD_alt [OF s ‹a ∈ s› ins, of "1/(u*C + 1)"] ‹C>0› ‹z ∉ s› Cpos u
by (simp add: * divide_simps algebra_simps)
} note contra = this
have "connected_component (- s) z (z + C *⇩R (z-a))"
apply (rule connected_componentI [OF connected_segment [of z "z + C *⇩R (z-a)"]])
apply (simp add: closed_segment_def)
using contra
apply auto
done
then have False
using zna B [of "z + C *⇩R (z-a)"] C
by (auto simp: divide_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 ⟹ 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 "⟦convex t; s ⊆ t⟧ ⟹ - t ⊆ outside s"
using outside_convex outside_mono by blast
lemma outside_Un_outside_Un:
fixes S :: "'a::real_normed_vector set"
assumes "S ∩ outside(T ∪ U) = {}"
shows "outside(T ∪ U) ⊆ outside(T ∪ S)"
proof
fix x
assume x: "x ∈ outside (T ∪ U)"
have "Y ⊆ - S" if "connected Y" "Y ⊆ - T" "Y ⊆ - U" "x ∈ Y" "u ∈ Y" for u Y
proof -
have "Y ⊆ connected_component_set (- (T ∪ U)) x"
by (simp add: connected_component_maximal that)
also have "… ⊆ outside(T ∪ U)"
by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x)
finally have "Y ⊆ outside(T ∪ U)" .
with assms show ?thesis by auto
qed
with x show "x ∈ outside (T ∪ 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) ⊆ - closure s"
unfolding outside_inside Lattices.boolean_algebra_class.compl_le_compl_iff
proof -
{ assume "interior s ⊆ inside (frontier s)"
hence "interior s ∪ inside (frontier s) = inside (frontier s)"
by (simp add: subset_Un_eq)
then have "closure s ⊆ frontier s ∪ inside (frontier s)"
using frontier_def by auto
}
then show "closure s ⊆ frontier s ∪ inside (frontier s)"
using interior_inside_frontier [OF assms] by blast
qed
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 "⟦bounded s; convex s⟧ ⟹ inside(frontier s) = interior s"
apply (simp add: inside_outside outside_frontier_eq_complement_closure)
using closure_subset interior_subset
apply (auto simp: frontier_def)
done
lemma open_inside:
fixes s :: "'a::real_normed_vector set"
assumes "closed s"
shows "open (inside s)"
proof -
{ fix x assume x: "x ∈ 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: "⋀y. dist y x < e ⟶ 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 "∃e>0. ball x e ⊆ 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 ∈ 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: "⋀y. dist y x < e ⟶ connected_component (- s) x y"
using dist_not_less_zero
apply (simp add: open_dist)
by (metis Int_iff outside_def connected_component_refl_eq x)
then have "∃e>0. ball x e ⊆ 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) ⊆ s ∪ 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) ⊆ s"
proof -
have "closure (inside s) ∩ - inside s = closure (inside s) - interior (inside s)"
by (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside)
moreover have "- inside s ∩ - outside s = s"
by (metis (no_types) compl_sup double_compl inside_Un_outside)
moreover have "closure (inside s) ⊆ - outside s"
by (metis (no_types) assms closure_inside_subset union_with_inside)
ultimately have "closure (inside s) - interior (inside s) ⊆ s"
by blast
then show ?thesis
by (simp add: frontier_def open_inside interior_open)
qed
lemma closure_outside_subset:
fixes s :: "'a::real_normed_vector set"
assumes "closed s"
shows "closure(outside s) ⊆ s ∪ outside s"
apply (rule closure_minimal, simp)
by (metis assms closed_open inside_outside open_inside)
lemma frontier_outside_subset:
fixes s :: "'a::real_normed_vector set"
assumes "closed s"
shows "frontier(outside s) ⊆ s"
apply (simp add: frontier_def open_outside interior_open)
by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup.commute)
lemma inside_complement_unbounded_connected_empty:
"⟦connected (- s); ¬ bounded (- s)⟧ ⟹ inside s = {}"
apply (simp add: inside_def)
by (meson Compl_iff bounded_subset connected_component_maximal order_refl)
lemma inside_bounded_complement_connected_empty:
fixes s :: "'a::{real_normed_vector, perfect_space} set"
shows "⟦connected (- s); bounded s⟧ ⟹ inside s = {}"
by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded)
lemma inside_inside:
assumes "s ⊆ inside t"
shows "inside s - t ⊆ inside t"
unfolding inside_def
proof clarify
fix x
assume x: "x ∉ t" "x ∉ s" and bo: "bounded (connected_component_set (- s) x)"
show "bounded (connected_component_set (- t) x)"
proof (cases "s ∩ connected_component_set (- t) x = {}")
case True show ?thesis
apply (rule bounded_subset [OF bo])
apply (rule connected_component_maximal)
using x True apply auto
done
next
case False then show ?thesis
using assms [unfolded inside_def] x
apply (simp add: disjoint_iff_not_equal, clarify)
apply (drule subsetD, assumption, auto)
by (metis (no_types, hide_lams) ComplI connected_component_eq_eq)
qed
qed
lemma inside_inside_subset: "inside(inside s) ⊆ s"
using inside_inside union_with_outside by fastforce
lemma inside_outside_intersect_connected:
"⟦connected t; inside s ∩ t ≠ {}; outside s ∩ t ≠ {}⟧ ⟹ s ∩ t ≠ {}"
apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify)
by (metis (no_types, hide_lams) Compl_anti_mono connected_component_eq connected_component_maximal contra_subsetD double_compl)
lemma outside_bounded_nonempty:
fixes s :: "'a :: {real_normed_vector, perfect_space} set"
assumes "bounded s" shows "outside s ≠ {}"
by (metis (no_types, lifting) Collect_empty_eq Collect_mem_eq Compl_eq_Diff_UNIV Diff_cancel
Diff_disjoint UNIV_I assms ball_eq_empty bounded_diff cobounded_outside convex_ball
double_complement order_refl outside_convex outside_def)
lemma outside_compact_in_open:
fixes s :: "'a :: {real_normed_vector,perfect_space} set"
assumes s: "compact s" and t: "open t" and "s ⊆ t" "t ≠ {}"
shows "outside s ∩ t ≠ {}"
proof -
have "outside s ≠ {}"
by (simp add: compact_imp_bounded outside_bounded_nonempty s)
with assms obtain a b where a: "a ∈ outside s" and b: "b ∈ t" by auto
show ?thesis
proof (cases "a ∈ t")
case True with a show ?thesis by blast
next
case False
have front: "frontier t ⊆ - s"
using ‹s ⊆ t› frontier_disjoint_eq t by auto
{ fix γ
assume "path γ" and pimg_sbs: "path_image γ - {pathfinish γ} ⊆ interior (- t)"
and pf: "pathfinish γ ∈ frontier t" and ps: "pathstart γ = a"
define c where "c = pathfinish γ"
have "c ∈ -s" unfolding c_def using front pf by blast
moreover have "open (-s)" using s compact_imp_closed by blast
ultimately obtain ε::real where "ε > 0" and ε: "cball c ε ⊆ -s"
using open_contains_cball[of "-s"] s by blast
then obtain d where "d ∈ t" and d: "dist d c < ε"
using closure_approachable [of c t] pf unfolding c_def
by (metis Diff_iff frontier_def)
then have "d ∈ -s" using ε
using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq)
have pimg_sbs_cos: "path_image γ ⊆ -s"
using pimg_sbs apply (auto simp: path_image_def)
apply (drule subsetD)
using ‹c ∈ - s› ‹s ⊆ t› interior_subset apply (auto simp: c_def)
done
have "closed_segment c d ≤ cball c ε"
apply (simp add: segment_convex_hull)
apply (rule hull_minimal)
using ‹ε > 0› d apply (auto simp: dist_commute)
done
with ε have "closed_segment c d ⊆ -s" by blast
moreover have con_gcd: "connected (path_image γ ∪ closed_segment c d)"
by (rule connected_Un) (auto simp: c_def ‹path γ› 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 ∩ t ≠ {}"
using outside_same_component [OF _ a] by (metis IntI ‹d ∈ t› empty_iff)
} note * = this
have pal: "pathstart (linepath a b) ∈ 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 ⊆ inside t"
shows "inside s ⊆ inside t"
proof (cases "inside t = {}")
case True with assms show ?thesis by auto
next
case False
consider "DIM('a) = 1" | "DIM('a) ≥ 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 coms: "compact s"
using assms apply (simp add: s compact_eq_bounded_closed)
by (meson bounded_inside bounded_subset compact_imp_bounded)
then have bst: "bounded (s ∪ t)"
by (simp add: compact_imp_bounded t)
then obtain r where "0 < r" and r: "s ∪ t ⊆ ball 0 r"
using bounded_subset_ballD by blast
have outst: "outside s ∩ outside t ≠ {}"
proof -
have "- ball 0 r ⊆ outside s"
apply (rule outside_subset_convex)
using r by auto
moreover have "- ball 0 r ⊆ outside t"
apply (rule outside_subset_convex)
using r by auto
ultimately show ?thesis
by (metis Compl_subset_Compl_iff Int_subset_iff bounded_ball inf.orderE outside_bounded_nonempty outside_no_overlap)
qed
have "s ∩ t = {}" using assms
by (metis disjoint_iff_not_equal inside_no_overlap subsetCE)
moreover have "outside s ∩ inside t ≠ {}"
by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open t)
ultimately have "inside s ∩ t = {}"
using inside_outside_intersect_connected [OF ‹connected t›, 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 ‹s ⊆ inside t›] 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 ∪ inside s)"
proof (cases "s ∪ inside s = UNIV")
case True with assms show ?thesis by auto
next
case False
then obtain b where b: "b ∉ s" "b ∉ inside s" by blast
have *: "∃y t. y ∈ s ∧ connected t ∧ a ∈ t ∧ y ∈ t ∧ t ⊆ (s ∪ inside s)" if "a ∈ (s ∪ inside s)" for a
using that proof
assume "a ∈ s" then show ?thesis
apply (rule_tac x=a in exI)
apply (rule_tac x="{a}" in exI, simp)
done
next
assume a: "a ∈ inside s"
show ?thesis
apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "inside s"])
using a apply (simp add: closure_def)
apply (simp add: b)
apply (rule_tac x="pathfinish h" in exI)
apply (rule_tac x="path_image h" in exI)
apply (simp add: pathfinish_in_path_image connected_path_image, auto)
using frontier_inside_subset s apply fastforce
by (metis (no_types, lifting) frontier_inside_subset insertE insert_Diff interior_eq open_inside pathfinish_in_path_image s subsetCE)
qed
show ?thesis
apply (simp add: connected_iff_connected_component)
apply (simp add: connected_component_def)
apply (clarify dest!: *)
apply (rename_tac u u' t t')
apply (rule_tac x="(s ∪ t ∪ t')" in exI)
apply (auto simp: intro!: connected_Un cons)
done
qed
text‹The proof is virtually the same as that above.›
lemma connected_with_outside:
fixes s :: "'a :: real_normed_vector set"
assumes s: "closed s" and cons: "connected s"
shows "connected(s ∪ outside s)"
proof (cases "s ∪ outside s = UNIV")
case True with assms show ?thesis by auto
next
case False
then obtain b where b: "b ∉ s" "b ∉ outside s" by blast
have *: "∃y t. y ∈ s ∧ connected t ∧ a ∈ t ∧ y ∈ t ∧ t ⊆ (s ∪ outside s)" if "a ∈ (s ∪ outside s)" for a
using that proof
assume "a ∈ s" then show ?thesis
apply (rule_tac x=a in exI)
apply (rule_tac x="{a}" in exI, simp)
done
next
assume a: "a ∈ outside s"
show ?thesis
apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "outside s"])
using a apply (simp add: closure_def)
apply (simp add: b)
apply (rule_tac x="pathfinish h" in exI)
apply (rule_tac x="path_image h" in exI)
apply (simp add: pathfinish_in_path_image connected_path_image, auto)
using frontier_outside_subset s apply fastforce
by (metis (no_types, lifting) frontier_outside_subset insertE insert_Diff interior_eq open_outside pathfinish_in_path_image s subsetCE)
qed
show ?thesis
apply (simp add: connected_iff_connected_component)
apply (simp add: connected_component_def)
apply (clarify dest!: *)
apply (rename_tac u u' t t')
apply (rule_tac x="(s ∪ t ∪ t')" in exI)
apply (auto simp: 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) = {}"
by (metis (no_types) unbounded_outside connected_with_outside [OF assms] bounded_Un
inside_complement_unbounded_connected_empty unbounded_outside union_with_outside)
lemma inside_in_components:
"inside s ∈ components (- s) ⟷ connected(inside s) ∧ inside s ≠ {}"
apply (simp add: in_components_maximal)
apply (auto intro: inside_same_component connected_componentI)
apply (metis IntI empty_iff inside_no_overlap)
done
text‹The proof is virtually the same as that above.›
lemma outside_in_components:
"outside s ∈ components (- s) ⟷ connected(outside s) ∧ outside s ≠ {}"
apply (simp add: in_components_maximal)
apply (auto intro: outside_same_component connected_componentI)
apply (metis IntI empty_iff outside_no_overlap)
done
lemma bounded_unique_outside:
fixes s :: "'a :: euclidean_space set"
shows "⟦bounded s; DIM('a) ≥ 2⟧ ⟹ (c ∈ components (- s) ∧ ~bounded c ⟷ c = outside s)"
apply (rule iffI)
apply (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside)
by (simp add: connected_outside outside_bounded_nonempty outside_in_components unbounded_outside)
subsection‹Condition for an open map's image to contain a ball›
proposition ball_subset_open_map_image:
fixes f :: "'a::heine_borel ⇒ 'b :: {real_normed_vector,heine_borel}"
assumes contf: "continuous_on (closure S) f"
and oint: "open (f ` interior S)"
and le_no: "⋀z. z ∈ frontier S ⟹ r ≤ norm(f z - f a)"
and "bounded S" "a ∈ S" "0 < r"
shows "ball (f a) r ⊆ f ` S"
proof (cases "f ` S = UNIV")
case True then show ?thesis by simp
next
case False
obtain w where w: "w ∈ frontier (f ` S)"
and dw_le: "⋀y. y ∈ frontier (f ` S) ⟹ norm (f a - w) ≤ norm (f a - y)"
apply (rule distance_attains_inf [of "frontier(f ` S)" "f a"])
using ‹a ∈ S› by (auto simp: frontier_eq_empty dist_norm False)
then obtain ξ where ξ: "⋀n. ξ n ∈ f ` S" and tendsw: "ξ ⇢ w"
by (metis Diff_iff frontier_def closure_sequential)
then have "⋀n. ∃x ∈ S. ξ n = f x" by force
then obtain z where zs: "⋀n. z n ∈ S" and fz: "⋀n. ξ n = f (z n)"
by metis
then obtain y K where y: "y ∈ closure S" and "strict_mono (K :: nat ⇒ nat)"
and Klim: "(z ∘ K) ⇢ y"
using ‹bounded S›
apply (simp add: compact_closure [symmetric] compact_def)
apply (drule_tac x=z in spec)
using closure_subset apply force
done
then have ftendsw: "((λn. f (z n)) ∘ K) ⇢ w"
by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
have zKs: "⋀n. (z ∘ K) n ∈ S" by (simp add: zs)
have fz: "f ∘ z = ξ" "(λn. f (z n)) = ξ"
using fz by auto
then have "(ξ ∘ K) ⇢ 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 rle: "r ≤ norm (f y - f a)"
apply (rule le_no)
using w wy oint
by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
have **: "(~(b ∩ (- S) = {}) ∧ ~(b - (- S) = {}) ⟹ (b ∩ f ≠ {}))
⟹ (b ∩ S ≠ {}) ⟹ b ∩ f = {} ⟹
b ⊆ S" for b f and S :: "'b set"
by blast
show ?thesis
apply (rule **)
apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball])
using ‹a ∈ S› ‹0 < r›
apply (auto simp: disjoint_iff_not_equal dist_norm)
by (metis dw_le norm_minus_commute not_less order_trans rle wy)
qed
section‹ Homotopy of maps p,q : X=>Y with property P of all intermediate maps›
text%important‹ We often just want to require that it fixes some subset, but to take in
the case of a loop homotopy, it's convenient to have a general property P.›
definition%important homotopic_with ::
"[('a::topological_space ⇒ 'b::topological_space) ⇒ bool, 'a set, 'b set, 'a ⇒ 'b, 'a ⇒ 'b] ⇒ bool"
where
"homotopic_with P X Y p q ≡
(∃h:: real × 'a ⇒ 'b.
continuous_on ({0..1} × X) h ∧
h ` ({0..1} × X) ⊆ Y ∧
(∀x. h(0, x) = p x) ∧
(∀x. h(1, x) = q x) ∧
(∀t ∈ {0..1}. P(λx. h(t, x))))"
text‹ We often want to just localize the ending function equality or whatever.›
proposition homotopic_with:
fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
assumes "⋀h k. (⋀x. x ∈ X ⟹ h x = k x) ⟹ (P h ⟷ P k)"
shows "homotopic_with P X Y p q ⟷
(∃h :: real × 'a ⇒ 'b.
continuous_on ({0..1} × X) h ∧
h ` ({0..1} × X) ⊆ Y ∧
(∀x ∈ X. h(0,x) = p x) ∧
(∀x ∈ X. h(1,x) = q x) ∧
(∀t ∈ {0..1}. P(λx. h(t, x))))"
unfolding homotopic_with_def
apply (rule iffI, blast, clarify)
apply (rule_tac x="λ(u,v). if v ∈ X then h(u,v) else if u = 0 then p v else q v" in exI)
apply auto
apply (force elim: continuous_on_eq)
apply (drule_tac x=t in bspec, force)
apply (subst assms; simp)
done
proposition homotopic_with_eq:
assumes h: "homotopic_with P X Y f g"
and f': "⋀x. x ∈ X ⟹ f' x = f x"
and g': "⋀x. x ∈ X ⟹ g' x = g x"
and P: "(⋀h k. (⋀x. x ∈ X ⟹ h x = k x) ⟹ (P h ⟷ P k))"
shows "homotopic_with P X Y f' g'"
using h unfolding homotopic_with_def
apply safe
apply (rule_tac x="λ(u,v). if v ∈ X then h(u,v) else if u = 0 then f' v else g' v" in exI)
apply (simp add: f' g', safe)
apply (fastforce intro: continuous_on_eq, fastforce)
apply (subst P; fastforce)
done
proposition homotopic_with_equal:
assumes contf: "continuous_on X f" and fXY: "f ` X ⊆ Y"
and gf: "⋀x. x ∈ X ⟹ g x = f x"
and P: "P f" "P g"
shows "homotopic_with P X Y f g"
unfolding homotopic_with_def
apply (rule_tac x="λ(u,v). if u = 1 then g v else f v" in exI)
using assms
apply (intro conjI)
apply (rule continuous_on_eq [where f = "f ∘ snd"])
apply (rule continuous_intros | force)+
apply clarify
apply (case_tac "t=1"; force)
done
lemma image_Pair_const: "(λx. (x, c)) ` A = A × {c}"
by auto
lemma homotopic_constant_maps:
"homotopic_with (λx. True) s t (λx. a) (λx. b) ⟷ s = {} ∨ path_component t a b"
proof (cases "s = {} ∨ t = {}")
case True with continuous_on_const show ?thesis
by (auto simp: homotopic_with path_component_def)
next
case False
then obtain c where "c ∈ s" by blast
show ?thesis
proof
assume "homotopic_with (λx. True) s t (λx. a) (λx. b)"
then obtain h :: "real × 'a ⇒ 'b"
where conth: "continuous_on ({0..1} × s) h"
and h: "h ` ({0..1} × s) ⊆ t" "(∀x∈s. h (0, x) = a)" "(∀x∈s. h (1, x) = b)"
by (auto simp: homotopic_with)
have "continuous_on {0..1} (h ∘ (λt. (t, c)))"
apply (rule continuous_intros conth | simp add: image_Pair_const)+
apply (blast intro: ‹c ∈ s› continuous_on_subset [OF conth])
done
with ‹c ∈ s› h show "s = {} ∨ path_component t a b"
apply (simp_all add: homotopic_with path_component_def, auto)
apply (drule_tac x="h ∘ (λt. (t, c))" in spec)
apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
done
next
assume "s = {} ∨ path_component t a b"
with False show "homotopic_with (λx. True) s t (λx. a) (λx. b)"
apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
apply (rule_tac x="g ∘ fst" in exI)
apply (rule conjI continuous_intros | force)+
done
qed
qed
subsection%unimportant‹Trivial properties›
lemma homotopic_with_imp_property: "homotopic_with P X Y f g ⟹ P f ∧ P g"
unfolding homotopic_with_def Ball_def
apply clarify
apply (frule_tac x=0 in spec)
apply (drule_tac x=1 in spec, auto)
done
lemma continuous_on_o_Pair: "⟦continuous_on (T × X) h; t ∈ T⟧ ⟹ continuous_on X (h ∘ Pair t)"
by (fast intro: continuous_intros elim!: continuous_on_subset)
lemma homotopic_with_imp_continuous:
assumes "homotopic_with P X Y f g"
shows "continuous_on X f ∧ continuous_on X g"
proof -
obtain h :: "real × 'a ⇒ 'b"
where conth: "continuous_on ({0..1} × X) h"
and h: "∀x. h (0, x) = f x" "∀x. h (1, x) = g x"
using assms by (auto simp: homotopic_with_def)
have *: "t ∈ {0..1} ⟹ continuous_on X (h ∘ (λx. (t,x)))" for t
by (rule continuous_intros continuous_on_subset [OF conth] | force)+
show ?thesis
using h *[of 0] *[of 1] by auto
qed
proposition homotopic_with_imp_subset1:
"homotopic_with P X Y f g ⟹ f ` X ⊆ Y"
by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
proposition homotopic_with_imp_subset2:
"homotopic_with P X Y f g ⟹ g ` X ⊆ Y"
by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
proposition homotopic_with_mono:
assumes hom: "homotopic_with P X Y f g"
and Q: "⋀h. ⟦continuous_on X h; image h X ⊆ Y ∧ P h⟧ ⟹ Q h"
shows "homotopic_with Q X Y f g"
using hom
apply (simp add: homotopic_with_def)
apply (erule ex_forward)
apply (force simp: intro!: Q dest: continuous_on_o_Pair)
done
proposition homotopic_with_subset_left:
"⟦homotopic_with P X Y f g; Z ⊆ X⟧ ⟹ homotopic_with P Z Y f g"
apply (simp add: homotopic_with_def)
apply (fast elim!: continuous_on_subset ex_forward)
done
proposition homotopic_with_subset_right:
"⟦homotopic_with P X Y f g; Y ⊆ Z⟧ ⟹ homotopic_with P X Z f g"
apply (simp add: homotopic_with_def)
apply (fast elim!: continuous_on_subset ex_forward)
done
proposition homotopic_with_compose_continuous_right:
"⟦homotopic_with (λf. p (f ∘ h)) X Y f g; continuous_on W h; h ` W ⊆ X⟧
⟹ homotopic_with p W Y (f ∘ h) (g ∘ h)"
apply (clarsimp simp add: homotopic_with_def)
apply (rename_tac k)
apply (rule_tac x="k ∘ (λy. (fst y, h (snd y)))" in exI)
apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
apply (erule continuous_on_subset)
apply (fastforce simp: o_def)+
done
proposition homotopic_compose_continuous_right:
"⟦homotopic_with (λf. True) X Y f g; continuous_on W h; h ` W ⊆ X⟧
⟹ homotopic_with (λf. True) W Y (f ∘ h) (g ∘ h)"
using homotopic_with_compose_continuous_right by fastforce
proposition homotopic_with_compose_continuous_left:
"⟦homotopic_with (λf. p (h ∘ f)) X Y f g; continuous_on Y h; h ` Y ⊆ Z⟧
⟹ homotopic_with p X Z (h ∘ f) (h ∘ g)"
apply (clarsimp simp add: homotopic_with_def)
apply (rename_tac k)
apply (rule_tac x="h ∘ k" in exI)
apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
apply (erule continuous_on_subset)
apply (fastforce simp: o_def)+
done
proposition homotopic_compose_continuous_left:
"⟦homotopic_with (λ_. True) X Y f g;
continuous_on Y h; h ` Y ⊆ Z⟧
⟹ homotopic_with (λf. True) X Z (h ∘ f) (h ∘ g)"
using homotopic_with_compose_continuous_left by fastforce
proposition homotopic_with_Pair:
assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
and q: "⋀f g. ⟦p f; p' g⟧ ⟹ q(λ(x,y). (f x, g y))"
shows "homotopic_with q (s × s') (t × t')
(λ(x,y). (f x, f' y)) (λ(x,y). (g x, g' y))"
using hom
apply (clarsimp simp add: homotopic_with_def)
apply (rename_tac k k')
apply (rule_tac x="λz. ((k ∘ (λx. (fst x, fst (snd x)))) z, (k' ∘ (λx. (fst x, snd (snd x)))) z)" in exI)
apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
apply (auto intro!: q [unfolded case_prod_unfold])
done
lemma homotopic_on_empty [simp]: "homotopic_with (λx. True) {} t f g"
by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
text‹Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
though this only affects reflexivity.›
proposition homotopic_with_refl:
"homotopic_with P X Y f f ⟷ continuous_on X f ∧ image f X ⊆ Y ∧ P f"
apply (rule iffI)
using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
apply (simp add: homotopic_with_def)
apply (rule_tac x="f ∘ snd" in exI)
apply (rule conjI continuous_intros | force)+
done
lemma homotopic_with_symD:
fixes X :: "'a::real_normed_vector set"
assumes "homotopic_with P X Y f g"
shows "homotopic_with P X Y g f"
using assms
apply (clarsimp simp add: homotopic_with_def)
apply (rename_tac h)
apply (rule_tac x="h ∘ (λy. (1 - fst y, snd y))" in exI)
apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
done
proposition homotopic_with_sym:
fixes X :: "'a::real_normed_vector set"
shows "homotopic_with P X Y f g ⟷ homotopic_with P X Y g f"
using homotopic_with_symD by blast
lemma split_01: "{0..1::real} = {0..1/2} ∪ {1/2..1}"
by force
lemma split_01_prod: "{0..1::real} × X = ({0..1/2} × X) ∪ ({1/2..1} × X)"
by force
proposition homotopic_with_trans:
fixes X :: "'a::real_normed_vector set"
assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
shows "homotopic_with P X Y f h"
proof -
have clo1: "closedin (subtopology euclidean ({0..1/2} × X ∪ {1/2..1} × X)) ({0..1/2::real} × X)"
apply (simp add: closedin_closed split_01_prod [symmetric])
apply (rule_tac x="{0..1/2} × UNIV" in exI)
apply (force simp: closed_Times)
done
have clo2: "closedin (subtopology euclidean ({0..1/2} × X ∪ {1/2..1} × X)) ({1/2..1::real} × X)"
apply (simp add: closedin_closed split_01_prod [symmetric])
apply (rule_tac x="{1/2..1} × UNIV" in exI)
apply (force simp: closed_Times)
done
{ fix k1 k2:: "real × 'a ⇒ 'b"
assume cont: "continuous_on ({0..1} × X) k1" "continuous_on ({0..1} × X) k2"
and Y: "k1 ` ({0..1} × X) ⊆ Y" "k2 ` ({0..1} × X) ⊆ Y"
and geq: "∀x. k1 (1, x) = g x" "∀x. k2 (0, x) = g x"
and k12: "∀x. k1 (0, x) = f x" "∀x. k2 (1, x) = h x"
and P: "∀t∈{0..1}. P (λx. k1 (t, x))" "∀t∈{0..1}. P (λx. k2 (t, x))"
define k where "k y =
(if fst y ≤ 1 / 2
then (k1 ∘ (λx. (2 *⇩R fst x, snd x))) y
else (k2 ∘ (λx. (2 *⇩R fst x -1, snd x))) y)" for y
have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2" for u v
by (simp add: geq that)
have "continuous_on ({0..1} × X) k"
using cont
apply (simp add: split_01_prod k_def)
apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
apply (force simp: keq)
done
moreover have "k ` ({0..1} × X) ⊆ Y"
using Y by (force simp: k_def)
moreover have "∀x. k (0, x) = f x"
by (simp add: k_def k12)
moreover have "(∀x. k (1, x) = h x)"
by (simp add: k_def k12)
moreover have "∀t∈{0..1}. P (λx. k (t, x))"
using P
apply (clarsimp simp add: k_def)
apply (case_tac "t ≤ 1/2", auto)
done
ultimately have *: "∃k :: real × 'a ⇒ 'b.
continuous_on ({0..1} × X) k ∧ k ` ({0..1} × X) ⊆ Y ∧
(∀x. k (0, x) = f x) ∧ (∀x. k (1, x) = h x) ∧ (∀t∈{0..1}. P (λx. k (t, x)))"
by blast
} note * = this
show ?thesis
using assms by (auto intro: * simp add: homotopic_with_def)
qed
proposition homotopic_compose:
fixes s :: "'a::real_normed_vector set"
shows "⟦homotopic_with (λx. True) s t f f'; homotopic_with (λx. True) t u g g'⟧
⟹ homotopic_with (λx. True) s u (g ∘ f) (g' ∘ f')"
apply (rule homotopic_with_trans [where g = "g ∘ f'"])
apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
text‹Homotopic triviality implicitly incorporates path-connectedness.›
lemma homotopic_triviality:
fixes S :: "'a::real_normed_vector set"
shows "(∀f g. continuous_on S f ∧ f ` S ⊆ T ∧
continuous_on S g ∧ g ` S ⊆ T
⟶ homotopic_with (λx. True) S T f g) ⟷
(S = {} ∨ path_connected T) ∧
(∀f. continuous_on S f ∧ f ` S ⊆ T ⟶ (∃c. homotopic_with (λx. True) S T f (λx. c)))"
(is "?lhs = ?rhs")
proof (cases "S = {} ∨ T = {}")
case True then show ?thesis by auto
next
case False show ?thesis
proof
assume LHS [rule_format]: ?lhs
have pab: "path_component T a b" if "a ∈ T" "b ∈ T" for a b
proof -
have "homotopic_with (λx. True) S T (λx. a) (λx. b)"
by (simp add: LHS continuous_on_const image_subset_iff that)
then show ?thesis
using False homotopic_constant_maps by blast
qed
moreover
have "∃c. homotopic_with (λx. True) S T f (λx. c)" if "continuous_on S f" "f ` S ⊆ T" for f
by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
ultimately show ?rhs
by (simp add: path_connected_component)
next
assume RHS: ?rhs
with False have T: "path_connected T"
by blast
show ?lhs
proof clarify
fix f g
assume "continuous_on S f" "f ` S ⊆ T" "continuous_on S g" "g ` S ⊆ T"
obtain c d where c: "homotopic_with (λx. True) S T f (λx. c)" and d: "homotopic_with (λx. True) S T g (λx. d)"
using False ‹continuous_on S f› ‹f ` S ⊆ T› RHS ‹continuous_on S g› ‹g ` S ⊆ T› by blast
then have "c ∈ T" "d ∈ T"
using False homotopic_with_imp_subset2 by fastforce+
with T have "path_component T c d"
using path_connected_component by blast
then have "homotopic_with (λx. True) S T (λx. c) (λx. d)"
by (simp add: homotopic_constant_maps)
with c d show "homotopic_with (λx. True) S T f g"
by (meson homotopic_with_symD homotopic_with_trans)
qed
qed
qed
subsection‹Homotopy of paths, maintaining the same endpoints›
definition%important homotopic_paths :: "['a set, real ⇒ 'a, real ⇒ 'a::topological_space] ⇒ bool"
where
"homotopic_paths s p q ≡
homotopic_with (λr. pathstart r = pathstart p ∧ pathfinish r = pathfinish p) {0..1} s p q"
lemma homotopic_paths:
"homotopic_paths s p q ⟷
(∃h. continuous_on ({0..1} × {0..1}) h ∧
h ` ({0..1} × {0..1}) ⊆ s ∧
(∀x ∈ {0..1}. h(0,x) = p x) ∧
(∀x ∈ {0..1}. h(1,x) = q x) ∧
(∀t ∈ {0..1::real}. pathstart(h ∘ Pair t) = pathstart p ∧
pathfinish(h ∘ Pair t) = pathfinish p))"
by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
proposition homotopic_paths_imp_pathstart:
"homotopic_paths s p q ⟹ pathstart p = pathstart q"
by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
proposition homotopic_paths_imp_pathfinish:
"homotopic_paths s p q ⟹ pathfinish p = pathfinish q"
by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
lemma homotopic_paths_imp_path:
"homotopic_paths s p q ⟹ path p ∧ path q"
using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
lemma homotopic_paths_imp_subset:
"homotopic_paths s p q ⟹ path_image p ⊆ s ∧ path_image q ⊆ s"
by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
proposition homotopic_paths_refl [simp]: "homotopic_paths s p p ⟷ path p ∧ path_image p ⊆ s"
by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
proposition homotopic_paths_sym: "homotopic_paths s p q ⟹ homotopic_paths s q p"
by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
proposition homotopic_paths_sym_eq: "homotopic_paths s p q ⟷ homotopic_paths s q p"
by (metis homotopic_paths_sym)
proposition homotopic_paths_trans [trans]:
"⟦homotopic_paths s p q; homotopic_paths s q r⟧ ⟹ homotopic_paths s p r"
apply (simp add: homotopic_paths_def)
apply (rule homotopic_with_trans, assumption)
by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
proposition homotopic_paths_eq:
"⟦path p; path_image p ⊆ s; ⋀t. t ∈ {0..1} ⟹ p t = q t⟧ ⟹ homotopic_paths s p q"
apply (simp add: homotopic_paths_def)
apply (rule homotopic_with_eq)
apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
done
proposition homotopic_paths_reparametrize:
assumes "path p"
and pips: "path_image p ⊆ s"
and contf: "continuous_on {0..1} f"
and f01:"f ` {0..1} ⊆ {0..1}"
and [simp]: "f(0) = 0" "f(1) = 1"
and q: "⋀t. t ∈ {0..1} ⟹ q(t) = p(f t)"
shows "homotopic_paths s p q"
proof -
have contp: "continuous_on {0..1} p"
by (metis ‹path p› path_def)
then have "continuous_on {0..1} (p ∘ f)"
using contf continuous_on_compose continuous_on_subset f01 by blast
then have "path q"
by (simp add: path_def) (metis q continuous_on_cong)
have piqs: "path_image q ⊆ s"
by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
have fb0: "⋀a b. ⟦0 ≤ a; a ≤ 1; 0 ≤ b; b ≤ 1⟧ ⟹ 0 ≤ (1 - a) * f b + a * b"
using f01 by force
have fb1: "⟦0 ≤ a; a ≤ 1; 0 ≤ b; b ≤ 1⟧ ⟹ (1 - a) * f b + a * b ≤ 1" for a b
using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
have "homotopic_paths s q p"
proof (rule homotopic_paths_trans)
show "homotopic_paths s q (p ∘ f)"
using q by (force intro: homotopic_paths_eq [OF ‹path q› piqs])
next
show "homotopic_paths s (p ∘ f) p"
apply (simp add: homotopic_paths_def homotopic_with_def)
apply (rule_tac x="p ∘ (λy. (1 - (fst y)) *⇩R ((f ∘ snd) y) + (fst y) *⇩R snd y)" in exI)
apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
using pips [unfolded path_image_def]
apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
done
qed
then show ?thesis
by (simp add: homotopic_paths_sym)
qed
lemma homotopic_paths_subset: "⟦homotopic_paths s p q; s ⊆ t⟧ ⟹ homotopic_paths t p q"
using homotopic_paths_def homotopic_with_subset_right by blast
text‹ A slightly ad-hoc but useful lemma in constructing homotopies.›
lemma homotopic_join_lemma:
fixes q :: "[real,real] ⇒ 'a::topological_space"
assumes p: "continuous_on ({0..1} × {0..1}) (λy. p (fst y) (snd y))"
and q: "continuous_on ({0..1} × {0..1}) (λy. q (fst y) (snd y))"
and pf: "⋀t. t ∈ {0..1} ⟹ pathfinish(p t) = pathstart(q t)"
shows "continuous_on ({0..1} × {0..1}) (λy. (p(fst y) +++ q(fst y)) (snd y))"
proof -
have 1: "(λy. p (fst y) (2 * snd y)) = (λy. p (fst y) (snd y)) ∘ (λy. (fst y, 2 * snd y))"
by (rule ext) (simp)
have 2: "(λy. q (fst y) (2 * snd y - 1)) = (λy. q (fst y) (snd y)) ∘ (λy. (fst y, 2 * snd y - 1))"
by (rule ext) (simp)
show ?thesis
apply (simp add: joinpaths_def)
apply (rule continuous_on_cases_le)
apply (simp_all only: 1 2)
apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
using pf
apply (auto simp: mult.commute pathstart_def pathfinish_def)
done
qed
text‹ Congruence properties of homotopy w.r.t. path-combining operations.›
lemma homotopic_paths_reversepath_D:
assumes "homotopic_paths s p q"
shows "homotopic_paths s (reversepath p) (reversepath q)"
using assms
apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
apply (rule_tac x="h ∘ (λx. (fst x, 1 - snd x))" in exI)
apply (rule conjI continuous_intros)+
apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
done
proposition homotopic_paths_reversepath:
"homotopic_paths s (reversepath p) (reversepath q) ⟷ homotopic_paths s p q"
using homotopic_paths_reversepath_D by force
proposition homotopic_paths_join:
"⟦homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q⟧ ⟹ homotopic_paths s (p +++ q) (p' +++ q')"
apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
apply (rename_tac k1 k2)
apply (rule_tac x="(λy. ((k1 ∘ Pair (fst y)) +++ (k2 ∘ Pair (fst y))) (snd y))" in exI)
apply (rule conjI continuous_intros homotopic_join_lemma)+
apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
done
proposition homotopic_paths_continuous_image:
"⟦homotopic_paths s f g; continuous_on s h; h ` s ⊆ t⟧ ⟹ homotopic_paths t (h ∘ f) (h ∘ g)"
unfolding homotopic_paths_def
apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
done
subsection‹Group properties for homotopy of paths›
text%important‹So taking equivalence classes under homotopy would give the fundamental group›
proposition homotopic_paths_rid:
"⟦path p; path_image p ⊆ s⟧ ⟹ homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
apply (subst homotopic_paths_sym)
apply (rule homotopic_paths_reparametrize [where f = "λt. if t ≤ 1 / 2 then 2 *⇩R t else 1"])
apply (simp_all del: le_divide_eq_numeral1)
apply (subst split_01)
apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
done
proposition homotopic_paths_lid:
"⟦path p; path_image p ⊆ s⟧ ⟹ homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
using homotopic_paths_rid [of "reversepath p" s]
by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
proposition homotopic_paths_assoc:
"⟦path p; path_image p ⊆ s; path q; path_image q ⊆ s; path r; path_image r ⊆ s; pathfinish p = pathstart q;
pathfinish q = pathstart r⟧
⟹ homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
apply (subst homotopic_paths_sym)
apply (rule homotopic_paths_reparametrize
[where f = "λt. if t ≤ 1 / 2 then inverse 2 *⇩R t
else if t ≤ 3 / 4 then t - (1 / 4)
else 2 *⇩R t - 1"])
apply (simp_all del: le_divide_eq_numeral1)
apply (simp add: subset_path_image_join)
apply (rule continuous_on_cases_1 continuous_intros)+
apply (auto simp: joinpaths_def)
done
proposition homotopic_paths_rinv:
assumes "path p" "path_image p ⊆ s"
shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
proof -
have "continuous_on ({0..1} × {0..1}) (λx. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
using assms
apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
apply (rule continuous_on_cases_le)
apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
apply (force elim!: continuous_on_subset simp add: mult_le_one)+
done
then show ?thesis
using assms
apply (subst homotopic_paths_sym_eq)
unfolding homotopic_paths_def homotopic_with_def
apply (rule_tac x="(λy. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
apply (force simp: mult_le_one)
done
qed
proposition homotopic_paths_linv:
assumes "path p" "path_image p ⊆ s"
shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
using homotopic_paths_rinv [of "reversepath p" s] assms by simp
subsection‹Homotopy of loops without requiring preservation of endpoints›
definition%important homotopic_loops :: "'a::topological_space set ⇒ (real ⇒ 'a) ⇒ (real ⇒ 'a) ⇒ bool" where
"homotopic_loops s p q ≡
homotopic_with (λr. pathfinish r = pathstart r) {0..1} s p q"
lemma homotopic_loops:
"homotopic_loops s p q ⟷
(∃h. continuous_on ({0..1::real} × {0..1}) h ∧
image h ({0..1} × {0..1}) ⊆ s ∧
(∀x ∈ {0..1}. h(0,x) = p x) ∧
(∀x ∈ {0..1}. h(1,x) = q x) ∧
(∀t ∈ {0..1}. pathfinish(h ∘ Pair t) = pathstart(h ∘ Pair t)))"
by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
proposition homotopic_loops_imp_loop:
"homotopic_loops s p q ⟹ pathfinish p = pathstart p ∧ pathfinish q = pathstart q"
using homotopic_with_imp_property homotopic_loops_def by blast
proposition homotopic_loops_imp_path:
"homotopic_loops s p q ⟹ path p ∧ path q"
unfolding homotopic_loops_def path_def
using homotopic_with_imp_continuous by blast
proposition homotopic_loops_imp_subset:
"homotopic_loops s p q ⟹ path_image p ⊆ s ∧ path_image q ⊆ s"
unfolding homotopic_loops_def path_image_def
by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
proposition homotopic_loops_refl:
"homotopic_loops s p p ⟷
path p ∧ path_image p ⊆ s ∧ pathfinish p = pathstart p"
by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
proposition homotopic_loops_sym: "homotopic_loops s p q ⟹ homotopic_loops s q p"
by (simp add: homotopic_loops_def homotopic_with_sym)
proposition homotopic_loops_sym_eq: "homotopic_loops s p q ⟷ homotopic_loops s q p"
by (metis homotopic_loops_sym)
proposition homotopic_loops_trans:
"⟦homotopic_loops s p q; homotopic_loops s q r⟧ ⟹ homotopic_loops s p r"
unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
proposition homotopic_loops_subset:
"⟦homotopic_loops s p q; s ⊆ t⟧ ⟹ homotopic_loops t p q"
by (simp add: homotopic_loops_def homotopic_with_subset_right)
proposition homotopic_loops_eq:
"⟦path p; path_image p ⊆ s; pathfinish p = pathstart p; ⋀t. t ∈ {0..1} ⟹ p(t) = q(t)⟧
⟹ homotopic_loops s p q"
unfolding homotopic_loops_def
apply (rule homotopic_with_eq)
apply (rule homotopic_with_refl [where f = p, THEN iffD2])
apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
done
proposition homotopic_loops_continuous_image:
"⟦homotopic_loops s f g; continuous_on s h; h ` s ⊆ t⟧ ⟹ homotopic_loops t (h ∘ f) (h ∘ g)"
unfolding homotopic_loops_def
apply (rule homotopic_with_compose_continuous_left)
apply (erule homotopic_with_mono)
by (simp add: pathfinish_def pathstart_def)
subsection‹Relations between the two variants of homotopy›
proposition homotopic_paths_imp_homotopic_loops:
"⟦homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p⟧ ⟹ homotopic_loops s p q"
by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
proposition homotopic_loops_imp_homotopic_paths_null:
assumes "homotopic_loops s p (linepath a a)"
shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
proof -
have "path p" by (metis assms homotopic_loops_imp_path)
have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
have pip: "path_image p ⊆ s" by (metis assms homotopic_loops_imp_subset)
obtain h where conth: "continuous_on ({0..1::real} × {0..1}) h"
and hs: "h ` ({0..1} × {0..1}) ⊆ s"
and [simp]: "⋀x. x ∈ {0..1} ⟹ h(0,x) = p x"
and [simp]: "⋀x. x ∈ {0..1} ⟹ h(1,x) = a"
and ends: "⋀t. t ∈ {0..1} ⟹ pathfinish (h ∘ Pair t) = pathstart (h ∘ Pair t)"
using assms by (auto simp: homotopic_loops homotopic_with)
have conth0: "path (λu. h (u, 0))"
unfolding path_def
apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
apply (force intro: continuous_intros continuous_on_subset [OF conth])+
done
have pih0: "path_image (λu. h (u, 0)) ⊆ s"
using hs by (force simp: path_image_def)
have c1: "continuous_on ({0..1} × {0..1}) (λx. h (fst x * snd x, 0))"
apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
done
have c2: "continuous_on ({0..1} × {0..1}) (λx. h (fst x - fst x * snd x, 0))"
apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
apply (rule continuous_on_subset [OF conth])
apply (auto simp: algebra_simps add_increasing2 mult_left_le)
done
have [simp]: "⋀t. ⟦0 ≤ t ∧ t ≤ 1⟧ ⟹ h (t, 1) = h (t, 0)"
using ends by (simp add: pathfinish_def pathstart_def)
have adhoc_le: "c * 4 ≤ 1 + c * (d * 4)" if "¬ d * 4 ≤ 3" "0 ≤ c" "c ≤ 1" for c d::real
proof -
have "c * 3 ≤ c * (d * 4)" using that less_eq_real_def by auto
with ‹c ≤ 1› show ?thesis by fastforce
qed
have *: "⋀p x. (path p ∧ path(reversepath p)) ∧
(path_image p ⊆ s ∧ path_image(reversepath p) ⊆ s) ∧
(pathfinish p = pathstart(linepath a a +++ reversepath p) ∧
pathstart(reversepath p) = a) ∧ pathstart p = x
⟹ homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
by (metis homotopic_paths_lid homotopic_paths_join
homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
using ‹path p› homotopic_paths_rid homotopic_paths_sym pip by blast
moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
(linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
apply (rule homotopic_paths_sym)
using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
((λu. h (u, 0)) +++ linepath a a +++ reversepath (λu. h (u, 0)))"
apply (simp add: homotopic_paths_def homotopic_with_def)
apply (rule_tac x="λy. (subpath 0 (fst y) (λu. h (u, 0)) +++ (λu. h (Pair (fst y) u)) +++ subpath (fst y) 0 (λu. h (u, 0))) (snd y)" in exI)
apply (simp add: subpath_reversepath)
apply (intro conjI homotopic_join_lemma)
using ploop
apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
done
moreover have "homotopic_paths s ((λu. h (u, 0)) +++ linepath a a +++ reversepath (λu. h (u, 0)))
(linepath (pathstart p) (pathstart p))"
apply (rule *)
apply (simp add: pih0 pathstart_def pathfinish_def conth0)
apply (simp add: reversepath_def joinpaths_def)
done
ultimately show ?thesis
by (blast intro: homotopic_paths_trans)
qed
proposition homotopic_loops_conjugate:
fixes s :: "'a::real_normed_vector set"
assumes "path p" "path q" and pip: "path_image p ⊆ s" and piq: "path_image q ⊆ s"
and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
shows "homotopic_loops s (p +++ q +++ reversepath p) q"
proof -
have contp: "continuous_on {0..1} p" using ‹path p› [unfolded path_def] by blast
have contq: "continuous_on {0..1} q" using ‹path q› [unfolded path_def] by blast
have c1: "continuous_on ({0..1} × {0..1}) (λx. p ((1 - fst x) * snd x + fst x))"
apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
apply (force simp: mult_le_one intro!: continuous_intros)
apply (rule continuous_on_subset [OF contp])
apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
done
have c2: "continuous_on ({0..1} × {0..1}) (λx. p ((fst x - 1) * snd x + 1))"
apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
apply (force simp: mult_le_one intro!: continuous_intros)
apply (rule continuous_on_subset [OF contp])
apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
done
have ps1: "⋀a b. ⟦b * 2 ≤ 1; 0 ≤ b; 0 ≤ a; a ≤ 1⟧ ⟹ p ((1 - a) * (2 * b) + a) ∈ s"
using sum_le_prod1
by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
have ps2: "⋀a b. ⟦¬ 4 * b ≤ 3; b ≤ 1; 0 ≤ a; a ≤ 1⟧ ⟹ p ((a - 1) * (4 * b - 3) + 1) ∈ s"
apply (rule pip [unfolded path_image_def, THEN subsetD])
apply (rule image_eqI, blast)
apply (simp add: algebra_simps)
by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
add.commute zero_le_numeral)
have qs: "⋀a b. ⟦4 * b ≤ 3; ¬ b * 2 ≤ 1⟧ ⟹ q (4 * b - 2) ∈ s"
using path_image_def piq by fastforce
have "homotopic_loops s (p +++ q +++ reversepath p)
(linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
apply (simp add: homotopic_loops_def homotopic_with_def)
apply (rule_tac x="(λy. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
apply (simp add: subpath_refl subpath_reversepath)
apply (intro conjI homotopic_join_lemma)
using papp qloop
apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
apply (auto simp: ps1 ps2 qs)
done
moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
proof -
have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
using ‹path q› homotopic_paths_lid qloop piq by auto
hence 1: "⋀f. homotopic_paths s f q ∨ ¬ homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
using homotopic_paths_trans by blast
hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
proof -
have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
by (simp add: ‹path q› homotopic_paths_rid piq)
thus ?thesis
by (metis (no_types) 1 ‹path q› homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
homotopic_paths_trans qloop pathfinish_linepath piq)
qed
thus ?thesis
by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
qed
ultimately show ?thesis
by (blast intro: homotopic_loops_trans)
qed
lemma homotopic_paths_loop_parts:
assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
shows "homotopic_paths S p q"
proof -
have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
then have "path p"
using ‹path q› homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
show ?thesis
proof (cases "pathfinish p = pathfinish q")
case True
have pipq: "path_image p ⊆ S" "path_image q ⊆ S"
by (metis Un_subset_iff paths ‹path p› ‹path q› homotopic_loops_imp_subset homotopic_paths_imp_path loops
path_image_join path_image_reversepath path_imp_reversepath path_join_eq)+
have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
using ‹path p› ‹path_image p ⊆ S› homotopic_paths_rid homotopic_paths_sym by blast
moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
by (simp add: True ‹path p› ‹path q› pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
by (simp add: True ‹path p› ‹path q› homotopic_paths_assoc pipq)
moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
by (simp add: ‹path q› homotopic_paths_join paths pipq)
moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
by (metis ‹path q› homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
ultimately show ?thesis
using homotopic_paths_trans by metis
next
case False
then show ?thesis
using ‹path q› homotopic_loops_imp_path loops path_join_path_ends by fastforce
qed
qed
subsection%unimportant‹Homotopy of "nearby" function, paths and loops›
lemma homotopic_with_linear:
fixes f g :: "_ ⇒ 'b::real_normed_vector"
assumes contf: "continuous_on s f"
and contg:"continuous_on s g"
and sub: "⋀x. x ∈ s ⟹ closed_segment (f x) (g x) ⊆ t"
shows "homotopic_with (λz. True) s t f g"
apply (simp add: homotopic_with_def)
apply (rule_tac x="λy. ((1 - (fst y)) *⇩R f(snd y) + (fst y) *⇩R g(snd y))" in exI)
apply (intro conjI)
apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
using sub closed_segment_def apply fastforce+
done
lemma homotopic_paths_linear:
fixes g h :: "real ⇒ 'a::real_normed_vector"
assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
"⋀t. t ∈ {0..1} ⟹ closed_segment (g t) (h t) ⊆ s"
shows "homotopic_paths s g h"
using assms
unfolding path_def
apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
apply (rule_tac x="λy. ((1 - (fst y)) *⇩R (g ∘ snd) y + (fst y) *⇩R (h ∘ snd) y)" in exI)
apply (intro conjI subsetI continuous_intros; force)
done
lemma homotopic_loops_linear:
fixes g h :: "real ⇒ 'a::real_normed_vector"
assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
"⋀t x. t ∈ {0..1} ⟹ closed_segment (g t) (h t) ⊆ s"
shows "homotopic_loops s g h"
using assms
unfolding path_def
apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
apply (rule_tac x="λy. ((1 - (fst y)) *⇩R g(snd y) + (fst y) *⇩R h(snd y))" in exI)
apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
apply (force simp: closed_segment_def)
done
lemma homotopic_paths_nearby_explicit:
assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
and no: "⋀t x. ⟦t ∈ {0..1}; x ∉ s⟧ ⟹ norm(h t - g t) < norm(g t - x)"
shows "homotopic_paths s g h"
apply (rule homotopic_paths_linear [OF assms(1-4)])
by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
lemma homotopic_loops_nearby_explicit:
assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
and no: "⋀t x. ⟦t ∈ {0..1}; x ∉ s⟧ ⟹ norm(h t - g t) < norm(g t - x)"
shows "homotopic_loops s g h"
apply (rule homotopic_loops_linear [OF assms(1-4)])
by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
lemma homotopic_nearby_paths:
fixes g h :: "real ⇒ 'a::euclidean_space"
assumes "path g" "open s" "path_image g ⊆ s"
shows "∃e. 0 < e ∧
(∀h. path h ∧
pathstart h = pathstart g ∧ pathfinish h = pathfinish g ∧
(∀t ∈ {0..1}. norm(h t - g t) < e) ⟶ homotopic_paths s g h)"
proof -
obtain e where "e > 0" and e: "⋀x y. x ∈ path_image g ⟹ y ∈ - s ⟹ e ≤ dist x y"
using separate_compact_closed [of "path_image g" "-s"] assms by force
show ?thesis
apply (intro exI conjI)
using e [unfolded dist_norm]
apply (auto simp: intro!: homotopic_paths_nearby_explicit assms ‹e > 0›)
by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
qed
lemma homotopic_nearby_loops:
fixes g h :: "real ⇒ 'a::euclidean_space"
assumes "path g" "open s" "path_image g ⊆ s" "pathfinish g = pathstart g"
shows "∃e. 0 < e ∧
(∀h. path h ∧ pathfinish h = pathstart h ∧
(∀t ∈ {0..1}. norm(h t - g t) < e) ⟶ homotopic_loops s g h)"
proof -
obtain e where "e > 0" and e: "⋀x y. x ∈ path_image g ⟹ y ∈ - s ⟹ e ≤ dist x y"
using separate_compact_closed [of "path_image g" "-s"] assms by force
show ?thesis
apply (intro exI conjI)
using e [unfolded dist_norm]
apply (auto simp: intro!: homotopic_loops_nearby_explicit assms ‹e > 0›)
by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
qed
subsection‹ Homotopy and subpaths›
lemma homotopic_join_subpaths1:
assumes "path g" and pag: "path_image g ⊆ s"
and u: "u ∈ {0..1}" and v: "v ∈ {0..1}" and w: "w ∈ {0..1}" "u ≤ v" "v ≤ w"
shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
proof -
have 1: "t * 2 ≤ 1 ⟹ u + t * (v * 2) ≤ v + t * (u * 2)" for t
using affine_ineq ‹u ≤ v› by fastforce
have 2: "t * 2 > 1 ⟹ u + (2*t - 1) * v ≤ v + (2*t - 1) * w" for t
by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono ‹u ≤ v› ‹v ≤ w›)
have t2: "⋀t::real. t*2 = 1 ⟹ t = 1/2" by auto
show ?thesis
apply (rule homotopic_paths_subset [OF _ pag])
using assms
apply (cases "w = u")
using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
apply (rule homotopic_paths_sym)
apply (rule homotopic_paths_reparametrize
[where f = "λt. if t ≤ 1 / 2
then inverse((w - u)) *⇩R (2 * (v - u)) *⇩R t
else inverse((w - u)) *⇩R ((v - u) + (w - v) *⇩R (2 *⇩R t - 1))"])
using ‹path g› path_subpath u w apply blast
using ‹path g› path_image_subpath_subset u w(1) apply blast
apply simp_all
apply (subst split_01)
apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
apply (simp_all add: field_simps not_le)
apply (force dest!: t2)
apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
apply (simp add: joinpaths_def subpath_def)
apply (force simp: algebra_simps)
done
qed
lemma homotopic_join_subpaths2:
assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
lemma homotopic_join_subpaths3:
assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
and "path g" and pag: "path_image g ⊆ s"
and u: "u ∈ {0..1}" and v: "v ∈ {0..1}" and w: "w ∈ {0..1}"
shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
proof -
have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
apply (rule homotopic_paths_join)
using hom homotopic_paths_sym_eq apply blast
apply (metis ‹path g› homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
done
also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
(subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
apply (rule homotopic_paths_join)
apply (metis ‹path g› homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
apply (metis (no_types, lifting) ‹path g› homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
apply simp
done
also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
apply (rule homotopic_paths_rid)
using ‹path g› path_subpath u v apply blast
apply (meson ‹path g› order.trans pag path_image_subpath_subset u v)
done
finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
then show ?thesis
using homotopic_join_subpaths2 by blast
qed
proposition homotopic_join_subpaths:
"⟦path g; path_image g ⊆ s; u ∈ {0..1}; v ∈ {0..1}; w ∈ {0..1}⟧
⟹ homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
apply (rule le_cases3 [of u v w])
using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
text‹Relating homotopy of trivial loops to path-connectedness.›
lemma path_component_imp_homotopic_points:
"path_component S a b ⟹ homotopic_loops S (linepath a a) (linepath b b)"
apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
pathstart_def pathfinish_def path_image_def path_def, clarify)
apply (rule_tac x="g ∘ fst" in exI)
apply (intro conjI continuous_intros continuous_on_compose)+
apply (auto elim!: continuous_on_subset)
done
lemma homotopic_loops_imp_path_component_value:
"⟦homotopic_loops S p q; 0 ≤ t; t ≤ 1⟧
⟹ path_component S (p t) (q t)"
apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
pathstart_def pathfinish_def path_image_def path_def, clarify)
apply (rule_tac x="h ∘ (λu. (u, t))" in exI)
apply (intro conjI continuous_intros continuous_on_compose)+
apply (auto elim!: continuous_on_subset)
done
lemma homotopic_points_eq_path_component:
"homotopic_loops S (linepath a a) (linepath b b) ⟷
path_component S a b"
by (auto simp: path_component_imp_homotopic_points
dest: homotopic_loops_imp_path_component_value [where t=1])
lemma path_connected_eq_homotopic_points:
"path_connected S ⟷
(∀a b. a ∈ S ∧ b ∈ S ⟶ homotopic_loops S (linepath a a) (linepath b b))"
by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
subsection‹Simply connected sets›
text%important‹defined as "all loops are homotopic (as loops)›
definition%important simply_connected where
"simply_connected S ≡
∀p q. path p ∧ pathfinish p = pathstart p ∧ path_image p ⊆ S ∧
path q ∧ pathfinish q = pathstart q ∧ path_image q ⊆ S
⟶ homotopic_loops S p q"
lemma simply_connected_empty [iff]: "simply_connected {}"
by (simp add: simply_connected_def)
lemma simply_connected_imp_path_connected:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟹ path_connected S"
by (simp add: simply_connected_def path_connected_eq_homotopic_points)
lemma simply_connected_imp_connected:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟹ connected S"
by (simp add: path_connected_imp_connected simply_connected_imp_path_connected)
lemma simply_connected_eq_contractible_loop_any:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟷
(∀p a. path p ∧ path_image p ⊆ S ∧
pathfinish p = pathstart p ∧ a ∈ S
⟶ homotopic_loops S p (linepath a a))"
apply (simp add: simply_connected_def)
apply (rule iffI, force, clarify)
apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
apply (fastforce simp add:)
using homotopic_loops_sym apply blast
done
lemma simply_connected_eq_contractible_loop_some:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟷
path_connected S ∧
(∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
⟶ (∃a. a ∈ S ∧ homotopic_loops S p (linepath a a)))"
apply (rule iffI)
apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
apply (drule_tac x=p in spec)
using homotopic_loops_trans path_connected_eq_homotopic_points
apply blast
done
lemma simply_connected_eq_contractible_loop_all:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟷
S = {} ∨
(∃a ∈ S. ∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
⟶ homotopic_loops S p (linepath a a))"
(is "?lhs = ?rhs")
proof (cases "S = {}")
case True then show ?thesis by force
next
case False
then obtain a where "a ∈ S" by blast
show ?thesis
proof
assume "simply_connected S"
then show ?rhs
using ‹a ∈ S› ‹simply_connected S› simply_connected_eq_contractible_loop_any
by blast
next
assume ?rhs
then show "simply_connected S"
apply (simp add: simply_connected_eq_contractible_loop_any False)
by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
path_component_imp_homotopic_points path_component_refl)
qed
qed
lemma simply_connected_eq_contractible_path:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟷
path_connected S ∧
(∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
⟶ homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
apply (rule iffI)
apply (simp add: simply_connected_imp_path_connected)
apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
simply_connected_eq_contractible_loop_some subset_iff)
lemma simply_connected_eq_homotopic_paths:
fixes S :: "_::real_normed_vector set"
shows "simply_connected S ⟷
path_connected S ∧
(∀p q. path p ∧ path_image p ⊆ S ∧
path q ∧ path_image q ⊆ S ∧
pathstart q = pathstart p ∧ pathfinish q = pathfinish p
⟶ homotopic_paths S p q)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have pc: "path_connected S"
and *: "⋀p. ⟦path p; path_image p ⊆ S;
pathfinish p = pathstart p⟧
⟹ homotopic_paths S p (linepath (pathstart p) (pathstart p))"
by (auto simp: simply_connected_eq_contractible_path)
have "homotopic_paths S p q"
if "path p" "path_image p ⊆ S" "path q"
"path_image q ⊆ S" "pathstart q = pathstart p"
"pathfinish q = pathfinish p" for p q
proof -
have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
by (simp add: homotopic_paths_rid homotopic_paths_sym that)
also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
(p +++ reversepath q +++ q)"
using that
by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
also have "homotopic_paths S (p +++ reversepath q +++ q)
((p +++ reversepath q) +++ q)"
by (simp add: that homotopic_paths_assoc)
also have "homotopic_paths S ((p +++ reversepath q) +++ q)
(linepath (pathstart q) (pathstart q) +++ q)"
using * [of "p +++ reversepath q"] that
by (simp add: homotopic_paths_join path_image_join)
also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q"
using that homotopic_paths_lid by blast
finally show ?thesis .
qed
then show ?rhs
by (blast intro: pc *)
next
assume ?rhs
then show ?lhs
by (force simp: simply_connected_eq_contractible_path)
qed
proposition simply_connected_Times:
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
assumes S: "simply_connected S" and T: "simply_connected T"
shows "simply_connected(S × T)"
proof -
have "homotopic_loops (S × T) p (linepath (a, b) (a, b))"
if "path p" "path_image p ⊆ S × T" "p 1 = p 0" "a ∈ S" "b ∈ T"
for p a b
proof -
have "path (fst ∘ p)"
apply (rule Path_Connected.path_continuous_image [OF ‹path p›])
apply (rule continuous_intros)+
done
moreover have "path_image (fst ∘ p) ⊆ S"
using that apply (simp add: path_image_def) by force
ultimately have p1: "homotopic_loops S (fst ∘ p) (linepath a a)"
using S that
apply (simp add: simply_connected_eq_contractible_loop_any)
apply (drule_tac x="fst ∘ p" in spec)
apply (drule_tac x=a in spec)
apply (auto simp: pathstart_def pathfinish_def)
done
have "path (snd ∘ p)"
apply (rule Path_Connected.path_continuous_image [OF ‹path p›])
apply (rule continuous_intros)+
done
moreover have "path_image (snd ∘ p) ⊆ T"
using that apply (simp add: path_image_def) by force
ultimately have p2: "homotopic_loops T (snd ∘ p) (linepath b b)"
using T that
apply (simp add: simply_connected_eq_contractible_loop_any)
apply (drule_tac x="snd ∘ p" in spec)
apply (drule_tac x=b in spec)
apply (auto simp: pathstart_def pathfinish_def)
done
show ?thesis
using p1 p2
apply (simp add: homotopic_loops, clarify)
apply (rename_tac h k)
apply (rule_tac x="λz. Pair (h z) (k z)" in exI)
apply (intro conjI continuous_intros | assumption)+
apply (auto simp: pathstart_def pathfinish_def)
done
qed
with assms show ?thesis
by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
qed
subsection‹Contractible sets›
definition%important contractible where
"contractible S ≡ ∃a. homotopic_with (λx. True) S S id (λx. a)"
proposition contractible_imp_simply_connected:
fixes S :: "_::real_normed_vector set"
assumes "contractible S" shows "simply_connected S"
proof (cases "S = {}")
case True then show ?thesis by force
next
case False
obtain a where a: "homotopic_with (λx. True) S S id (λx. a)"
using assms by (force simp: contractible_def)
then have "a ∈ S"
by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
show ?thesis
apply (simp add: simply_connected_eq_contractible_loop_all False)
apply (rule bexI [OF _ ‹a ∈ S›])
using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
apply (rule_tac x="(h ∘ (λy. (fst y, (p ∘ snd) y)))" in exI)
apply (intro conjI continuous_on_compose continuous_intros)
apply (erule continuous_on_subset | force)+
done
qed
corollary contractible_imp_connected:
fixes S :: "_::real_normed_vector set"
shows "contractible S ⟹ connected S"
by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)
lemma contractible_imp_path_connected:
fixes S :: "_::real_normed_vector set"
shows "contractible S ⟹ path_connected S"
by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)
lemma nullhomotopic_through_contractible:
fixes S :: "_::topological_space set"
assumes f: "continuous_on S f" "f ` S ⊆ T"
and g: "continuous_on T g" "g ` T ⊆ U"
and T: "contractible T"
obtains c where "homotopic_with (λh. True) S U (g ∘ f) (λx. c)"
proof -
obtain b where b: "homotopic_with (λx. True) T T id (λx. b)"
using assms by (force simp: contractible_def)
have "homotopic_with (λf. True) T U (g ∘ id) (g ∘ (λx. b))"
by (rule homotopic_compose_continuous_left [OF b g])
then have "homotopic_with (λf. True) S U (g ∘ id ∘ f) (g ∘ (λx. b) ∘ f)"
by (rule homotopic_compose_continuous_right [OF _ f])
then show ?thesis
by (simp add: comp_def that)
qed
lemma nullhomotopic_into_contractible:
assumes f: "continuous_on S f" "f ` S ⊆ T"
and T: "contractible T"
obtains c where "homotopic_with (λh. True) S T f (λx. c)"
apply (rule nullhomotopic_through_contractible [OF f, of id T])
using assms
apply (auto simp: continuous_on_id)
done
lemma nullhomotopic_from_contractible:
assumes f: "continuous_on S f" "f ` S ⊆ T"
and S: "contractible S"
obtains c where "homotopic_with (λh. True) S T f (λx. c)"
apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
using assms
apply (auto simp: comp_def)
done
lemma homotopic_through_contractible:
fixes S :: "_::real_normed_vector set"
assumes "continuous_on S f1" "f1 ` S ⊆ T"
"continuous_on T g1" "g1 ` T ⊆ U"
"continuous_on S f2" "f2 ` S ⊆ T"
"continuous_on T g2" "g2 ` T ⊆ U"
"contractible T" "path_connected U"
shows "homotopic_with (λh. True) S U (g1 ∘ f1) (g2 ∘ f2)"
proof -
obtain c1 where c1: "homotopic_with (λh. True) S U (g1 ∘ f1) (λx. c1)"
apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
using assms apply auto
done
obtain c2 where c2: "homotopic_with (λh. True) S U (g2 ∘ f2) (λx. c2)"
apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
using assms apply auto
done
have *: "S = {} ∨ (∃t. path_connected t ∧ t ⊆ U ∧ c2 ∈ t ∧ c1 ∈ t)"
proof (cases "S = {}")
case True then show ?thesis by force
next
case False
with c1 c2 have "c1 ∈ U" "c2 ∈ U"
using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
with ‹path_connected U› show ?thesis by blast
qed
show ?thesis
apply (rule homotopic_with_trans [OF c1])
apply (rule homotopic_with_symD)
apply (rule homotopic_with_trans [OF c2])
apply (simp add: path_component homotopic_constant_maps *)
done
qed
lemma homotopic_into_contractible:
fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
assumes f: "continuous_on S f" "f ` S ⊆ T"
and g: "continuous_on S g" "g ` S ⊆ T"
and T: "contractible T"
shows "homotopic_with (λh. True) S T f g"
using homotopic_through_contractible [of S f T id T g id]
by (simp add: assms contractible_imp_path_connected continuous_on_id)
lemma homotopic_from_contractible:
fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
assumes f: "continuous_on S f" "f ` S ⊆ T"
and g: "continuous_on S g" "g ` S ⊆ T"
and "contractible S" "path_connected T"
shows "homotopic_with (λh. True) S T f g"
using homotopic_through_contractible [of S id S f T id g]
by (simp add: assms contractible_imp_path_connected continuous_on_id)
lemma starlike_imp_contractible_gen:
fixes S :: "'a::real_normed_vector set"
assumes S: "starlike S"
and P: "⋀a T. ⟦a ∈ S; 0 ≤ T; T ≤ 1⟧ ⟹ P(λx. (1 - T) *⇩R x + T *⇩R a)"
obtains a where "homotopic_with P S S (λx. x) (λx. a)"
proof -
obtain a where "a ∈ S" and a: "⋀x. x ∈ S ⟹ closed_segment a x ⊆ S"
using S by (auto simp: starlike_def)
have "(λy. (1 - fst y) *⇩R snd y + fst y *⇩R a) ` ({0..1} × S) ⊆ S"
apply clarify
apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
done
then show ?thesis
apply (rule_tac a=a in that)
using ‹a ∈ S›
apply (simp add: homotopic_with_def)
apply (rule_tac x="λy. (1 - (fst y)) *⇩R snd y + (fst y) *⇩R a" in exI)
apply (intro conjI ballI continuous_on_compose continuous_intros)
apply (simp_all add: P)
done
qed
lemma starlike_imp_contractible:
fixes S :: "'a::real_normed_vector set"
shows "starlike S ⟹ contractible S"
using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)
lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
by (simp add: starlike_imp_contractible)
lemma starlike_imp_simply_connected:
fixes S :: "'a::real_normed_vector set"
shows "starlike S ⟹ simply_connected S"
by (simp add: contractible_imp_simply_connected starlike_imp_contractible)
lemma convex_imp_simply_connected:
fixes S :: "'a::real_normed_vector set"
shows "convex S ⟹ simply_connected S"
using convex_imp_starlike starlike_imp_simply_connected by blast
lemma starlike_imp_path_connected:
fixes S :: "'a::real_normed_vector set"
shows "starlike S ⟹ path_connected S"
by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)
lemma starlike_imp_connected:
fixes S :: "'a::real_normed_vector set"
shows "starlike S ⟹ connected S"
by (simp add: path_connected_imp_connected starlike_imp_path_connected)
lemma is_interval_simply_connected_1:
fixes S :: "real set"
shows "is_interval S ⟷ simply_connected S"
using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
lemma contractible_empty [simp]: "contractible {}"
by (simp add: contractible_def homotopic_with)
lemma contractible_convex_tweak_boundary_points:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and TS: "rel_interior S ⊆ T" "T ⊆ closure S"
shows "contractible T"
proof (cases "S = {}")
case True
with assms show ?thesis
by (simp add: subsetCE)
next
case False
show ?thesis
apply (rule starlike_imp_contractible)
apply (rule starlike_convex_tweak_boundary_points [OF ‹convex S› False TS])
done
qed
lemma convex_imp_contractible:
fixes S :: "'a::real_normed_vector set"
shows "convex S ⟹ contractible S"
using contractible_empty convex_imp_starlike starlike_imp_contractible by blast
lemma contractible_sing [simp]:
fixes a :: "'a::real_normed_vector"
shows "contractible {a}"
by (rule convex_imp_contractible [OF convex_singleton])
lemma is_interval_contractible_1:
fixes S :: "real set"
shows "is_interval S ⟷ contractible S"
using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
is_interval_simply_connected_1 by auto
lemma contractible_Times:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes S: "contractible S" and T: "contractible T"
shows "contractible (S × T)"
proof -
obtain a h where conth: "continuous_on ({0..1} × S) h"
and hsub: "h ` ({0..1} × S) ⊆ S"
and [simp]: "⋀x. x ∈ S ⟹ h (0, x) = x"
and [simp]: "⋀x. x ∈ S ⟹ h (1::real, x) = a"
using S by (auto simp: contractible_def homotopic_with)
obtain b k where contk: "continuous_on ({0..1} × T) k"
and ksub: "k ` ({0..1} × T) ⊆ T"
and [simp]: "⋀x. x ∈ T ⟹ k (0, x) = x"
and [simp]: "⋀x. x ∈ T ⟹ k (1::real, x) = b"
using T by (auto simp: contractible_def homotopic_with)
show ?thesis
apply (simp add: contractible_def homotopic_with)
apply (rule exI [where x=a])
apply (rule exI [where x=b])
apply (rule exI [where x = "λz. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
using hsub ksub
apply auto
done
qed
lemma homotopy_dominated_contractibility:
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
assumes S: "contractible S"
and f: "continuous_on S f" "image f S ⊆ T"
and g: "continuous_on T g" "image g T ⊆ S"
and hom: "homotopic_with (λx. True) T T (f ∘ g) id"
shows "contractible T"
proof -
obtain b where "homotopic_with (λh. True) S T f (λx. b)"
using nullhomotopic_from_contractible [OF f S] .
then have homg: "homotopic_with (λx. True) T T ((λx. b) ∘ g) (f ∘ g)"
by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
show ?thesis
apply (simp add: contractible_def)
apply (rule exI [where x = b])
apply (rule homotopic_with_symD)
apply (rule homotopic_with_trans [OF _ hom])
using homg apply (simp add: o_def)
done
qed
subsection‹Local versions of topological properties in general›
definition%important locally :: "('a::topological_space set ⇒ bool) ⇒ 'a set ⇒ bool"
where
"locally P S ≡
∀w x. openin (subtopology euclidean S) w ∧ x ∈ w
⟶ (∃u v. openin (subtopology euclidean S) u ∧ P v ∧
x ∈ u ∧ u ⊆ v ∧ v ⊆ w)"
lemma locallyI:
assumes "⋀w x. ⟦openin (subtopology euclidean S) w; x ∈ w⟧
⟹ ∃u v. openin (subtopology euclidean S) u ∧ P v ∧
x ∈ u ∧ u ⊆ v ∧ v ⊆ w"
shows "locally P S"
using assms by (force simp: locally_def)
lemma locallyE:
assumes "locally P S" "openin (subtopology euclidean S) w" "x ∈ w"
obtains u v where "openin (subtopology euclidean S) u"
"P v" "x ∈ u" "u ⊆ v" "v ⊆ w"
using assms unfolding locally_def by meson
lemma locally_mono:
assumes "locally P S" "⋀t. P t ⟹ Q t"
shows "locally Q S"
by (metis assms locally_def)
lemma locally_open_subset:
assumes "locally P S" "openin (subtopology euclidean S) t"
shows "locally P t"
using assms
apply (simp add: locally_def)
apply (erule all_forward)+
apply (rule impI)
apply (erule impCE)
using openin_trans apply blast
apply (erule ex_forward)
by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
lemma locally_diff_closed:
"⟦locally P S; closedin (subtopology euclidean S) t⟧ ⟹ locally P (S - t)"
using locally_open_subset closedin_def by fastforce
lemma locally_empty [iff]: "locally P {}"
by (simp add: locally_def openin_subtopology)
lemma locally_singleton [iff]:
fixes a :: "'a::metric_space"
shows "locally P {a} ⟷ P {a}"
apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
using zero_less_one by blast
lemma locally_iff:
"locally P S ⟷
(∀T x. open T ∧ x ∈ S ∩ T ⟶ (∃U. open U ∧ (∃v. P v ∧ x ∈ S ∩ U ∧ S ∩ U ⊆ v ∧ v ⊆ S ∩ T)))"
apply (simp add: le_inf_iff locally_def openin_open, safe)
apply (metis IntE IntI le_inf_iff)
apply (metis IntI Int_subset_iff)
done
lemma locally_Int:
assumes S: "locally P S" and t: "locally P t"
and P: "⋀S t. P S ∧ P t ⟹ P(S ∩ t)"
shows "locally P (S ∩ t)"
using S t unfolding locally_iff
apply clarify
apply (drule_tac x=T in spec)+
apply (drule_tac x=x in spec)+
apply clarsimp
apply (rename_tac U1 U2 V1 V2)
apply (rule_tac x="U1 ∩ U2" in exI)
apply (simp add: open_Int)
apply (rule_tac x="V1 ∩ V2" in exI)
apply (auto intro: P)
done
lemma locally_Times:
fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
assumes PS: "locally P S" and QT: "locally Q T" and R: "⋀S T. P S ∧ Q T ⟹ R(S × T)"
shows "locally R (S × T)"
unfolding locally_def
proof (clarify)
fix W x y
assume W: "openin (subtopology euclidean (S × T)) W" and xy: "(x, y) ∈ W"
then obtain U V where "openin (subtopology euclidean S) U" "x ∈ U"
"openin (subtopology euclidean T) V" "y ∈ V" "U × V ⊆ W"
using Times_in_interior_subtopology by metis
then obtain U1 U2 V1 V2
where opeS: "openin (subtopology euclidean S) U1 ∧ P U2 ∧ x ∈ U1 ∧ U1 ⊆ U2 ∧ U2 ⊆ U"
and opeT: "openin (subtopology euclidean T) V1 ∧ Q V2 ∧ y ∈ V1 ∧ V1 ⊆ V2 ∧ V2 ⊆ V"
by (meson PS QT locallyE)
with ‹U × V ⊆ W› show "∃u v. openin (subtopology euclidean (S × T)) u ∧ R v ∧ (x,y) ∈ u ∧ u ⊆ v ∧ v ⊆ W"
apply (rule_tac x="U1 × V1" in exI)
apply (rule_tac x="U2 × V2" in exI)
apply (auto simp: openin_Times R)
done
qed
proposition homeomorphism_locally_imp:
fixes S :: "'a::metric_space set" and t :: "'b::t2_space set"
assumes S: "locally P S" and hom: "homeomorphism S t f g"
and Q: "⋀S S'. ⟦P S; homeomorphism S S' f g⟧ ⟹ Q S'"
shows "locally Q t"
proof (clarsimp simp: locally_def)
fix W y
assume "y ∈ W" and "openin (subtopology euclidean t) W"
then obtain T where T: "open T" "W = t ∩ T"
by (force simp: openin_open)
then have "W ⊆ t" by auto
have f: "⋀x. x ∈ S ⟹ g(f x) = x" "f ` S = t" "continuous_on S f"
and g: "⋀y. y ∈ t ⟹ f(g y) = y" "g ` t = S" "continuous_on t g"
using hom by (auto simp: homeomorphism_def)
have gw: "g ` W = S ∩ f -` W"
using ‹W ⊆ t›
apply auto
using ‹g ` t = S› ‹W ⊆ t› apply blast
using g ‹W ⊆ t› apply auto[1]
by (simp add: f rev_image_eqI)
have ∘: "openin (subtopology euclidean S) (g ` W)"
proof -
have "continuous_on S f"
using f(3) by blast
then show "openin (subtopology euclidean S) (g ` W)"
by (simp add: gw Collect_conj_eq ‹openin (subtopology euclidean t) W› continuous_on_open f(2))
qed
then obtain u v
where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y ∈ u" "u ⊆ v" "v ⊆ g ` W"
using S [unfolded locally_def, rule_format, of "g ` W" "g y"] ‹y ∈ W› by force
have "v ⊆ S" using uv by (simp add: gw)
have fv: "f ` v = t ∩ {x. g x ∈ v}"
using ‹f ` S = t› f ‹v ⊆ S› by auto
have "f ` v ⊆ W"
using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
have contvf: "continuous_on v f"
using ‹v ⊆ S› continuous_on_subset f(3) by blast
have contvg: "continuous_on (f ` v) g"
using ‹f ` v ⊆ W› ‹W ⊆ t› continuous_on_subset [OF g(3)] by blast
have homv: "homeomorphism v (f ` v) f g"
using ‹v ⊆ S› ‹W ⊆ t› f
apply (simp add: homeomorphism_def contvf contvg, auto)
by (metis f(1) rev_image_eqI rev_subsetD)
have 1: "openin (subtopology euclidean t) (t ∩ g -` u)"
apply (rule continuous_on_open [THEN iffD1, rule_format])
apply (rule ‹continuous_on t g›)
using ‹g ` t = S› apply (simp add: osu)
done
have 2: "∃V. Q V ∧ y ∈ (t ∩ g -` u) ∧ (t ∩ g -` u) ⊆ V ∧ V ⊆ W"
apply (rule_tac x="f ` v" in exI)
apply (intro conjI Q [OF ‹P v› homv])
using ‹W ⊆ t› ‹y ∈ W› ‹f ` v ⊆ W› uv apply (auto simp: fv)
done
show "∃U. openin (subtopology euclidean t) U ∧ (∃v. Q v ∧ y ∈ U ∧ U ⊆ v ∧ v ⊆ W)"
by (meson 1 2)
qed
lemma homeomorphism_locally:
fixes f:: "'a::metric_space ⇒ 'b::metric_space"
assumes hom: "homeomorphism S t f g"
and eq: "⋀S t. homeomorphism S t f g ⟹ (P S ⟷ Q t)"
shows "locally P S ⟷ locally Q t"
apply (rule iffI)
apply (erule homeomorphism_locally_imp [OF _ hom])
apply (simp add: eq)
apply (erule homeomorphism_locally_imp)
using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
done
lemma homeomorphic_locally:
fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
assumes hom: "S homeomorphic T"
and iff: "⋀X Y. X homeomorphic Y ⟹ (P X ⟷ Q Y)"
shows "locally P S ⟷ locally Q T"
proof -
obtain f g where hom: "homeomorphism S T f g"
using assms by (force simp: homeomorphic_def)
then show ?thesis
using homeomorphic_def local.iff
by (blast intro!: homeomorphism_locally)
qed
lemma homeomorphic_local_compactness:
fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
shows "S homeomorphic T ⟹ locally compact S ⟷ locally compact T"
by (simp add: homeomorphic_compactness homeomorphic_locally)
lemma locally_translation:
fixes P :: "'a :: real_normed_vector set ⇒ bool"
shows
"(⋀S. P (image (λx. a + x) S) ⟷ P S)
⟹ locally P (image (λx. a + x) S) ⟷ locally P S"
apply (rule homeomorphism_locally [OF homeomorphism_translation])
apply (simp add: homeomorphism_def)
by metis
lemma locally_injective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "linear f" "inj f" and iff: "⋀S. P (f ` S) ⟷ Q S"
shows "locally P (f ` S) ⟷ locally Q S"
apply (rule linear_homeomorphism_image [OF f])
apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
by (metis iff homeomorphism_def)
lemma locally_open_map_image:
fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes P: "locally P S"
and f: "continuous_on S f"
and oo: "⋀t. openin (subtopology euclidean S) t
⟹ openin (subtopology euclidean (f ` S)) (f ` t)"
and Q: "⋀t. ⟦t ⊆ S; P t⟧ ⟹ Q(f ` t)"
shows "locally Q (f ` S)"
proof (clarsimp simp add: locally_def)
fix W y
assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y ∈ W"
then have "W ⊆ f ` S" by (simp add: openin_euclidean_subtopology_iff)
have oivf: "openin (subtopology euclidean S) (S ∩ f -` W)"
by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
then obtain x where "x ∈ S" "f x = y"
using ‹W ⊆ f ` S› ‹y ∈ W› by blast
then obtain U V
where "openin (subtopology euclidean S) U" "P V" "x ∈ U" "U ⊆ V" "V ⊆ S ∩ f -` W"
using P [unfolded locally_def, rule_format, of "(S ∩ f -` W)" x] oivf ‹y ∈ W›
by auto
then show "∃X. openin (subtopology euclidean (f ` S)) X ∧ (∃Y. Q Y ∧ y ∈ X ∧ X ⊆ Y ∧ Y ⊆ W)"
apply (rule_tac x="f ` U" in exI)
apply (rule conjI, blast intro!: oo)
apply (rule_tac x="f ` V" in exI)
apply (force simp: ‹f x = y› rev_image_eqI intro: Q)
done
qed
subsection‹Sort of induction principle for connected sets›
proposition connected_induction:
assumes "connected S"
and opD: "⋀T a. ⟦openin (subtopology euclidean S) T; a ∈ T⟧ ⟹ ∃z. z ∈ T ∧ P z"
and opI: "⋀a. a ∈ S
⟹ ∃T. openin (subtopology euclidean S) T ∧ a ∈ T ∧
(∀x ∈ T. ∀y ∈ T. P x ∧ P y ∧ Q x ⟶ Q y)"
and etc: "a ∈ S" "b ∈ S" "P a" "P b" "Q a"
shows "Q b"
proof -
have 1: "openin (subtopology euclidean S)
{b. ∃T. openin (subtopology euclidean S) T ∧
b ∈ T ∧ (∀x∈T. P x ⟶ Q x)}"
apply (subst openin_subopen, clarify)
apply (rule_tac x=T in exI, auto)
done
have 2: "openin (subtopology euclidean S)
{b. ∃T. openin (subtopology euclidean S) T ∧
b ∈ T ∧ (∀x∈T. P x ⟶ ~ Q x)}"
apply (subst openin_subopen, clarify)
apply (rule_tac x=T in exI, auto)
done
show ?thesis
using ‹connected S›
apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
apply (elim disjE allE)
apply (blast intro: 1)
apply (blast intro: 2, simp_all)
apply clarify apply (metis opI)
using opD apply (blast intro: etc elim: dest:)
using opI etc apply meson+
done
qed
lemma connected_equivalence_relation_gen:
assumes "connected S"
and etc: "a ∈ S" "b ∈ S" "P a" "P b"
and trans: "⋀x y z. ⟦R x y; R y z⟧ ⟹ R x z"
and opD: "⋀T a. ⟦openin (subtopology euclidean S) T; a ∈ T⟧ ⟹ ∃z. z ∈ T ∧ P z"
and opI: "⋀a. a ∈ S
⟹ ∃T. openin (subtopology euclidean S) T ∧ a ∈ T ∧
(∀x ∈ T. ∀y ∈ T. P x ∧ P y ⟶ R x y)"
shows "R a b"
proof -
have "⋀a b c. ⟦a ∈ S; P a; b ∈ S; c ∈ S; P b; P c; R a b⟧ ⟹ R a c"
apply (rule connected_induction [OF ‹connected S› opD], simp_all)
by (meson trans opI)
then show ?thesis by (metis etc opI)
qed
lemma connected_induction_simple:
assumes "connected S"
and etc: "a ∈ S" "b ∈ S" "P a"
and opI: "⋀a. a ∈ S
⟹ ∃T. openin (subtopology euclidean S) T ∧ a ∈ T ∧
(∀x ∈ T. ∀y ∈ T. P x ⟶ P y)"
shows "P b"
apply (rule connected_induction [OF ‹connected S› _, where P = "λx. True"], blast)
apply (frule opI)
using etc apply simp_all
done
lemma connected_equivalence_relation:
assumes "connected S"
and etc: "a ∈ S" "b ∈ S"
and sym: "⋀x y. ⟦R x y; x ∈ S; y ∈ S⟧ ⟹ R y x"
and trans: "⋀x y z. ⟦R x y; R y z; x ∈ S; y ∈ S; z ∈ S⟧ ⟹ R x z"
and opI: "⋀a. a ∈ S ⟹ ∃T. openin (subtopology euclidean S) T ∧ a ∈ T ∧ (∀x ∈ T. R a x)"
shows "R a b"
proof -
have "⋀a b c. ⟦a ∈ S; b ∈ S; c ∈ S; R a b⟧ ⟹ R a c"
apply (rule connected_induction_simple [OF ‹connected S›], simp_all)
by (meson local.sym local.trans opI openin_imp_subset subsetCE)
then show ?thesis by (metis etc opI)
qed
lemma locally_constant_imp_constant:
assumes "connected S"
and opI: "⋀a. a ∈ S
⟹ ∃T. openin (subtopology euclidean S) T ∧ a ∈ T ∧ (∀x ∈ T. f x = f a)"
shows "f constant_on S"
proof -
have "⋀x y. x ∈ S ⟹ y ∈ S ⟹ f x = f y"
apply (rule connected_equivalence_relation [OF ‹connected S›], simp_all)
by (metis opI)
then show ?thesis
by (metis constant_on_def)
qed
lemma locally_constant:
"connected S ⟹ locally (λU. f constant_on U) S ⟷ f constant_on S"
apply (simp add: locally_def)
apply (rule iffI)
apply (rule locally_constant_imp_constant, assumption)
apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
by (meson constant_on_subset openin_imp_subset order_refl)
subsection‹Basic properties of local compactness›
proposition locally_compact:
fixes s :: "'a :: metric_space set"
shows
"locally compact s ⟷
(∀x ∈ s. ∃u v. x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∧
openin (subtopology euclidean s) u ∧ compact v)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply clarify
apply (erule_tac w = "s ∩ ball x 1" in locallyE)
by auto
next
assume r [rule_format]: ?rhs
have *: "∃u v.
openin (subtopology euclidean s) u ∧
compact v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∩ T"
if "open T" "x ∈ s" "x ∈ T" for x T
proof -
obtain u v where uv: "x ∈ u" "u ⊆ v" "v ⊆ s" "compact v" "openin (subtopology euclidean s) u"
using r [OF ‹x ∈ s›] by auto
obtain e where "e>0" and e: "cball x e ⊆ T"
using open_contains_cball ‹open T› ‹x ∈ T› by blast
show ?thesis
apply (rule_tac x="(s ∩ ball x e) ∩ u" in exI)
apply (rule_tac x="cball x e ∩ v" in exI)
using that ‹e > 0› e uv
apply auto
done
qed
show ?lhs
apply (rule locallyI)
apply (subst (asm) openin_open)
apply (blast intro: *)
done
qed
lemma locally_compactE:
fixes s :: "'a :: metric_space set"
assumes "locally compact s"
obtains u v where "⋀x. x ∈ s ⟹ x ∈ u x ∧ u x ⊆ v x ∧ v x ⊆ s ∧
openin (subtopology euclidean s) (u x) ∧ compact (v x)"
using assms
unfolding locally_compact by metis
lemma locally_compact_alt:
fixes s :: "'a :: heine_borel set"
shows "locally compact s ⟷
(∀x ∈ s. ∃u. x ∈ u ∧
openin (subtopology euclidean s) u ∧ compact(closure u) ∧ closure u ⊆ s)"
apply (simp add: locally_compact)
apply (intro ball_cong ex_cong refl iffI)
apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
by (meson closure_subset compact_closure)
lemma locally_compact_Int_cball:
fixes s :: "'a :: heine_borel set"
shows "locally compact s ⟷ (∀x ∈ s. ∃e. 0 < e ∧ closed(cball x e ∩ s))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (simp add: locally_compact openin_contains_cball)
apply (clarify | assumption | drule bspec)+
by (metis (no_types, lifting) compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
next
assume ?rhs
then show ?lhs
apply (simp add: locally_compact openin_contains_cball)
apply (clarify | assumption | drule bspec)+
apply (rule_tac x="ball x e ∩ s" in exI, simp)
apply (rule_tac x="cball x e ∩ s" in exI)
using compact_eq_bounded_closed
apply auto
apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
done
qed
lemma locally_compact_compact:
fixes s :: "'a :: heine_borel set"
shows "locally compact s ⟷
(∀k. k ⊆ s ∧ compact k
⟶ (∃u v. k ⊆ u ∧ u ⊆ v ∧ v ⊆ s ∧
openin (subtopology euclidean s) u ∧ compact v))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain u v where
uv: "⋀x. x ∈ s ⟹ x ∈ u x ∧ u x ⊆ v x ∧ v x ⊆ s ∧
openin (subtopology euclidean s) (u x) ∧ compact (v x)"
by (metis locally_compactE)
have *: "∃u v. k ⊆ u ∧ u ⊆ v ∧ v ⊆ s ∧ openin (subtopology euclidean s) u ∧ compact v"
if "k ⊆ s" "compact k" for k
proof -
have "⋀C. (∀c∈C. openin (subtopology euclidean k) c) ∧ k ⊆ ⋃C ⟹
∃D⊆C. finite D ∧ k ⊆ ⋃D"
using that by (simp add: compact_eq_openin_cover)
moreover have "∀c ∈ (λx. k ∩ u x) ` k. openin (subtopology euclidean k) c"
using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
moreover have "k ⊆ ⋃((λx. k ∩ u x) ` k)"
using that by clarsimp (meson subsetCE uv)
ultimately obtain D where "D ⊆ (λx. k ∩ u x) ` k" "finite D" "k ⊆ ⋃D"
by metis
then obtain T where T: "T ⊆ k" "finite T" "k ⊆ ⋃((λx. k ∩ u x) ` T)"
by (metis finite_subset_image)
have Tuv: "UNION T u ⊆ UNION T v"
using T that by (force simp: dest!: uv)
show ?thesis
apply (rule_tac x="⋃(u ` T)" in exI)
apply (rule_tac x="⋃(v ` T)" in exI)
apply (simp add: Tuv)
using T that
apply (auto simp: dest!: uv)
done
qed
show ?rhs
by (blast intro: *)
next
assume ?rhs
then show ?lhs
apply (clarsimp simp add: locally_compact)
apply (drule_tac x="{x}" in spec, simp)
done
qed
lemma open_imp_locally_compact:
fixes s :: "'a :: heine_borel set"
assumes "open s"
shows "locally compact s"
proof -
have *: "∃u v. x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∧ openin (subtopology euclidean s) u ∧ compact v"
if "x ∈ s" for x
proof -
obtain e where "e>0" and e: "cball x e ⊆ s"
using open_contains_cball assms ‹x ∈ s› by blast
have ope: "openin (subtopology euclidean s) (ball x e)"
by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
show ?thesis
apply (rule_tac x="ball x e" in exI)
apply (rule_tac x="cball x e" in exI)
using ‹e > 0› e apply (auto simp: ope)
done
qed
show ?thesis
unfolding locally_compact
by (blast intro: *)
qed
lemma closed_imp_locally_compact:
fixes s :: "'a :: heine_borel set"
assumes "closed s"
shows "locally compact s"
proof -
have *: "∃u v. x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∧
openin (subtopology euclidean s) u ∧ compact v"
if "x ∈ s" for x
proof -
show ?thesis
apply (rule_tac x = "s ∩ ball x 1" in exI)
apply (rule_tac x = "s ∩ cball x 1" in exI)
using ‹x ∈ s› assms apply auto
done
qed
show ?thesis
unfolding locally_compact
by (blast intro: *)
qed
lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
by (simp add: closed_imp_locally_compact)
lemma locally_compact_Int:
fixes s :: "'a :: t2_space set"
shows "⟦locally compact s; locally compact t⟧ ⟹ locally compact (s ∩ t)"
by (simp add: compact_Int locally_Int)
lemma locally_compact_closedin:
fixes s :: "'a :: heine_borel set"
shows "⟦closedin (subtopology euclidean s) t; locally compact s⟧
⟹ locally compact t"
unfolding closedin_closed
using closed_imp_locally_compact locally_compact_Int by blast
lemma locally_compact_delete:
fixes s :: "'a :: t1_space set"
shows "locally compact s ⟹ locally compact (s - {a})"
by (auto simp: openin_delete locally_open_subset)
lemma locally_closed:
fixes s :: "'a :: heine_borel set"
shows "locally closed s ⟷ locally compact s"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (simp only: locally_def)
apply (erule all_forward imp_forward asm_rl exE)+
apply (rule_tac x = "u ∩ ball x 1" in exI)
apply (rule_tac x = "v ∩ cball x 1" in exI)
apply (force intro: openin_trans)
done
next
assume ?rhs then show ?lhs
using compact_eq_bounded_closed locally_mono by blast
qed
lemma locally_compact_openin_Un:
fixes S :: "'a::euclidean_space set"
assumes LCS: "locally compact S" and LCT:"locally compact T"
and opS: "openin (subtopology euclidean (S ∪ T)) S"
and opT: "openin (subtopology euclidean (S ∪ T)) T"
shows "locally compact (S ∪ T)"
proof -
have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ S" for x
proof -
obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
moreover obtain e2 where "e2 > 0" and e2: "cball x e2 ∩ (S ∪ T) ⊆ S"
by (meson ‹x ∈ S› opS openin_contains_cball)
then have "cball x e2 ∩ (S ∪ T) = cball x e2 ∩ S"
by force
ultimately show ?thesis
apply (rule_tac x="min e1 e2" in exI)
apply (auto simp: cball_min_Int ‹e2 > 0› inf_assoc closed_Int)
by (metis closed_Int closed_cball inf_left_commute)
qed
moreover have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ T" for x
proof -
obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ T)"
using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
moreover obtain e2 where "e2 > 0" and e2: "cball x e2 ∩ (S ∪ T) ⊆ T"
by (meson ‹x ∈ T› opT openin_contains_cball)
then have "cball x e2 ∩ (S ∪ T) = cball x e2 ∩ T"
by force
ultimately show ?thesis
apply (rule_tac x="min e1 e2" in exI)
apply (auto simp: cball_min_Int ‹e2 > 0› inf_assoc closed_Int)
by (metis closed_Int closed_cball inf_left_commute)
qed
ultimately show ?thesis
by (force simp: locally_compact_Int_cball)
qed
lemma locally_compact_closedin_Un:
fixes S :: "'a::euclidean_space set"
assumes LCS: "locally compact S" and LCT:"locally compact T"
and clS: "closedin (subtopology euclidean (S ∪ T)) S"
and clT: "closedin (subtopology euclidean (S ∪ T)) T"
shows "locally compact (S ∪ T)"
proof -
have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ S" "x ∈ T" for x
proof -
obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
moreover
obtain e2 where "e2 > 0" and e2: "closed (cball x e2 ∩ T)"
using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
ultimately show ?thesis
apply (rule_tac x="min e1 e2" in exI)
apply (auto simp: cball_min_Int ‹e2 > 0› inf_assoc closed_Int Int_Un_distrib)
by (metis closed_Int closed_Un closed_cball inf_left_commute)
qed
moreover
have "∃e>0. closed (cball x e ∩ (S ∪ T))" if x: "x ∈ S" "x ∉ T" for x
proof -
obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
moreover
obtain e2 where "e2>0" and "cball x e2 ∩ (S ∪ T) ⊆ S - T"
using clT x by (fastforce simp: openin_contains_cball closedin_def)
then have "closed (cball x e2 ∩ T)"
proof -
have "{} = T - (T - cball x e2)"
using Diff_subset Int_Diff ‹cball x e2 ∩ (S ∪ T) ⊆ S - T› by auto
then show ?thesis
by (simp add: Diff_Diff_Int inf_commute)
qed
ultimately show ?thesis
apply (rule_tac x="min e1 e2" in exI)
apply (auto simp: cball_min_Int ‹e2 > 0› inf_assoc closed_Int Int_Un_distrib)
by (metis closed_Int closed_Un closed_cball inf_left_commute)
qed
moreover
have "∃e>0. closed (cball x e ∩ (S ∪ T))" if x: "x ∉ S" "x ∈ T" for x
proof -
obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ T)"
using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
moreover
obtain e2 where "e2>0" and "cball x e2 ∩ (S ∪ T) ⊆ S ∪ T - S"
using clS x by (fastforce simp: openin_contains_cball closedin_def)
then have "closed (cball x e2 ∩ S)"
by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
ultimately show ?thesis
apply (rule_tac x="min e1 e2" in exI)
apply (auto simp: cball_min_Int ‹e2 > 0› inf_assoc closed_Int Int_Un_distrib)
by (metis closed_Int closed_Un closed_cball inf_left_commute)
qed
ultimately show ?thesis
by (auto simp: locally_compact_Int_cball)
qed
lemma locally_compact_Times:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
shows "⟦locally compact S; locally compact T⟧ ⟹ locally compact (S × T)"
by (auto simp: compact_Times locally_Times)
lemma locally_compact_compact_subopen:
fixes S :: "'a :: heine_borel set"
shows
"locally compact S ⟷
(∀K T. K ⊆ S ∧ compact K ∧ open T ∧ K ⊆ T
⟶ (∃U V. K ⊆ U ∧ U ⊆ V ∧ U ⊆ T ∧ V ⊆ S ∧
openin (subtopology euclidean S) U ∧ compact V))"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
show ?rhs
proof clarify
fix K :: "'a set" and T :: "'a set"
assume "K ⊆ S" and "compact K" and "open T" and "K ⊆ T"
obtain U V where "K ⊆ U" "U ⊆ V" "V ⊆ S" "compact V"
and ope: "openin (subtopology euclidean S) U"
using L unfolding locally_compact_compact by (meson ‹K ⊆ S› ‹compact K›)
show "∃U V. K ⊆ U ∧ U ⊆ V ∧ U ⊆ T ∧ V ⊆ S ∧
openin (subtopology euclidean S) U ∧ compact V"
proof (intro exI conjI)
show "K ⊆ U ∩ T"
by (simp add: ‹K ⊆ T› ‹K ⊆ U›)
show "U ∩ T ⊆ closure(U ∩ T)"
by (rule closure_subset)
show "closure (U ∩ T) ⊆ S"
by (metis ‹U ⊆ V› ‹V ⊆ S› ‹compact V› closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
show "openin (subtopology euclidean S) (U ∩ T)"
by (simp add: ‹open T› ope openin_Int_open)
show "compact (closure (U ∩ T))"
by (meson Int_lower1 ‹U ⊆ V› ‹compact V› bounded_subset compact_closure compact_eq_bounded_closed)
qed auto
qed
next
assume ?rhs then show ?lhs
unfolding locally_compact_compact
by (metis open_openin openin_topspace subtopology_superset top.extremum topspace_euclidean_subtopology)
qed
subsection‹Sura-Bura's results about compact components of sets›
proposition Sura_Bura_compact:
fixes S :: "'a::euclidean_space set"
assumes "compact S" and C: "C ∈ components S"
shows "C = ⋂{T. C ⊆ T ∧ openin (subtopology euclidean S) T ∧
closedin (subtopology euclidean S) T}"
(is "C = ⋂?𝒯")
proof
obtain x where x: "C = connected_component_set S x" and "x ∈ S"
using C by (auto simp: components_def)
have "C ⊆ S"
by (simp add: C in_components_subset)
have "⋂?𝒯 ⊆ connected_component_set S x"
proof (rule connected_component_maximal)
have "x ∈ C"
by (simp add: ‹x ∈ S› x)
then show "x ∈ ⋂?𝒯"
by blast
have clo: "closed (⋂?𝒯)"
by (simp add: ‹compact S› closed_Inter closedin_compact_eq compact_imp_closed)
have False
if K1: "closedin (subtopology euclidean (⋂?𝒯)) K1" and
K2: "closedin (subtopology euclidean (⋂?𝒯)) K2" and
K12_Int: "K1 ∩ K2 = {}" and K12_Un: "K1 ∪ K2 = ⋂?𝒯" and "K1 ≠ {}" "K2 ≠ {}"
for K1 K2
proof -
have "closed K1" "closed K2"
using closedin_closed_trans clo K1 K2 by blast+
then obtain V1 V2 where "open V1" "open V2" "K1 ⊆ V1" "K2 ⊆ V2" and V12: "V1 ∩ V2 = {}"
using separation_normal ‹K1 ∩ K2 = {}› by metis
have SV12_ne: "(S - (V1 ∪ V2)) ∩ (⋂?𝒯) ≠ {}"
proof (rule compact_imp_fip)
show "compact (S - (V1 ∪ V2))"
by (simp add: ‹open V1› ‹open V2› ‹compact S› compact_diff open_Un)
show clo𝒯: "closed T" if "T ∈ ?𝒯" for T
using that ‹compact S›
by (force intro: closedin_closed_trans simp add: compact_imp_closed)
show "(S - (V1 ∪ V2)) ∩ ⋂ℱ ≠ {}" if "finite ℱ" and ℱ: "ℱ ⊆ ?𝒯" for ℱ
proof
assume djo: "(S - (V1 ∪ V2)) ∩ ⋂ℱ = {}"
obtain D where opeD: "openin (subtopology euclidean S) D"
and cloD: "closedin (subtopology euclidean S) D"
and "C ⊆ D" and DV12: "D ⊆ V1 ∪ V2"
proof (cases "ℱ = {}")
case True
with ‹C ⊆ S› djo that show ?thesis
by force
next
case False show ?thesis
proof
show ope: "openin (subtopology euclidean S) (⋂ℱ)"
using openin_Inter ‹finite ℱ› False ℱ by blast
then show "closedin (subtopology euclidean S) (⋂ℱ)"
by (meson clo𝒯 ℱ closed_Inter closed_subset openin_imp_subset subset_eq)
show "C ⊆ ⋂ℱ"
using ℱ by auto
show "⋂ℱ ⊆ V1 ∪ V2"
using ope djo openin_imp_subset by fastforce
qed
qed
have "connected C"
by (simp add: x)
have "closed D"
using ‹compact S› cloD closedin_closed_trans compact_imp_closed by blast
have cloV1: "closedin (subtopology euclidean D) (D ∩ closure V1)"
and cloV2: "closedin (subtopology euclidean D) (D ∩ closure V2)"
by (simp_all add: closedin_closed_Int)
moreover have "D ∩ closure V1 = D ∩ V1" "D ∩ closure V2 = D ∩ V2"
apply safe
using ‹D ⊆ V1 ∪ V2› ‹open V1› ‹open V2› V12
apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
done
ultimately have cloDV1: "closedin (subtopology euclidean D) (D ∩ V1)"
and cloDV2: "closedin (subtopology euclidean D) (D ∩ V2)"
by metis+
then obtain U1 U2 where "closed U1" "closed U2"
and D1: "D ∩ V1 = D ∩ U1" and D2: "D ∩ V2 = D ∩ U2"
by (auto simp: closedin_closed)
have "D ∩ U1 ∩ C ≠ {}"
proof
assume "D ∩ U1 ∩ C = {}"
then have *: "C ⊆ D ∩ V2"
using D1 DV12 ‹C ⊆ D› by auto
have "⋂?𝒯 ⊆ D ∩ V2"
apply (rule Inter_lower)
using * apply simp
by (meson cloDV2 ‹open V2› cloD closedin_trans le_inf_iff opeD openin_Int_open)
then show False
using K1 V12 ‹K1 ≠ {}› ‹K1 ⊆ V1› closedin_imp_subset by blast
qed
moreover have "D ∩ U2 ∩ C ≠ {}"
proof
assume "D ∩ U2 ∩ C = {}"
then have *: "C ⊆ D ∩ V1"
using D2 DV12 ‹C ⊆ D› by auto
have "⋂?𝒯 ⊆ D ∩ V1"
apply (rule Inter_lower)
using * apply simp
by (meson cloDV1 ‹open V1› cloD closedin_trans le_inf_iff opeD openin_Int_open)
then show False
using K2 V12 ‹K2 ≠ {}› ‹K2 ⊆ V2› closedin_imp_subset by blast
qed
ultimately show False
using ‹connected C› unfolding connected_closed
apply (simp only: not_ex)
apply (drule_tac x="D ∩ U1" in spec)
apply (drule_tac x="D ∩ U2" in spec)
using ‹C ⊆ D› D1 D2 V12 DV12 ‹closed U1› ‹closed U2› ‹closed D›
by blast
qed
qed
show False
by (metis (full_types) DiffE UnE Un_upper2 SV12_ne ‹K1 ⊆ V1› ‹K2 ⊆ V2› disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
qed
then show "connected (⋂?𝒯)"
by (auto simp: connected_closedin_eq)
show "⋂?𝒯 ⊆ S"
by (fastforce simp: C in_components_subset)
qed
with x show "⋂?𝒯 ⊆ C" by simp
qed auto
corollary Sura_Bura_clopen_subset:
fixes S :: "'a::euclidean_space set"
assumes S: "locally compact S" and C: "C ∈ components S" and "compact C"
and U: "open U" "C ⊆ U"
obtains K where "openin (subtopology euclidean S) K" "compact K" "C ⊆ K" "K ⊆ U"
proof (rule ccontr)
assume "¬ thesis"
with that have neg: "∄K. openin (subtopology euclidean S) K ∧ compact K ∧ C ⊆ K ∧ K ⊆ U"
by metis
obtain V K where "C ⊆ V" "V ⊆ U" "V ⊆ K" "K ⊆ S" "compact K"
and opeSV: "openin (subtopology euclidean S) V"
using S U ‹compact C›
apply (simp add: locally_compact_compact_subopen)
by (meson C in_components_subset)
let ?𝒯 = "{T. C ⊆ T ∧ openin (subtopology euclidean K) T ∧ compact T ∧ T ⊆ K}"
have CK: "C ∈ components K"
by (meson C ‹C ⊆ V› ‹K ⊆ S› ‹V ⊆ K› components_intermediate_subset subset_trans)
with ‹compact K›
have "C = ⋂{T. C ⊆ T ∧ openin (subtopology euclidean K) T ∧ closedin (subtopology euclidean K) T}"
by (simp add: Sura_Bura_compact)
then have Ceq: "C = ⋂?𝒯"
by (simp add: closedin_compact_eq ‹compact K›)
obtain W where "open W" and W: "V = S ∩ W"
using opeSV by (auto simp: openin_open)
have "-(U ∩ W) ∩ ⋂?𝒯 ≠ {}"
proof (rule closed_imp_fip_compact)
show "- (U ∩ W) ∩ ⋂ℱ ≠ {}"
if "finite ℱ" and ℱ: "ℱ ⊆ ?𝒯" for ℱ
proof (cases "ℱ = {}")
case True
have False if "U = UNIV" "W = UNIV"
proof -
have "V = S"
by (simp add: W ‹W = UNIV›)
with neg show False
using ‹C ⊆ V› ‹K ⊆ S› ‹V ⊆ K› ‹V ⊆ U› ‹compact K› by auto
qed
with True show ?thesis
by auto
next
case False
show ?thesis
proof
assume "- (U ∩ W) ∩ ⋂ℱ = {}"
then have FUW: "⋂ℱ ⊆ U ∩ W"
by blast
have "C ⊆ ⋂ℱ"
using ℱ by auto
moreover have "compact (⋂ℱ)"
by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE ℱ)
moreover have "⋂ℱ ⊆ K"
using False that(2) by fastforce
moreover have opeKF: "openin (subtopology euclidean K) (⋂ℱ)"
using False ℱ ‹finite ℱ› by blast
then have opeVF: "openin (subtopology euclidean V) (⋂ℱ)"
using W ‹K ⊆ S› ‹V ⊆ K› opeKF ‹⋂ℱ ⊆ K› FUW openin_subset_trans by fastforce
then have "openin (subtopology euclidean S) (⋂ℱ)"
by (metis opeSV openin_trans)
moreover have "⋂ℱ ⊆ U"
by (meson ‹V ⊆ U› opeVF dual_order.trans openin_imp_subset)
ultimately show False
using neg by blast
qed
qed
qed (use ‹open W› ‹open U› in auto)
with W Ceq ‹C ⊆ V› ‹C ⊆ U› show False
by auto
qed
corollary Sura_Bura_clopen_subset_alt:
fixes S :: "'a::euclidean_space set"
assumes S: "locally compact S" and C: "C ∈ components S" and "compact C"
and opeSU: "openin (subtopology euclidean S) U" and "C ⊆ U"
obtains K where "openin (subtopology euclidean S) K" "compact K" "C ⊆ K" "K ⊆ U"
proof -
obtain V where "open V" "U = S ∩ V"
using opeSU by (auto simp: openin_open)
with ‹C ⊆ U› have "C ⊆ V"
by auto
then show ?thesis
using Sura_Bura_clopen_subset [OF S C ‹compact C› ‹open V›]
by (metis ‹U = S ∩ V› inf.bounded_iff openin_imp_subset that)
qed
corollary Sura_Bura:
fixes S :: "'a::euclidean_space set"
assumes "locally compact S" "C ∈ components S" "compact C"
shows "C = ⋂ {K. C ⊆ K ∧ compact K ∧ openin (subtopology euclidean S) K}"
(is "C = ?rhs")
proof
show "?rhs ⊆ C"
proof (clarsimp, rule ccontr)
fix x
assume *: "∀X. C ⊆ X ∧ compact X ∧ openin (subtopology euclidean S) X ⟶ x ∈ X"
and "x ∉ C"
obtain U V where "open U" "open V" "{x} ⊆ U" "C ⊆ V" "U ∩ V = {}"
using separation_normal [of "{x}" C]
by (metis Int_empty_left ‹x ∉ C› ‹compact C› closed_empty closed_insert compact_imp_closed insert_disjoint(1))
have "x ∉ V"
using ‹U ∩ V = {}› ‹{x} ⊆ U› by blast
then show False
by (meson "*" Sura_Bura_clopen_subset ‹C ⊆ V› ‹open V› assms(1) assms(2) assms(3) subsetCE)
qed
qed blast
subsection‹Important special cases of local connectedness and path connectedness›
lemma locally_connected_1:
assumes
"⋀v x. ⟦openin (subtopology euclidean S) v; x ∈ v⟧
⟹ ∃u. openin (subtopology euclidean S) u ∧
connected u ∧ x ∈ u ∧ u ⊆ v"
shows "locally connected S"
apply (clarsimp simp add: locally_def)
apply (drule assms; blast)
done
lemma locally_connected_2:
assumes "locally connected S"
"openin (subtopology euclidean S) t"
"x ∈ t"
shows "openin (subtopology euclidean S) (connected_component_set t x)"
proof -
{ fix y :: 'a
let ?SS = "subtopology euclidean S"
assume 1: "openin ?SS t"
"∀w x. openin ?SS w ∧ x ∈ w ⟶ (∃u. openin ?SS u ∧ (∃v. connected v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ w))"
and "connected_component t x y"
then have "y ∈ t" and y: "y ∈ connected_component_set t x"
using connected_component_subset by blast+
obtain F where
"∀x y. (∃w. openin ?SS w ∧ (∃u. connected u ∧ x ∈ w ∧ w ⊆ u ∧ u ⊆ y)) = (openin ?SS (F x y) ∧ (∃u. connected u ∧ x ∈ F x y ∧ F x y ⊆ u ∧ u ⊆ y))"
by moura
then obtain G where
"∀a A. (∃U. openin ?SS U ∧ (∃V. connected V ∧ a ∈ U ∧ U ⊆ V ∧ V ⊆ A)) = (openin ?SS (F a A) ∧ connected (G a A) ∧ a ∈ F a A ∧ F a A ⊆ G a A ∧ G a A ⊆ A)"
by moura
then have *: "openin ?SS (F y t) ∧ connected (G y t) ∧ y ∈ F y t ∧ F y t ⊆ G y t ∧ G y t ⊆ t"
using 1 ‹y ∈ t› by presburger
have "G y t ⊆ connected_component_set t y"
by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
then have "∃A. openin ?SS A ∧ y ∈ A ∧ A ⊆ connected_component_set t x"
by (metis (no_types) * connected_component_eq dual_order.trans y)
}
then show ?thesis
using assms openin_subopen by (force simp: locally_def)
qed
lemma locally_connected_3:
assumes "⋀t x. ⟦openin (subtopology euclidean S) t; x ∈ t⟧
⟹ openin (subtopology euclidean S)
(connected_component_set t x)"
"openin (subtopology euclidean S) v" "x ∈ v"
shows "∃u. openin (subtopology euclidean S) u ∧ connected u ∧ x ∈ u ∧ u ⊆ v"
using assms connected_component_subset by fastforce
lemma locally_connected:
"locally connected S ⟷
(∀v x. openin (subtopology euclidean S) v ∧ x ∈ v
⟶ (∃u. openin (subtopology euclidean S) u ∧ connected u ∧ x ∈ u ∧ u ⊆ v))"
by (metis locally_connected_1 locally_connected_2 locally_connected_3)
lemma locally_connected_open_connected_component:
"locally connected S ⟷
(∀t x. openin (subtopology euclidean S) t ∧ x ∈ t
⟶ openin (subtopology euclidean S) (connected_component_set t x))"
by (metis locally_connected_1 locally_connected_2 locally_connected_3)
lemma locally_path_connected_1:
assumes
"⋀v x. ⟦openin (subtopology euclidean S) v; x ∈ v⟧
⟹ ∃u. openin (subtopology euclidean S) u ∧ path_connected u ∧ x ∈ u ∧ u ⊆ v"
shows "locally path_connected S"
apply (clarsimp simp add: locally_def)
apply (drule assms; blast)
done
lemma locally_path_connected_2:
assumes "locally path_connected S"
"openin (subtopology euclidean S) t"
"x ∈ t"
shows "openin (subtopology euclidean S) (path_component_set t x)"
proof -
{ fix y :: 'a
let ?SS = "subtopology euclidean S"
assume 1: "openin ?SS t"
"∀w x. openin ?SS w ∧ x ∈ w ⟶ (∃u. openin ?SS u ∧ (∃v. path_connected v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ w))"
and "path_component t x y"
then have "y ∈ t" and y: "y ∈ path_component_set t x"
using path_component_mem(2) by blast+
obtain F where
"∀x y. (∃w. openin ?SS w ∧ (∃u. path_connected u ∧ x ∈ w ∧ w ⊆ u ∧ u ⊆ y)) = (openin ?SS (F x y) ∧ (∃u. path_connected u ∧ x ∈ F x y ∧ F x y ⊆ u ∧ u ⊆ y))"
by moura
then obtain G where
"∀a A. (∃U. openin ?SS U ∧ (∃V. path_connected V ∧ a ∈ U ∧ U ⊆ V ∧ V ⊆ A)) = (openin ?SS (F a A) ∧ path_connected (G a A) ∧ a ∈ F a A ∧ F a A ⊆ G a A ∧ G a A ⊆ A)"
by moura
then have *: "openin ?SS (F y t) ∧ path_connected (G y t) ∧ y ∈ F y t ∧ F y t ⊆ G y t ∧ G y t ⊆ t"
using 1 ‹y ∈ t› by presburger
have "G y t ⊆ path_component_set t y"
using * path_component_maximal set_rev_mp by blast
then have "∃A. openin ?SS A ∧ y ∈ A ∧ A ⊆ path_component_set t x"
by (metis "*" ‹G y t ⊆ path_component_set t y› dual_order.trans path_component_eq y)
}
then show ?thesis
using assms openin_subopen by (force simp: locally_def)
qed
lemma locally_path_connected_3:
assumes "⋀t x. ⟦openin (subtopology euclidean S) t; x ∈ t⟧
⟹ openin (subtopology euclidean S) (path_component_set t x)"
"openin (subtopology euclidean S) v" "x ∈ v"
shows "∃u. openin (subtopology euclidean S) u ∧ path_connected u ∧ x ∈ u ∧ u ⊆ v"
proof -
have "path_component v x x"
by (meson assms(3) path_component_refl)
then show ?thesis
by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
qed
proposition locally_path_connected:
"locally path_connected S ⟷
(∀v x. openin (subtopology euclidean S) v ∧ x ∈ v
⟶ (∃u. openin (subtopology euclidean S) u ∧ path_connected u ∧ x ∈ u ∧ u ⊆ v))"
by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
proposition locally_path_connected_open_path_component:
"locally path_connected S ⟷
(∀t x. openin (subtopology euclidean S) t ∧ x ∈ t
⟶ openin (subtopology euclidean S) (path_component_set t x))"
by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
lemma locally_connected_open_component:
"locally connected S ⟷
(∀t c. openin (subtopology euclidean S) t ∧ c ∈ components t
⟶ openin (subtopology euclidean S) c)"
by (metis components_iff locally_connected_open_connected_component)
proposition locally_connected_im_kleinen:
"locally connected S ⟷
(∀v x. openin (subtopology euclidean S) v ∧ x ∈ v
⟶ (∃u. openin (subtopology euclidean S) u ∧
x ∈ u ∧ u ⊆ v ∧
(∀y. y ∈ u ⟶ (∃c. connected c ∧ c ⊆ v ∧ x ∈ c ∧ y ∈ c))))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (fastforce simp add: locally_connected)
next
assume ?rhs
have *: "∃T. openin (subtopology euclidean S) T ∧ x ∈ T ∧ T ⊆ c"
if "openin (subtopology euclidean S) t" and c: "c ∈ components t" and "x ∈ c" for t c x
proof -
from that ‹?rhs› [rule_format, of t x]
obtain u where u:
"openin (subtopology euclidean S) u ∧ x ∈ u ∧ u ⊆ t ∧
(∀y. y ∈ u ⟶ (∃c. connected c ∧ c ⊆ t ∧ x ∈ c ∧ y ∈ c))"
using in_components_subset by auto
obtain F :: "'a set ⇒ 'a set ⇒ 'a" where
"∀x y. (∃z. z ∈ x ∧ y = connected_component_set x z) = (F x y ∈ x ∧ y = connected_component_set x (F x y))"
by moura
then have F: "F t c ∈ t ∧ c = connected_component_set t (F t c)"
by (meson components_iff c)
obtain G :: "'a set ⇒ 'a set ⇒ 'a" where
G: "∀x y. (∃z. z ∈ y ∧ z ∉ x) = (G x y ∈ y ∧ G x y ∉ x)"
by moura
have "G c u ∉ u ∨ G c u ∈ c"
using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3))
then show ?thesis
using G u by auto
qed
show ?lhs
apply (clarsimp simp add: locally_connected_open_component)
apply (subst openin_subopen)
apply (blast intro: *)
done
qed
proposition locally_path_connected_im_kleinen:
"locally path_connected S ⟷
(∀v x. openin (subtopology euclidean S) v ∧ x ∈ v
⟶ (∃u. openin (subtopology euclidean S) u ∧
x ∈ u ∧ u ⊆ v ∧
(∀y. y ∈ u ⟶ (∃p. path p ∧ path_image p ⊆ v ∧
pathstart p = x ∧ pathfinish p = y))))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (simp add: locally_path_connected path_connected_def)
apply (erule all_forward ex_forward imp_forward conjE | simp)+
by (meson dual_order.trans)
next
assume ?rhs
have *: "∃T. openin (subtopology euclidean S) T ∧
x ∈ T ∧ T ⊆ path_component_set u z"
if "openin (subtopology euclidean S) u" and "z ∈ u" and c: "path_component u z x" for u z x
proof -
have "x ∈ u"
by (meson c path_component_mem(2))
with that ‹?rhs› [rule_format, of u x]
obtain U where U:
"openin (subtopology euclidean S) U ∧ x ∈ U ∧ U ⊆ u ∧
(∀y. y ∈ U ⟶ (∃p. path p ∧ path_image p ⊆ u ∧ pathstart p = x ∧ pathfinish p = y))"
by blast
show ?thesis
apply (rule_tac x=U in exI)
apply (auto simp: U)
apply (metis U c path_component_trans path_component_def)
done
qed
show ?lhs
apply (clarsimp simp add: locally_path_connected_open_path_component)
apply (subst openin_subopen)
apply (blast intro: *)
done
qed
lemma locally_path_connected_imp_locally_connected:
"locally path_connected S ⟹ locally connected S"
using locally_mono path_connected_imp_connected by blast
lemma locally_connected_components:
"⟦locally connected S; c ∈ components S⟧ ⟹ locally connected c"
by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
lemma locally_path_connected_components:
"⟦locally path_connected S; c ∈ components S⟧ ⟹ locally path_connected c"
by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)
lemma locally_path_connected_connected_component:
"locally path_connected S ⟹ locally path_connected (connected_component_set S x)"
by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)
lemma open_imp_locally_path_connected:
fixes S :: "'a :: real_normed_vector set"
shows "open S ⟹ locally path_connected S"
apply (rule locally_mono [of convex])
apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
apply (meson open_ball centre_in_ball convex_ball openE order_trans)
done
lemma open_imp_locally_connected:
fixes S :: "'a :: real_normed_vector set"
shows "open S ⟹ locally connected S"
by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)
lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
by (simp add: open_imp_locally_path_connected)
lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
by (simp add: open_imp_locally_connected)
lemma openin_connected_component_locally_connected:
"locally connected S
⟹ openin (subtopology euclidean S) (connected_component_set S x)"
apply (simp add: locally_connected_open_connected_component)
by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
lemma openin_components_locally_connected:
"⟦locally connected S; c ∈ components S⟧ ⟹ openin (subtopology euclidean S) c"
using locally_connected_open_component openin_subtopology_self by blast
lemma openin_path_component_locally_path_connected:
"locally path_connected S
⟹ openin (subtopology euclidean S) (path_component_set S x)"
by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
lemma closedin_path_component_locally_path_connected:
"locally path_connected S
⟹ closedin (subtopology euclidean S) (path_component_set S x)"
apply (simp add: closedin_def path_component_subset complement_path_component_Union)
apply (rule openin_Union)
using openin_path_component_locally_path_connected by auto
lemma convex_imp_locally_path_connected:
fixes S :: "'a:: real_normed_vector set"
shows "convex S ⟹ locally path_connected S"
apply (clarsimp simp add: locally_path_connected)
apply (subst (asm) openin_open)
apply clarify
apply (erule (1) openE)
apply (rule_tac x = "S ∩ ball x e" in exI)
apply (force simp: convex_Int convex_imp_path_connected)
done
lemma convex_imp_locally_connected:
fixes S :: "'a:: real_normed_vector set"
shows "convex S ⟹ locally connected S"
by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
subsection‹Relations between components and path components›
lemma path_component_eq_connected_component:
assumes "locally path_connected S"
shows "(path_component S x = connected_component S x)"
proof (cases "x ∈ S")
case True
have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
apply (rule openin_subset_trans [of S])
apply (intro conjI openin_path_component_locally_path_connected [OF assms])
using path_component_subset_connected_component apply (auto simp: connected_component_subset)
done
moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
apply (rule closedin_subset_trans [of S])
apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
using path_component_subset_connected_component apply (auto simp: connected_component_subset)
done
ultimately have *: "path_component_set S x = connected_component_set S x"
by (metis connected_connected_component connected_clopen True path_component_eq_empty)
then show ?thesis
by blast
next
case False then show ?thesis
by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
qed
lemma path_component_eq_connected_component_set:
"locally path_connected S ⟹ (path_component_set S x = connected_component_set S x)"
by (simp add: path_component_eq_connected_component)
lemma locally_path_connected_path_component:
"locally path_connected S ⟹ locally path_connected (path_component_set S x)"
using locally_path_connected_connected_component path_component_eq_connected_component by fastforce
lemma open_path_connected_component:
fixes S :: "'a :: real_normed_vector set"
shows "open S ⟹ path_component S x = connected_component S x"
by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)
lemma open_path_connected_component_set:
fixes S :: "'a :: real_normed_vector set"
shows "open S ⟹ path_component_set S x = connected_component_set S x"
by (simp add: open_path_connected_component)
proposition locally_connected_quotient_image:
assumes lcS: "locally connected S"
and oo: "⋀T. T ⊆ f ` S
⟹ openin (subtopology euclidean S) (S ∩ f -` T) ⟷
openin (subtopology euclidean (f ` S)) T"
shows "locally connected (f ` S)"
proof (clarsimp simp: locally_connected_open_component)
fix U C
assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C ∈ components U"
then have "C ⊆ U" "U ⊆ f ` S"
by (meson in_components_subset openin_imp_subset)+
then have "openin (subtopology euclidean (f ` S)) C ⟷
openin (subtopology euclidean S) (S ∩ f -` C)"
by (auto simp: oo)
moreover have "openin (subtopology euclidean S) (S ∩ f -` C)"
proof (subst openin_subopen, clarify)
fix x
assume "x ∈ S" "f x ∈ C"
show "∃T. openin (subtopology euclidean S) T ∧ x ∈ T ∧ T ⊆ (S ∩ f -` C)"
proof (intro conjI exI)
show "openin (subtopology euclidean S) (connected_component_set (S ∩ f -` U) x)"
proof (rule ccontr)
assume **: "¬ openin (subtopology euclidean S) (connected_component_set (S ∩ f -` U) x)"
then have "x ∉ (S ∩ f -` U)"
using ‹U ⊆ f ` S› opefSU lcS locally_connected_2 oo by blast
with ** show False
by (metis (no_types) connected_component_eq_empty empty_iff openin_subopen)
qed
next
show "x ∈ connected_component_set (S ∩ f -` U) x"
using ‹C ⊆ U› ‹f x ∈ C› ‹x ∈ S› by auto
next
have contf: "continuous_on S f"
by (simp add: continuous_on_open oo openin_imp_subset)
then have "continuous_on (connected_component_set (S ∩ f -` U) x) f"
apply (rule continuous_on_subset)
using connected_component_subset apply blast
done
then have "connected (f ` connected_component_set (S ∩ f -` U) x)"
by (rule connected_continuous_image [OF _ connected_connected_component])
moreover have "f ` connected_component_set (S ∩ f -` U) x ⊆ U"
using connected_component_in by blast
moreover have "C ∩ f ` connected_component_set (S ∩ f -` U) x ≠ {}"
using ‹C ⊆ U› ‹f x ∈ C› ‹x ∈ S› by fastforce
ultimately have fC: "f ` (connected_component_set (S ∩ f -` U) x) ⊆ C"
by (rule components_maximal [OF ‹C ∈ components U›])
have cUC: "connected_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` C)"
using connected_component_subset fC by blast
have "connected_component_set (S ∩ f -` U) x ⊆ connected_component_set (S ∩ f -` C) x"
proof -
{ assume "x ∈ connected_component_set (S ∩ f -` U) x"
then have ?thesis
using cUC connected_component_idemp connected_component_mono by blast }
then show ?thesis
using connected_component_eq_empty by auto
qed
also have "… ⊆ (S ∩ f -` C)"
by (rule connected_component_subset)
finally show "connected_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` C)" .
qed
qed
ultimately show "openin (subtopology euclidean (f ` S)) C"
by metis
qed
text‹The proof resembles that above but is not identical!›
proposition locally_path_connected_quotient_image:
assumes lcS: "locally path_connected S"
and oo: "⋀T. T ⊆ f ` S
⟹ openin (subtopology euclidean S) (S ∩ f -` T) ⟷ openin (subtopology euclidean (f ` S)) T"
shows "locally path_connected (f ` S)"
proof (clarsimp simp: locally_path_connected_open_path_component)
fix U y
assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "y ∈ U"
then have "path_component_set U y ⊆ U" "U ⊆ f ` S"
by (meson path_component_subset openin_imp_subset)+
then have "openin (subtopology euclidean (f ` S)) (path_component_set U y) ⟷
openin (subtopology euclidean S) (S ∩ f -` path_component_set U y)"
proof -
have "path_component_set U y ⊆ f ` S"
using ‹U ⊆ f ` S› ‹path_component_set U y ⊆ U› by blast
then show ?thesis
using oo by blast
qed
moreover have "openin (subtopology euclidean S) (S ∩ f -` path_component_set U y)"
proof (subst openin_subopen, clarify)
fix x
assume "x ∈ S" and Uyfx: "path_component U y (f x)"
then have "f x ∈ U"
using path_component_mem by blast
show "∃T. openin (subtopology euclidean S) T ∧ x ∈ T ∧ T ⊆ (S ∩ f -` path_component_set U y)"
proof (intro conjI exI)
show "openin (subtopology euclidean S) (path_component_set (S ∩ f -` U) x)"
proof (rule ccontr)
assume **: "¬ openin (subtopology euclidean S) (path_component_set (S ∩ f -` U) x)"
then have "x ∉ (S ∩ f -` U)"
by (metis (no_types, lifting) ‹U ⊆ f ` S› opefSU lcS oo locally_path_connected_open_path_component)
then show False
using ** ‹path_component_set U y ⊆ U› ‹x ∈ S› ‹path_component U y (f x)› by blast
qed
next
show "x ∈ path_component_set (S ∩ f -` U) x"
by (simp add: ‹f x ∈ U› ‹x ∈ S› path_component_refl)
next
have contf: "continuous_on S f"
by (simp add: continuous_on_open oo openin_imp_subset)
then have "continuous_on (path_component_set (S ∩ f -` U) x) f"
apply (rule continuous_on_subset)
using path_component_subset apply blast
done
then have "path_connected (f ` path_component_set (S ∩ f -` U) x)"
by (simp add: path_connected_continuous_image)
moreover have "f ` path_component_set (S ∩ f -` U) x ⊆ U"
using path_component_mem by fastforce
moreover have "f x ∈ f ` path_component_set (S ∩ f -` U) x"
by (force simp: ‹x ∈ S› ‹f x ∈ U› path_component_refl_eq)
ultimately have "f ` (path_component_set (S ∩ f -` U) x) ⊆ path_component_set U (f x)"
by (meson path_component_maximal)
also have "… ⊆ path_component_set U y"
by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
finally have fC: "f ` (path_component_set (S ∩ f -` U) x) ⊆ path_component_set U y" .
have cUC: "path_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` path_component_set U y)"
using path_component_subset fC by blast
have "path_component_set (S ∩ f -` U) x ⊆ path_component_set (S ∩ f -` path_component_set U y) x"
proof -
have "⋀a. path_component_set (path_component_set (S ∩ f -` U) x) a ⊆ path_component_set (S ∩ f -` path_component_set U y) a"
using cUC path_component_mono by blast
then show ?thesis
using path_component_path_component by blast
qed
also have "… ⊆ (S ∩ f -` path_component_set U y)"
by (rule path_component_subset)
finally show "path_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` path_component_set U y)" .
qed
qed
ultimately show "openin (subtopology euclidean (f ` S)) (path_component_set U y)"
by metis
qed
subsection%unimportant‹Components, continuity, openin, closedin›
lemma continuous_on_components_gen:
fixes f :: "'a::topological_space ⇒ 'b::topological_space"
assumes "⋀c. c ∈ components S ⟹
openin (subtopology euclidean S) c ∧ continuous_on c f"
shows "continuous_on S f"
proof (clarsimp simp: continuous_openin_preimage_eq)
fix t :: "'b set"
assume "open t"
have *: "S ∩ f -` t = (⋃c ∈ components S. c ∩ f -` t)"
by auto
show "openin (subtopology euclidean S) (S ∩ f -` t)"
unfolding * using ‹open t› assms continuous_openin_preimage_gen openin_trans openin_Union by blast
qed
lemma continuous_on_components:
fixes f :: "'a::topological_space ⇒ 'b::topological_space"
assumes "locally connected S "
"⋀c. c ∈ components S ⟹ continuous_on c f"
shows "continuous_on S f"
apply (rule continuous_on_components_gen)
apply (auto simp: assms intro: openin_components_locally_connected)
done
lemma continuous_on_components_eq:
"locally connected S
⟹ (continuous_on S f ⟷ (∀c ∈ components S. continuous_on c f))"
by (meson continuous_on_components continuous_on_subset in_components_subset)
lemma continuous_on_components_open:
fixes S :: "'a::real_normed_vector set"
assumes "open S "
"⋀c. c ∈ components S ⟹ continuous_on c f"
shows "continuous_on S f"
using continuous_on_components open_imp_locally_connected assms by blast
lemma continuous_on_components_open_eq:
fixes S :: "'a::real_normed_vector set"
shows "open S ⟹ (continuous_on S f ⟷ (∀c ∈ components S. continuous_on c f))"
using continuous_on_subset in_components_subset
by (blast intro: continuous_on_components_open)
lemma closedin_union_complement_components:
assumes u: "locally connected u"
and S: "closedin (subtopology euclidean u) S"
and cuS: "c ⊆ components(u - S)"
shows "closedin (subtopology euclidean u) (S ∪ ⋃c)"
proof -
have di: "(⋀S t. S ∈ c ∧ t ∈ c' ⟹ disjnt S t) ⟹ disjnt (⋃ c) (⋃ c')" for c'
by (simp add: disjnt_def) blast
have "S ⊆ u"
using S closedin_imp_subset by blast
moreover have "u - S = ⋃c ∪ ⋃(components (u - S) - c)"
by (metis Diff_partition Union_components Union_Un_distrib assms(3))
moreover have "disjnt (⋃c) (⋃(components (u - S) - c))"
apply (rule di)
by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
ultimately have eq: "S ∪ ⋃c = u - (⋃(components(u - S) - c))"
by (auto simp: disjnt_def)
have *: "openin (subtopology euclidean u) (⋃(components (u - S) - c))"
apply (rule openin_Union)
apply (rule openin_trans [of "u - S"])
apply (simp add: u S locally_diff_closed openin_components_locally_connected)
apply (simp add: openin_diff S)
done
have "openin (subtopology euclidean u) (u - (u - ⋃(components (u - S) - c)))"
apply (rule openin_diff, simp)
apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
done
then show ?thesis
by (force simp: eq closedin_def)
qed
lemma closed_union_complement_components:
fixes S :: "'a::real_normed_vector set"
assumes S: "closed S" and c: "c ⊆ components(- S)"
shows "closed(S ∪ ⋃ c)"
proof -
have "closedin (subtopology euclidean UNIV) (S ∪ ⋃c)"
apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
using S c apply (simp_all add: Compl_eq_Diff_UNIV)
done
then show ?thesis by simp
qed
lemma closedin_Un_complement_component:
fixes S :: "'a::real_normed_vector set"
assumes u: "locally connected u"
and S: "closedin (subtopology euclidean u) S"
and c: " c ∈ components(u - S)"
shows "closedin (subtopology euclidean u) (S ∪ c)"
proof -
have "closedin (subtopology euclidean u) (S ∪ ⋃{c})"
using c by (blast intro: closedin_union_complement_components [OF u S])
then show ?thesis
by simp
qed
lemma closed_Un_complement_component:
fixes S :: "'a::real_normed_vector set"
assumes S: "closed S" and c: " c ∈ components(-S)"
shows "closed (S ∪ c)"
by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
locally_connected_UNIV subtopology_UNIV)
subsection‹Existence of isometry between subspaces of same dimension›
lemma isometry_subset_subspace:
fixes S :: "'a::euclidean_space set"
and T :: "'b::euclidean_space set"
assumes S: "subspace S"
and T: "subspace T"
and d: "dim S ≤ dim T"
obtains f where "linear f" "f ` S ⊆ T" "⋀x. x ∈ S ⟹ norm(f x) = norm x"
proof -
obtain B where "B ⊆ S" and Borth: "pairwise orthogonal B"
and B1: "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "finite B" "card B = dim S" "span B = S"
by (metis orthonormal_basis_subspace [OF S] independent_finite)
obtain C where "C ⊆ T" and Corth: "pairwise orthogonal C"
and C1:"⋀x. x ∈ C ⟹ norm x = 1"
and "independent C" "finite C" "card C = dim T" "span C = T"
by (metis orthonormal_basis_subspace [OF T] independent_finite)
obtain fb where "fb ` B ⊆ C" "inj_on fb B"
by (metis ‹card B = dim S› ‹card C = dim T› ‹finite B› ‹finite C› card_le_inj d)
then have pairwise_orth_fb: "pairwise (λv j. orthogonal (fb v) (fb j)) B"
using Corth
apply (auto simp: pairwise_def orthogonal_clauses)
by (meson subsetD image_eqI inj_on_def)
obtain f where "linear f" and ffb: "⋀x. x ∈ B ⟹ f x = fb x"
using linear_independent_extend ‹independent B› by fastforce
have "span (f ` B) ⊆ span C"
by (metis ‹fb ` B ⊆ C› ffb image_cong span_mono)
then have "f ` S ⊆ T"
unfolding ‹span B = S› ‹span C = T› span_linear_image[OF ‹linear f›] .
have [simp]: "⋀x. x ∈ B ⟹ norm (fb x) = norm x"
using B1 C1 ‹fb ` B ⊆ C› by auto
have "norm (f x) = norm x" if "x ∈ S" for x
proof -
interpret linear f by fact
obtain a where x: "x = (∑v ∈ B. a v *⇩R v)"
using ‹finite B› ‹span B = S› ‹x ∈ S› span_finite by fastforce
have "norm (f x)^2 = norm (∑v∈B. a v *⇩R fb v)^2" by (simp add: sum scale ffb x)
also have "… = (∑v∈B. norm ((a v *⇩R fb v))^2)"
apply (rule norm_sum_Pythagorean [OF ‹finite B›])
apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
done
also have "… = norm x ^2"
by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF ‹finite B›])
finally show ?thesis
by (simp add: norm_eq_sqrt_inner)
qed
then show ?thesis
by (rule that [OF ‹linear f› ‹f ` S ⊆ T›])
qed
proposition isometries_subspaces:
fixes S :: "'a::euclidean_space set"
and T :: "'b::euclidean_space set"
assumes S: "subspace S"
and T: "subspace T"
and d: "dim S = dim T"
obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
"⋀x. x ∈ S ⟹ norm(f x) = norm x"
"⋀x. x ∈ T ⟹ norm(g x) = norm x"
"⋀x. x ∈ S ⟹ g(f x) = x"
"⋀x. x ∈ T ⟹ f(g x) = x"
proof -
obtain B where "B ⊆ S" and Borth: "pairwise orthogonal B"
and B1: "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "finite B" "card B = dim S" "span B = S"
by (metis orthonormal_basis_subspace [OF S] independent_finite)
obtain C where "C ⊆ T" and Corth: "pairwise orthogonal C"
and C1:"⋀x. x ∈ C ⟹ norm x = 1"
and "independent C" "finite C" "card C = dim T" "span C = T"
by (metis orthonormal_basis_subspace [OF T] independent_finite)
obtain fb where "bij_betw fb B C"
by (metis ‹finite B› ‹finite C› bij_betw_iff_card ‹card B = dim S› ‹card C = dim T› d)
then have pairwise_orth_fb: "pairwise (λv j. orthogonal (fb v) (fb j)) B"
using Corth
apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
by (meson subsetD image_eqI inj_on_def)
obtain f where "linear f" and ffb: "⋀x. x ∈ B ⟹ f x = fb x"
using linear_independent_extend ‹independent B› by fastforce
interpret f: linear f by fact
define gb where "gb ≡ inv_into B fb"
then have pairwise_orth_gb: "pairwise (λv j. orthogonal (gb v) (gb j)) C"
using Borth
apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
by (metis ‹bij_betw fb B C› bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
obtain g where "linear g" and ggb: "⋀x. x ∈ C ⟹ g x = gb x"
using linear_independent_extend ‹independent C› by fastforce
interpret g: linear g by fact
have "span (f ` B) ⊆ span C"
by (metis ‹bij_betw fb B C› bij_betw_imp_surj_on eq_iff ffb image_cong)
then have "f ` S ⊆ T"
unfolding ‹span B = S› ‹span C = T›
span_linear_image[OF ‹linear f›] .
have [simp]: "⋀x. x ∈ B ⟹ norm (fb x) = norm x"
using B1 C1 ‹bij_betw fb B C› bij_betw_imp_surj_on by fastforce
have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x ∈ S" for x
proof -
obtain a where x: "x = (∑v ∈ B. a v *⇩R v)"
using ‹finite B› ‹span B = S› ‹x ∈ S› span_finite by fastforce
have "f x = (∑v ∈ B. f (a v *⇩R v))"
using linear_sum [OF ‹linear f›] x by auto
also have "… = (∑v ∈ B. a v *⇩R f v)"
by (simp add: f.sum f.scale)
also have "… = (∑v ∈ B. a v *⇩R fb v)"
by (simp add: ffb cong: sum.cong)
finally have *: "f x = (∑v∈B. a v *⇩R fb v)" .
then have "(norm (f x))⇧2 = (norm (∑v∈B. a v *⇩R fb v))⇧2" by simp
also have "… = (∑v∈B. norm ((a v *⇩R fb v))^2)"
apply (rule norm_sum_Pythagorean [OF ‹finite B›])
apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
done
also have "… = (norm x)⇧2"
by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF ‹finite B›])
finally show "norm (f x) = norm x"
by (simp add: norm_eq_sqrt_inner)
have "g (f x) = g (∑v∈B. a v *⇩R fb v)" by (simp add: *)
also have "… = (∑v∈B. g (a v *⇩R fb v))"
by (simp add: g.sum g.scale)
also have "… = (∑v∈B. a v *⇩R g (fb v))"
by (simp add: g.scale)
also have "… = (∑v∈B. a v *⇩R v)"
apply (rule sum.cong [OF refl])
using ‹bij_betw fb B C› gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
also have "… = x"
using x by blast
finally show "g (f x) = x" .
qed
have [simp]: "⋀x. x ∈ C ⟹ norm (gb x) = norm x"
by (metis B1 C1 ‹bij_betw fb B C› bij_betw_imp_surj_on gb_def inv_into_into)
have g [simp]: "f (g x) = x" if "x ∈ T" for x
proof -
obtain a where x: "x = (∑v ∈ C. a v *⇩R v)"
using ‹finite C› ‹span C = T› ‹x ∈ T› span_finite by fastforce
have "g x = (∑v ∈ C. g (a v *⇩R v))"
by (simp add: x g.sum)
also have "… = (∑v ∈ C. a v *⇩R g v)"
by (simp add: g.scale)
also have "… = (∑v ∈ C. a v *⇩R gb v)"
by (simp add: ggb cong: sum.cong)
finally have "f (g x) = f (∑v∈C. a v *⇩R gb v)" by simp
also have "… = (∑v∈C. f (a v *⇩R gb v))"
by (simp add: f.scale f.sum)
also have "… = (∑v∈C. a v *⇩R f (gb v))"
by (simp add: f.scale f.sum)
also have "… = (∑v∈C. a v *⇩R v)"
using ‹bij_betw fb B C›
by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
also have "… = x"
using x by blast
finally show "f (g x) = x" .
qed
have gim: "g ` T = S"
by (metis (full_types) S T ‹f ` S ⊆ T› d dim_eq_span dim_image_le f(2) g.linear_axioms
image_iff linear_subspace_image span_eq_iff subset_iff)
have fim: "f ` S = T"
using ‹g ` T = S› image_iff by fastforce
have [simp]: "norm (g x) = norm x" if "x ∈ T" for x
using fim that by auto
show ?thesis
apply (rule that [OF ‹linear f› ‹linear g›])
apply (simp_all add: fim gim)
done
qed
corollary isometry_subspaces:
fixes S :: "'a::euclidean_space set"
and T :: "'b::euclidean_space set"
assumes S: "subspace S"
and T: "subspace T"
and d: "dim S = dim T"
obtains f where "linear f" "f ` S = T" "⋀x. x ∈ S ⟹ norm(f x) = norm x"
using isometries_subspaces [OF assms]
by metis
corollary isomorphisms_UNIV_UNIV:
assumes "DIM('M) = DIM('N)"
obtains f::"'M::euclidean_space ⇒'N::euclidean_space" and g
where "linear f" "linear g"
"⋀x. norm(f x) = norm x" "⋀y. norm(g y) = norm y"
"⋀x. g (f x) = x" "⋀y. f(g y) = y"
using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])
lemma homeomorphic_subspaces:
fixes S :: "'a::euclidean_space set"
and T :: "'b::euclidean_space set"
assumes S: "subspace S"
and T: "subspace T"
and d: "dim S = dim T"
shows "S homeomorphic T"
proof -
obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
"⋀x. x ∈ S ⟹ g(f x) = x" "⋀x. x ∈ T ⟹ f(g x) = x"
by (blast intro: isometries_subspaces [OF assms])
then show ?thesis
apply (simp add: homeomorphic_def homeomorphism_def)
apply (rule_tac x=f in exI)
apply (rule_tac x=g in exI)
apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
done
qed
lemma homeomorphic_affine_sets:
assumes "affine S" "affine T" "aff_dim S = aff_dim T"
shows "S homeomorphic T"
proof (cases "S = {} ∨ T = {}")
case True with assms aff_dim_empty homeomorphic_empty show ?thesis
by metis
next
case False
then obtain a b where ab: "a ∈ S" "b ∈ T" by auto
then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
using affine_diffs_subspace assms by blast+
have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
using assms ab by (simp add: aff_dim_eq_dim [OF hull_inc] image_def)
have "S homeomorphic ((+) (- a) ` S)"
by (simp add: homeomorphic_translation)
also have "… homeomorphic ((+) (- b) ` T)"
by (rule homeomorphic_subspaces [OF ss dd])
also have "… homeomorphic T"
using homeomorphic_sym homeomorphic_translation by auto
finally show ?thesis .
qed
subsection‹Retracts, in a general sense, preserve (co)homotopic triviality)›
locale%important Retracts =
fixes s h t k
assumes conth: "continuous_on s h"
and imh: "h ` s = t"
and contk: "continuous_on t k"
and imk: "k ` t ⊆ s"
and idhk: "⋀y. y ∈ t ⟹ h(k y) = y"
begin
lemma homotopically_trivial_retraction_gen:
assumes P: "⋀f. ⟦continuous_on u f; f ` u ⊆ t; Q f⟧ ⟹ P(k ∘ f)"
and Q: "⋀f. ⟦continuous_on u f; f ` u ⊆ s; P f⟧ ⟹ Q(h ∘ f)"
and Qeq: "⋀h k. (⋀x. x ∈ u ⟹ h x = k x) ⟹ Q h = Q k"
and hom: "⋀f g. ⟦continuous_on u f; f ` u ⊆ s; P f;
continuous_on u g; g ` u ⊆ s; P g⟧
⟹ homotopic_with P u s f g"
and contf: "continuous_on u f" and imf: "f ` u ⊆ t" and Qf: "Q f"
and contg: "continuous_on u g" and img: "g ` u ⊆ t" and Qg: "Q g"
shows "homotopic_with Q u t f g"
proof -
have feq: "⋀x. x ∈ u ⟹ (h ∘ (k ∘ f)) x = f x" using idhk imf by auto
have geq: "⋀x. x ∈ u ⟹ (h ∘ (k ∘ g)) x = g x" using idhk img by auto
have "continuous_on u (k ∘ f)"
using contf continuous_on_compose continuous_on_subset contk imf by blast
moreover have "(k ∘ f) ` u ⊆ s"
using imf imk by fastforce
moreover have "P (k ∘ f)"
by (simp add: P Qf contf imf)
moreover have "continuous_on u (k ∘ g)"
using contg continuous_on_compose continuous_on_subset contk img by blast
moreover have "(k ∘ g) ` u ⊆ s"
using img imk by fastforce
moreover have "P (k ∘ g)"
by (simp add: P Qg contg img)
ultimately have "homotopic_with P u s (k ∘ f) (k ∘ g)"
by (rule hom)
then have "homotopic_with Q u t (h ∘ (k ∘ f)) (h ∘ (k ∘ g))"
apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
using Q by (auto simp: conth imh)
then show ?thesis
apply (rule homotopic_with_eq)
apply (metis feq)
apply (metis geq)
apply (metis Qeq)
done
qed
lemma homotopically_trivial_retraction_null_gen:
assumes P: "⋀f. ⟦continuous_on u f; f ` u ⊆ t; Q f⟧ ⟹ P(k ∘ f)"
and Q: "⋀f. ⟦continuous_on u f; f ` u ⊆ s; P f⟧ ⟹ Q(h ∘ f)"
and Qeq: "⋀h k. (⋀x. x ∈ u ⟹ h x = k x) ⟹ Q h = Q k"
and hom: "⋀f. ⟦continuous_on u f; f ` u ⊆ s; P f⟧
⟹ ∃c. homotopic_with P u s f (λx. c)"
and contf: "continuous_on u f" and imf:"f ` u ⊆ t" and Qf: "Q f"
obtains c where "homotopic_with Q u t f (λx. c)"
proof -
have feq: "⋀x. x ∈ u ⟹ (h ∘ (k ∘ f)) x = f x" using idhk imf by auto
have "continuous_on u (k ∘ f)"
using contf continuous_on_compose continuous_on_subset contk imf by blast
moreover have "(k ∘ f) ` u ⊆ s"
using imf imk by fastforce
moreover have "P (k ∘ f)"
by (simp add: P Qf contf imf)
ultimately obtain c where "homotopic_with P u s (k ∘ f) (λx. c)"
by (metis hom)
then have "homotopic_with Q u t (h ∘ (k ∘ f)) (h ∘ (λx. c))"
apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
using Q by (auto simp: conth imh)
then show ?thesis
apply (rule_tac c = "h c" in that)
apply (erule homotopic_with_eq)
apply (metis feq, simp)
apply (metis Qeq)
done
qed
lemma cohomotopically_trivial_retraction_gen:
assumes P: "⋀f. ⟦continuous_on t f; f ` t ⊆ u; Q f⟧ ⟹ P(f ∘ h)"
and Q: "⋀f. ⟦continuous_on s f; f ` s ⊆ u; P f⟧ ⟹ Q(f ∘ k)"
and Qeq: "⋀h k. (⋀x. x ∈ t ⟹ h x = k x) ⟹ Q h = Q k"
and hom: "⋀f g. ⟦continuous_on s f; f ` s ⊆ u; P f;
continuous_on s g; g ` s ⊆ u; P g⟧
⟹ homotopic_with P s u f g"
and contf: "continuous_on t f" and imf: "f ` t ⊆ u" and Qf: "Q f"
and contg: "continuous_on t g" and img: "g ` t ⊆ u" and Qg: "Q g"
shows "homotopic_with Q t u f g"
proof -
have feq: "⋀x. x ∈ t ⟹ (f ∘ h ∘ k) x = f x" using idhk imf by auto
have geq: "⋀x. x ∈ t ⟹ (g ∘ h ∘ k) x = g x" using idhk img by auto
have "continuous_on s (f ∘ h)"
using contf conth continuous_on_compose imh by blast
moreover have "(f ∘ h) ` s ⊆ u"
using imf imh by fastforce
moreover have "P (f ∘ h)"
by (simp add: P Qf contf imf)
moreover have "continuous_on s (g ∘ h)"
using contg continuous_on_compose continuous_on_subset conth imh by blast
moreover have "(g ∘ h) ` s ⊆ u"
using img imh by fastforce
moreover have "P (g ∘ h)"
by (simp add: P Qg contg img)
ultimately have "homotopic_with P s u (f ∘ h) (g ∘ h)"
by (rule hom)
then have "homotopic_with Q t u (f ∘ h ∘ k) (g ∘ h ∘ k)"
apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
using Q by (auto simp: contk imk)
then show ?thesis
apply (rule homotopic_with_eq)
apply (metis feq)
apply (metis geq)
apply (metis Qeq)
done
qed
lemma cohomotopically_trivial_retraction_null_gen:
assumes P: "⋀f. ⟦continuous_on t f; f ` t ⊆ u; Q f⟧ ⟹ P(f ∘ h)"
and Q: "⋀f. ⟦continuous_on s f; f ` s ⊆ u; P f⟧ ⟹ Q(f ∘ k)"
and Qeq: "⋀h k. (⋀x. x ∈ t ⟹ h x = k x) ⟹ Q h = Q k"
and hom: "⋀f g. ⟦continuous_on s f; f ` s ⊆ u; P f⟧
⟹ ∃c. homotopic_with P s u f (λx. c)"
and contf: "continuous_on t f" and imf: "f ` t ⊆ u" and Qf: "Q f"
obtains c where "homotopic_with Q t u f (λx. c)"
proof -
have feq: "⋀x. x ∈ t ⟹ (f ∘ h ∘ k) x = f x" using idhk imf by auto
have "continuous_on s (f ∘ h)"
using contf conth continuous_on_compose imh by blast
moreover have "(f ∘ h) ` s ⊆ u"
using imf imh by fastforce
moreover have "P (f ∘ h)"
by (simp add: P Qf contf imf)
ultimately obtain c where "homotopic_with P s u (f ∘ h) (λx. c)"
by (metis hom)
then have "homotopic_with Q t u (f ∘ h ∘ k) ((λx. c) ∘ k)"
apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
using Q by (auto simp: contk imk)
then show ?thesis
apply (rule_tac c = c in that)
apply (erule homotopic_with_eq)
apply (metis feq, simp)
apply (metis Qeq)
done
qed
end
lemma simply_connected_retraction_gen:
shows "⟦simply_connected S; continuous_on S h; h ` S = T;
continuous_on T k; k ` T ⊆ S; ⋀y. y ∈ T ⟹ h(k y) = y⟧
⟹ simply_connected T"
apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
apply (rule Retracts.homotopically_trivial_retraction_gen
[of S h _ k _ "λp. pathfinish p = pathstart p" "λp. pathfinish p = pathstart p"])
apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
done
lemma homeomorphic_simply_connected:
"⟦S homeomorphic T; simply_connected S⟧ ⟹ simply_connected T"
by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)
lemma homeomorphic_simply_connected_eq:
"S homeomorphic T ⟹ (simply_connected S ⟷ simply_connected T)"
by (metis homeomorphic_simply_connected homeomorphic_sym)
subsection‹Homotopy equivalence›
definition%important homotopy_eqv :: "'a::topological_space set ⇒ 'b::topological_space set ⇒ bool"
(infix "homotopy'_eqv" 50)
where "S homotopy_eqv T ≡
∃f g. continuous_on S f ∧ f ` S ⊆ T ∧
continuous_on T g ∧ g ` T ⊆ S ∧
homotopic_with (λx. True) S S (g ∘ f) id ∧
homotopic_with (λx. True) T T (f ∘ g) id"
lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T ⟹ S homotopy_eqv T"
unfolding homeomorphic_def homotopy_eqv_def homeomorphism_def
by (fastforce intro!: homotopic_with_equal continuous_on_compose)
lemma homotopy_eqv_refl: "S homotopy_eqv S"
by (rule homeomorphic_imp_homotopy_eqv homeomorphic_refl)+
lemma homotopy_eqv_sym: "S homotopy_eqv T ⟷ T homotopy_eqv S"
by (auto simp: homotopy_eqv_def)
lemma homotopy_eqv_trans [trans]:
fixes S :: "'a::real_normed_vector set" and U :: "'c::real_normed_vector set"
assumes ST: "S homotopy_eqv T" and TU: "T homotopy_eqv U"
shows "S homotopy_eqv U"
proof -
obtain f1 g1 where f1: "continuous_on S f1" "f1 ` S ⊆ T"
and g1: "continuous_on T g1" "g1 ` T ⊆ S"
and hom1: "homotopic_with (λx. True) S S (g1 ∘ f1) id"
"homotopic_with (λx. True) T T (f1 ∘ g1) id"
using ST by (auto simp: homotopy_eqv_def)
obtain f2 g2 where f2: "continuous_on T f2" "f2 ` T ⊆ U"
and g2: "continuous_on U g2" "g2 ` U ⊆ T"
and hom2: "homotopic_with (λx. True) T T (g2 ∘ f2) id"
"homotopic_with (λx. True) U U (f2 ∘ g2) id"
using TU by (auto simp: homotopy_eqv_def)
have "homotopic_with (λf. True) S T (g2 ∘ f2 ∘ f1) (id ∘ f1)"
by (rule homotopic_with_compose_continuous_right hom2 f1)+
then have "homotopic_with (λf. True) S T (g2 ∘ (f2 ∘ f1)) (id ∘ f1)"
by (simp add: o_assoc)
then have "homotopic_with (λx. True) S S
(g1 ∘ (g2 ∘ (f2 ∘ f1))) (g1 ∘ (id ∘ f1))"
by (simp add: g1 homotopic_with_compose_continuous_left)
moreover have "homotopic_with (λx. True) S S (g1 ∘ id ∘ f1) id"
using hom1 by simp
ultimately have SS: "homotopic_with (λx. True) S S (g1 ∘ g2 ∘ (f2 ∘ f1)) id"
apply (simp add: o_assoc)
apply (blast intro: homotopic_with_trans)
done
have "homotopic_with (λf. True) U T (f1 ∘ g1 ∘ g2) (id ∘ g2)"
by (rule homotopic_with_compose_continuous_right hom1 g2)+
then have "homotopic_with (λf. True) U T (f1 ∘ (g1 ∘ g2)) (id ∘ g2)"
by (simp add: o_assoc)
then have "homotopic_with (λx. True) U U
(f2 ∘ (f1 ∘ (g1 ∘ g2))) (f2 ∘ (id ∘ g2))"
by (simp add: f2 homotopic_with_compose_continuous_left)
moreover have "homotopic_with (λx. True) U U (f2 ∘ id ∘ g2) id"
using hom2 by simp
ultimately have UU: "homotopic_with (λx. True) U U (f2 ∘ f1 ∘ (g1 ∘ g2)) id"
apply (simp add: o_assoc)
apply (blast intro: homotopic_with_trans)
done
show ?thesis
unfolding homotopy_eqv_def
apply (rule_tac x = "f2 ∘ f1" in exI)
apply (rule_tac x = "g1 ∘ g2" in exI)
apply (intro conjI continuous_on_compose SS UU)
using f1 f2 g1 g2 apply (force simp: elim!: continuous_on_subset)+
done
qed
lemma homotopy_eqv_inj_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "(f ` S) homotopy_eqv S"
apply (rule homeomorphic_imp_homotopy_eqv)
using assms homeomorphic_sym linear_homeomorphic_image by auto
lemma homotopy_eqv_translation:
fixes S :: "'a::real_normed_vector set"
shows "(+) a ` S homotopy_eqv S"
apply (rule homeomorphic_imp_homotopy_eqv)
using homeomorphic_translation homeomorphic_sym by blast
lemma homotopy_eqv_homotopic_triviality_imp:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
and f: "continuous_on U f" "f ` U ⊆ T"
and g: "continuous_on U g" "g ` U ⊆ T"
and homUS: "⋀f g. ⟦continuous_on U f; f ` U ⊆ S;
continuous_on U g; g ` U ⊆ S⟧
⟹ homotopic_with (λx. True) U S f g"
shows "homotopic_with (λx. True) U T f g"
proof -
obtain h k where h: "continuous_on S h" "h ` S ⊆ T"
and k: "continuous_on T k" "k ` T ⊆ S"
and hom: "homotopic_with (λx. True) S S (k ∘ h) id"
"homotopic_with (λx. True) T T (h ∘ k) id"
using assms by (auto simp: homotopy_eqv_def)
have "homotopic_with (λf. True) U S (k ∘ f) (k ∘ g)"
apply (rule homUS)
using f g k
apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
apply (force simp: o_def)+
done
then have "homotopic_with (λx. True) U T (h ∘ (k ∘ f)) (h ∘ (k ∘ g))"
apply (rule homotopic_with_compose_continuous_left)
apply (simp_all add: h)
done
moreover have "homotopic_with (λx. True) U T (h ∘ k ∘ f) (id ∘ f)"
apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
apply (auto simp: hom f)
done
moreover have "homotopic_with (λx. True) U T (h ∘ k ∘ g) (id ∘ g)"
apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
apply (auto simp: hom g)
done
ultimately show "homotopic_with (λx. True) U T f g"
apply (simp add: o_assoc)
using homotopic_with_trans homotopic_with_sym by blast
qed
lemma homotopy_eqv_homotopic_triviality:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
shows "(∀f g. continuous_on U f ∧ f ` U ⊆ S ∧
continuous_on U g ∧ g ` U ⊆ S
⟶ homotopic_with (λx. True) U S f g) ⟷
(∀f g. continuous_on U f ∧ f ` U ⊆ T ∧
continuous_on U g ∧ g ` U ⊆ T
⟶ homotopic_with (λx. True) U T f g)"
apply (rule iffI)
apply (metis assms homotopy_eqv_homotopic_triviality_imp)
by (metis (no_types) assms homotopy_eqv_homotopic_triviality_imp homotopy_eqv_sym)
lemma homotopy_eqv_cohomotopic_triviality_null_imp:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
and f: "continuous_on T f" "f ` T ⊆ U"
and homSU: "⋀f. ⟦continuous_on S f; f ` S ⊆ U⟧
⟹ ∃c. homotopic_with (λx. True) S U f (λx. c)"
obtains c where "homotopic_with (λx. True) T U f (λx. c)"
proof -
obtain h k where h: "continuous_on S h" "h ` S ⊆ T"
and k: "continuous_on T k" "k ` T ⊆ S"
and hom: "homotopic_with (λx. True) S S (k ∘ h) id"
"homotopic_with (λx. True) T T (h ∘ k) id"
using assms by (auto simp: homotopy_eqv_def)
obtain c where "homotopic_with (λx. True) S U (f ∘ h) (λx. c)"
apply (rule exE [OF homSU [of "f ∘ h"]])
apply (intro continuous_on_compose h)
using h f apply (force elim!: continuous_on_subset)+
done
then have "homotopic_with (λx. True) T U ((f ∘ h) ∘ k) ((λx. c) ∘ k)"
apply (rule homotopic_with_compose_continuous_right [where X=S])
using k by auto
moreover have "homotopic_with (λx. True) T U (f ∘ id) (f ∘ (h ∘ k))"
apply (rule homotopic_with_compose_continuous_left [where Y=T])
apply (simp add: hom homotopic_with_symD)
using f apply auto
done
ultimately show ?thesis
apply (rule_tac c=c in that)
apply (simp add: o_def)
using homotopic_with_trans by blast
qed
lemma homotopy_eqv_cohomotopic_triviality_null:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
shows "(∀f. continuous_on S f ∧ f ` S ⊆ U
⟶ (∃c. homotopic_with (λx. True) S U f (λx. c))) ⟷
(∀f. continuous_on T f ∧ f ` T ⊆ U
⟶ (∃c. homotopic_with (λx. True) T U f (λx. c)))"
apply (rule iffI)
apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
lemma homotopy_eqv_homotopic_triviality_null_imp:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
and f: "continuous_on U f" "f ` U ⊆ T"
and homSU: "⋀f. ⟦continuous_on U f; f ` U ⊆ S⟧
⟹ ∃c. homotopic_with (λx. True) U S f (λx. c)"
shows "∃c. homotopic_with (λx. True) U T f (λx. c)"
proof -
obtain h k where h: "continuous_on S h" "h ` S ⊆ T"
and k: "continuous_on T k" "k ` T ⊆ S"
and hom: "homotopic_with (λx. True) S S (k ∘ h) id"
"homotopic_with (λx. True) T T (h ∘ k) id"
using assms by (auto simp: homotopy_eqv_def)
obtain c::'a where "homotopic_with (λx. True) U S (k ∘ f) (λx. c)"
apply (rule exE [OF homSU [of "k ∘ f"]])
apply (intro continuous_on_compose h)
using k f apply (force elim!: continuous_on_subset)+
done
then have "homotopic_with (λx. True) U T (h ∘ (k ∘ f)) (h ∘ (λx. c))"
apply (rule homotopic_with_compose_continuous_left [where Y=S])
using h by auto
moreover have "homotopic_with (λx. True) U T (id ∘ f) ((h ∘ k) ∘ f)"
apply (rule homotopic_with_compose_continuous_right [where X=T])
apply (simp add: hom homotopic_with_symD)
using f apply auto
done
ultimately show ?thesis
using homotopic_with_trans by (fastforce simp add: o_def)
qed
lemma homotopy_eqv_homotopic_triviality_null:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
and U :: "'c::real_normed_vector set"
assumes "S homotopy_eqv T"
shows "(∀f. continuous_on U f ∧ f ` U ⊆ S
⟶ (∃c. homotopic_with (λx. True) U S f (λx. c))) ⟷
(∀f. continuous_on U f ∧ f ` U ⊆ T
⟶ (∃c. homotopic_with (λx. True) U T f (λx. c)))"
apply (rule iffI)
apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
lemma homotopy_eqv_contractible_sets:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
assumes "contractible S" "contractible T" "S = {} ⟷ T = {}"
shows "S homotopy_eqv T"
proof (cases "S = {}")
case True with assms show ?thesis
by (simp add: homeomorphic_imp_homotopy_eqv)
next
case False
with assms obtain a b where "a ∈ S" "b ∈ T"
by auto
then show ?thesis
unfolding homotopy_eqv_def
apply (rule_tac x="λx. b" in exI)
apply (rule_tac x="λx. a" in exI)
apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
apply (auto simp: o_def continuous_on_const)
done
qed
lemma homotopy_eqv_empty1 [simp]:
fixes S :: "'a::real_normed_vector set"
shows "S homotopy_eqv ({}::'b::real_normed_vector set) ⟷ S = {}"
apply (rule iffI)
using homotopy_eqv_def apply fastforce
by (simp add: homotopy_eqv_contractible_sets)
lemma homotopy_eqv_empty2 [simp]:
fixes S :: "'a::real_normed_vector set"
shows "({}::'b::real_normed_vector set) homotopy_eqv S ⟷ S = {}"
by (metis homotopy_eqv_empty1 homotopy_eqv_sym)
lemma homotopy_eqv_contractibility:
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
shows "S homotopy_eqv T ⟹ (contractible S ⟷ contractible T)"
unfolding homotopy_eqv_def
by (blast intro: homotopy_dominated_contractibility)
lemma homotopy_eqv_sing:
fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
shows "S homotopy_eqv {a} ⟷ S ≠ {} ∧ contractible S"
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False then show ?thesis
by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
qed
lemma homeomorphic_contractible_eq:
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
shows "S homeomorphic T ⟹ (contractible S ⟷ contractible T)"
by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)
lemma homeomorphic_contractible:
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
shows "⟦contractible S; S homeomorphic T⟧ ⟹ contractible T"
by (metis homeomorphic_contractible_eq)
subsection%unimportant‹Misc other results›
lemma bounded_connected_Compl_real:
fixes S :: "real set"
assumes "bounded S" and conn: "connected(- S)"
shows "S = {}"
proof -
obtain a b where "S ⊆ box a b"
by (meson assms bounded_subset_box_symmetric)
then have "a ∉ S" "b ∉ S"
by auto
then have "∀x. a ≤ x ∧ x ≤ b ⟶ x ∈ - S"
by (meson Compl_iff conn connected_iff_interval)
then show ?thesis
using ‹S ⊆ box a b› by auto
qed
lemma bounded_connected_Compl_1:
fixes S :: "'a::{euclidean_space} set"
assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
shows "S = {}"
proof -
have "DIM('a) = DIM(real)"
by (simp add: "1")
then obtain f::"'a ⇒ real" and g
where "linear f" "⋀x. norm(f x) = norm x" "⋀x. g(f x) = x" "⋀y. f(g y) = y"
by (rule isomorphisms_UNIV_UNIV) blast
with ‹bounded S› have "bounded (f ` S)"
using bounded_linear_image linear_linear by blast
have "connected (f ` (-S))"
using connected_linear_image assms ‹linear f› by blast
moreover have "f ` (-S) = - (f ` S)"
apply (rule bij_image_Compl_eq)
apply (auto simp: bij_def)
apply (metis ‹⋀x. g (f x) = x› injI)
by (metis UNIV_I ‹⋀y. f (g y) = y› image_iff)
finally have "connected (- (f ` S))"
by simp
then have "f ` S = {}"
using ‹bounded (f ` S)› bounded_connected_Compl_real by blast
then show ?thesis
by blast
qed
subsection%unimportant‹Some Uncountable Sets›
lemma uncountable_closed_segment:
fixes a :: "'a::real_normed_vector"
assumes "a ≠ b" shows "uncountable (closed_segment a b)"
unfolding path_image_linepath [symmetric] path_image_def
using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
countable_image_inj_on by auto
lemma uncountable_open_segment:
fixes a :: "'a::real_normed_vector"
assumes "a ≠ b" shows "uncountable (open_segment a b)"
by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
lemma uncountable_convex:
fixes a :: "'a::real_normed_vector"
assumes "convex S" "a ∈ S" "b ∈ S" "a ≠ b"
shows "uncountable S"
proof -
have "uncountable (closed_segment a b)"
by (simp add: uncountable_closed_segment assms)
then show ?thesis
by (meson assms convex_contains_segment countable_subset)
qed
lemma uncountable_ball:
fixes a :: "'a::euclidean_space"
assumes "r > 0"
shows "uncountable (ball a r)"
proof -
have "uncountable (open_segment a (a + r *⇩R (SOME i. i ∈ Basis)))"
by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
moreover have "open_segment a (a + r *⇩R (SOME i. i ∈ Basis)) ⊆ ball a r"
using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
ultimately show ?thesis
by (metis countable_subset)
qed
lemma ball_minus_countable_nonempty:
assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
shows "ball z r - A ≠ {}"
proof
assume *: "ball z r - A = {}"
have "uncountable (ball z r - A)"
by (intro uncountable_minus_countable assms uncountable_ball)
thus False by (subst (asm) *) auto
qed
lemma uncountable_cball:
fixes a :: "'a::euclidean_space"
assumes "r > 0"
shows "uncountable (cball a r)"
using assms countable_subset uncountable_ball by auto
lemma pairwise_disjnt_countable:
fixes 𝒩 :: "nat set set"
assumes "pairwise disjnt 𝒩"
shows "countable 𝒩"
proof -
have "inj_on (λX. SOME n. n ∈ X) (𝒩 - {{}})"
apply (clarsimp simp add: inj_on_def)
by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
then show ?thesis
by (metis countable_Diff_eq countable_def)
qed
lemma pairwise_disjnt_countable_Union:
assumes "countable (⋃𝒩)" and pwd: "pairwise disjnt 𝒩"
shows "countable 𝒩"
proof -
obtain f :: "_ ⇒ nat" where f: "inj_on f (⋃𝒩)"
using assms by blast
then have "pairwise disjnt (⋃ X ∈ 𝒩. {f ` X})"
using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
then have "countable (⋃ X ∈ 𝒩. {f ` X})"
using pairwise_disjnt_countable by blast
then show ?thesis
by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
qed
lemma connected_uncountable:
fixes S :: "'a::metric_space set"
assumes "connected S" "a ∈ S" "b ∈ S" "a ≠ b" shows "uncountable S"
proof -
have "continuous_on S (dist a)"
by (intro continuous_intros)
then have "connected (dist a ` S)"
by (metis connected_continuous_image ‹connected S›)
then have "closed_segment 0 (dist a b) ⊆ (dist a ` S)"
by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
then have "uncountable (dist a ` S)"
by (metis ‹a ≠ b› countable_subset dist_eq_0_iff uncountable_closed_segment)
then show ?thesis
by blast
qed
lemma path_connected_uncountable:
fixes S :: "'a::metric_space set"
assumes "path_connected S" "a ∈ S" "b ∈ S" "a ≠ b" shows "uncountable S"
using path_connected_imp_connected assms connected_uncountable by metis
lemma connected_finite_iff_sing:
fixes S :: "'a::metric_space set"
assumes "connected S"
shows "finite S ⟷ S = {} ∨ (∃a. S = {a})" (is "_ = ?rhs")
proof -
have "uncountable S" if "¬ ?rhs"
using connected_uncountable assms that by blast
then show ?thesis
using uncountable_infinite by auto
qed
lemma connected_card_eq_iff_nontrivial:
fixes S :: "'a::metric_space set"
shows "connected S ⟹ uncountable S ⟷ ~(∃a. S ⊆ {a})"
apply (auto simp: countable_finite finite_subset)
by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
lemma simple_path_image_uncountable:
fixes g :: "real ⇒ 'a::metric_space"
assumes "simple_path g"
shows "uncountable (path_image g)"
proof -
have "g 0 ∈ path_image g" "g (1/2) ∈ path_image g"
by (simp_all add: path_defs)
moreover have "g 0 ≠ g (1/2)"
using assms by (fastforce simp add: simple_path_def)
ultimately show ?thesis
apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
by blast
qed
lemma arc_image_uncountable:
fixes g :: "real ⇒ 'a::metric_space"
assumes "arc g"
shows "uncountable (path_image g)"
by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
subsection%unimportant‹ Some simple positive connection theorems›
proposition path_connected_convex_diff_countable:
fixes U :: "'a::euclidean_space set"
assumes "convex U" "~ collinear U" "countable S"
shows "path_connected(U - S)"
proof (clarsimp simp add: path_connected_def)
fix a b
assume "a ∈ U" "a ∉ S" "b ∈ U" "b ∉ S"
let ?m = "midpoint a b"
show "∃g. path g ∧ path_image g ⊆ U - S ∧ pathstart g = a ∧ pathfinish g = b"
proof (cases "a = b")
case True
then show ?thesis
by (metis DiffI ‹a ∈ U› ‹a ∉ S› path_component_def path_component_refl)
next
case False
then have "a ≠ ?m" "b ≠ ?m"
using midpoint_eq_endpoint by fastforce+
have "?m ∈ U"
using ‹a ∈ U› ‹b ∈ U› ‹convex U› convex_contains_segment by force
obtain c where "c ∈ U" and nc_abc: "¬ collinear {a,b,c}"
by (metis False ‹a ∈ U› ‹b ∈ U› ‹~ collinear U› collinear_triples insert_absorb)
have ncoll_mca: "¬ collinear {?m,c,a}"
by (metis (full_types) ‹a ≠ ?m› collinear_3_trans collinear_midpoint insert_commute nc_abc)
have ncoll_mcb: "¬ collinear {?m,c,b}"
by (metis (full_types) ‹b ≠ ?m› collinear_3_trans collinear_midpoint insert_commute nc_abc)
have "c ≠ ?m"
by (metis collinear_midpoint insert_commute nc_abc)
then have "closed_segment ?m c ⊆ U"
by (simp add: ‹c ∈ U› ‹?m ∈ U› ‹convex U› closed_segment_subset)
then obtain z where z: "z ∈ closed_segment ?m c"
and disjS: "(closed_segment a z ∪ closed_segment z b) ∩ S = {}"
proof -
have False if "closed_segment ?m c ⊆ {z. (closed_segment a z ∪ closed_segment z b) ∩ S ≠ {}}"
proof -
have closb: "closed_segment ?m c ⊆
{z ∈ closed_segment ?m c. closed_segment a z ∩ S ≠ {}} ∪ {z ∈ closed_segment ?m c. closed_segment z b ∩ S ≠ {}}"
using that by blast
have *: "countable {z ∈ closed_segment ?m c. closed_segment z u ∩ S ≠ {}}"
if "u ∈ U" "u ∉ S" and ncoll: "¬ collinear {?m, c, u}" for u
proof -
have **: False if x1: "x1 ∈ closed_segment ?m c" and x2: "x2 ∈ closed_segment ?m c"
and "x1 ≠ x2" "x1 ≠ u"
and w: "w ∈ closed_segment x1 u" "w ∈ closed_segment x2 u"
and "w ∈ S" for x1 x2 w
proof -
have "x1 ∈ affine hull {?m,c}" "x2 ∈ affine hull {?m,c}"
using segment_as_ball x1 x2 by auto
then have coll_x1: "collinear {x1, ?m, c}" and coll_x2: "collinear {?m, c, x2}"
by (simp_all add: affine_hull_3_imp_collinear) (metis affine_hull_3_imp_collinear insert_commute)
have "¬ collinear {x1, u, x2}"
proof
assume "collinear {x1, u, x2}"
then have "collinear {?m, c, u}"
by (metis (full_types) ‹c ≠ ?m› coll_x1 coll_x2 collinear_3_trans insert_commute ncoll ‹x1 ≠ x2›)
with ncoll show False ..
qed
then have "closed_segment x1 u ∩ closed_segment u x2 = {u}"
by (blast intro!: Int_closed_segment)
then have "w = u"
using closed_segment_commute w by auto
show ?thesis
using ‹u ∉ S› ‹w = u› that(7) by auto
qed
then have disj: "disjoint ((⋃z∈closed_segment ?m c. {closed_segment z u ∩ S}))"
by (fastforce simp: pairwise_def disjnt_def)
have cou: "countable ((⋃z ∈ closed_segment ?m c. {closed_segment z u ∩ S}) - {{}})"
apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
apply (rule countable_subset [OF _ ‹countable S›], auto)
done
define f where "f ≡ λX. (THE z. z ∈ closed_segment ?m c ∧ X = closed_segment z u ∩ S)"
show ?thesis
proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
fix x
assume x: "x ∈ closed_segment ?m c" "closed_segment x u ∩ S ≠ {}"
show "x ∈ f ` ((⋃z∈closed_segment ?m c. {closed_segment z u ∩ S}) - {{}})"
proof (rule_tac x="closed_segment x u ∩ S" in image_eqI)
show "x = f (closed_segment x u ∩ S)"
unfolding f_def
apply (rule the_equality [symmetric])
using x apply (auto simp: dest: **)
done
qed (use x in auto)
qed
qed
have "uncountable (closed_segment ?m c)"
by (metis ‹c ≠ ?m› uncountable_closed_segment)
then show False
using closb * [OF ‹a ∈ U› ‹a ∉ S› ncoll_mca] * [OF ‹b ∈ U› ‹b ∉ S› ncoll_mcb]
apply (simp add: closed_segment_commute)
by (simp add: countable_subset)
qed
then show ?thesis
by (force intro: that)
qed
show ?thesis
proof (intro exI conjI)
have "path_image (linepath a z +++ linepath z b) ⊆ U"
by (metis ‹a ∈ U› ‹b ∈ U› ‹closed_segment ?m c ⊆ U› z ‹convex U› closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
with disjS show "path_image (linepath a z +++ linepath z b) ⊆ U - S"
by (force simp: path_image_join)
qed auto
qed
qed
corollary connected_convex_diff_countable:
fixes U :: "'a::euclidean_space set"
assumes "convex U" "~ collinear U" "countable S"
shows "connected(U - S)"
by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)
lemma path_connected_punctured_convex:
assumes "convex S" and aff: "aff_dim S ≠ 1"
shows "path_connected(S - {a})"
proof -
consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S ≥ 2"
using assms aff_dim_geq [of S] by linarith
then show ?thesis
proof cases
assume "aff_dim S = -1"
then show ?thesis
by (metis aff_dim_empty empty_Diff path_connected_empty)
next
assume "aff_dim S = 0"
then show ?thesis
by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
next
assume ge2: "aff_dim S ≥ 2"
then have "¬ collinear S"
proof (clarsimp simp add: collinear_affine_hull)
fix u v
assume "S ⊆ affine hull {u, v}"
then have "aff_dim S ≤ aff_dim {u, v}"
by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
with ge2 show False
by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
qed
then show ?thesis
apply (rule path_connected_convex_diff_countable [OF ‹convex S›])
by simp
qed
qed
lemma connected_punctured_convex:
shows "⟦convex S; aff_dim S ≠ 1⟧ ⟹ connected(S - {a})"
using path_connected_imp_connected path_connected_punctured_convex by blast
lemma path_connected_complement_countable:
fixes S :: "'a::euclidean_space set"
assumes "2 ≤ DIM('a)" "countable S"
shows "path_connected(- S)"
proof -
have "path_connected(UNIV - S)"
apply (rule path_connected_convex_diff_countable)
using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
then show ?thesis
by (simp add: Compl_eq_Diff_UNIV)
qed
proposition path_connected_openin_diff_countable:
fixes S :: "'a::euclidean_space set"
assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
and "~ collinear S" "countable T"
shows "path_connected(S - T)"
proof (clarsimp simp add: path_connected_component)
fix x y
assume xy: "x ∈ S" "x ∉ T" "y ∈ S" "y ∉ T"
show "path_component (S - T) x y"
proof (rule connected_equivalence_relation_gen [OF ‹connected S›, where P = "λx. x ∉ T"])
show "∃z. z ∈ U ∧ z ∉ T" if opeU: "openin (subtopology euclidean S) U" and "x ∈ U" for U x
proof -
have "openin (subtopology euclidean (affine hull S)) U"
using opeU ope openin_trans by blast
with ‹x ∈ U› obtain r where Usub: "U ⊆ affine hull S" and "r > 0"
and subU: "ball x r ∩ affine hull S ⊆ U"
by (auto simp: openin_contains_ball)
with ‹x ∈ U› have x: "x ∈ ball x r ∩ affine hull S"
by auto
have "~ S ⊆ {x}"
using ‹~ collinear S› collinear_subset by blast
then obtain x' where "x' ≠ x" "x' ∈ S"
by blast
obtain y where y: "y ≠ x" "y ∈ ball x r ∩ affine hull S"
proof
show "x + (r / 2 / norm(x' - x)) *⇩R (x' - x) ≠ x"
using ‹x' ≠ x› ‹r > 0› by auto
show "x + (r / 2 / norm (x' - x)) *⇩R (x' - x) ∈ ball x r ∩ affine hull S"
using ‹x' ≠ x› ‹r > 0› ‹x' ∈ S› x
by (simp add: dist_norm mem_affine_3_minus hull_inc)
qed
have "convex (ball x r ∩ affine hull S)"
by (simp add: affine_imp_convex convex_Int)
with x y subU have "uncountable U"
by (meson countable_subset uncountable_convex)
then have "¬ U ⊆ T"
using ‹countable T› countable_subset by blast
then show ?thesis by blast
qed
show "∃U. openin (subtopology euclidean S) U ∧ x ∈ U ∧
(∀x∈U. ∀y∈U. x ∉ T ∧ y ∉ T ⟶ path_component (S - T) x y)"
if "x ∈ S" for x
proof -
obtain r where Ssub: "S ⊆ affine hull S" and "r > 0"
and subS: "ball x r ∩ affine hull S ⊆ S"
using ope ‹x ∈ S› by (auto simp: openin_contains_ball)
then have conv: "convex (ball x r ∩ affine hull S)"
by (simp add: affine_imp_convex convex_Int)
have "¬ aff_dim (affine hull S) ≤ 1"
using ‹¬ collinear S› collinear_aff_dim by auto
then have "¬ collinear (ball x r ∩ affine hull S)"
apply (simp add: collinear_aff_dim)
by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball ‹0 < r› aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
then have *: "path_connected ((ball x r ∩ affine hull S) - T)"
by (rule path_connected_convex_diff_countable [OF conv _ ‹countable T›])
have ST: "ball x r ∩ affine hull S - T ⊆ S - T"
using subS by auto
show ?thesis
proof (intro exI conjI)
show "x ∈ ball x r ∩ affine hull S"
using ‹x ∈ S› ‹r > 0› by (simp add: hull_inc)
have "openin (subtopology euclidean (affine hull S)) (ball x r ∩ affine hull S)"
by (simp add: inf.commute openin_Int_open)
then show "openin (subtopology euclidean S) (ball x r ∩ affine hull S)"
by (rule openin_subset_trans [OF _ subS Ssub])
qed (use * path_component_trans in ‹auto simp: path_connected_component path_component_of_subset [OF ST]›)
qed
qed (use xy path_component_trans in auto)
qed
corollary connected_openin_diff_countable:
fixes S :: "'a::euclidean_space set"
assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
and "~ collinear S" "countable T"
shows "connected(S - T)"
by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])
corollary path_connected_open_diff_countable:
fixes S :: "'a::euclidean_space set"
assumes "2 ≤ DIM('a)" "open S" "connected S" "countable T"
shows "path_connected(S - T)"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: path_connected_empty)
next
case False
show ?thesis
proof (rule path_connected_openin_diff_countable)
show "openin (subtopology euclidean (affine hull S)) S"
by (simp add: assms hull_subset open_subset)
show "¬ collinear S"
using assms False by (simp add: collinear_aff_dim aff_dim_open)
qed (simp_all add: assms)
qed
corollary connected_open_diff_countable:
fixes S :: "'a::euclidean_space set"
assumes "2 ≤ DIM('a)" "open S" "connected S" "countable T"
shows "connected(S - T)"
by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)
subsection‹Self-homeomorphisms shuffling points about in various ways›
subsubsection%unimportant‹The theorem ‹homeomorphism_moving_points_exists››
lemma homeomorphism_moving_point_1:
fixes a :: "'a::euclidean_space"
assumes "affine T" "a ∈ T" and u: "u ∈ ball a r ∩ T"
obtains f g where "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
"f a = u" "⋀x. x ∈ sphere a r ⟹ f x = x"
proof -
have nou: "norm (u - a) < r" and "u ∈ T"
using u by (auto simp: dist_norm norm_minus_commute)
then have "0 < r"
by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
define f where "f ≡ λx. (1 - norm(x - a) / r) *⇩R (u - a) + x"
have *: "False" if eq: "x + (norm y / r) *⇩R u = y + (norm x / r) *⇩R u"
and nou: "norm u < r" and yx: "norm y < norm x" for x y and u::'a
proof -
have "x = y + (norm x / r - (norm y / r)) *⇩R u"
using eq by (simp add: algebra_simps)
then have "norm x = norm (y + ((norm x - norm y) / r) *⇩R u)"
by (metis diff_divide_distrib)
also have "… ≤ norm y + norm(((norm x - norm y) / r) *⇩R u)"
using norm_triangle_ineq by blast
also have "… = norm y + (norm x - norm y) * (norm u / r)"
using yx ‹r > 0›
by (simp add: divide_simps)
also have "… < norm y + (norm x - norm y) * 1"
apply (subst add_less_cancel_left)
apply (rule mult_strict_left_mono)
using nou ‹0 < r› yx
apply (simp_all add: field_simps)
done
also have "… = norm x"
by simp
finally show False by simp
qed
have "inj f"
unfolding f_def
proof (clarsimp simp: inj_on_def)
fix x y
assume "(1 - norm (x - a) / r) *⇩R (u - a) + x =
(1 - norm (y - a) / r) *⇩R (u - a) + y"
then have eq: "(x - a) + (norm (y - a) / r) *⇩R (u - a) = (y - a) + (norm (x - a) / r) *⇩R (u - a)"
by (auto simp: algebra_simps)
show "x=y"
proof (cases "norm (x - a) = norm (y - a)")
case True
then show ?thesis
using eq by auto
next
case False
then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
by linarith
then have "False"
proof cases
case 1 show False
using * [OF _ nou 1] eq by simp
next
case 2 with * [OF eq nou] show False
by auto
qed
then show "x=y" ..
qed
qed
then have inj_onf: "inj_on f (cball a r ∩ T)"
using inj_on_Int by fastforce
have contf: "continuous_on (cball a r ∩ T) f"
unfolding f_def using ‹0 < r› by (intro continuous_intros) blast
have fim: "f ` (cball a r ∩ T) = cball a r ∩ T"
proof
have *: "norm (y + (1 - norm y / r) *⇩R u) ≤ r" if "norm y ≤ r" "norm u < r" for y u::'a
proof -
have "norm (y + (1 - norm y / r) *⇩R u) ≤ norm y + norm((1 - norm y / r) *⇩R u)"
using norm_triangle_ineq by blast
also have "… = norm y + abs(1 - norm y / r) * norm u"
by simp
also have "… ≤ r"
proof -
have "(r - norm u) * (r - norm y) ≥ 0"
using that by auto
then have "r * norm u + r * norm y ≤ r * r + norm u * norm y"
by (simp add: algebra_simps)
then show ?thesis
using that ‹0 < r› by (simp add: abs_if field_simps)
qed
finally show ?thesis .
qed
have "f ` (cball a r) ⊆ cball a r"
apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
moreover have "f ` T ⊆ T"
unfolding f_def using ‹affine T› ‹a ∈ T› ‹u ∈ T›
by (force simp: add.commute mem_affine_3_minus)
ultimately show "f ` (cball a r ∩ T) ⊆ cball a r ∩ T"
by blast
next
show "cball a r ∩ T ⊆ f ` (cball a r ∩ T)"
proof (clarsimp simp add: dist_norm norm_minus_commute)
fix x
assume x: "norm (x - a) ≤ r" and "x ∈ T"
have "∃v ∈ {0..1}. ((1 - v) * r - norm ((x - a) - v *⇩R (u - a))) ∙ 1 = 0"
by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
then obtain v where "0≤v" "v≤1" and v: "(1 - v) * r = norm ((x - a) - v *⇩R (u - a))"
by auto
show "x ∈ f ` (cball a r ∩ T)"
proof (rule image_eqI)
show "x = f (x - v *⇩R (u - a))"
using ‹r > 0› v by (simp add: f_def field_simps)
have "x - v *⇩R (u - a) ∈ cball a r"
using ‹r > 0› v ‹0 ≤ v›
apply (simp add: field_simps dist_norm norm_minus_commute)
by (metis le_add_same_cancel2 order.order_iff_strict zero_le_mult_iff)
moreover have "x - v *⇩R (u - a) ∈ T"
by (simp add: f_def ‹affine T› ‹u ∈ T› ‹x ∈ T› assms mem_affine_3_minus2)
ultimately show "x - v *⇩R (u - a) ∈ cball a r ∩ T"
by blast
qed
qed
qed
have "∃g. homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
apply (simp add: affine_closed compact_Int_closed ‹affine T›)
done
then show ?thesis
apply (rule exE)
apply (erule_tac f=f in that)
using ‹r > 0›
apply (simp_all add: f_def dist_norm norm_minus_commute)
done
qed
corollary homeomorphism_moving_point_2:
fixes a :: "'a::euclidean_space"
assumes "affine T" "a ∈ T" and u: "u ∈ ball a r ∩ T" and v: "v ∈ ball a r ∩ T"
obtains f g where "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
"f u = v" "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ f x = x"
proof -
have "0 < r"
by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
obtain f1 g1 where hom1: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f1 g1"
and "f1 a = u" and f1: "⋀x. x ∈ sphere a r ⟹ f1 x = x"
using homeomorphism_moving_point_1 [OF ‹affine T› ‹a ∈ T› u] by blast
obtain f2 g2 where hom2: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f2 g2"
and "f2 a = v" and f2: "⋀x. x ∈ sphere a r ⟹ f2 x = x"
using homeomorphism_moving_point_1 [OF ‹affine T› ‹a ∈ T› v] by blast
show ?thesis
proof
show "homeomorphism (cball a r ∩ T) (cball a r ∩ T) (f2 ∘ g1) (f1 ∘ g2)"
by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
have "g1 u = a"
using ‹0 < r› ‹f1 a = u› assms hom1 homeomorphism_apply1 by fastforce
then show "(f2 ∘ g1) u = v"
by (simp add: ‹f2 a = v›)
show "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ (f2 ∘ g1) x = x"
using f1 f2 hom1 homeomorphism_apply1 by fastforce
qed
qed
corollary homeomorphism_moving_point_3:
fixes a :: "'a::euclidean_space"
assumes "affine T" "a ∈ T" and ST: "ball a r ∩ T ⊆ S" "S ⊆ T"
and u: "u ∈ ball a r ∩ T" and v: "v ∈ ball a r ∩ T"
obtains f g where "homeomorphism S S f g"
"f u = v" "{x. ~ (f x = x ∧ g x = x)} ⊆ ball a r ∩ T"
proof -
obtain f g where hom: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
and "f u = v" and fid: "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ f x = x"
using homeomorphism_moving_point_2 [OF ‹affine T› ‹a ∈ T› u v] by blast
have gid: "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ g x = x"
using fid hom homeomorphism_apply1 by fastforce
define ff where "ff ≡ λx. if x ∈ ball a r ∩ T then f x else x"
define gg where "gg ≡ λx. if x ∈ ball a r ∩ T then g x else x"
show ?thesis
proof
show "homeomorphism S S ff gg"
proof (rule homeomorphismI)
have "continuous_on ((cball a r ∩ T) ∪ (T - ball a r)) ff"
apply (simp add: ff_def)
apply (rule continuous_on_cases)
using homeomorphism_cont1 [OF hom]
apply (auto simp: affine_closed ‹affine T› continuous_on_id fid)
done
then show "continuous_on S ff"
apply (rule continuous_on_subset)
using ST by auto
have "continuous_on ((cball a r ∩ T) ∪ (T - ball a r)) gg"
apply (simp add: gg_def)
apply (rule continuous_on_cases)
using homeomorphism_cont2 [OF hom]
apply (auto simp: affine_closed ‹affine T› continuous_on_id gid)
done
then show "continuous_on S gg"
apply (rule continuous_on_subset)
using ST by auto
show "ff ` S ⊆ S"
proof (clarsimp simp add: ff_def)
fix x
assume "x ∈ S" and x: "dist a x < r" and "x ∈ T"
then have "f x ∈ cball a r ∩ T"
using homeomorphism_image1 [OF hom] by force
then show "f x ∈ S"
using ST(1) ‹x ∈ T› gid hom homeomorphism_def x by fastforce
qed
show "gg ` S ⊆ S"
proof (clarsimp simp add: gg_def)
fix x
assume "x ∈ S" and x: "dist a x < r" and "x ∈ T"
then have "g x ∈ cball a r ∩ T"
using homeomorphism_image2 [OF hom] by force
then have "g x ∈ ball a r"
using homeomorphism_apply2 [OF hom]
by (metis Diff_Diff_Int Diff_iff ‹x ∈ T› cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
then show "g x ∈ S"
using ST(1) ‹g x ∈ cball a r ∩ T› by force
qed
show "⋀x. x ∈ S ⟹ gg (ff x) = x"
apply (simp add: ff_def gg_def)
using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
apply auto
apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
done
show "⋀x. x ∈ S ⟹ ff (gg x) = x"
apply (simp add: ff_def gg_def)
using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
apply auto
apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
done
qed
show "ff u = v"
using u by (auto simp: ff_def ‹f u = v›)
show "{x. ¬ (ff x = x ∧ gg x = x)} ⊆ ball a r ∩ T"
by (auto simp: ff_def gg_def)
qed
qed
proposition homeomorphism_moving_point:
fixes a :: "'a::euclidean_space"
assumes ope: "openin (subtopology euclidean (affine hull S)) S"
and "S ⊆ T"
and TS: "T ⊆ affine hull S"
and S: "connected S" "a ∈ S" "b ∈ S"
obtains f g where "homeomorphism T T f g" "f a = b"
"{x. ~ (f x = x ∧ g x = x)} ⊆ S"
"bounded {x. ~ (f x = x ∧ g x = x)}"
proof -
have 1: "∃h k. homeomorphism T T h k ∧ h (f d) = d ∧
{x. ~ (h x = x ∧ k x = x)} ⊆ S ∧ bounded {x. ~ (h x = x ∧ k x = x)}"
if "d ∈ S" "f d ∈ S" and homfg: "homeomorphism T T f g"
and S: "{x. ~ (f x = x ∧ g x = x)} ⊆ S"
and bo: "bounded {x. ~ (f x = x ∧ g x = x)}" for d f g
proof (intro exI conjI)
show homgf: "homeomorphism T T g f"
by (metis homeomorphism_symD homfg)
then show "g (f d) = d"
by (meson ‹S ⊆ T› homeomorphism_def subsetD ‹d ∈ S›)
show "{x. ¬ (g x = x ∧ f x = x)} ⊆ S"
using S by blast
show "bounded {x. ¬ (g x = x ∧ f x = x)}"
using bo by (simp add: conj_commute)
qed
have 2: "∃f g. homeomorphism T T f g ∧ f x = f2 (f1 x) ∧
{x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
if "x ∈ S" "f1 x ∈ S" "f2 (f1 x) ∈ S"
and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
and sub: "{x. ¬ (f1 x = x ∧ g1 x = x)} ⊆ S" "{x. ¬ (f2 x = x ∧ g2 x = x)} ⊆ S"
and bo: "bounded {x. ¬ (f1 x = x ∧ g1 x = x)}" "bounded {x. ¬ (f2 x = x ∧ g2 x = x)}"
for x f1 f2 g1 g2
proof (intro exI conjI)
show homgf: "homeomorphism T T (f2 ∘ f1) (g1 ∘ g2)"
by (metis homeomorphism_compose hom)
then show "(f2 ∘ f1) x = f2 (f1 x)"
by force
show "{x. ¬ ((f2 ∘ f1) x = x ∧ (g1 ∘ g2) x = x)} ⊆ S"
using sub by force
have "bounded ({x. ~(f1 x = x ∧ g1 x = x)} ∪ {x. ~(f2 x = x ∧ g2 x = x)})"
using bo by simp
then show "bounded {x. ¬ ((f2 ∘ f1) x = x ∧ (g1 ∘ g2) x = x)}"
by (rule bounded_subset) auto
qed
have 3: "∃U. openin (subtopology euclidean S) U ∧
d ∈ U ∧
(∀x∈U.
∃f g. homeomorphism T T f g ∧ f d = x ∧
{x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧
bounded {x. ¬ (f x = x ∧ g x = x)})"
if "d ∈ S" for d
proof -
obtain r where "r > 0" and r: "ball d r ∩ affine hull S ⊆ S"
by (metis ‹d ∈ S› ope openin_contains_ball)
have *: "∃f g. homeomorphism T T f g ∧ f d = e ∧
{x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧
bounded {x. ¬ (f x = x ∧ g x = x)}" if "e ∈ S" "e ∈ ball d r" for e
apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
using r ‹S ⊆ T› TS that
apply (auto simp: ‹d ∈ S› ‹0 < r› hull_inc)
using bounded_subset by blast
show ?thesis
apply (rule_tac x="S ∩ ball d r" in exI)
apply (intro conjI)
apply (simp add: openin_open_Int)
apply (simp add: ‹0 < r› that)
apply (blast intro: *)
done
qed
have "∃f g. homeomorphism T T f g ∧ f a = b ∧
{x. ~ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ~ (f x = x ∧ g x = x)}"
apply (rule connected_equivalence_relation [OF S], safe)
apply (blast intro: 1 2 3)+
done
then show ?thesis
using that by auto
qed
lemma homeomorphism_moving_points_exists_gen:
assumes K: "finite K" "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
"pairwise (λi j. (x i ≠ x j) ∧ (y i ≠ y j)) K"
and "2 ≤ aff_dim S"
and ope: "openin (subtopology euclidean (affine hull S)) S"
and "S ⊆ T" "T ⊆ affine hull S" "connected S"
shows "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(x i) = y i) ∧
{x. ~ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ~ (f x = x ∧ g x = x)}"
using assms
proof (induction K)
case empty
then show ?case
by (force simp: homeomorphism_ident)
next
case (insert i K)
then have xney: "⋀j. ⟦j ∈ K; j ≠ i⟧ ⟹ x i ≠ x j ∧ y i ≠ y j"
and pw: "pairwise (λi j. x i ≠ x j ∧ y i ≠ y j) K"
and "x i ∈ S" "y i ∈ S"
and xyS: "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
by (simp_all add: pairwise_insert)
obtain f g where homfg: "homeomorphism T T f g" and feq: "⋀i. i ∈ K ⟹ f(x i) = y i"
and fg_sub: "{x. ~ (f x = x ∧ g x = x)} ⊆ S"
and bo_fg: "bounded {x. ~ (f x = x ∧ g x = x)}"
using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
then have "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(x i) = y i) ∧
{x. ~ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ~ (f x = x ∧ g x = x)}"
using insert by blast
have aff_eq: "affine hull (S - y ` K) = affine hull S"
apply (rule affine_hull_Diff)
apply (auto simp: insert)
using ‹y i ∈ S› insert.hyps(2) xney xyS by fastforce
have f_in_S: "f x ∈ S" if "x ∈ S" for x
using homfg fg_sub homeomorphism_apply1 ‹S ⊆ T›
proof -
have "(f (f x) ≠ f x ∨ g (f x) ≠ f x) ∨ f x ∈ S"
by (metis ‹S ⊆ T› homfg subsetD homeomorphism_apply1 that)
then show ?thesis
using fg_sub by force
qed
obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
and hk_sub: "{x. ¬ (h x = x ∧ k x = x)} ⊆ S - y ` K"
and bo_hk: "bounded {x. ¬ (h x = x ∧ k x = x)}"
proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
show "openin (subtopology euclidean (affine hull (S - y ` K))) (S - y ` K)"
by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
show "S - y ` K ⊆ T"
using ‹S ⊆ T› by auto
show "T ⊆ affine hull (S - y ` K)"
using insert by (simp add: aff_eq)
show "connected (S - y ` K)"
proof (rule connected_openin_diff_countable [OF ‹connected S› ope])
show "¬ collinear S"
using collinear_aff_dim ‹2 ≤ aff_dim S› by force
show "countable (y ` K)"
using countable_finite insert.hyps(1) by blast
qed
show "f (x i) ∈ S - y ` K"
apply (auto simp: f_in_S ‹x i ∈ S›)
by (metis feq homfg ‹x i ∈ S› homeomorphism_def ‹S ⊆ T› ‹i ∉ K› subsetCE xney xyS)
show "y i ∈ S - y ` K"
using insert.hyps xney by (auto simp: ‹y i ∈ S›)
qed blast
show ?case
proof (intro exI conjI)
show "homeomorphism T T (h ∘ f) (g ∘ k)"
using homfg homhk homeomorphism_compose by blast
show "∀i ∈ insert i K. (h ∘ f) (x i) = y i"
using feq hk_sub by (auto simp: heq)
show "{x. ¬ ((h ∘ f) x = x ∧ (g ∘ k) x = x)} ⊆ S"
using fg_sub hk_sub by force
have "bounded ({x. ~(f x = x ∧ g x = x)} ∪ {x. ~(h x = x ∧ k x = x)})"
using bo_fg bo_hk bounded_Un by blast
then show "bounded {x. ¬ ((h ∘ f) x = x ∧ (g ∘ k) x = x)}"
by (rule bounded_subset) auto
qed
qed
proposition homeomorphism_moving_points_exists:
fixes S :: "'a::euclidean_space set"
assumes 2: "2 ≤ DIM('a)" "open S" "connected S" "S ⊆ T" "finite K"
and KS: "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
and pw: "pairwise (λi j. (x i ≠ x j) ∧ (y i ≠ y j)) K"
and S: "S ⊆ T" "T ⊆ affine hull S" "connected S"
obtains f g where "homeomorphism T T f g" "⋀i. i ∈ K ⟹ f(x i) = y i"
"{x. ~ (f x = x ∧ g x = x)} ⊆ S" "bounded {x. (~ (f x = x ∧ g x = x))}"
proof (cases "S = {}")
case True
then show ?thesis
using KS homeomorphism_ident that by fastforce
next
case False
then have affS: "affine hull S = UNIV"
by (simp add: affine_hull_open ‹open S›)
then have ope: "openin (subtopology euclidean (affine hull S)) S"
using ‹open S› open_openin by auto
have "2 ≤ DIM('a)" by (rule 2)
also have "… = aff_dim (UNIV :: 'a set)"
by simp
also have "… ≤ aff_dim S"
by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
finally have "2 ≤ aff_dim S"
by linarith
then show ?thesis
using homeomorphism_moving_points_exists_gen [OF ‹finite K› KS pw _ ope S] that by fastforce
qed
subsubsection%unimportant‹The theorem ‹homeomorphism_grouping_points_exists››
lemma homeomorphism_grouping_point_1:
fixes a::real and c::real
assumes "a < b" "c < d"
obtains f g where "homeomorphism (cbox a b) (cbox c d) f g" "f a = c" "f b = d"
proof -
define f where "f ≡ λx. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
have "∃g. homeomorphism (cbox a b) (cbox c d) f g"
proof (rule homeomorphism_compact)
show "continuous_on (cbox a b) f"
apply (simp add: f_def)
apply (intro continuous_intros)
using assms by auto
have "f ` {a..b} = {c..d}"
unfolding f_def image_affinity_atLeastAtMost
using assms sum_sqs_eq by (auto simp: divide_simps algebra_simps)
then show "f ` cbox a b = cbox c d"
by auto
show "inj_on f (cbox a b)"
unfolding f_def inj_on_def using assms by auto
qed auto
then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
then show ?thesis
proof
show "f a = c"
by (simp add: f_def)
show "f b = d"
using assms sum_sqs_eq [of a b] by (auto simp: f_def divide_simps algebra_simps)
qed
qed
lemma homeomorphism_grouping_point_2:
fixes a::real and w::real
assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
and "b ∈ cbox a c" "v ∈ cbox u w"
and eq: "f1 a = u" "f1 b = v" "f2 b = v" "f2 c = w"
obtains f g where "homeomorphism (cbox a c) (cbox u w) f g" "f a = u" "f c = w"
"⋀x. x ∈ cbox a b ⟹ f x = f1 x" "⋀x. x ∈ cbox b c ⟹ f x = f2 x"
proof -
have le: "a ≤ b" "b ≤ c" "u ≤ v" "v ≤ w"
using assms by simp_all
then have ac: "cbox a c = cbox a b ∪ cbox b c" and uw: "cbox u w = cbox u v ∪ cbox v w"
by auto
define f where "f ≡ λx. if x ≤ b then f1 x else f2 x"
have "∃g. homeomorphism (cbox a c) (cbox u w) f g"
proof (rule homeomorphism_compact)
have cf1: "continuous_on (cbox a b) f1"
using hom_ab homeomorphism_cont1 by blast
have cf2: "continuous_on (cbox b c) f2"
using hom_bc homeomorphism_cont1 by blast
show "continuous_on (cbox a c) f"
apply (simp add: f_def)
apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
using le eq apply (force simp: continuous_on_id)+
done
have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
unfolding f_def using eq by force+
then show "f ` cbox a c = cbox u w"
apply (simp only: ac uw image_Un)
by (metis hom_ab hom_bc homeomorphism_def)
have neq12: "f1 x ≠ f2 y" if x: "a ≤ x" "x ≤ b" and y: "b < y" "y ≤ c" for x y
proof -
have "f1 x ∈ cbox u v"
by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
moreover have "f2 y ∈ cbox v w"
by (metis (full_types) hom_bc homeomorphism_def image_subset_iff mem_box_real(2) not_le not_less_iff_gr_or_eq order_refl y)
moreover have "f2 y ≠ f2 b"
by (metis cancel_comm_monoid_add_class.diff_cancel diff_gt_0_iff_gt hom_bc homeomorphism_def le(2) less_imp_le less_numeral_extra(3) mem_box_real(2) order_refl y)
ultimately show ?thesis
using le eq by simp
qed
have "inj_on f1 (cbox a b)"
by (metis (full_types) hom_ab homeomorphism_def inj_onI)
moreover have "inj_on f2 (cbox b c)"
by (metis (full_types) hom_bc homeomorphism_def inj_onI)
ultimately show "inj_on f (cbox a c)"
apply (simp (no_asm) add: inj_on_def)
apply (simp add: f_def inj_on_eq_iff)
using neq12 apply force
done
qed auto
then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
then show ?thesis
apply (rule that)
using eq le by (auto simp: f_def)
qed
lemma homeomorphism_grouping_point_3:
fixes a::real
assumes cbox_sub: "cbox c d ⊆ box a b" "cbox u v ⊆ box a b"
and box_ne: "box c d ≠ {}" "box u v ≠ {}"
obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
"⋀x. x ∈ cbox c d ⟹ f x ∈ cbox u v"
proof -
have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d ≠ {}"
using assms
by (simp_all add: cbox_sub subset_eq)
obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
and f1_eq: "f1 a = a" "f1 c = u"
using homeomorphism_grouping_point_1 [OF ‹a < c› ‹a < u›] .
obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
and f2_eq: "f2 c = u" "f2 d = v"
using homeomorphism_grouping_point_1 [OF ‹c < d› ‹u < v›] .
obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
and f3_eq: "f3 d = v" "f3 b = b"
using homeomorphism_grouping_point_1 [OF ‹d < b› ‹v < b›] .
obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
and f4_eq: "⋀x. x ∈ cbox a c ⟹ f4 x = f1 x" "⋀x. x ∈ cbox c d ⟹ f4 x = f2 x"
using homeomorphism_grouping_point_2 [OF 1 2] less by (auto simp: f1_eq f2_eq)
obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
and f_eq: "⋀x. x ∈ cbox a d ⟹ f x = f4 x" "⋀x. x ∈ cbox d b ⟹ f x = f3 x"
using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
show ?thesis
apply (rule that [OF fg])
using f4_eq f_eq homeomorphism_image1 [OF 2]
apply simp
by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
qed
lemma homeomorphism_grouping_point_4:
fixes T :: "real set"
assumes "open U" "open S" "connected S" "U ≠ {}" "finite K" "K ⊆ S" "U ⊆ S" "S ⊆ T"
obtains f g where "homeomorphism T T f g"
"⋀x. x ∈ K ⟹ f x ∈ U" "{x. (~ (f x = x ∧ g x = x))} ⊆ S"
"bounded {x. (~ (f x = x ∧ g x = x))}"
proof -
obtain c d where "box c d ≠ {}" "cbox c d ⊆ U"
proof -
obtain u where "u ∈ U"
using ‹U ≠ {}› by blast
then obtain e where "e > 0" "cball u e ⊆ U"
using ‹open U› open_contains_cball by blast
then show ?thesis
by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
qed
have "compact K"
by (simp add: ‹finite K› finite_imp_compact)
obtain a b where "box a b ≠ {}" "K ⊆ cbox a b" "cbox a b ⊆ S"
proof (cases "K = {}")
case True then show ?thesis
using ‹box c d ≠ {}› ‹cbox c d ⊆ U› ‹U ⊆ S› that by blast
next
case False
then obtain a b where "a ∈ K" "b ∈ K"
and a: "⋀x. x ∈ K ⟹ a ≤ x" and b: "⋀x. x ∈ K ⟹ x ≤ b"
using compact_attains_inf compact_attains_sup by (metis ‹compact K›)+
obtain e where "e > 0" "cball b e ⊆ S"
using ‹open S› open_contains_cball
by (metis ‹b ∈ K› ‹K ⊆ S› subsetD)
show ?thesis
proof
show "box a (b + e) ≠ {}"
using ‹0 < e› ‹b ∈ K› a by force
show "K ⊆ cbox a (b + e)"
using ‹0 < e› a b by fastforce
have "a ∈ S"
using ‹a ∈ K› assms(6) by blast
have "b + e ∈ S"
using ‹0 < e› ‹cball b e ⊆ S› by (force simp: dist_norm)
show "cbox a (b + e) ⊆ S"
using ‹a ∈ S› ‹b + e ∈ S› ‹connected S› connected_contains_Icc by auto
qed
qed
obtain w z where "cbox w z ⊆ S" and sub_wz: "cbox a b ∪ cbox c d ⊆ box w z"
proof -
have "a ∈ S" "b ∈ S"
using ‹box a b ≠ {}› ‹cbox a b ⊆ S› by auto
moreover have "c ∈ S" "d ∈ S"
using ‹box c d ≠ {}› ‹cbox c d ⊆ U› ‹U ⊆ S› by force+
ultimately have "min a c ∈ S" "max b d ∈ S"
by linarith+
then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 ⊆ S" "e2 > 0" "cball (max b d) e2 ⊆ S"
using ‹open S› open_contains_cball by metis
then have *: "min a c - e1 ∈ S" "max b d + e2 ∈ S"
by (auto simp: dist_norm)
show ?thesis
proof
show "cbox (min a c - e1) (max b d+ e2) ⊆ S"
using * ‹connected S› connected_contains_Icc by auto
show "cbox a b ∪ cbox c d ⊆ box (min a c - e1) (max b d + e2)"
using ‹0 < e1› ‹0 < e2› by auto
qed
qed
then
obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
and "f w = w" "f z = z"
and fin: "⋀x. x ∈ cbox a b ⟹ f x ∈ cbox c d"
using homeomorphism_grouping_point_3 [of a b w z c d]
using ‹box a b ≠ {}› ‹box c d ≠ {}› by blast
have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
using hom homeomorphism_def by blast+
define f' where "f' ≡ λx. if x ∈ cbox w z then f x else x"
define g' where "g' ≡ λx. if x ∈ cbox w z then g x else x"
show ?thesis
proof
have T: "cbox w z ∪ (T - box w z) = T"
using ‹cbox w z ⊆ S› ‹S ⊆ T› by auto
show "homeomorphism T T f' g'"
proof
have clo: "closedin (subtopology euclidean (cbox w z ∪ (T - box w z))) (T - box w z)"
by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
have "continuous_on (cbox w z ∪ (T - box w z)) f'" "continuous_on (cbox w z ∪ (T - box w z)) g'"
unfolding f'_def g'_def
apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
apply (simp_all add: closed_subset)
using ‹f w = w› ‹f z = z› apply force
by (metis ‹f w = w› ‹f z = z› hom homeomorphism_def less_eq_real_def mem_box_real(2))
then show "continuous_on T f'" "continuous_on T g'"
by (simp_all only: T)
show "f' ` T ⊆ T"
unfolding f'_def
by clarsimp (metis ‹cbox w z ⊆ S› ‹S ⊆ T› subsetD hom homeomorphism_def imageI mem_box_real(2))
show "g' ` T ⊆ T"
unfolding g'_def
by clarsimp (metis ‹cbox w z ⊆ S› ‹S ⊆ T› subsetD hom homeomorphism_def imageI mem_box_real(2))
show "⋀x. x ∈ T ⟹ g' (f' x) = x"
unfolding f'_def g'_def
using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom] by fastforce
show "⋀y. y ∈ T ⟹ f' (g' y) = y"
unfolding f'_def g'_def
using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom] by fastforce
qed
show "⋀x. x ∈ K ⟹ f' x ∈ U"
using fin sub_wz ‹K ⊆ cbox a b› ‹cbox c d ⊆ U› by (force simp: f'_def)
show "{x. ¬ (f' x = x ∧ g' x = x)} ⊆ S"
using ‹cbox w z ⊆ S› by (auto simp: f'_def g'_def)
show "bounded {x. ¬ (f' x = x ∧ g' x = x)}"
apply (rule bounded_subset [of "cbox w z"])
using bounded_cbox apply blast
apply (auto simp: f'_def g'_def)
done
qed
qed
proposition homeomorphism_grouping_points_exists:
fixes S :: "'a::euclidean_space set"
assumes "open U" "open S" "connected S" "U ≠ {}" "finite K" "K ⊆ S" "U ⊆ S" "S ⊆ T"
obtains f g where "homeomorphism T T f g" "{x. (~ (f x = x ∧ g x = x))} ⊆ S"
"bounded {x. (~ (f x = x ∧ g x = x))}" "⋀x. x ∈ K ⟹ f x ∈ U"
proof (cases "2 ≤ DIM('a)")
case True
have TS: "T ⊆ affine hull S"
using affine_hull_open assms by blast
have "infinite U"
using ‹open U› ‹U ≠ {}› finite_imp_not_open by blast
then obtain P where "P ⊆ U" "finite P" "card K = card P"
using infinite_arbitrarily_large by metis
then obtain γ where γ: "bij_betw γ K P"
using ‹finite K› finite_same_card_bij by blast
obtain f g where "homeomorphism T T f g" "⋀i. i ∈ K ⟹ f (id i) = γ i" "{x. ¬ (f x = x ∧ g x = x)} ⊆ S" "bounded {x. ¬ (f x = x ∧ g x = x)}"
proof (rule homeomorphism_moving_points_exists [OF True ‹open S› ‹connected S› ‹S ⊆ T› ‹finite K›])
show "⋀i. i ∈ K ⟹ id i ∈ S ∧ γ i ∈ S"
using ‹P ⊆ U› ‹bij_betw γ K P› ‹K ⊆ S› ‹U ⊆ S› bij_betwE by blast
show "pairwise (λi j. id i ≠ id j ∧ γ i ≠ γ j) K"
using γ by (auto simp: pairwise_def bij_betw_def inj_on_def)
qed (use affine_hull_open assms that in auto)
then show ?thesis
using γ ‹P ⊆ U› bij_betwE by (fastforce simp add: intro!: that)
next
case False
with DIM_positive have "DIM('a) = 1"
by (simp add: dual_order.antisym)
then obtain h::"'a ⇒real" and j
where "linear h" "linear j"
and noh: "⋀x. norm(h x) = norm x" and noj: "⋀y. norm(j y) = norm y"
and hj: "⋀x. j(h x) = x" "⋀y. h(j y) = y"
and ranh: "surj h"
using isomorphisms_UNIV_UNIV
by (metis (mono_tags, hide_lams) DIM_real UNIV_eq_I range_eqI)
obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
and f: "⋀x. x ∈ h ` K ⟹ f x ∈ h ` U"
and sub: "{x. ¬ (f x = x ∧ g x = x)} ⊆ h ` S"
and bou: "bounded {x. ¬ (f x = x ∧ g x = x)}"
apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
by (simp_all add: assms image_mono ‹linear h› open_surjective_linear_image connected_linear_image ranh)
have jf: "j (f (h x)) = x ⟷ f (h x) = h x" for x
by (metis hj)
have jg: "j (g (h x)) = x ⟷ g (h x) = h x" for x
by (metis hj)
have cont_hj: "continuous_on X h" "continuous_on Y j" for X Y
by (simp_all add: ‹linear h› ‹linear j› linear_linear linear_continuous_on)
show ?thesis
proof
show "homeomorphism T T (j ∘ f ∘ h) (j ∘ g ∘ h)"
proof
show "continuous_on T (j ∘ f ∘ h)" "continuous_on T (j ∘ g ∘ h)"
using hom homeomorphism_def
by (blast intro: continuous_on_compose cont_hj)+
show "(j ∘ f ∘ h) ` T ⊆ T" "(j ∘ g ∘ h) ` T ⊆ T"
by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
show "⋀x. x ∈ T ⟹ (j ∘ g ∘ h) ((j ∘ f ∘ h) x) = x"
using hj hom homeomorphism_apply1 by fastforce
show "⋀y. y ∈ T ⟹ (j ∘ f ∘ h) ((j ∘ g ∘ h) y) = y"
using hj hom homeomorphism_apply2 by fastforce
qed
show "{x. ¬ ((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)} ⊆ S"
apply (clarsimp simp: jf jg hj)
using sub hj
apply (drule_tac c="h x" in subsetD, force)
by (metis imageE)
have "bounded (j ` {x. (~ (f x = x ∧ g x = x))})"
by (rule bounded_linear_image [OF bou]) (use ‹linear j› linear_conv_bounded_linear in auto)
moreover
have *: "{x. ~((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)} = j ` {x. (~ (f x = x ∧ g x = x))}"
using hj by (auto simp: jf jg image_iff, metis+)
ultimately show "bounded {x. ¬ ((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)}"
by metis
show "⋀x. x ∈ K ⟹ (j ∘ f ∘ h) x ∈ U"
using f hj by fastforce
qed
qed
proposition homeomorphism_grouping_points_exists_gen:
fixes S :: "'a::euclidean_space set"
assumes opeU: "openin (subtopology euclidean S) U"
and opeS: "openin (subtopology euclidean (affine hull S)) S"
and "U ≠ {}" "finite K" "K ⊆ S" and S: "S ⊆ T" "T ⊆ affine hull S" "connected S"
obtains f g where "homeomorphism T T f g" "{x. (~ (f x = x ∧ g x = x))} ⊆ S"
"bounded {x. (~ (f x = x ∧ g x = x))}" "⋀x. x ∈ K ⟹ f x ∈ U"
proof (cases "2 ≤ aff_dim S")
case True
have opeU': "openin (subtopology euclidean (affine hull S)) U"
using opeS opeU openin_trans by blast
obtain u where "u ∈ U" "u ∈ S"
using ‹U ≠ {}› opeU openin_imp_subset by fastforce+
have "infinite U"
apply (rule infinite_openin [OF opeU ‹u ∈ U›])
apply (rule connected_imp_perfect_aff_dim [OF ‹connected S› _ ‹u ∈ S›])
using True apply simp
done
then obtain P where "P ⊆ U" "finite P" "card K = card P"
using infinite_arbitrarily_large by metis
then obtain γ where γ: "bij_betw γ K P"
using ‹finite K› finite_same_card_bij by blast
have "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(id i) = γ i) ∧
{x. ~ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ~ (f x = x ∧ g x = x)}"
proof (rule homeomorphism_moving_points_exists_gen [OF ‹finite K› _ _ True opeS S])
show "⋀i. i ∈ K ⟹ id i ∈ S ∧ γ i ∈ S"
by (metis id_apply opeU openin_contains_cball subsetCE ‹P ⊆ U› ‹bij_betw γ K P› ‹K ⊆ S› bij_betwE)
show "pairwise (λi j. id i ≠ id j ∧ γ i ≠ γ j) K"
using γ by (auto simp: pairwise_def bij_betw_def inj_on_def)
qed
then show ?thesis
using γ ‹P ⊆ U› bij_betwE by (fastforce simp add: intro!: that)
next
case False
with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
then show ?thesis
proof cases
assume "aff_dim S = -1"
then have "S = {}"
using aff_dim_empty by blast
then have "False"
using ‹U ≠ {}› ‹K ⊆ S› openin_imp_subset [OF opeU] by blast
then show ?thesis ..
next
assume "aff_dim S = 0"
then obtain a where "S = {a}"
using aff_dim_eq_0 by blast
then have "K ⊆ U"
using ‹U ≠ {}› ‹K ⊆ S› openin_imp_subset [OF opeU] by blast
show ?thesis
apply (rule that [of id id])
using ‹K ⊆ U› by (auto simp: continuous_on_id intro: homeomorphismI)
next
assume "aff_dim S = 1"
then have "affine hull S homeomorphic (UNIV :: real set)"
by (auto simp: homeomorphic_affine_sets)
then obtain h::"'a⇒real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
using homeomorphic_def by blast
then have h: "⋀x. x ∈ affine hull S ⟹ j(h(x)) = x" and j: "⋀y. j y ∈ affine hull S ∧ h(j y) = y"
by (auto simp: homeomorphism_def)
have connh: "connected (h ` S)"
by (meson Topological_Spaces.connected_continuous_image ‹connected S› homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
have hUS: "h ` U ⊆ h ` S"
by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
have opn: "openin (subtopology euclidean (affine hull S)) U ⟹ open (h ` U)" for U
using homeomorphism_imp_open_map [OF homhj] by simp
have "open (h ` U)" "open (h ` S)"
by (auto intro: opeS opeU openin_trans opn)
then obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
and f: "⋀x. x ∈ h ` K ⟹ f x ∈ h ` U"
and sub: "{x. ¬ (f x = x ∧ g x = x)} ⊆ h ` S"
and bou: "bounded {x. ¬ (f x = x ∧ g x = x)}"
apply (rule homeomorphism_grouping_points_exists [of "h ` U" "h ` S" "h ` K" "h ` T"])
using assms by (auto simp: connh hUS)
have jf: "⋀x. x ∈ affine hull S ⟹ j (f (h x)) = x ⟷ f (h x) = h x"
by (metis h j)
have jg: "⋀x. x ∈ affine hull S ⟹ j (g (h x)) = x ⟷ g (h x) = h x"
by (metis h j)
have cont_hj: "continuous_on T h" "continuous_on Y j" for Y
apply (rule continuous_on_subset [OF _ ‹T ⊆ affine hull S›])
using homeomorphism_def homhj apply blast
by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
define f' where "f' ≡ λx. if x ∈ affine hull S then (j ∘ f ∘ h) x else x"
define g' where "g' ≡ λx. if x ∈ affine hull S then (j ∘ g ∘ h) x else x"
show ?thesis
proof
show "homeomorphism T T f' g'"
proof
have "continuous_on T (j ∘ f ∘ h)"
apply (intro continuous_on_compose cont_hj)
using hom homeomorphism_def by blast
then show "continuous_on T f'"
apply (rule continuous_on_eq)
using ‹T ⊆ affine hull S› f'_def by auto
have "continuous_on T (j ∘ g ∘ h)"
apply (intro continuous_on_compose cont_hj)
using hom homeomorphism_def by blast
then show "continuous_on T g'"
apply (rule continuous_on_eq)
using ‹T ⊆ affine hull S› g'_def by auto
show "f' ` T ⊆ T"
proof (clarsimp simp: f'_def)
fix x assume "x ∈ T"
then have "f (h x) ∈ h ` T"
by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
then show "j (f (h x)) ∈ T"
using ‹T ⊆ affine hull S› h by auto
qed
show "g' ` T ⊆ T"
proof (clarsimp simp: g'_def)
fix x assume "x ∈ T"
then have "g (h x) ∈ h ` T"
by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
then show "j (g (h x)) ∈ T"
using ‹T ⊆ affine hull S› h by auto
qed
show "⋀x. x ∈ T ⟹ g' (f' x) = x"
using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
show "⋀y. y ∈ T ⟹ f' (g' y) = y"
using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
qed
next
show "{x. ¬ (f' x = x ∧ g' x = x)} ⊆ S"
apply (clarsimp simp: f'_def g'_def jf jg)
apply (rule imageE [OF subsetD [OF sub]], force)
by (metis h hull_inc)
next
have "compact (j ` closure {x. ¬ (f x = x ∧ g x = x)})"
using bou by (auto simp: compact_continuous_image cont_hj)
then have "bounded (j ` {x. (~ (f x = x ∧ g x = x))})"
by (rule bounded_closure_image [OF compact_imp_bounded])
moreover
have *: "{x ∈ affine hull S. j (f (h x)) ≠ x ∨ j (g (h x)) ≠ x} = j ` {x. (~ (f x = x ∧ g x = x))}"
using h j by (auto simp: image_iff; metis)
ultimately have "bounded {x ∈ affine hull S. j (f (h x)) ≠ x ∨ j (g (h x)) ≠ x}"
by metis
then show "bounded {x. ¬ (f' x = x ∧ g' x = x)}"
by (simp add: f'_def g'_def Collect_mono bounded_subset)
next
show "f' x ∈ U" if "x ∈ K" for x
proof -
have "U ⊆ S"
using opeU openin_imp_subset by blast
then have "j (f (h x)) ∈ U"
using f h hull_subset that by fastforce
then show "f' x ∈ U"
using ‹K ⊆ S› S f'_def that by auto
qed
qed
qed
qed
subsection‹nullhomotopic mappings›
text‹ A mapping out of a sphere is nullhomotopic iff it extends to the ball.
This even works out in the degenerate cases when the radius is ‹≤› 0, and
we also don't need to explicitly assume continuity since it's already implicit
in both sides of the equivalence.›
lemma nullhomotopic_from_lemma:
assumes contg: "continuous_on (cball a r - {a}) g"
and fa: "⋀e. 0 < e
⟹ ∃d. 0 < d ∧ (∀x. x ≠ a ∧ norm(x - a) < d ⟶ norm(g x - f a) < e)"
and r: "⋀x. x ∈ cball a r ∧ x ≠ a ⟹ f x = g x"
shows "continuous_on (cball a r) f"
proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
fix x
assume x: "dist a x ≤ r"
show "continuous (at x within cball a r) f"
proof (cases "x=a")
case True
then show ?thesis
by (metis continuous_within_eps_delta fa dist_norm dist_self r)
next
case False
show ?thesis
proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
have "∃d>0. ∀x'∈cball a r.
dist x' x < d ⟶ dist (g x') (g x) < e" if "e>0" for e
proof -
obtain d where "d > 0"
and d: "⋀x'. ⟦dist x' a ≤ r; x' ≠ a; dist x' x < d⟧ ⟹
dist (g x') (g x) < e"
using contg False x ‹e>0›
unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
show ?thesis
using ‹d > 0› ‹x ≠ a›
by (rule_tac x="min d (norm(x - a))" in exI)
(auto simp: dist_commute dist_norm [symmetric] intro!: d)
qed
then show "continuous (at x within cball a r) g"
using contg False by (auto simp: continuous_within_eps_delta)
show "0 < norm (x - a)"
using False by force
show "x ∈ cball a r"
by (simp add: x)
show "⋀x'. ⟦x' ∈ cball a r; dist x' x < norm (x - a)⟧
⟹ g x' = f x'"
by (metis dist_commute dist_norm less_le r)
qed
qed
qed
proposition nullhomotopic_from_sphere_extension:
fixes f :: "'M::euclidean_space ⇒ 'a::real_normed_vector"
shows "(∃c. homotopic_with (λx. True) (sphere a r) S f (λx. c)) ⟷
(∃g. continuous_on (cball a r) g ∧ g ` (cball a r) ⊆ S ∧
(∀x ∈ sphere a r. g x = f x))"
(is "?lhs = ?rhs")
proof (cases r "0::real" rule: linorder_cases)
case equal
then show ?thesis
apply (auto simp: homotopic_with)
apply (rule_tac x="λx. h (0, a)" in exI)
apply (fastforce simp add:)
using continuous_on_const by blast
next
case greater
let ?P = "continuous_on {x. norm(x - a) = r} f ∧ f ` {x. norm(x - a) = r} ⊆ S"
have ?P if ?lhs using that
proof
fix c
assume c: "homotopic_with (λx. True) (sphere a r) S f (λx. c)"
then have contf: "continuous_on (sphere a r) f" and fim: "f ` sphere a r ⊆ S"
by (auto simp: homotopic_with_imp_subset1 homotopic_with_imp_continuous)
show ?P
using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
qed
moreover have ?P if ?rhs using that
proof
fix g
assume g: "continuous_on (cball a r) g ∧ g ` cball a r ⊆ S ∧ (∀xa∈sphere a r. g xa = f xa)"
then
show ?P
apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
apply (auto simp: dist_norm norm_minus_commute)
by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
qed
moreover have ?thesis if ?P
proof
assume ?lhs
then obtain c where "homotopic_with (λx. True) (sphere a r) S (λx. c) f"
using homotopic_with_sym by blast
then obtain h where conth: "continuous_on ({0..1::real} × sphere a r) h"
and him: "h ` ({0..1} × sphere a r) ⊆ S"
and h: "⋀x. h(0, x) = c" "⋀x. h(1, x) = f x"
by (auto simp: homotopic_with_def)
obtain b1::'M where "b1 ∈ Basis"
using SOME_Basis by auto
have "c ∈ S"
apply (rule him [THEN subsetD])
apply (rule_tac x = "(0, a + r *⇩R b1)" in image_eqI)
using h greater ‹b1 ∈ Basis›
apply (auto simp: dist_norm)
done
have uconth: "uniformly_continuous_on ({0..1::real} × (sphere a r)) h"
by (force intro: compact_Times conth compact_uniformly_continuous)
let ?g = "λx. h (norm (x - a)/r,
a + (if x = a then r *⇩R b1 else (r / norm(x - a)) *⇩R (x - a)))"
let ?g' = "λx. h (norm (x - a)/r, a + (r / norm(x - a)) *⇩R (x - a))"
show ?rhs
proof (intro exI conjI)
have "continuous_on (cball a r - {a}) ?g'"
apply (rule continuous_on_compose2 [OF conth])
apply (intro continuous_intros)
using greater apply (auto simp: dist_norm norm_minus_commute)
done
then show "continuous_on (cball a r) ?g"
proof (rule nullhomotopic_from_lemma)
show "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d ⟶ norm (?g' x - ?g a) < e" if "0 < e" for e
proof -
obtain d where "0 < d"
and d: "⋀x x'. ⟦x ∈ {0..1} × sphere a r; x' ∈ {0..1} × sphere a r; dist x' x < d⟧
⟹ dist (h x') (h x) < e"
using uniformly_continuous_onE [OF uconth ‹0 < e›] by auto
have *: "norm (h (norm (x - a) / r,
a + (r / norm (x - a)) *⇩R (x - a)) - h (0, a + r *⇩R b1)) < e"
if "x ≠ a" "norm (x - a) < r" "norm (x - a) < d * r" for x
proof -
have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *⇩R (x - a)) - h (0, a + r *⇩R b1)) =
norm (h (norm (x - a) / r, a + (r / norm (x - a)) *⇩R (x - a)) - h (0, a + (r / norm (x - a)) *⇩R (x - a)))"
by (simp add: h)
also have "… < e"
apply (rule d [unfolded dist_norm])
using greater ‹0 < d› ‹b1 ∈ Basis› that
by (auto simp: dist_norm divide_simps)
finally show ?thesis .
qed
show ?thesis
apply (rule_tac x = "min r (d * r)" in exI)
using greater ‹0 < d› by (auto simp: *)
qed
show "⋀x. x ∈ cball a r ∧ x ≠ a ⟹ ?g x = ?g' x"
by auto
qed
next
show "?g ` cball a r ⊆ S"
using greater him ‹c ∈ S›
by (force simp: h dist_norm norm_minus_commute)
next
show "∀x∈sphere a r. ?g x = f x"
using greater by (auto simp: h dist_norm norm_minus_commute)
qed
next
assume ?rhs
then obtain g where contg: "continuous_on (cball a r) g"
and gim: "g ` cball a r ⊆ S"
and gf: "∀x ∈ sphere a r. g x = f x"
by auto
let ?h = "λy. g (a + (fst y) *⇩R (snd y - a))"
have "continuous_on ({0..1} × sphere a r) ?h"
apply (rule continuous_on_compose2 [OF contg])
apply (intro continuous_intros)
apply (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
done
moreover
have "?h ` ({0..1} × sphere a r) ⊆ S"
by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
moreover
have "∀x∈sphere a r. ?h (0, x) = g a" "∀x∈sphere a r. ?h (1, x) = f x"
by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
ultimately
show ?lhs
apply (subst homotopic_with_sym)
apply (rule_tac x="g a" in exI)
apply (auto simp: homotopic_with)
done
qed
ultimately
show ?thesis by meson
qed simp
end