(* Title: HOL/Analysis/Path_Connected.thy Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light *) 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)" ― ‹massive duplication with the proof above› { 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 **) (*such a horrible mess*) 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