(* Title: HOL/Multivariate_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 Convex_Euclidean_Space begin subsection \Paths and Arcs\ definition path :: "(real \ 'a::topological_space) \ bool" where "path g \ continuous_on {0..1} g" definition pathstart :: "(real \ 'a::topological_space) \ 'a" where "pathstart g = g 0" definition pathfinish :: "(real \ 'a::topological_space) \ 'a" where "pathfinish g = g 1" definition path_image :: "(real \ 'a::topological_space) \ 'a set" where "path_image g = g ` {0 .. 1}" definition reversepath :: "(real \ 'a::topological_space) \ real \ 'a" where "reversepath g = (\x. g(1 - x))" definition joinpaths :: "(real \ 'a::topological_space) \ (real \ 'a) \ real \ 'a" (infixr "+++" 75) where "g1 +++ g2 = (\x. if x \ 1/2 then g1 (2 * x) else g2 (2 * x - 1))" definition simple_path :: "(real \ 'a::topological_space) \ bool" where "simple_path g \ path g \ (\x\{0..1}. \y\{0..1}. g x = g y \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0)" definition arc :: "(real \ 'a :: topological_space) \ bool" where "arc g \ path g \ inj_on g {0..1}" subsection\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 o 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((\x. a + x) o g) = path g" proof - have g: "g = (\x. -a + x) o ((\x. a + x) o 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 o 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 o (f o 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((\x. a + x) o g) = a + pathstart g" by (simp add: pathstart_def) lemma pathstart_linear_image_eq: "linear f \ pathstart(f o g) = f(pathstart g)" by (simp add: pathstart_def) lemma pathfinish_translation: "pathfinish((\x. a + x) o g) = a + pathfinish g" by (simp add: pathfinish_def) lemma pathfinish_linear_image: "linear f \ pathfinish(f o g) = f(pathfinish g)" by (simp add: pathfinish_def) lemma path_image_translation: "path_image((\x. a + x) o g) = (\x. a + x) ` (path_image g)" by (simp add: image_comp path_image_def) lemma path_image_linear_image: "linear f \ path_image(f o g) = f ` (path_image g)" by (simp add: image_comp path_image_def) lemma reversepath_translation: "reversepath((\x. a + x) o g) = (\x. a + x) o reversepath g" by (rule ext) (simp add: reversepath_def) lemma reversepath_linear_image: "linear f \ reversepath(f o g) = f o reversepath g" by (rule ext) (simp add: reversepath_def) lemma joinpaths_translation: "((\x. a + x) o g1) +++ ((\x. a + x) o g2) = (\x. a + x) o (g1 +++ g2)" by (rule ext) (simp add: joinpaths_def) lemma joinpaths_linear_image: "linear f \ (f o g1) +++ (f o g2) = f o (g1 +++ g2)" by (rule ext) (simp add: joinpaths_def) lemma simple_path_translation_eq: fixes g :: "real \ 'a::euclidean_space" shows "simple_path((\x. a + x) o 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 o 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((\x. a + x) o 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 o g) = arc g" using assms inj_on_eq_iff [of f] by (auto simp: arc_def inj_on_def path_linear_image_eq) subsection\Basic lemmas about paths\ 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 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_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 _ "\x. 1 - x"]) apply (intro continuous_intros) apply (rule continuous_on_subset[of "{0..1}"]) apply assumption apply auto done show ?thesis using *[of "reversepath g"] *[of g] unfolding reversepath_reversepath by (rule iffI) qed 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 apply (auto simp: arc_def inj_on_def path_reversepath) apply (simp add: arc_imp_path assms) apply (rule **) apply (rule inj_onD [OF injg]) apply (auto simp: reversepath_def) done 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) \ (\x. 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) \ (\x. x / 2 + 1/2))" using assms by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def) show "continuous_on {0..1} g1" and "continuous_on {0..1} g2" unfolding g1 g2 by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply) next assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2" have 01: "{0 .. 1} = {0..1/2} \ {1/2 .. 1::real}" by auto { fix x :: real assume "0 \ x" and "x \ 1" then have "x \ (\x. 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 \ (\x. x * 2 - 1) ` {1 / 2..1}" by (intro image_eqI[where x="x/2 + 1/2"]) auto } note 2 = this show "continuous_on {0..1} (g1 +++ g2)" using assms unfolding joinpaths_def 01 apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros) apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2) done qed section \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 o p) = f(pathstart p)" by (simp add: pathstart_def) lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)" by (simp add: pathfinish_def) lemma path_image_compose: "path_image (f o p) = f ` (path_image p)" by (simp add: image_comp path_image_def) lemma path_compose_join: "f o (p +++ q) = (f o p) +++ (f o q)" by (rule ext) (simp add: joinpaths_def) lemma path_compose_reversepath: "f o reversepath p = reversepath(f o 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\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\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}" "\x. 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 o (\x. 2*x)"]) apply (rule continuous_intros | simp add: joinpaths_def assms)+ done have "continuous_on {1/2..1} (g2 o (\x. 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 o (\x. 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\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) def 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\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) section\Choosing a subpath of an existing path\ definition subpath :: "real \ real \ (real \ 'a) \ real \ 'a::real_normed_vector" where "subpath a b g \ \x. 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_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 o (\x. ((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 ((\x. a + x) o g) = (\x. a + x) o subpath u v g" by (rule ext) (simp add: subpath_def) lemma subpath_linear_image: "linear f \ subpath u v (f o g) = f o 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 (\x. 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 (\x. 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 add: 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: "\path g; 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\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 \Reparametrizing a closed curve to start at some chosen point\ definition shiftpath :: "real \ (real \ 'a::topological_space) \ real \ 'a" where "shiftpath a f = (\x. if (a + x) \ 1 then f (a + x) else f (a + x - 1))" lemma pathstart_shiftpath: "a \ 1 \ pathstart (shiftpath a g) = g a" unfolding pathstart_def shiftpath_def by auto lemma pathfinish_shiftpath: assumes "0 \ a" and "pathfinish g = pathstart g" shows "pathfinish (shiftpath a g) = g a" using assms unfolding pathstart_def pathfinish_def shiftpath_def by auto lemma endpoints_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0 .. 1}" shows "pathfinish (shiftpath a g) = g a" and "pathstart (shiftpath a g) = g a" using assms by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) lemma closed_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0..1}" shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" using endpoints_shiftpath[OF assms] by auto lemma path_shiftpath: assumes "path g" and "pathfinish g = pathstart g" and "a \ {0..1}" shows "path (shiftpath a g)" proof - have *: "{0 .. 1} = {0 .. 1-a} \ {1-a .. 1}" using assms(3) by auto have **: "\x. x + a = 1 \ g (x + a - 1) = g (x + a)" using assms(2)[unfolded pathfinish_def pathstart_def] by auto show ?thesis unfolding path_def shiftpath_def * apply (rule continuous_on_closed_Un) apply (rule closed_real_atLeastAtMost)+ apply (rule continuous_on_eq[of _ "g \ (\x. a + x)"]) prefer 3 apply (rule continuous_on_eq[of _ "g \ (\x. a - 1 + x)"]) prefer 3 apply (rule continuous_intros)+ prefer 2 apply (rule continuous_intros)+ apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) using assms(3) and ** apply auto apply (auto simp add: field_simps) done qed lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" and "a \ {0..1}" and "x \ {0..1}" shows "shiftpath (1 - a) (shiftpath a g) x = g x" using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto lemma path_image_shiftpath: assumes "a \ {0..1}" and "pathfinish g = pathstart g" shows "path_image (shiftpath a g) = path_image g" proof - { fix x assume as: "g 1 = g 0" "x \ {0..1::real}" " \y\{0..1} \ {x. \ a + x \ 1}. g x \ g (a + y - 1)" then have "\y\{0..1} \ {x. a + x \ 1}. g x = g (a + y)" proof (cases "a \ x") case False then show ?thesis apply (rule_tac x="1 + x - a" in bexI) using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) apply (auto simp add: field_simps atomize_not) done next case True then show ?thesis using as(1-2) and assms(1) apply (rule_tac x="x - a" in bexI) apply (auto simp add: field_simps) done qed } then show ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def by (auto simp add: image_iff) qed subsection \Special case of straight-line paths\ definition linepath :: "'a::real_normed_vector \ 'a \ real \ 'a" where "linepath a b = (\x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" unfolding pathstart_def linepath_def by auto lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" unfolding pathfinish_def linepath_def by auto lemma 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[intro]: "path (linepath a b)" unfolding path_def by (rule continuous_on_linepath) lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" unfolding path_image_def segment linepath_def by auto lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" unfolding reversepath_def linepath_def by auto lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b" by (simp add: linepath_def) lemma arc_linepath: assumes "a \ b" shows "arc (linepath a b)" proof - { fix x y :: "real" assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps) with assms have "x = y" by simp } then show ?thesis unfolding arc_def inj_on_def by (simp add: path_linepath) (force 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 arc_linepath) lemma linepath_trivial [simp]: "linepath a a x = a" by (simp add: linepath_def real_vector.scale_left_diff_distrib) lemma subpath_refl: "subpath a a g = linepath (g a) (g a)" by (simp add: subpath_def linepath_def algebra_simps) subsection \Bounding a point away from a path\ lemma not_on_path_ball: fixes g :: "real \ 'a::heine_borel" assumes "path g" and "z \ path_image g" shows "\e > 0. ball z e \ path_image g = {}" proof - obtain a where "a \ path_image g" "\y \ path_image g. dist z a \ dist z y" apply (rule distance_attains_inf[OF _ path_image_nonempty, of g z]) using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto then show ?thesis apply (rule_tac x="dist z a" in exI) using assms(2) apply (auto intro!: dist_pos_lt) done qed lemma not_on_path_cball: fixes g :: "real \ 'a::heine_borel" assumes "path g" and "z \ path_image g" shows "\e>0. cball z e \ (path_image g) = {}" proof - obtain e where "ball z e \ path_image g = {}" "e > 0" using not_on_path_ball[OF assms] by auto moreover have "cball z (e/2) \ ball z e" using \e > 0\ by auto ultimately show ?thesis apply (rule_tac x="e/2" in exI) apply auto done qed section \Path component, considered as a "joinability" relation (from Tom Hales)\ definition "path_component s x y \ (\g. path g \ path_image g \ s \ pathstart g = x \ pathfinish g = y)" abbreviation "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="\u. x" in exI) using assms apply (auto intro!: continuous_intros) done lemma path_component_refl_eq: "path_component s x x \ x \ s" by (auto intro!: path_component_mem path_component_refl) lemma path_component_sym: "path_component s x y \ path_component s y x" using assms unfolding path_component_def apply (erule exE) apply (rule_tac x="reversepath g" in exI) apply auto done lemma path_component_trans: assumes "path_component s x y" and "path_component s y z" shows "path_component s x z" using assms unfolding path_component_def apply (elim exE) apply (rule_tac x="g +++ ga" in exI) apply (auto simp add: path_image_join) done lemma path_component_of_subset: "s \ t \ path_component s x y \ path_component t x y" unfolding path_component_def by auto lemma path_connected_linepath: fixes s :: "'a::real_normed_vector set" shows "closed_segment a b \ s \ path_component s a b" apply (simp add: path_component_def) apply (rule_tac x="linepath a b" in exI, auto) done text \Can also consider it as a set, as the name suggests.\ 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 add: 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 "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) subsection \Some useful lemmas about path-connectedness\ lemma convex_imp_path_connected: fixes s :: "'a::real_normed_vector set" assumes "convex s" shows "path_connected s" unfolding path_connected_def apply rule apply rule apply (rule_tac x = "linepath x y" in exI) unfolding path_image_linepath using assms [unfolded convex_contains_segment] apply auto done lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s" unfolding connected_def not_ex apply rule apply rule apply (rule ccontr) unfolding not_not apply (elim conjE) proof - fix e1 e2 assume as: "open e1" "open e2" "s \ e1 \ e2" "e1 \ e2 \ s = {}" "e1 \ s \ {}" "e2 \ s \ {}" then obtain x1 x2 where obt:"x1 \ e1 \ s" "x2 \ e2 \ s" by auto then obtain g where g: "path g" "path_image g \ s" "pathstart g = x1" "pathfinish g = x2" using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto have *: "connected {0..1::real}" by (auto intro!: convex_connected convex_real_interval) have "{0..1} \ {x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2}" using as(3) g(2)[unfolded path_defs] by blast moreover have "{x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto moreover have "{x \ {0..1}. g x \ e1} \ {} \ {x \ {0..1}. g x \ e2} \ {}" using g(3,4)[unfolded path_defs] using obt by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) ultimately show False using *[unfolded connected_local not_ex, rule_format, of "{x\{0..1}. g x \ e1}" "{x\{0..1}. g x \ e2}"] using continuous_openin_preimage[OF g(1)[unfolded path_def] as(1)] using continuous_openin_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed lemma open_path_component: fixes s :: "'a::real_normed_vector set" assumes "open s" shows "open (path_component_set s x)" unfolding open_contains_ball proof fix y assume as: "y \ path_component_set s x" then have "y \ s" apply - apply (rule path_component_mem(2)) unfolding mem_Collect_eq apply auto done then obtain e where e: "e > 0" "ball y e \ s" using assms[unfolded open_contains_ball] by auto show "\e > 0. ball y e \ path_component_set s x" apply (rule_tac x=e in exI) apply (rule,rule \e>0\) apply rule unfolding mem_ball mem_Collect_eq proof - fix z assume "dist y z < e" then show "path_component s x z" apply (rule_tac path_component_trans[of _ _ y]) defer apply (rule path_component_of_subset[OF e(2)]) apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using \e > 0\ as apply auto done qed qed lemma open_non_path_component: fixes s :: "'a::real_normed_vector set" assumes "open s" shows "open (s - path_component_set s x)" unfolding open_contains_ball proof fix y assume as: "y \ s - path_component_set s x" then obtain e where e: "e > 0" "ball y e \ s" using assms [unfolded open_contains_ball] by auto show "\e>0. ball y e \ s - path_component_set s x" apply (rule_tac x=e in exI) apply rule apply (rule \e>0\) apply rule apply rule defer proof (rule ccontr) fix z assume "z \ ball y e" "\ z \ path_component_set s x" then have "y \ path_component_set s x" unfolding not_not mem_Collect_eq using \e>0\ apply - apply (rule path_component_trans, assumption) apply (rule path_component_of_subset[OF e(2)]) apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) apply auto done then show False using as by auto qed (insert e(2), auto) qed lemma connected_open_path_connected: fixes s :: "'a::real_normed_vector set" assumes "open s" and "connected s" shows "path_connected s" unfolding path_connected_component_set proof (rule, rule, rule path_component_subset, rule) fix x y assume "x \ s" and "y \ s" show "y \ 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_segment: 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: 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: "path_connected {}" unfolding path_connected_def by auto lemma path_connected_singleton: "path_connected {a}" unfolding path_connected_def pathstart_def pathfinish_def path_image_def apply clarify apply (rule_tac x="\x. a" in exI) apply (simp add: image_constant_conv) apply (simp add: path_def continuous_on_const) done lemma path_connected_Un: assumes "path_connected s" and "path_connected t" and "s \ t \ {}" shows "path_connected (s \ t)" unfolding path_connected_component proof (rule, rule) fix x y assume as: "x \ s \ t" "y \ s \ t" from assms(3) obtain z where "z \ s \ t" by auto then show "path_component (s \ t) x y" using as and assms(1-2)[unfolded path_connected_component] apply - apply (erule_tac[!] UnE)+ apply (rule_tac[2-3] path_component_trans[of _ _ z]) apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) done qed lemma path_connected_UNION: assumes "\i. i \ A \ path_connected (S i)" and "\i. i \ A \ z \ S i" shows "path_connected (\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" using x proof (clarsimp simp add: path_image_def) fix y assume "x = p y" and y: "0 \ y" "y \ 1" show "path_component (p ` {0..1}) (pathstart p) (p y)" proof (cases "y=0") case True then show ?thesis by (simp add: path_component_refl_eq pathstart_def) next case False have "continuous_on {0..1} (p o (op*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 have "path (\u. p (y * u))" by (simp add: path_def) then show ?thesis apply (simp add: path_component_def) apply (rule_tac x = "\u. p (y * u)" in exI) apply (intro conjI) using y False apply (auto simp: mult_le_one pathstart_def pathfinish_def path_image_def) done qed 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: "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 path_connected_path_component) done next case False then show ?thesis using path_component_eq_empty by auto 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)*\<^sub>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) *\<^sub>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 lemma path_connected_sphere: assumes "2 \ DIM('a::euclidean_space)" shows "path_connected {x::'a. norm (x - a) = r}" proof (rule linorder_cases [of r 0]) assume "r < 0" then have "{x::'a. norm(x - a) = r} = {}" by auto then show ?thesis using path_connected_empty by simp next assume "r = 0" then show ?thesis using path_connected_singleton by simp next assume r: "0 < r" have *: "{x::'a. norm(x - a) = r} = (\x. a + r *\<^sub>R x) ` {x. norm x = 1}" apply (rule set_eqI) apply rule unfolding image_iff apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR using r apply (auto simp add: scaleR_right_diff_distrib) done have **: "{x::'a. norm x = 1} = (\x. (1/norm x) *\<^sub>R x) ` (- {0})" apply (rule set_eqI) apply rule unfolding image_iff apply (rule_tac x=x in bexI) unfolding mem_Collect_eq apply (auto split: if_split_asm) done have "continuous_on (- {0}) (\x::'a. 1 / norm x)" by (auto intro!: continuous_intros) then show ?thesis unfolding * ** using path_connected_punctured_universe[OF assms] by (auto intro!: path_connected_continuous_image continuous_intros) qed corollary connected_sphere: "2 \ DIM('a::euclidean_space) \ connected {x::'a. norm (x - a) = r}" using path_connected_sphere path_connected_imp_connected by auto 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) def C == "B / norm(x - a)" { fix u assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>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) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)" by (simp add: algebra_simps) have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) = (1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x" by (simp add: algebra_simps) also have "... = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x" using CC by (simp add: field_simps) also have "... = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x" by (simp add: algebra_simps) also have "... = x + ((1 / (1 + C * u - u)) *\<^sub>R a + ((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))" using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x" by (simp add: algebra_simps) have False using \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) *\<^sub>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 *\<^sub>R (x - a))" by (force simp: closed_segment_def intro!: path_connected_linepath) def D == "B / norm(y - a)" \\massive duplication with the proof above\ { fix u assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>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) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)" by (simp add: algebra_simps) have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) = (1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y" by (simp add: algebra_simps) also have "... = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y" using DD by (simp add: field_simps) also have "... = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y" by (simp add: algebra_simps) also have "... = y + ((1 / (1 + D * u - u)) *\<^sub>R a + ((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))" using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y" by (simp add: algebra_simps) have False using \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) *\<^sub>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 *\<^sub>R (y - a))" by (force simp: closed_segment_def intro!: path_connected_linepath) have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))" apply (rule path_component_of_subset [of "{x. norm(x - 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: 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 \ where "\ > 0" and \: "cball a \ \ S" using open_contains_cball_eq by blast have "dist a (a + \ *\<^sub>R (SOME i. i \ Basis)) = \" by (simp add: dist_norm SOME_Basis \0 < \\ less_imp_le) with \ have "\{S - ball a r |r. 0 < r \ r < \} \ {} \ False" apply (drule_tac c="a + scaleR (\) ((SOME i. i \ Basis))" in subsetD) by auto then have nonemp: "(\{S - ball a r |r. 0 < r \ r < \}) = {} \ False" by auto have con: "\r. r < \ \ connected (S - ball a r)" using \ by (force intro: connected_diff_ball [OF \connected S\ _ 2]) have "x \ \{S - ball a r |r. 0 < r \ r < \}" if "x \ S - {a}" for x apply (rule UnionI [of "S - ball a (min \ (dist a x) / 2)"]) using that \0 < \\ apply (simp_all add:) apply (rule_tac x="min \ (dist a x) / 2" in exI) apply auto done then have "S - {a} = \{S - ball a r | r. 0 < r \ r < \}" 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) lemma connected_punctured_ball: "2 \ DIM('N::euclidean_space) \ connected(ball a r - {a::'N})" by (simp add: connected_open_delete) subsection\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 subsection\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 add: 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\) *\<^sub>R i" in exI) apply simp using i apply (auto simp: algebra_simps) done have **: "{x. B \ i \ x} \ connected_component_set s (B *\<^sub>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 *\<^sub>R i \ s" by (rule *) (simp add: norm_Basis [OF i]) then show ?thesis apply (rule_tac x="B *\<^sub>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 add: 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" shows "\bounded (- s); ~connected s; 2 \ DIM('a)\ \ \c. c \ components s \ bounded c" by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq) section\The "inside" and "outside" of a set\ text\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 inside where "inside s \ {x. (x \ s) \ bounded(connected_component_set ( - s) x)}" definition 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_inter_outside [simp]: "inside s \ outside s = {}" by (auto simp: inside_def outside_def) lemma inside_union_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 add: 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 show ?thesis apply (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) apply (simp add: connected_component_def) apply (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) apply simp apply (rule_tac y="- ball 0 B" in order_trans) prefer 2 apply force apply (simp add: closed_segment_def ball_def dist_norm, clarify) apply (simp add: 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) qed } then show ?thesis apply (simp add: outside_def assms) apply (rule bounded_subset [OF bounded_ball [of 0 B]]) apply (force simp add: 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 (simp add: connected_iff_connected_component, clarify) apply (simp add: 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_interiors: "frontier s = - interior(s) - interior(-s)" by (simp add: Int_commute frontier_def interior_closure) lemma frontier_interior_subset: "frontier(interior S) \ frontier S" by (simp add: Diff_mono frontier_interiors interior_mono interior_subset) lemma connected_Int_frontier: "\connected s; s \ t \ {}; s - t \ {}\ \ (s \ frontier t \ {})" apply (simp add: frontier_interiors connected_open_in, safe) apply (drule_tac x="s \ interior t" in spec, safe) apply (drule_tac [2] x="s \ interior (-t)" in spec) apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD]) done 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 1: "y \ closure S" using y1 closure_mono connected_component_subset by blast 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 (no_types, hide_lams) connected_component_eq_eq connected_component_in subsetD dist_commute mem_Collect_eq mem_ball mem_interior \0 < e\ y2) qed then have 2: "y \ interior S" using y2 by (force simp: open_contains_ball_eq [OF open_interior]) note 1 2 } then show ?thesis by (auto simp: frontier_def) qed lemma connected_component_UNIV: 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_union_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) def 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 *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\ \ norm z" by (metis add_diff_cancel norm_triangle_ineq3) moreover have "norm (C *\<^sub>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 *\<^sub>R (z-a)) > B" by linarith { fix u::real assume u: "0\u" "u\1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>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)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>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 *\<^sub>R (z-a))" apply (rule connected_componentI [OF connected_segment [of z "z + C *\<^sub>R (z-a)"]]) apply (simp add: closed_segment_def) using contra apply auto done then have False using zna B [of "z + C *\<^sub>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_union_outside subsetI sup.cobounded2) 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 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_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 add: 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_union_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 \ assume "path \" and pimg_sbs: "path_image \ - {pathfinish \} \ interior (- t)" and pf: "pathfinish \ \ frontier t" and ps: "pathstart \ = a" def c \ "pathfinish \" have "c \ -s" unfolding c_def using front pf by blast moreover have "open (-s)" using s compact_imp_closed by blast ultimately obtain \::real where "\ > 0" and \: "cball c \ \ -s" using open_contains_cball[of "-s"] s by blast then obtain d where "d \ t" and d: "dist d c < \" using closure_approachable [of c t] pf unfolding c_def by (metis Diff_iff frontier_def) then have "d \ -s" using \ using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq) have pimg_sbs_cos: "path_image \ \ -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 \" apply (simp add: segment_convex_hull) apply (rule hull_minimal) using \\ > 0\ d apply (auto simp: dist_commute) done with \ have "closed_segment c d \ -s" by blast moreover have con_gcd: "connected (path_image \ \ closed_segment c d)" by (rule connected_Un) (auto simp: c_def \path \\ 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) apply (simp add:) 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) apply (simp add:) 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\ lemma 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 \ where \: "\n. \ n \ f ` S" and tendsw: "\ \ w" by (metis Diff_iff frontier_def closure_sequential) then have "\n. \x \ S. \ n = f x" by force then obtain z where zs: "\n. z n \ S" and fz: "\n. \ n = f (z n)" by metis then obtain y K where y: "y \ closure S" and "subseq K" 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: "((\n. f (z n)) \ K) \ w" by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw) have zKs: "\n. (z o K) n \ S" by (simp add: zs) have "f \ z = \" "(\n. f (z n)) = \" using fz by auto moreover then have "(\ \ K) \ f y" by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially) ultimately 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\ 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 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(\x. 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(\x. h(t, x))))" unfolding homotopic_with_def apply (rule iffI, blast, clarify) apply (rule_tac x="\(u,v). if v \ X then h(u,v) else if u = 0 then p v else q v" in exI) apply (auto simp:) 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="\(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) apply 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="\(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 o snd"]) apply (rule continuous_intros | force)+ apply clarify apply (case_tac "t=1"; force) done lemma image_Pair_const: "(\x. (x, c)) ` A = A \ {c}" by (auto simp:) lemma homotopic_constant_maps: "homotopic_with (\x. True) s t (\x. a) (\x. 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 (\x. True) s t (\x. a) (\x. 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 \ (\t. (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) apply (auto simp:) apply (drule_tac x="h o (\t. (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 (\x. True) s t (\x. a) (\x. b)" apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def) apply (rule_tac x="g o fst" in exI) apply (rule conjI continuous_intros | force)+ done qed qed subsection\ 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) apply (auto simp:) done lemma continuous_on_o_Pair: "\continuous_on (T \ X) h; t \ T\ \ continuous_on X (h o 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 o (\x. (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 (\f. p (f \ h)) X Y f g; continuous_on W h; h ` W \ X\ \ homotopic_with p W Y (f o h) (g o h)" apply (clarsimp simp add: homotopic_with_def) apply (rename_tac k) apply (rule_tac x="k o (\y. (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 (\f. True) X Y f g; continuous_on W h; h ` W \ X\ \ homotopic_with (\f. True) W Y (f o h) (g o h)" using homotopic_with_compose_continuous_right by fastforce proposition homotopic_with_compose_continuous_left: "\homotopic_with (\f. p (h \ f)) X Y f g; continuous_on Y h; h ` Y \ Z\ \ homotopic_with p X Z (h o f) (h o g)" apply (clarsimp simp add: homotopic_with_def) apply (rename_tac k) apply (rule_tac x="h o 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 (\f. True) X Y f g \ continuous_on Y h \ image h Y \ Z \ homotopic_with (\f. True) X Z (h o f) (h o 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(\(x,y). (f x, g y))" shows "homotopic_with q (s \ s') (t \ t') (\(x,y). (f x, f' y)) (\(x,y). (g x, g' y))" using hom apply (clarsimp simp add: homotopic_with_def) apply (rename_tac k k') apply (rule_tac x="\z. ((k o (\x. (fst x, fst (snd x)))) z, (k' o (\x. (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: "homotopic_with (\x. 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 o 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 o (\y. (1 - fst y, snd y))" in exI) apply (rule conjI continuous_intros | erule continuous_on_subset | force simp add: 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 add: 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 add: 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 (\x. k1 (t, x))" "\t\{0..1}. P (\x. k2 (t, x))" def k \ "\y. if fst y \ 1 / 2 then (k1 o (\x. (2 *\<^sub>R fst x, snd x))) y else (k2 o (\x. (2 *\<^sub>R fst x -1, snd x))) 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 add: keq) done moreover have "k ` ({0..1} \ X) \ Y" using Y by (force simp add: 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 (\x. k (t, x))" using P apply (clarsimp simp add: k_def) apply (case_tac "t \ 1/2") apply (auto simp:) 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 (\x. 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 (\x. True) s t f f'; homotopic_with (\x. True) t u g g'\ \ homotopic_with (\x. True) s u (g o f) (g' o f')" apply (rule homotopic_with_trans [where g = "g o 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) subsection\Homotopy of paths, maintaining the same endpoints.\ definition homotopic_paths :: "['a set, real \ 'a, real \ 'a::topological_space] \ bool" where "homotopic_paths s p q \ homotopic_with (\r. 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 o Pair t) = pathstart p \ pathfinish(h o 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 o 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 o (\y. (1 - (fst y)) *\<^sub>R ((f o snd) y) + (fst y) *\<^sub>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}) (\y. p (fst y) (snd y))" and q: "continuous_on ({0..1} \ {0..1}) (\y. q (fst y) (snd y))" and pf: "\t. t \ {0..1} \ pathfinish(p t) = pathstart(q t)" shows "continuous_on ({0..1} \ {0..1}) (\y. (p(fst y) +++ q(fst y)) (snd y))" proof - have 1: "(\y. p (fst y) (2 * snd y)) = (\y. p (fst y) (snd y)) o (\y. (fst y, 2 * snd y))" by (rule ext) (simp ) have 2: "(\y. q (fst y) (2 * snd y - 1)) = (\y. q (fst y) (snd y)) o (\y. (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 o (\x. (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="(\y. ((k1 o Pair (fst y)) +++ (k2 o 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 o f) (h o 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\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 = "\t. if t \ 1 / 2 then 2 *\<^sub>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 = "\t. if t \ 1 / 2 then inverse 2 *\<^sub>R t else if t \ 3 / 4 then t - (1 / 4) else 2 *\<^sub>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}) (\x. (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="(\y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI) apply (simp add: path_defs joinpaths_def subpath_def reversepath_def) apply (force simp: mult_le_one) done qed proposition homotopic_paths_linv: assumes "path p" "path_image p \ s" shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))" using homotopic_paths_rinv [of "reversepath p" s] assms by simp subsection\ Homotopy of loops without requiring preservation of endpoints.\ definition homotopic_loops :: "'a::topological_space set \ (real \ 'a) \ (real \ 'a) \ bool" where "homotopic_loops s p q \ homotopic_with (\r. pathfinish r = pathstart r) {0..1} s p q" lemma homotopic_loops: "homotopic_loops s p q \ (\h. continuous_on ({0..1::real} \ {0..1}) h \ image h ({0..1} \ {0..1}) \ s \ (\x \ {0..1}. h(0,x) = p x) \ (\x \ {0..1}. h(1,x) = q x) \ (\t \ {0..1}. pathfinish(h o Pair t) = pathstart(h o Pair t)))" by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with) proposition homotopic_loops_imp_loop: "homotopic_loops s p q \ pathfinish p = pathstart p \ pathfinish q = pathstart q" using homotopic_with_imp_property homotopic_loops_def by blast proposition homotopic_loops_imp_path: "homotopic_loops s p q \ path p \ path q" unfolding homotopic_loops_def path_def using homotopic_with_imp_continuous by blast proposition homotopic_loops_imp_subset: "homotopic_loops s p q \ path_image p \ s \ path_image q \ s" unfolding homotopic_loops_def path_image_def by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2) proposition homotopic_loops_refl: "homotopic_loops s p p \ path p \ path_image p \ s \ pathfinish p = pathstart p" by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def) proposition homotopic_loops_sym: "homotopic_loops s p q \ homotopic_loops s q p" by (simp add: homotopic_loops_def homotopic_with_sym) proposition homotopic_loops_sym_eq: "homotopic_loops s p q \ homotopic_loops s q p" by (metis homotopic_loops_sym) proposition homotopic_loops_trans: "\homotopic_loops s p q; homotopic_loops s q r\ \ homotopic_loops s p r" unfolding homotopic_loops_def by (blast intro: homotopic_with_trans) proposition homotopic_loops_subset: "\homotopic_loops s p q; s \ t\ \ homotopic_loops t p q" by (simp add: homotopic_loops_def homotopic_with_subset_right) proposition homotopic_loops_eq: "\path p; path_image p \ s; pathfinish p = pathstart p; \t. t \ {0..1} \ p(t) = q(t)\ \ homotopic_loops s p q" unfolding homotopic_loops_def apply (rule homotopic_with_eq) apply (rule homotopic_with_refl [where f = p, THEN iffD2]) apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def) done proposition homotopic_loops_continuous_image: "\homotopic_loops s f g; continuous_on s h; h ` s \ t\ \ homotopic_loops t (h \ f) (h \ g)" unfolding homotopic_loops_def apply (rule homotopic_with_compose_continuous_left) apply (erule homotopic_with_mono) by (simp add: pathfinish_def pathstart_def) subsection\Relations between the two variants of homotopy\ proposition homotopic_paths_imp_homotopic_loops: "\homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\ \ homotopic_loops s p q" by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono) proposition homotopic_loops_imp_homotopic_paths_null: assumes "homotopic_loops s p (linepath a a)" shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))" proof - have "path p" by (metis assms homotopic_loops_imp_path) have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop) have pip: "path_image p \ s" by (metis assms homotopic_loops_imp_subset) obtain h where conth: "continuous_on ({0..1::real} \ {0..1}) h" and hs: "h ` ({0..1} \ {0..1}) \ s" and [simp]: "\x. x \ {0..1} \ h(0,x) = p x" and [simp]: "\x. x \ {0..1} \ h(1,x) = a" and ends: "\t. t \ {0..1} \ pathfinish (h \ Pair t) = pathstart (h \ Pair t)" using assms by (auto simp: homotopic_loops homotopic_with) have conth0: "path (\u. h (u, 0))" unfolding path_def apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) apply (force intro: continuous_intros continuous_on_subset [OF conth])+ done have pih0: "path_image (\u. h (u, 0)) \ s" using hs by (force simp: path_image_def) have c1: "continuous_on ({0..1} \ {0..1}) (\x. h (fst x * snd x, 0))" apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+ done have c2: "continuous_on ({0..1} \ {0..1}) (\x. h (fst x - fst x * snd x, 0))" apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+ apply (rule continuous_on_subset [OF conth]) apply (auto simp: algebra_simps add_increasing2 mult_left_le) done have [simp]: "\t. \0 \ t \ t \ 1\ \ h (t, 1) = h (t, 0)" using ends by (simp add: pathfinish_def pathstart_def) have adhoc_le: "c * 4 \ 1 + c * (d * 4)" if "\ d * 4 \ 3" "0 \ c" "c \ 1" for c d::real proof - have "c * 3 \ c * (d * 4)" using that less_eq_real_def by auto with \c \ 1\ show ?thesis by fastforce qed have *: "\p x. (path p \ path(reversepath p)) \ (path_image p \ s \ path_image(reversepath p) \ s) \ (pathfinish p = pathstart(linepath a a +++ reversepath p) \ pathstart(reversepath p) = a) \ pathstart p = x \ homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)" by (metis homotopic_paths_lid homotopic_paths_join homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv) have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))" using \path p\ homotopic_paths_rid homotopic_paths_sym pip by blast moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))" apply (rule homotopic_paths_sym) using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s] by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset) moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p)) ((\u. h (u, 0)) +++ linepath a a +++ reversepath (\u. h (u, 0)))" apply (simp add: homotopic_paths_def homotopic_with_def) apply (rule_tac x="\y. (subpath 0 (fst y) (\u. h (u, 0)) +++ (\u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\u. h (u, 0))) (snd y)" in exI) apply (simp add: subpath_reversepath) apply (intro conjI homotopic_join_lemma) using ploop apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2) apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le) done moreover have "homotopic_paths s ((\u. h (u, 0)) +++ linepath a a +++ reversepath (\u. h (u, 0))) (linepath (pathstart p) (pathstart p))" apply (rule *) apply (simp add: pih0 pathstart_def pathfinish_def conth0) apply (simp add: reversepath_def joinpaths_def) done ultimately show ?thesis by (blast intro: homotopic_paths_trans) qed proposition homotopic_loops_conjugate: fixes s :: "'a::real_normed_vector set" assumes "path p" "path q" and pip: "path_image p \ s" and piq: "path_image q \ s" and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q" shows "homotopic_loops s (p +++ q +++ reversepath p) q" proof - have contp: "continuous_on {0..1} p" using \path p\ [unfolded path_def] by blast have contq: "continuous_on {0..1} q" using \path q\ [unfolded path_def] by blast have c1: "continuous_on ({0..1} \ {0..1}) (\x. p ((1 - fst x) * snd x + fst x))" apply (rule continuous_on_compose [of _ _ p, unfolded o_def]) apply (force simp: mult_le_one intro!: continuous_intros) apply (rule continuous_on_subset [OF contp]) apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1) done have c2: "continuous_on ({0..1} \ {0..1}) (\x. p ((fst x - 1) * snd x + 1))" apply (rule continuous_on_compose [of _ _ p, unfolded o_def]) apply (force simp: mult_le_one intro!: continuous_intros) apply (rule continuous_on_subset [OF contp]) apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le) done have ps1: "\a b. \b * 2 \ 1; 0 \ b; 0 \ a; a \ 1\ \ p ((1 - a) * (2 * b) + a) \ s" using sum_le_prod1 by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD]) have ps2: "\a b. \\ 4 * b \ 3; b \ 1; 0 \ a; a \ 1\ \ p ((a - 1) * (4 * b - 3) + 1) \ s" apply (rule pip [unfolded path_image_def, THEN subsetD]) apply (rule image_eqI, blast) apply (simp add: algebra_simps) by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le add.commute zero_le_numeral) have qs: "\a b. \4 * b \ 3; \ b * 2 \ 1\ \ q (4 * b - 2) \ s" using path_image_def piq by fastforce have "homotopic_loops s (p +++ q +++ reversepath p) (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))" apply (simp add: homotopic_loops_def homotopic_with_def) apply (rule_tac x="(\y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI) apply (simp add: subpath_refl subpath_reversepath) apply (intro conjI homotopic_join_lemma) using papp qloop apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2) apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd) apply (auto simp: ps1 ps2 qs) done moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q" proof - have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q" using \path q\ homotopic_paths_lid qloop piq by auto hence 1: "\f. homotopic_paths s f q \ \ homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)" using homotopic_paths_trans by blast hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q" proof - have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q" by (simp add: \path q\ homotopic_paths_rid piq) thus ?thesis by (metis (no_types) 1 \path q\ homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym homotopic_paths_trans qloop pathfinish_linepath piq) qed thus ?thesis by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym) qed ultimately show ?thesis by (blast intro: homotopic_loops_trans) qed subsection\ Homotopy of "nearby" function, paths and loops.\ lemma homotopic_with_linear: fixes f g :: "_ \ 'b::real_normed_vector" assumes contf: "continuous_on s f" and contg:"continuous_on s g" and sub: "\x. x \ s \ closed_segment (f x) (g x) \ t" shows "homotopic_with (\z. True) s t f g" apply (simp add: homotopic_with_def) apply (rule_tac x="\y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI) apply (intro conjI) apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f] continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+ using sub closed_segment_def apply fastforce+ done lemma homotopic_paths_linear: fixes g h :: "real \ 'a::real_normed_vector" assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g" "\t x. t \ {0..1} \ closed_segment (g t) (h t) \ s" shows "homotopic_paths s g h" using assms unfolding path_def apply (simp add: pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def) apply (rule_tac x="\y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI) apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h] ) apply (force simp: closed_segment_def) apply (simp_all add: algebra_simps) done lemma homotopic_loops_linear: fixes g h :: "real \ 'a::real_normed_vector" assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h" "\t x. t \ {0..1} \ closed_segment (g t) (h t) \ s" shows "homotopic_loops s g h" using assms unfolding path_def apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def) apply (rule_tac x="\y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI) apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h]) apply (force simp: closed_segment_def) done lemma homotopic_paths_nearby_explicit: assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g" and no: "\t x. \t \ {0..1}; x \ s\ \ norm(h t - g t) < norm(g t - x)" shows "homotopic_paths s g h" apply (rule homotopic_paths_linear [OF assms(1-4)]) by (metis no segment_bound(1) subsetI norm_minus_commute not_le) lemma homotopic_loops_nearby_explicit: assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h" and no: "\t x. \t \ {0..1}; x \ s\ \ norm(h t - g t) < norm(g t - x)" shows "homotopic_loops s g h" apply (rule homotopic_loops_linear [OF assms(1-4)]) by (metis no segment_bound(1) subsetI norm_minus_commute not_le) lemma homotopic_nearby_paths: fixes g h :: "real \ 'a::euclidean_space" assumes "path g" "open s" "path_image g \ s" shows "\e. 0 < e \ (\h. path h \ pathstart h = pathstart g \ pathfinish h = pathfinish g \ (\t \ {0..1}. norm(h t - g t) < e) \ homotopic_paths s g h)" proof - obtain e where "e > 0" and e: "\x y. x \ path_image g \ y \ - s \ e \ dist x y" using separate_compact_closed [of "path_image g" "-s"] assms by force show ?thesis apply (intro exI conjI) using e [unfolded dist_norm] apply (auto simp: intro!: homotopic_paths_nearby_explicit assms \e > 0\) by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def) qed lemma homotopic_nearby_loops: fixes g h :: "real \ 'a::euclidean_space" assumes "path g" "open s" "path_image g \ s" "pathfinish g = pathstart g" shows "\e. 0 < e \ (\h. path h \ pathfinish h = pathstart h \ (\t \ {0..1}. norm(h t - g t) < e) \ homotopic_loops s g h)" proof - obtain e where "e > 0" and e: "\x y. x \ path_image g \ y \ - s \ e \ dist x y" using separate_compact_closed [of "path_image g" "-s"] assms by force show ?thesis apply (intro exI conjI) using e [unfolded dist_norm] apply (auto simp: intro!: homotopic_loops_nearby_explicit assms \e > 0\) by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def) qed subsection\ Homotopy and subpaths\ lemma homotopic_join_subpaths1: assumes "path g" and pag: "path_image g \ s" and u: "u \ {0..1}" and v: "v \ {0..1}" and w: "w \ {0..1}" "u \ v" "v \ w" shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" proof - have 1: "t * 2 \ 1 \ u + t * (v * 2) \ v + t * (u * 2)" for t using affine_ineq \u \ v\ by fastforce have 2: "t * 2 > 1 \ u + (2*t - 1) * v \ v + (2*t - 1) * w" for t by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \u \ v\ \v \ w\) have t2: "\t::real. t*2 = 1 \ t = 1/2" by auto show ?thesis apply (rule homotopic_paths_subset [OF _ pag]) using assms apply (cases "w = u") using homotopic_paths_rinv [of "subpath u v g" "path_image g"] apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl) apply (rule homotopic_paths_sym) apply (rule homotopic_paths_reparametrize [where f = "\t. if t \ 1 / 2 then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"]) using \path g\ path_subpath u w apply blast using \path g\ path_image_subpath_subset u w(1) apply blast apply simp_all apply (subst split_01) apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+ apply (simp_all add: field_simps not_le) apply (force dest!: t2) apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2) apply (simp add: joinpaths_def subpath_def) apply (force simp: algebra_simps) done qed lemma homotopic_join_subpaths2: assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)" by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath) lemma homotopic_join_subpaths3: assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" and "path g" and pag: "path_image g \ s" and u: "u \ {0..1}" and v: "v \ {0..1}" and w: "w \ {0..1}" shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)" proof - have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)" apply (rule homotopic_paths_join) using hom homotopic_paths_sym_eq apply blast apply (metis \path g\ homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w) apply (simp add:) done also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)" apply (rule homotopic_paths_sym [OF homotopic_paths_assoc]) using assms by (simp_all add: path_image_subpath_subset [THEN order_trans]) also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g) (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))" apply (rule homotopic_paths_join) apply (metis \path g\ homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v) apply (metis (no_types, lifting) \path g\ homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w) apply (simp add:) done also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)" apply (rule homotopic_paths_rid) using \path g\ path_subpath u v apply blast apply (meson \path g\ order.trans pag path_image_subpath_subset u v) done finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" . then show ?thesis using homotopic_join_subpaths2 by blast qed proposition homotopic_join_subpaths: "\path g; path_image g \ s; u \ {0..1}; v \ {0..1}; w \ {0..1}\ \ homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" apply (rule le_cases3 [of u v w]) using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+ end