author | hoelzl |
Mon, 21 Jun 2010 19:33:51 +0200 | |
changeset 37489 | 44e42d392c6e |
parent 36583 | 68ce5760c585 |
child 37674 | f86de9c00c47 |
permissions | -rw-r--r-- |
36583 | 1 |
(* Title: Multivariate_Analysis/Path_Connected.thy |
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Author: Robert Himmelmann, TU Muenchen |
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*) |
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header {* Continuous paths and path-connected sets *} |
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theory Path_Connected |
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37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
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imports Cartesian_Euclidean_Space |
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begin |
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subsection {* Paths. *} |
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definition |
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path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "path g \<longleftrightarrow> continuous_on {0 .. 1} g" |
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definition |
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pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathstart g = g 0" |
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definition |
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pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathfinish g = g 1" |
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definition |
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path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set" |
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where "path_image g = g ` {0 .. 1}" |
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definition |
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reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)" |
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where "reversepath g = (\<lambda>x. g(1 - x))" |
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definition |
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joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)" |
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(infixr "+++" 75) |
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where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))" |
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definition |
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simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "simple_path g \<longleftrightarrow> |
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(\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)" |
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definition |
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injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "injective_path g \<longleftrightarrow> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)" |
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subsection {* Some lemmas about these concepts. *} |
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lemma injective_imp_simple_path: |
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"injective_path g \<Longrightarrow> simple_path g" |
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unfolding injective_path_def simple_path_def by auto |
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lemma path_image_nonempty: "path_image g \<noteq> {}" |
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unfolding path_image_def image_is_empty interval_eq_empty by auto |
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lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g" |
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unfolding pathstart_def path_image_def by auto |
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lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g" |
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unfolding pathfinish_def path_image_def by auto |
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lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)" |
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unfolding path_def path_image_def |
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apply (erule connected_continuous_image) |
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by(rule convex_connected, rule convex_real_interval) |
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lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)" |
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unfolding path_def path_image_def |
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37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
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by (erule compact_continuous_image, rule compact_interval) |
36583 | 70 |
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lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g" |
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unfolding reversepath_def by auto |
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lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g" |
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unfolding pathstart_def reversepath_def pathfinish_def by auto |
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lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g" |
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unfolding pathstart_def reversepath_def pathfinish_def by auto |
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lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1" |
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unfolding pathstart_def joinpaths_def pathfinish_def by auto |
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lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2" |
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unfolding pathstart_def joinpaths_def pathfinish_def by auto |
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lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof- |
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have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g" |
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unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE) |
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apply(rule_tac x="1 - xa" in bexI) by auto |
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show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed |
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lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof- |
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have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def |
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apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"]) |
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apply(rule continuous_on_sub, rule continuous_on_const, rule continuous_on_id) |
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apply(rule continuous_on_subset[of "{0..1}"], assumption) by auto |
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show ?thesis using *[of "reversepath g"] *[of g] unfolding reversepath_reversepath by (rule iffI) qed |
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lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
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lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
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unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof- |
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assume as:"continuous_on {0..1} (g1 +++ g2)" |
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have *:"g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)" |
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"g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))" |
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unfolding o_def by (auto simp add: add_divide_distrib) |
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have "op *\<^sub>R (1 / 2) ` {0::real..1} \<subseteq> {0..1}" "(\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {(0::real)..1} \<subseteq> {0..1}" |
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by auto |
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thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule |
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apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose) |
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apply (rule continuous_on_cmul, rule continuous_on_add, rule continuous_on_id, rule continuous_on_const) defer |
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apply (rule continuous_on_cmul, rule continuous_on_id) apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3 |
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apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption) |
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apply(rule) defer apply rule proof- |
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fix x assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real..1}" |
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hence "x \<le> 1 / 2" unfolding image_iff by auto |
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thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto next |
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fix x assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real..1}" |
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hence "x \<ge> 1 / 2" unfolding image_iff by auto |
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thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" proof(cases "x = 1 / 2") |
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37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
121 |
case True hence "x = (1/2) *\<^sub>R 1" by auto |
36583 | 122 |
thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by (auto simp add: mult_ac) |
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qed (auto simp add:le_less joinpaths_def) qed |
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next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2" |
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have *:"{0 .. 1::real} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by auto |
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have **:"op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real}" apply(rule set_ext, rule) unfolding image_iff |
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defer apply(rule_tac x="(1/2)*\<^sub>R x" in bexI) by auto |
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have ***:"(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real}" |
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apply (auto simp add: image_def) |
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apply (rule_tac x="(x + 1) / 2" in bexI) |
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apply (auto simp add: add_divide_distrib) |
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done |
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show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply (rule closed_real_atLeastAtMost)+ proof- |
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show "continuous_on {0..(1 / 2) *\<^sub>R 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *\<^sub>R x)"]) defer |
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unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply(rule continuous_on_cmul, rule continuous_on_id) |
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unfolding ** apply(rule as(1)) unfolding joinpaths_def by auto next |
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show "continuous_on {(1/2)*\<^sub>R1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *\<^sub>R x - 1)"]) defer |
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apply(rule continuous_on_compose) apply(rule continuous_on_sub, rule continuous_on_cmul, rule continuous_on_id, rule continuous_on_const) |
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unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def] |
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by (auto simp add: mult_ac) qed qed |
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lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof |
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fix x assume "x \<in> path_image (g1 +++ g2)" |
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then obtain y where y:"y\<in>{0..1}" "x = (if y \<le> 1 / 2 then g1 (2 *\<^sub>R y) else g2 (2 *\<^sub>R y - 1))" |
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unfolding path_image_def image_iff joinpaths_def by auto |
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thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "y \<le> 1/2") |
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apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1) |
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by(auto intro!: imageI) qed |
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lemma subset_path_image_join: |
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assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s" |
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using path_image_join_subset[of g1 g2] and assms by auto |
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lemma path_image_join: |
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assumes "path g1" "path g2" "pathfinish g1 = pathstart g2" |
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shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)" |
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apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE) |
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fix x assume "x \<in> path_image g1" |
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then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto |
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thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff |
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apply(rule_tac x="(1/2) *\<^sub>R y" in bexI) by auto next |
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fix x assume "x \<in> path_image g2" |
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then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto |
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then show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff |
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apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def] |
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by (auto simp add: add_divide_distrib) qed |
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lemma not_in_path_image_join: |
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assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)" |
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using assms and path_image_join_subset[of g1 g2] by auto |
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lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)" |
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using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+ |
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apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE) |
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by auto |
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lemma simple_path_join_loop: |
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assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1" |
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"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}" |
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shows "simple_path(g1 +++ g2)" |
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unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2" |
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note inj = assms(1,2)[unfolded injective_path_def, rule_format] |
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fix x y::"real" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y" |
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show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" proof(case_tac "x \<le> 1/2",case_tac[!] "y \<le> 1/2", unfold not_le) |
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assume as:"x \<le> 1 / 2" "y \<le> 1 / 2" |
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hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" using xy(3) unfolding joinpaths_def by auto |
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moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as |
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by auto |
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ultimately show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto |
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next assume as:"x > 1 / 2" "y > 1 / 2" |
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hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" using xy(3) unfolding joinpaths_def by auto |
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moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" using xy(1,2) as by auto |
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ultimately show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto |
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next assume as:"x \<le> 1 / 2" "y > 1 / 2" |
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hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def |
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using xy(1,2) by auto |
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moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def |
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using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2) |
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by (auto simp add: field_simps) |
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ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto |
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hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1) |
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using inj(1)[of "2 *\<^sub>R x" 0] by auto |
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moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym] |
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unfolding joinpaths_def pathfinish_def using as(2) and xy(2) |
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using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto |
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ultimately show ?thesis by auto |
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next assume as:"x > 1 / 2" "y \<le> 1 / 2" |
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hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def |
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using xy(1,2) by auto |
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moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def |
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using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1) |
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by (auto simp add: field_simps) |
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ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto |
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hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2) |
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using inj(1)[of "2 *\<^sub>R y" 0] by auto |
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moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym] |
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unfolding joinpaths_def pathfinish_def using as(1) and xy(1) |
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using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto |
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ultimately show ?thesis by auto qed qed |
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lemma injective_path_join: |
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assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2" |
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"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}" |
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shows "injective_path(g1 +++ g2)" |
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unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2" |
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note inj = assms(1,2)[unfolded injective_path_def, rule_format] |
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fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y" |
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show "x = y" proof(cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le) |
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assume "x \<le> 1 / 2" "y \<le> 1 / 2" thus ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy |
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unfolding joinpaths_def by auto |
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next assume "x > 1 / 2" "y > 1 / 2" thus ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy |
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unfolding joinpaths_def by auto |
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next assume as:"x \<le> 1 / 2" "y > 1 / 2" |
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hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def |
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using xy(1,2) by auto |
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hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto |
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thus ?thesis using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2) |
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unfolding pathstart_def pathfinish_def joinpaths_def |
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by auto |
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next assume as:"x > 1 / 2" "y \<le> 1 / 2" |
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hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def |
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using xy(1,2) by auto |
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hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto |
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thus ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2) |
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unfolding pathstart_def pathfinish_def joinpaths_def |
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by auto qed qed |
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lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join |
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subsection {* Reparametrizing a closed curve to start at some chosen point. *} |
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definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) = |
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(\<lambda>x. if (a + x) \<le> 1 then f(a + x) else f(a + x - 1))" |
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lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a" |
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unfolding pathstart_def shiftpath_def by auto |
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lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g" |
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shows "pathfinish(shiftpath a g) = g a" |
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using assms unfolding pathstart_def pathfinish_def shiftpath_def |
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by auto |
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lemma endpoints_shiftpath: |
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assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" |
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shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a" |
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using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath) |
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||
268 |
lemma closed_shiftpath: |
|
269 |
assumes "pathfinish g = pathstart g" "a \<in> {0..1}" |
|
270 |
shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)" |
|
271 |
using endpoints_shiftpath[OF assms] by auto |
|
272 |
||
273 |
lemma path_shiftpath: |
|
274 |
assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
|
275 |
shows "path(shiftpath a g)" proof- |
|
276 |
have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto |
|
277 |
have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
|
278 |
using assms(2)[unfolded pathfinish_def pathstart_def] by auto |
|
279 |
show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union) |
|
280 |
apply(rule closed_real_atLeastAtMost)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3 |
|
281 |
apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3 |
|
282 |
apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+ |
|
283 |
apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) |
|
284 |
using assms(3) and ** by(auto, auto simp add: field_simps) qed |
|
285 |
||
286 |
lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" |
|
287 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
|
288 |
using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto |
|
289 |
||
290 |
lemma path_image_shiftpath: |
|
291 |
assumes "a \<in> {0..1}" "pathfinish g = pathstart g" |
|
292 |
shows "path_image(shiftpath a g) = path_image g" proof- |
|
293 |
{ fix x assume as:"g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)" |
|
294 |
hence "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x") |
|
295 |
case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI) |
|
296 |
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) |
|
297 |
by(auto simp add: field_simps atomize_not) next |
|
298 |
case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI) |
|
299 |
by(auto simp add: field_simps) qed } |
|
300 |
thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
|
301 |
by(auto simp add: image_iff) qed |
|
302 |
||
303 |
subsection {* Special case of straight-line paths. *} |
|
304 |
||
305 |
definition |
|
306 |
linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" where |
|
307 |
"linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
|
308 |
||
309 |
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a" |
|
310 |
unfolding pathstart_def linepath_def by auto |
|
311 |
||
312 |
lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b" |
|
313 |
unfolding pathfinish_def linepath_def by auto |
|
314 |
||
315 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
|
316 |
unfolding linepath_def by (intro continuous_intros) |
|
317 |
||
318 |
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" |
|
319 |
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on) |
|
320 |
||
321 |
lemma path_linepath[intro]: "path(linepath a b)" |
|
322 |
unfolding path_def by(rule continuous_on_linepath) |
|
323 |
||
324 |
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)" |
|
325 |
unfolding path_image_def segment linepath_def apply (rule set_ext, rule) defer |
|
326 |
unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI) |
|
327 |
by auto |
|
328 |
||
329 |
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a" |
|
330 |
unfolding reversepath_def linepath_def by(rule ext, auto) |
|
331 |
||
332 |
lemma injective_path_linepath: |
|
333 |
assumes "a \<noteq> b" shows "injective_path(linepath a b)" |
|
334 |
proof - |
|
335 |
{ fix x y :: "real" |
|
336 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
|
337 |
hence "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps) |
|
338 |
with assms have "x = y" by simp } |
|
339 |
thus ?thesis unfolding injective_path_def linepath_def by(auto simp add: algebra_simps) qed |
|
340 |
||
341 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath) |
|
342 |
||
343 |
subsection {* Bounding a point away from a path. *} |
|
344 |
||
345 |
lemma not_on_path_ball: |
|
346 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
347 |
assumes "path g" "z \<notin> path_image g" |
|
348 |
shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof- |
|
349 |
obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y" |
|
350 |
using distance_attains_inf[OF _ path_image_nonempty, of g z] |
|
351 |
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto |
|
352 |
thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed |
|
353 |
||
354 |
lemma not_on_path_cball: |
|
355 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
356 |
assumes "path g" "z \<notin> path_image g" |
|
357 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof- |
|
358 |
obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto |
|
359 |
moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto |
|
360 |
ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed |
|
361 |
||
362 |
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *} |
|
363 |
||
364 |
definition "path_component s x y \<longleftrightarrow> (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
365 |
||
366 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
|
367 |
||
368 |
lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s" |
|
369 |
using assms unfolding path_defs by auto |
|
370 |
||
371 |
lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x" |
|
372 |
unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms |
|
373 |
by(auto intro!:continuous_on_intros) |
|
374 |
||
375 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
376 |
by(auto intro!: path_component_mem path_component_refl) |
|
377 |
||
378 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
379 |
using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI) |
|
380 |
by auto |
|
381 |
||
382 |
lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z" |
|
383 |
using assms unfolding path_component_def apply- apply(erule exE)+ apply(rule_tac x="g +++ ga" in exI) by(auto simp add: path_image_join) |
|
384 |
||
385 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
|
386 |
unfolding path_component_def by auto |
|
387 |
||
388 |
subsection {* Can also consider it as a set, as the name suggests. *} |
|
389 |
||
390 |
lemma path_component_set: "path_component s x = { y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y )}" |
|
391 |
apply(rule set_ext) unfolding mem_Collect_eq unfolding mem_def path_component_def by auto |
|
392 |
||
393 |
lemma mem_path_component_set:"x \<in> path_component s y \<longleftrightarrow> path_component s y x" unfolding mem_def by auto |
|
394 |
||
395 |
lemma path_component_subset: "(path_component s x) \<subseteq> s" |
|
396 |
apply(rule, rule path_component_mem(2)) by(auto simp add:mem_def) |
|
397 |
||
398 |
lemma path_component_eq_empty: "path_component s x = {} \<longleftrightarrow> x \<notin> s" |
|
399 |
apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_path_component_set |
|
400 |
apply(drule path_component_mem(1)) using path_component_refl by auto |
|
401 |
||
402 |
subsection {* Path connectedness of a space. *} |
|
403 |
||
404 |
definition "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
405 |
||
406 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
|
407 |
unfolding path_connected_def path_component_def by auto |
|
408 |
||
409 |
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component s x = s)" |
|
410 |
unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset) |
|
411 |
unfolding subset_eq mem_path_component_set Ball_def mem_def by auto |
|
412 |
||
413 |
subsection {* Some useful lemmas about path-connectedness. *} |
|
414 |
||
415 |
lemma convex_imp_path_connected: |
|
416 |
fixes s :: "'a::real_normed_vector set" |
|
417 |
assumes "convex s" shows "path_connected s" |
|
418 |
unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI) |
|
419 |
unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto |
|
420 |
||
421 |
lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s" |
|
422 |
unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof- |
|
423 |
fix e1 e2 assume as:"open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}" |
|
424 |
then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto |
|
425 |
then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" |
|
426 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
|
427 |
have *:"connected {0..1::real}" by(auto intro!: convex_connected convex_real_interval) |
|
428 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" using as(3) g(2)[unfolded path_defs] by blast |
|
429 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto |
|
430 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" using g(3,4)[unfolded path_defs] using obt |
|
431 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
|
432 |
ultimately show False using *[unfolded connected_local not_ex,rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"] |
|
433 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)] |
|
434 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed |
|
435 |
||
436 |
lemma open_path_component: |
|
437 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
438 |
assumes "open s" shows "open(path_component s x)" |
|
439 |
unfolding open_contains_ball proof |
|
440 |
fix y assume as:"y \<in> path_component s x" |
|
441 |
hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_def by auto |
|
442 |
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto |
|
443 |
show "\<exists>e>0. ball y e \<subseteq> path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule) unfolding mem_ball mem_path_component_set proof- |
|
444 |
fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer |
|
445 |
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` |
|
446 |
using as[unfolded mem_def] by auto qed qed |
|
447 |
||
448 |
lemma open_non_path_component: |
|
449 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
450 |
assumes "open s" shows "open(s - path_component s x)" |
|
451 |
unfolding open_contains_ball proof |
|
452 |
fix y assume as:"y\<in>s - path_component s x" |
|
453 |
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto |
|
454 |
show "\<exists>e>0. ball y e \<subseteq> s - path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule,rule) defer proof(rule ccontr) |
|
455 |
fix z assume "z\<in>ball y e" "\<not> z \<notin> path_component s x" |
|
456 |
hence "y \<in> path_component s x" unfolding not_not mem_path_component_set using `e>0` |
|
457 |
apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)]) |
|
458 |
apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto |
|
459 |
thus False using as by auto qed(insert e(2), auto) qed |
|
460 |
||
461 |
lemma connected_open_path_connected: |
|
462 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
463 |
assumes "open s" "connected s" shows "path_connected s" |
|
464 |
unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule) |
|
465 |
fix x y assume "x \<in> s" "y \<in> s" show "y \<in> path_component s x" proof(rule ccontr) |
|
466 |
assume "y \<notin> path_component s x" moreover |
|
467 |
have "path_component s x \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto |
|
468 |
ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
|
469 |
using assms(2)[unfolded connected_def not_ex, rule_format, of"path_component s x" "s - path_component s x"] by auto |
|
470 |
qed qed |
|
471 |
||
472 |
lemma path_connected_continuous_image: |
|
473 |
assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)" |
|
474 |
unfolding path_connected_def proof(rule,rule) |
|
475 |
fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s" |
|
476 |
then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto |
|
477 |
guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] .. |
|
478 |
thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" |
|
479 |
unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs |
|
480 |
using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed |
|
481 |
||
482 |
lemma homeomorphic_path_connectedness: |
|
483 |
"s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)" |
|
484 |
unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule |
|
485 |
apply(drule_tac f=f in path_connected_continuous_image) prefer 3 |
|
486 |
apply(drule_tac f=g in path_connected_continuous_image) by auto |
|
487 |
||
488 |
lemma path_connected_empty: "path_connected {}" |
|
489 |
unfolding path_connected_def by auto |
|
490 |
||
491 |
lemma path_connected_singleton: "path_connected {a}" |
|
492 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
|
493 |
apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv) |
|
494 |
apply (simp add: path_def continuous_on_const) |
|
495 |
done |
|
496 |
||
497 |
lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}" |
|
498 |
shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule) |
|
499 |
fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t" |
|
500 |
from assms(3) obtain z where "z \<in> s \<inter> t" by auto |
|
501 |
thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply- |
|
502 |
apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z]) |
|
503 |
by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed |
|
504 |
||
505 |
subsection {* sphere is path-connected. *} |
|
506 |
||
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
507 |
(** TODO covert this to ordered_euclidean_space **) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
508 |
|
36583 | 509 |
lemma path_connected_punctured_universe: |
510 |
assumes "2 \<le> CARD('n::finite)" shows "path_connected((UNIV::(real^'n) set) - {a})" proof- |
|
511 |
obtain \<psi> where \<psi>:"bij_betw \<psi> {1..CARD('n)} (UNIV::'n set)" using ex_bij_betw_nat_finite_1[OF finite_UNIV] by auto |
|
512 |
let ?U = "UNIV::(real^'n) set" let ?u = "?U - {0}" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
513 |
let ?basis = "\<lambda>k. cart_basis (\<psi> k)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
514 |
let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. inner (cart_basis (\<psi> i)) x \<noteq> 0}" |
36583 | 515 |
have "\<forall>k\<in>{2..CARD('n)}. path_connected (?A k)" proof |
516 |
have *:"\<And>k. ?A (Suc k) = {x. inner (?basis (Suc k)) x < 0} \<union> {x. inner (?basis (Suc k)) x > 0} \<union> ?A k" apply(rule set_ext,rule) defer |
|
517 |
apply(erule UnE)+ unfolding mem_Collect_eq apply(rule_tac[1-2] x="Suc k" in bexI) |
|
518 |
by(auto elim!: ballE simp add: not_less le_Suc_eq) |
|
519 |
fix k assume "k \<in> {2..CARD('n)}" thus "path_connected (?A k)" proof(induct k) |
|
520 |
case (Suc k) show ?case proof(cases "k = 1") |
|
521 |
case False from Suc have d:"k \<in> {1..CARD('n)}" "Suc k \<in> {1..CARD('n)}" by auto |
|
522 |
hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto |
|
523 |
hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < inner (?basis (Suc k)) x} \<inter> (?A k)" |
|
524 |
"?basis k - ?basis (Suc k) \<in> {x. 0 > inner (?basis (Suc k)) x} \<inter> ({x. 0 < inner (?basis (Suc k)) x} \<union> (?A k))" using d |
|
525 |
by(auto simp add: inner_basis intro!:bexI[where x=k]) |
|
526 |
show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un) |
|
527 |
prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt) |
|
528 |
apply(rule Suc(1)) using d ** False by auto |
|
529 |
next case True hence d:"1\<in>{1..CARD('n)}" "2\<in>{1..CARD('n)}" using Suc(2) by auto |
|
530 |
have ***:"Suc 1 = 2" by auto |
|
531 |
have **:"\<And>s t P Q. s \<union> t \<union> {x. P x \<or> Q x} = (s \<union> {x. P x}) \<union> (t \<union> {x. Q x})" by auto |
|
532 |
have nequals0I:"\<And>x A. x\<in>A \<Longrightarrow> A \<noteq> {}" by auto |
|
533 |
have "\<psi> 2 \<noteq> \<psi> (Suc 0)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=2]] using assms by auto |
|
534 |
thus ?thesis unfolding * True unfolding ** neq_iff bex_disj_distrib apply - |
|
535 |
apply(rule path_connected_Un, rule_tac[1-2] path_connected_Un) defer 3 apply(rule_tac[1-4] convex_imp_path_connected) |
|
536 |
apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I) |
|
537 |
apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I) |
|
538 |
apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I) |
|
539 |
using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add: inner_basis) |
|
540 |
qed qed auto qed note lem = this |
|
541 |
||
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
542 |
have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. inner (cart_basis (\<psi> i)) x \<noteq> 0) \<longleftrightarrow> (\<exists>i. inner (cart_basis i) x \<noteq> 0)" |
36583 | 543 |
apply rule apply(erule bexE) apply(rule_tac x="\<psi> i" in exI) defer apply(erule exE) proof- |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
544 |
fix x::"real^'n" and i assume as:"inner (cart_basis i) x \<noteq> 0" |
36583 | 545 |
have "i\<in>\<psi> ` {1..CARD('n)}" using \<psi>[unfolded bij_betw_def, THEN conjunct2] by auto |
546 |
then obtain j where "j\<in>{1..CARD('n)}" "\<psi> j = i" by auto |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
547 |
thus "\<exists>i\<in>{1..CARD('n)}. inner (cart_basis (\<psi> i)) x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto |
36583 | 548 |
have *:"?U - {a} = (\<lambda>x. x + a) ` {x. x \<noteq> 0}" apply(rule set_ext) unfolding image_iff |
549 |
apply rule apply(rule_tac x="x - a" in bexI) by auto |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
550 |
have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. inner (cart_basis i) x \<noteq> 0)" unfolding Cart_eq by(auto simp add: inner_basis) |
36583 | 551 |
show ?thesis unfolding * apply(rule path_connected_continuous_image) apply(rule continuous_on_intros)+ |
552 |
unfolding ** apply(rule lem[THEN bspec[where x="CARD('n)"], unfolded ***]) using assms by auto qed |
|
553 |
||
554 |
lemma path_connected_sphere: assumes "2 \<le> CARD('n::finite)" shows "path_connected {x::real^'n. norm(x - a) = r}" proof(cases "r\<le>0") |
|
555 |
case True thus ?thesis proof(cases "r=0") |
|
556 |
case False hence "{x::real^'n. norm(x - a) = r} = {}" using True by auto |
|
557 |
thus ?thesis using path_connected_empty by auto |
|
558 |
qed(auto intro!:path_connected_singleton) next |
|
559 |
case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule) |
|
560 |
unfolding image_iff apply(rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR by (auto simp add: scaleR_right_diff_distrib) |
|
561 |
have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" apply(rule set_ext,rule) |
|
562 |
unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto split:split_if_asm) |
|
563 |
have "continuous_on (UNIV - {0}) (\<lambda>x::real^'n. 1 / norm x)" unfolding o_def continuous_on_eq_continuous_within |
|
564 |
apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within) |
|
565 |
apply(rule continuous_at_norm[unfolded o_def]) by auto |
|
566 |
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms] |
|
567 |
by(auto intro!: path_connected_continuous_image continuous_on_intros) qed |
|
568 |
||
569 |
lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n. norm(x - a) = r}" |
|
570 |
using path_connected_sphere path_connected_imp_connected by auto |
|
571 |
||
572 |
end |