author | hoelzl |
Thu, 17 Jan 2013 11:57:17 +0100 | |
changeset 50935 | cfdf19d3ca32 |
parent 50884 | 2b21b4e2d7cb |
child 51478 | 270b21f3ae0a |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Multivariate_Analysis/Path_Connected.thy |
36583 | 2 |
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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imports Convex_Euclidean_Space |
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begin |
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subsection {* Paths. *} |
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definition 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 pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathstart g = g 0" |
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definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathfinish g = g 1" |
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definition 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 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 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 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 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: "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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apply (rule convex_connected, rule convex_real_interval) |
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done |
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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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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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by (erule compact_continuous_image, rule compact_interval) |
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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" |
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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 |
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apply(rule,rule,erule bexE) |
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apply(rule_tac x="1 - xa" in bexI) |
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apply auto |
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done |
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show ?thesis |
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using *[of g] *[of "reversepath g"] |
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unfolding reversepath_reversepath by auto |
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qed |
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lemma path_reversepath[simp]: "path (reversepath g) \<longleftrightarrow> path g" |
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proof - |
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have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)" |
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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 (intro continuous_on_intros) |
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apply (rule continuous_on_subset[of "{0..1}"], assumption) |
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apply auto |
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done |
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show ?thesis |
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using *[of "reversepath g"] *[of g] |
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unfolding reversepath_reversepath |
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by (rule iffI) |
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qed |
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lemmas reversepath_simps = |
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path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
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lemma path_join[simp]: |
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assumes "pathfinish g1 = pathstart g2" |
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shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
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unfolding path_def pathfinish_def pathstart_def |
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apply rule defer |
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apply(erule conjE) |
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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}" |
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"(\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {(0::real)..1} \<subseteq> {0..1}" |
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by auto |
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then show "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" |
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apply - |
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apply rule |
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apply (subst *) defer |
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apply (subst *) |
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apply (rule_tac[!] continuous_on_compose) |
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apply (intro continuous_on_intros) defer |
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apply (intro continuous_on_intros) |
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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}"]) |
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apply (rule as, assumption, rule as, assumption) |
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apply rule defer |
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apply rule |
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proof - |
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fix x |
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assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real..1}" |
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49654 | 140 |
then have "x \<le> 1 / 2" unfolding image_iff by auto |
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then show "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto |
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next |
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fix x |
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assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real..1}" |
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then have "x \<ge> 1 / 2" unfolding image_iff by auto |
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then show "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" |
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proof (cases "x = 1 / 2") |
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case True |
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then have "x = (1/2) *\<^sub>R 1" by auto |
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then show ?thesis |
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49653 | 151 |
unfolding joinpaths_def |
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using assms[unfolded pathstart_def pathfinish_def] |
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by (auto simp add: mult_ac) |
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qed (auto simp add:le_less joinpaths_def) |
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qed |
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next |
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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}" |
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apply (rule set_eqI, rule) |
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unfolding image_iff |
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defer |
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apply (rule_tac x="(1/2)*\<^sub>R x" in bexI) |
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apply auto |
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done |
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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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36583 | 167 |
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)" |
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unfolding * |
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apply (rule continuous_on_union) |
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apply (rule closed_real_atLeastAtMost)+ |
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proof - |
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show "continuous_on {0..(1 / 2) *\<^sub>R 1} (g1 +++ g2)" |
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apply (rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *\<^sub>R x)"]) defer |
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unfolding o_def[THEN sym] |
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apply (rule continuous_on_compose) |
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apply (intro continuous_on_intros) |
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unfolding ** |
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apply (rule as(1)) |
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unfolding joinpaths_def |
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apply auto |
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done |
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next |
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show "continuous_on {(1/2)*\<^sub>R1..1} (g1 +++ g2)" |
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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) |
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apply (intro continuous_on_intros) |
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unfolding *** o_def joinpaths_def |
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apply (rule as(2)) |
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using assms[unfolded pathstart_def pathfinish_def] |
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apply (auto simp add: mult_ac) |
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done |
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qed |
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qed |
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lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" |
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proof |
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fix x |
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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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then show "x \<in> path_image g1 \<union> path_image g2" |
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apply (cases "y \<le> 1/2") |
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apply (rule_tac UnI1) defer |
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apply (rule_tac UnI2) |
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unfolding y(2) path_image_def |
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using y(1) |
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apply (auto intro!: imageI) |
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done |
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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" |
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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) |
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unfolding Un_iff |
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proof (erule disjE) |
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fix x |
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assume "x \<in> path_image g1" |
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then obtain y where y: "y\<in>{0..1}" "x = g1 y" |
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unfolding path_image_def image_iff by auto |
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then show "x \<in> path_image (g1 +++ g2)" |
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unfolding joinpaths_def path_image_def image_iff |
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apply (rule_tac x="(1/2) *\<^sub>R y" in bexI) |
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apply auto |
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done |
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next |
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fix x |
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assume "x \<in> path_image g2" |
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then obtain y where y: "y\<in>{0..1}" "x = g2 y" |
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unfolding path_image_def image_iff by auto |
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then show "x \<in> path_image (g1 +++ g2)" |
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unfolding joinpaths_def path_image_def image_iff |
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apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) |
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using assms(3)[unfolded pathfinish_def pathstart_def] |
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apply (auto simp add: add_divide_distrib) |
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done |
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qed |
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36583 | 247 |
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lemma not_in_path_image_join: |
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49653 | 249 |
assumes "x \<notin> path_image g1" "x \<notin> path_image g2" |
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shows "x \<notin> path_image(g1 +++ g2)" |
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36583 | 251 |
using assms and path_image_join_subset[of g1 g2] by auto |
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lemma simple_path_reversepath: |
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assumes "simple_path g" |
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shows "simple_path (reversepath g)" |
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using assms |
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unfolding simple_path_def reversepath_def |
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apply - |
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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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apply auto |
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done |
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36583 | 263 |
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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 |
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proof ((rule ballI)+, rule impI) |
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let ?g = "g1 +++ g2" |
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36583 | 271 |
note inj = assms(1,2)[unfolded injective_path_def, rule_format] |
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fix x y :: real |
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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" |
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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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49654 | 277 |
then have "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" |
49653 | 278 |
using xy(3) unfolding joinpaths_def by auto |
279 |
moreover |
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280 |
have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as |
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36583 | 281 |
by auto |
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ultimately |
283 |
show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto |
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284 |
next |
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assume as:"x > 1 / 2" "y > 1 / 2" |
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49654 | 286 |
then have "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" |
49653 | 287 |
using xy(3) unfolding joinpaths_def by auto |
288 |
moreover |
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289 |
have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" |
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290 |
using xy(1,2) as by auto |
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291 |
ultimately |
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show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto |
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293 |
next |
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assume as:"x \<le> 1 / 2" "y > 1 / 2" |
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49654 | 295 |
then have "?g x \<in> path_image g1" "?g y \<in> path_image g2" |
49653 | 296 |
unfolding path_image_def joinpaths_def |
36583 | 297 |
using xy(1,2) by auto |
49653 | 298 |
moreover |
299 |
have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def |
|
36583 | 300 |
using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2) |
301 |
by (auto simp add: field_simps) |
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49653 | 302 |
ultimately |
303 |
have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto |
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49654 | 304 |
then have "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1) |
36583 | 305 |
using inj(1)[of "2 *\<^sub>R x" 0] by auto |
49653 | 306 |
moreover |
307 |
have "y = 1" using * unfolding xy(3) assms(3)[THEN sym] |
|
36583 | 308 |
unfolding joinpaths_def pathfinish_def using as(2) and xy(2) |
309 |
using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto |
|
310 |
ultimately show ?thesis by auto |
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49653 | 311 |
next |
312 |
assume as: "x > 1 / 2" "y \<le> 1 / 2" |
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49654 | 313 |
then have "?g x \<in> path_image g2" "?g y \<in> path_image g1" |
49653 | 314 |
unfolding path_image_def joinpaths_def |
36583 | 315 |
using xy(1,2) by auto |
49653 | 316 |
moreover |
317 |
have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def |
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36583 | 318 |
using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1) |
319 |
by (auto simp add: field_simps) |
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49653 | 320 |
ultimately |
321 |
have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto |
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49654 | 322 |
then have "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2) |
36583 | 323 |
using inj(1)[of "2 *\<^sub>R y" 0] by auto |
49653 | 324 |
moreover |
325 |
have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym] |
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36583 | 326 |
unfolding joinpaths_def pathfinish_def using as(1) and xy(1) |
327 |
using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto |
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49653 | 328 |
ultimately show ?thesis by auto |
329 |
qed |
|
330 |
qed |
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36583 | 331 |
|
332 |
lemma injective_path_join: |
|
333 |
assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2" |
|
49653 | 334 |
"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}" |
36583 | 335 |
shows "injective_path(g1 +++ g2)" |
49653 | 336 |
unfolding injective_path_def |
337 |
proof (rule, rule, rule) |
|
338 |
let ?g = "g1 +++ g2" |
|
36583 | 339 |
note inj = assms(1,2)[unfolded injective_path_def, rule_format] |
49653 | 340 |
fix x y |
341 |
assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y" |
|
342 |
show "x = y" |
|
343 |
proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le) |
|
344 |
assume "x \<le> 1 / 2" "y \<le> 1 / 2" |
|
49654 | 345 |
then show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy |
36583 | 346 |
unfolding joinpaths_def by auto |
49653 | 347 |
next |
348 |
assume "x > 1 / 2" "y > 1 / 2" |
|
49654 | 349 |
then show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy |
36583 | 350 |
unfolding joinpaths_def by auto |
49653 | 351 |
next |
352 |
assume as: "x \<le> 1 / 2" "y > 1 / 2" |
|
49654 | 353 |
then have "?g x \<in> path_image g1" "?g y \<in> path_image g2" |
49653 | 354 |
unfolding path_image_def joinpaths_def |
36583 | 355 |
using xy(1,2) by auto |
49654 | 356 |
then have "?g x = pathfinish g1" "?g y = pathstart g2" |
49653 | 357 |
using assms(4) unfolding assms(3) xy(3) by auto |
49654 | 358 |
then show ?thesis |
49653 | 359 |
using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2) |
36583 | 360 |
unfolding pathstart_def pathfinish_def joinpaths_def |
361 |
by auto |
|
49653 | 362 |
next |
363 |
assume as:"x > 1 / 2" "y \<le> 1 / 2" |
|
49654 | 364 |
then have "?g x \<in> path_image g2" "?g y \<in> path_image g1" |
49653 | 365 |
unfolding path_image_def joinpaths_def |
36583 | 366 |
using xy(1,2) by auto |
49654 | 367 |
then have "?g x = pathstart g2" "?g y = pathfinish g1" |
49653 | 368 |
using assms(4) unfolding assms(3) xy(3) by auto |
49654 | 369 |
then show ?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) |
36583 | 370 |
unfolding pathstart_def pathfinish_def joinpaths_def |
49653 | 371 |
by auto |
372 |
qed |
|
373 |
qed |
|
36583 | 374 |
|
375 |
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join |
|
376 |
||
49653 | 377 |
|
36583 | 378 |
subsection {* Reparametrizing a closed curve to start at some chosen point. *} |
379 |
||
380 |
definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) = |
|
381 |
(\<lambda>x. if (a + x) \<le> 1 then f(a + x) else f(a + x - 1))" |
|
382 |
||
383 |
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a" |
|
384 |
unfolding pathstart_def shiftpath_def by auto |
|
385 |
||
49653 | 386 |
lemma pathfinish_shiftpath: |
387 |
assumes "0 \<le> a" "pathfinish g = pathstart g" |
|
36583 | 388 |
shows "pathfinish(shiftpath a g) = g a" |
389 |
using assms unfolding pathstart_def pathfinish_def shiftpath_def |
|
390 |
by auto |
|
391 |
||
392 |
lemma endpoints_shiftpath: |
|
393 |
assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" |
|
394 |
shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a" |
|
395 |
using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath) |
|
396 |
||
397 |
lemma closed_shiftpath: |
|
398 |
assumes "pathfinish g = pathstart g" "a \<in> {0..1}" |
|
399 |
shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)" |
|
400 |
using endpoints_shiftpath[OF assms] by auto |
|
401 |
||
402 |
lemma path_shiftpath: |
|
403 |
assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
|
49653 | 404 |
shows "path(shiftpath a g)" |
405 |
proof - |
|
406 |
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto |
|
407 |
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
|
36583 | 408 |
using assms(2)[unfolded pathfinish_def pathstart_def] by auto |
49653 | 409 |
show ?thesis |
410 |
unfolding path_def shiftpath_def * |
|
411 |
apply (rule continuous_on_union) |
|
412 |
apply (rule closed_real_atLeastAtMost)+ |
|
413 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3 |
|
414 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3 |
|
415 |
apply (rule continuous_on_intros)+ prefer 2 |
|
416 |
apply (rule continuous_on_intros)+ |
|
417 |
apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) |
|
418 |
using assms(3) and ** |
|
419 |
apply (auto, auto simp add: field_simps) |
|
420 |
done |
|
421 |
qed |
|
36583 | 422 |
|
49653 | 423 |
lemma shiftpath_shiftpath: |
424 |
assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" |
|
36583 | 425 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
426 |
using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto |
|
427 |
||
428 |
lemma path_image_shiftpath: |
|
429 |
assumes "a \<in> {0..1}" "pathfinish g = pathstart g" |
|
49653 | 430 |
shows "path_image(shiftpath a g) = path_image g" |
431 |
proof - |
|
432 |
{ fix x |
|
433 |
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)" |
|
49654 | 434 |
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" |
49653 | 435 |
proof (cases "a \<le> x") |
436 |
case False |
|
49654 | 437 |
then show ?thesis |
49653 | 438 |
apply (rule_tac x="1 + x - a" in bexI) |
36583 | 439 |
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) |
49653 | 440 |
apply (auto simp add: field_simps atomize_not) |
441 |
done |
|
442 |
next |
|
443 |
case True |
|
49654 | 444 |
then show ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI) |
49653 | 445 |
by(auto simp add: field_simps) |
446 |
qed |
|
447 |
} |
|
49654 | 448 |
then show ?thesis |
49653 | 449 |
using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
450 |
by(auto simp add: image_iff) |
|
451 |
qed |
|
452 |
||
36583 | 453 |
|
454 |
subsection {* Special case of straight-line paths. *} |
|
455 |
||
49653 | 456 |
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" |
457 |
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
|
36583 | 458 |
|
459 |
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a" |
|
460 |
unfolding pathstart_def linepath_def by auto |
|
461 |
||
462 |
lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b" |
|
463 |
unfolding pathfinish_def linepath_def by auto |
|
464 |
||
465 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
|
466 |
unfolding linepath_def by (intro continuous_intros) |
|
467 |
||
468 |
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" |
|
469 |
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on) |
|
470 |
||
471 |
lemma path_linepath[intro]: "path(linepath a b)" |
|
472 |
unfolding path_def by(rule continuous_on_linepath) |
|
473 |
||
474 |
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)" |
|
49653 | 475 |
unfolding path_image_def segment linepath_def |
476 |
apply (rule set_eqI, rule) defer |
|
477 |
unfolding mem_Collect_eq image_iff |
|
478 |
apply(erule exE) |
|
479 |
apply(rule_tac x="u *\<^sub>R 1" in bexI) |
|
480 |
apply auto |
|
481 |
done |
|
482 |
||
483 |
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a" |
|
484 |
unfolding reversepath_def linepath_def |
|
36583 | 485 |
by auto |
486 |
||
487 |
lemma injective_path_linepath: |
|
49653 | 488 |
assumes "a \<noteq> b" |
489 |
shows "injective_path (linepath a b)" |
|
36583 | 490 |
proof - |
491 |
{ fix x y :: "real" |
|
492 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
|
49654 | 493 |
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps) |
36583 | 494 |
with assms have "x = y" by simp } |
49654 | 495 |
then show ?thesis |
49653 | 496 |
unfolding injective_path_def linepath_def |
497 |
by (auto simp add: algebra_simps) |
|
498 |
qed |
|
36583 | 499 |
|
49653 | 500 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" |
501 |
by(auto intro!: injective_imp_simple_path injective_path_linepath) |
|
502 |
||
36583 | 503 |
|
504 |
subsection {* Bounding a point away from a path. *} |
|
505 |
||
506 |
lemma not_on_path_ball: |
|
507 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
508 |
assumes "path g" "z \<notin> path_image g" |
|
49653 | 509 |
shows "\<exists>e > 0. ball z e \<inter> (path_image g) = {}" |
510 |
proof - |
|
511 |
obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y" |
|
36583 | 512 |
using distance_attains_inf[OF _ path_image_nonempty, of g z] |
513 |
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto |
|
49654 | 514 |
then show ?thesis |
49653 | 515 |
apply (rule_tac x="dist z a" in exI) |
516 |
using assms(2) |
|
517 |
apply (auto intro!: dist_pos_lt) |
|
518 |
done |
|
519 |
qed |
|
36583 | 520 |
|
521 |
lemma not_on_path_cball: |
|
522 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
523 |
assumes "path g" "z \<notin> path_image g" |
|
49653 | 524 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" |
525 |
proof - |
|
526 |
obtain e where "ball z e \<inter> path_image g = {}" "e>0" |
|
527 |
using not_on_path_ball[OF assms] by auto |
|
36583 | 528 |
moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto |
49653 | 529 |
ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto |
530 |
qed |
|
531 |
||
36583 | 532 |
|
533 |
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *} |
|
534 |
||
49653 | 535 |
definition "path_component s x y \<longleftrightarrow> |
536 |
(\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
36583 | 537 |
|
538 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
|
539 |
||
49653 | 540 |
lemma path_component_mem: |
541 |
assumes "path_component s x y" |
|
542 |
shows "x \<in> s" "y \<in> s" |
|
36583 | 543 |
using assms unfolding path_defs by auto |
544 |
||
49653 | 545 |
lemma path_component_refl: |
546 |
assumes "x \<in> s" |
|
547 |
shows "path_component s x x" |
|
548 |
unfolding path_defs |
|
549 |
apply (rule_tac x="\<lambda>u. x" in exI) |
|
550 |
using assms apply (auto intro!:continuous_on_intros) done |
|
36583 | 551 |
|
552 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
49653 | 553 |
by (auto intro!: path_component_mem path_component_refl) |
36583 | 554 |
|
555 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
49653 | 556 |
using assms |
557 |
unfolding path_component_def |
|
558 |
apply (erule exE) |
|
559 |
apply (rule_tac x="reversepath g" in exI) |
|
560 |
apply auto |
|
561 |
done |
|
36583 | 562 |
|
49653 | 563 |
lemma path_component_trans: |
564 |
assumes "path_component s x y" "path_component s y z" |
|
565 |
shows "path_component s x z" |
|
566 |
using assms |
|
567 |
unfolding path_component_def |
|
568 |
apply - |
|
569 |
apply (erule exE)+ |
|
570 |
apply (rule_tac x="g +++ ga" in exI) |
|
571 |
apply (auto simp add: path_image_join) |
|
572 |
done |
|
36583 | 573 |
|
574 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
|
575 |
unfolding path_component_def by auto |
|
576 |
||
49653 | 577 |
|
36583 | 578 |
subsection {* Can also consider it as a set, as the name suggests. *} |
579 |
||
49653 | 580 |
lemma path_component_set: |
581 |
"{y. path_component s x y} = |
|
582 |
{y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}" |
|
583 |
apply (rule set_eqI) |
|
584 |
unfolding mem_Collect_eq |
|
585 |
unfolding path_component_def |
|
586 |
apply auto |
|
587 |
done |
|
36583 | 588 |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
41959
diff
changeset
|
589 |
lemma path_component_subset: "{y. path_component s x y} \<subseteq> s" |
49653 | 590 |
apply (rule, rule path_component_mem(2)) |
591 |
apply auto |
|
592 |
done |
|
36583 | 593 |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
41959
diff
changeset
|
594 |
lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s" |
49653 | 595 |
apply rule |
596 |
apply (drule equals0D[of _ x]) defer |
|
597 |
apply (rule equals0I) |
|
598 |
unfolding mem_Collect_eq |
|
599 |
apply (drule path_component_mem(1)) |
|
600 |
using path_component_refl |
|
601 |
apply auto |
|
602 |
done |
|
603 |
||
36583 | 604 |
|
605 |
subsection {* Path connectedness of a space. *} |
|
606 |
||
49653 | 607 |
definition "path_connected s \<longleftrightarrow> |
608 |
(\<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)" |
|
36583 | 609 |
|
610 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
|
611 |
unfolding path_connected_def path_component_def by auto |
|
612 |
||
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
41959
diff
changeset
|
613 |
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)" |
49653 | 614 |
unfolding path_connected_component |
615 |
apply (rule, rule, rule, rule path_component_subset) |
|
616 |
unfolding subset_eq mem_Collect_eq Ball_def |
|
617 |
apply auto |
|
618 |
done |
|
619 |
||
36583 | 620 |
|
621 |
subsection {* Some useful lemmas about path-connectedness. *} |
|
622 |
||
623 |
lemma convex_imp_path_connected: |
|
624 |
fixes s :: "'a::real_normed_vector set" |
|
625 |
assumes "convex s" shows "path_connected s" |
|
49653 | 626 |
unfolding path_connected_def |
627 |
apply (rule, rule, rule_tac x = "linepath x y" in exI) |
|
628 |
unfolding path_image_linepath |
|
629 |
using assms [unfolded convex_contains_segment] |
|
630 |
apply auto |
|
631 |
done |
|
36583 | 632 |
|
49653 | 633 |
lemma path_connected_imp_connected: |
634 |
assumes "path_connected s" |
|
635 |
shows "connected s" |
|
636 |
unfolding connected_def not_ex |
|
637 |
apply (rule, rule, rule ccontr) |
|
638 |
unfolding not_not |
|
639 |
apply (erule conjE)+ |
|
640 |
proof - |
|
641 |
fix e1 e2 |
|
642 |
assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}" |
|
36583 | 643 |
then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto |
644 |
then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" |
|
645 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
|
49653 | 646 |
have *: "connected {0..1::real}" |
647 |
by (auto intro!: convex_connected convex_real_interval) |
|
648 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" |
|
649 |
using as(3) g(2)[unfolded path_defs] by blast |
|
650 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" |
|
651 |
using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto |
|
652 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" |
|
653 |
using g(3,4)[unfolded path_defs] using obt |
|
36583 | 654 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
49653 | 655 |
ultimately show False |
656 |
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}"] |
|
36583 | 657 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)] |
49653 | 658 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] |
659 |
by auto |
|
660 |
qed |
|
36583 | 661 |
|
662 |
lemma open_path_component: |
|
663 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
49653 | 664 |
assumes "open s" |
665 |
shows "open {y. path_component s x y}" |
|
666 |
unfolding open_contains_ball |
|
667 |
proof |
|
668 |
fix y |
|
669 |
assume as: "y \<in> {y. path_component s x y}" |
|
49654 | 670 |
then have "y \<in> s" |
49653 | 671 |
apply - |
672 |
apply (rule path_component_mem(2)) |
|
673 |
unfolding mem_Collect_eq |
|
674 |
apply auto |
|
675 |
done |
|
676 |
then obtain e where e:"e>0" "ball y e \<subseteq> s" |
|
677 |
using assms[unfolded open_contains_ball] by auto |
|
678 |
show "\<exists>e > 0. ball y e \<subseteq> {y. path_component s x y}" |
|
679 |
apply (rule_tac x=e in exI) |
|
680 |
apply (rule,rule `e>0`, rule) |
|
681 |
unfolding mem_ball mem_Collect_eq |
|
682 |
proof - |
|
683 |
fix z |
|
684 |
assume "dist y z < e" |
|
49654 | 685 |
then show "path_component s x z" |
49653 | 686 |
apply (rule_tac path_component_trans[of _ _ y]) defer |
687 |
apply (rule path_component_of_subset[OF e(2)]) |
|
688 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
689 |
using `e>0` as |
|
690 |
apply auto |
|
691 |
done |
|
692 |
qed |
|
693 |
qed |
|
36583 | 694 |
|
695 |
lemma open_non_path_component: |
|
696 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
49653 | 697 |
assumes "open s" |
698 |
shows "open(s - {y. path_component s x y})" |
|
699 |
unfolding open_contains_ball |
|
700 |
proof |
|
701 |
fix y |
|
702 |
assume as: "y\<in>s - {y. path_component s x y}" |
|
703 |
then obtain e where e:"e>0" "ball y e \<subseteq> s" |
|
704 |
using assms [unfolded open_contains_ball] by auto |
|
705 |
show "\<exists>e>0. ball y e \<subseteq> s - {y. path_component s x y}" |
|
706 |
apply (rule_tac x=e in exI) |
|
707 |
apply (rule, rule `e>0`, rule, rule) defer |
|
708 |
proof (rule ccontr) |
|
709 |
fix z |
|
710 |
assume "z \<in> ball y e" "\<not> z \<notin> {y. path_component s x y}" |
|
49654 | 711 |
then have "y \<in> {y. path_component s x y}" |
49653 | 712 |
unfolding not_not mem_Collect_eq using `e>0` |
713 |
apply - |
|
714 |
apply (rule path_component_trans, assumption) |
|
715 |
apply (rule path_component_of_subset[OF e(2)]) |
|
716 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
717 |
apply auto |
|
718 |
done |
|
49654 | 719 |
then show False using as by auto |
49653 | 720 |
qed (insert e(2), auto) |
721 |
qed |
|
36583 | 722 |
|
723 |
lemma connected_open_path_connected: |
|
724 |
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) |
|
49653 | 725 |
assumes "open s" "connected s" |
726 |
shows "path_connected s" |
|
727 |
unfolding path_connected_component_set |
|
728 |
proof (rule, rule, rule path_component_subset, rule) |
|
729 |
fix x y |
|
730 |
assume "x \<in> s" "y \<in> s" |
|
731 |
show "y \<in> {y. path_component s x y}" |
|
732 |
proof (rule ccontr) |
|
733 |
assume "y \<notin> {y. path_component s x y}" |
|
734 |
moreover |
|
735 |
have "{y. path_component s x y} \<inter> s \<noteq> {}" |
|
736 |
using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto |
|
737 |
ultimately |
|
738 |
show False |
|
739 |
using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
|
740 |
using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"] |
|
741 |
by auto |
|
742 |
qed |
|
743 |
qed |
|
36583 | 744 |
|
745 |
lemma path_connected_continuous_image: |
|
49653 | 746 |
assumes "continuous_on s f" "path_connected s" |
747 |
shows "path_connected (f ` s)" |
|
748 |
unfolding path_connected_def |
|
749 |
proof (rule, rule) |
|
750 |
fix x' y' |
|
751 |
assume "x' \<in> f ` s" "y' \<in> f ` s" |
|
752 |
then obtain x y where xy: "x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto |
|
753 |
guess g using assms(2)[unfolded path_connected_def, rule_format, OF xy(1,2)] .. |
|
49654 | 754 |
then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" |
49653 | 755 |
unfolding xy |
756 |
apply (rule_tac x="f \<circ> g" in exI) |
|
757 |
unfolding path_defs |
|
758 |
using assms(1) |
|
759 |
apply (auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) |
|
760 |
done |
|
761 |
qed |
|
36583 | 762 |
|
763 |
lemma homeomorphic_path_connectedness: |
|
764 |
"s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)" |
|
49653 | 765 |
unfolding homeomorphic_def homeomorphism_def |
766 |
apply (erule exE|erule conjE)+ |
|
767 |
apply rule |
|
768 |
apply (drule_tac f=f in path_connected_continuous_image) prefer 3 |
|
769 |
apply (drule_tac f=g in path_connected_continuous_image) |
|
770 |
apply auto |
|
771 |
done |
|
36583 | 772 |
|
773 |
lemma path_connected_empty: "path_connected {}" |
|
774 |
unfolding path_connected_def by auto |
|
775 |
||
776 |
lemma path_connected_singleton: "path_connected {a}" |
|
777 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
|
778 |
apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv) |
|
779 |
apply (simp add: path_def continuous_on_const) |
|
780 |
done |
|
781 |
||
49653 | 782 |
lemma path_connected_Un: |
783 |
assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}" |
|
784 |
shows "path_connected (s \<union> t)" |
|
785 |
unfolding path_connected_component |
|
786 |
proof (rule, rule) |
|
787 |
fix x y |
|
788 |
assume as: "x \<in> s \<union> t" "y \<in> s \<union> t" |
|
36583 | 789 |
from assms(3) obtain z where "z \<in> s \<inter> t" by auto |
49654 | 790 |
then show "path_component (s \<union> t) x y" |
49653 | 791 |
using as and assms(1-2)[unfolded path_connected_component] |
792 |
apply - |
|
793 |
apply (erule_tac[!] UnE)+ |
|
794 |
apply (rule_tac[2-3] path_component_trans[of _ _ z]) |
|
795 |
apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) |
|
796 |
done |
|
797 |
qed |
|
36583 | 798 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
799 |
lemma path_connected_UNION: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
800 |
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" |
49653 | 801 |
and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
802 |
shows "path_connected (\<Union>i\<in>A. S i)" |
49653 | 803 |
unfolding path_connected_component |
804 |
proof clarify |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
805 |
fix x i y j |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
806 |
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" |
49654 | 807 |
then have "path_component (S i) x z" and "path_component (S j) z y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
808 |
using assms by (simp_all add: path_connected_component) |
49654 | 809 |
then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y" |
48125
602dc0215954
tuned proofs -- prefer direct "rotated" instead of old-style COMP;
wenzelm
parents:
44647
diff
changeset
|
810 |
using *(1,3) by (auto elim!: path_component_of_subset [rotated]) |
49654 | 811 |
then show "path_component (\<Union>i\<in>A. S i) x y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
812 |
by (rule path_component_trans) |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
813 |
qed |
36583 | 814 |
|
49653 | 815 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
816 |
subsection {* sphere is path-connected. *} |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
817 |
|
36583 | 818 |
lemma path_connected_punctured_universe: |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
819 |
assumes "2 \<le> DIM('a::euclidean_space)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
820 |
shows "path_connected((UNIV::'a::euclidean_space set) - {a})" |
49653 | 821 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
822 |
let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
823 |
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}" |
36583 | 824 |
|
49653 | 825 |
have A: "path_connected ?A" |
826 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
827 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
828 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
829 |
assume "i \<in> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
830 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
831 |
show "path_connected {x. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
832 |
using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
833 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
834 |
qed |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
835 |
have B: "path_connected ?B" unfolding Collect_bex_eq |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
836 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
837 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
838 |
assume "i \<in> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
839 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
840 |
show "path_connected {x. a \<bullet> i < x \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
841 |
using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
842 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
843 |
qed |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
844 |
obtain S :: "'a set" where "S \<subseteq> Basis" "card S = Suc (Suc 0)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
845 |
using ex_card[OF assms] by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
846 |
then obtain b0 b1 :: 'a where "b0 \<in> Basis" "b1 \<in> Basis" "b0 \<noteq> b1" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
847 |
unfolding card_Suc_eq by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
848 |
then have "a + b0 - b1 \<in> ?A \<inter> ?B" by (auto simp: inner_simps inner_Basis) |
49654 | 849 |
then have "?A \<inter> ?B \<noteq> {}" by fast |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
850 |
with A B have "path_connected (?A \<union> ?B)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
851 |
by (rule path_connected_Un) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
852 |
also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
853 |
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
854 |
also have "\<dots> = {x. x \<noteq> a}" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
855 |
unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
856 |
also have "\<dots> = UNIV - {a}" by auto |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
857 |
finally show ?thesis . |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
858 |
qed |
36583 | 859 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
860 |
lemma path_connected_sphere: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
861 |
assumes "2 \<le> DIM('a::euclidean_space)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
862 |
shows "path_connected {x::'a::euclidean_space. norm(x - a) = r}" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
863 |
proof (rule linorder_cases [of r 0]) |
49653 | 864 |
assume "r < 0" |
49654 | 865 |
then have "{x::'a. norm(x - a) = r} = {}" by auto |
866 |
then show ?thesis using path_connected_empty by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
867 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
868 |
assume "r = 0" |
49654 | 869 |
then show ?thesis using path_connected_singleton by simp |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
870 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
871 |
assume r: "0 < r" |
49654 | 872 |
then have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" |
49653 | 873 |
apply - |
874 |
apply (rule set_eqI, rule) |
|
875 |
unfolding image_iff |
|
876 |
apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) |
|
877 |
unfolding mem_Collect_eq norm_scaleR |
|
878 |
apply (auto simp add: scaleR_right_diff_distrib) |
|
879 |
done |
|
880 |
have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" |
|
881 |
apply (rule set_eqI,rule) |
|
882 |
unfolding image_iff |
|
883 |
apply (rule_tac x=x in bexI) |
|
884 |
unfolding mem_Collect_eq |
|
885 |
apply (auto split:split_if_asm) |
|
886 |
done |
|
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44531
diff
changeset
|
887 |
have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)" |
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44531
diff
changeset
|
888 |
unfolding field_divide_inverse by (simp add: continuous_on_intros) |
49654 | 889 |
then show ?thesis unfolding * ** using path_connected_punctured_universe[OF assms] |
49653 | 890 |
by (auto intro!: path_connected_continuous_image continuous_on_intros) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
891 |
qed |
36583 | 892 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
893 |
lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x - a) = r}" |
36583 | 894 |
using path_connected_sphere path_connected_imp_connected by auto |
895 |
||
896 |
end |