author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 44647  e4de7750cdeb 
child 48125  602dc0215954 
permissions  rwrr 
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(* Title: HOL/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 pathconnected 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 

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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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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" 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(intro continuous_on_intros) 
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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 (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[12] 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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case True hence "x = (1/2) *\<^sub>R 1" by auto 
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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_eqI, 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 (intro continuous_on_intros) 
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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 (intro continuous_on_intros) 
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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="1x" in ballE, erule_tac x="1y" 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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lemma closed_shiftpath: 

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assumes "pathfinish g = pathstart g" "a \<in> {0..1}" 

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shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)" 

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using endpoints_shiftpath[OF assms] by auto 

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lemma path_shiftpath: 

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assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}" 

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shows "path(shiftpath a g)" proof 

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have *:"{0 .. 1} = {0 .. 1a} \<union> {1a .. 1}" using assms(3) by auto 

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have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a  1) = g (x + a)" 

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using assms(2)[unfolded pathfinish_def pathstart_def] by auto 

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show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union) 

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apply(rule closed_real_atLeastAtMost)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3 

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apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a  1 + x)"]) defer prefer 3 

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apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+ 

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apply(rule_tac[12] continuous_on_subset[OF assms(1)[unfolded path_def]]) 

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using assms(3) and ** by(auto, auto simp add: field_simps) qed 

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lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" 

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shows "shiftpath (1  a) (shiftpath a g) x = g x" 

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using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto 

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lemma path_image_shiftpath: 

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assumes "a \<in> {0..1}" "pathfinish g = pathstart g" 

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shows "path_image(shiftpath a g) = path_image g" proof 

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{ 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)" 

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hence "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x") 

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case False thus ?thesis apply(rule_tac x="1 + x  a" in bexI) 

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using as(1,2) and as(3)[THEN bspec[where x="1 + x  a"]] and assms(1) 

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by(auto simp add: field_simps atomize_not) next 

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case True thus ?thesis using as(12) and assms(1) apply(rule_tac x="x  a" in bexI) 

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by(auto simp add: field_simps) qed } 

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thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def 

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by(auto simp add: image_iff) qed 

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subsection {* Special case of straightline paths. *} 

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definition 

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linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" where 

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"linepath a b = (\<lambda>x. (1  x) *\<^sub>R a + x *\<^sub>R b)" 

309 

310 
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a" 

311 
unfolding pathstart_def linepath_def by auto 

312 

313 
lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b" 

314 
unfolding pathfinish_def linepath_def by auto 

315 

316 
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" 

317 
unfolding linepath_def by (intro continuous_intros) 

318 

319 
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" 

320 
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on) 

321 

322 
lemma path_linepath[intro]: "path(linepath a b)" 

323 
unfolding path_def by(rule continuous_on_linepath) 

324 

325 
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)" 

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326 
unfolding path_image_def segment linepath_def apply (rule set_eqI, rule) defer 
36583  327 
unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI) 
328 
by auto 

329 

330 
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a" 

331 
unfolding reversepath_def linepath_def by(rule ext, auto) 

332 

333 
lemma injective_path_linepath: 

334 
assumes "a \<noteq> b" shows "injective_path(linepath a b)" 

335 
proof  

336 
{ fix x y :: "real" 

337 
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" 

338 
hence "(x  y) *\<^sub>R a = (x  y) *\<^sub>R b" by (simp add: algebra_simps) 

339 
with assms have "x = y" by simp } 

340 
thus ?thesis unfolding injective_path_def linepath_def by(auto simp add: algebra_simps) qed 

341 

342 
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath) 

343 

344 
subsection {* Bounding a point away from a path. *} 

345 

346 
lemma not_on_path_ball: 

347 
fixes g :: "real \<Rightarrow> 'a::heine_borel" 

348 
assumes "path g" "z \<notin> path_image g" 

349 
shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof 

350 
obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y" 

351 
using distance_attains_inf[OF _ path_image_nonempty, of g z] 

352 
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto 

353 
thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed 

354 

355 
lemma not_on_path_cball: 

356 
fixes g :: "real \<Rightarrow> 'a::heine_borel" 

357 
assumes "path g" "z \<notin> path_image g" 

358 
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof 

359 
obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto 

360 
moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto 

361 
ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed 

362 

363 
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *} 

364 

365 
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)" 

366 

367 
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def 

368 

369 
lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s" 

370 
using assms unfolding path_defs by auto 

371 

372 
lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x" 

373 
unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms 

374 
by(auto intro!:continuous_on_intros) 

375 

376 
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" 

377 
by(auto intro!: path_component_mem path_component_refl) 

378 

379 
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" 

380 
using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI) 

381 
by auto 

382 

383 
lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z" 

384 
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) 

385 

386 
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" 

387 
unfolding path_component_def by auto 

388 

389 
subsection {* Can also consider it as a set, as the name suggests. *} 

390 

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391 
lemma path_component_set: "{y. path_component s x y} = { y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y )}" 
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392 
apply(rule set_eqI) unfolding mem_Collect_eq unfolding path_component_def by auto 
36583  393 

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394 
lemma path_component_subset: "{y. path_component s x y} \<subseteq> s" 
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395 
apply(rule, rule path_component_mem(2)) by auto 
36583  396 

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397 
lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s" 
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398 
apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_Collect_eq 
36583  399 
apply(drule path_component_mem(1)) using path_component_refl by auto 
400 

401 
subsection {* Path connectedness of a space. *} 

402 

403 
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)" 

404 

405 
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" 

406 
unfolding path_connected_def path_component_def by auto 

407 

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408 
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)" 
36583  409 
unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset) 
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410 
unfolding subset_eq mem_Collect_eq Ball_def by auto 
36583  411 

412 
subsection {* Some useful lemmas about pathconnectedness. *} 

413 

414 
lemma convex_imp_path_connected: 

415 
fixes s :: "'a::real_normed_vector set" 

416 
assumes "convex s" shows "path_connected s" 

417 
unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI) 

418 
unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto 

419 

420 
lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s" 

421 
unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof 

422 
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> {}" 

423 
then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto 

424 
then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" 

425 
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto 

426 
have *:"connected {0..1::real}" by(auto intro!: convex_connected convex_real_interval) 

427 
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 

428 
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 

429 
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 

430 
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) 

431 
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}"] 

432 
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)] 

433 
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed 

434 

435 
lemma open_path_component: 

436 
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) 

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437 
assumes "open s" shows "open {y. path_component s x y}" 
36583  438 
unfolding open_contains_ball proof 
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439 
fix y assume as:"y \<in> {y. path_component s x y}" 
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440 
hence "y\<in>s" apply apply(rule path_component_mem(2)) unfolding mem_Collect_eq by auto 
36583  441 
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto 
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442 
show "\<exists>e>0. ball y e \<subseteq> {y. path_component s x y}" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule) unfolding mem_ball mem_Collect_eq proof 
36583  443 
fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer 
444 
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` 

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445 
using as by auto qed qed 
36583  446 

447 
lemma open_non_path_component: 

448 
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) 

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449 
assumes "open s" shows "open(s  {y. path_component s x y})" 
36583  450 
unfolding open_contains_ball proof 
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451 
fix y assume as:"y\<in>s  {y. path_component s x y}" 
36583  452 
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto 
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453 
show "\<exists>e>0. ball y e \<subseteq> s  {y. path_component s x y}" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule,rule) defer proof(rule ccontr) 
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454 
fix z assume "z\<in>ball y e" "\<not> z \<notin> {y. path_component s x y}" 
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455 
hence "y \<in> {y. path_component s x y}" unfolding not_not mem_Collect_eq using `e>0` 
36583  456 
apply apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)]) 
457 
apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto 

458 
thus False using as by auto qed(insert e(2), auto) qed 

459 

460 
lemma connected_open_path_connected: 

461 
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*) 

462 
assumes "open s" "connected s" shows "path_connected s" 

463 
unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule) 

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464 
fix x y assume "x \<in> s" "y \<in> s" show "y \<in> {y. path_component s x y}" proof(rule ccontr) 
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465 
assume "y \<notin> {y. path_component s x y}" moreover 
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466 
have "{y. path_component s x y} \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto 
36583  467 
ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] 
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468 
using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s  {y. path_component s x y}"] by auto 
36583  469 
qed qed 
470 

471 
lemma path_connected_continuous_image: 

472 
assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)" 

473 
unfolding path_connected_def proof(rule,rule) 

474 
fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s" 

475 
then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto 

476 
guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] .. 

477 
thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" 

478 
unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs 

479 
using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed 

480 

481 
lemma homeomorphic_path_connectedness: 

482 
"s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)" 

483 
unfolding homeomorphic_def homeomorphism_def apply(erule exEerule conjE)+ apply rule 

484 
apply(drule_tac f=f in path_connected_continuous_image) prefer 3 

485 
apply(drule_tac f=g in path_connected_continuous_image) by auto 

486 

487 
lemma path_connected_empty: "path_connected {}" 

488 
unfolding path_connected_def by auto 

489 

490 
lemma path_connected_singleton: "path_connected {a}" 

491 
unfolding path_connected_def pathstart_def pathfinish_def path_image_def 

492 
apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv) 

493 
apply (simp add: path_def continuous_on_const) 

494 
done 

495 

496 
lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}" 

497 
shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule) 

498 
fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t" 

499 
from assms(3) obtain z where "z \<in> s \<inter> t" by auto 

500 
thus "path_component (s \<union> t) x y" using as using assms(12)[unfolded path_connected_component] apply 

501 
apply(erule_tac[!] UnE)+ apply(rule_tac[23] path_component_trans[of _ _ z]) 

502 
by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed 

503 

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504 
lemma path_connected_UNION: 
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505 
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" 
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506 
assumes "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i" 
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507 
shows "path_connected (\<Union>i\<in>A. S i)" 
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508 
unfolding path_connected_component proof(clarify) 
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509 
fix x i y j 
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510 
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" 
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511 
hence "path_component (S i) x z" and "path_component (S j) z y" 
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512 
using assms by (simp_all add: path_connected_component) 
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513 
hence "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y" 
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514 
using *(1,3) by (auto elim!: path_component_of_subset [COMP swap_prems_rl]) 
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515 
thus "path_component (\<Union>i\<in>A. S i) x y" 
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516 
by (rule path_component_trans) 
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517 
qed 
36583  518 

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519 
subsection {* sphere is pathconnected. *} 
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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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520 

36583  521 
lemma path_connected_punctured_universe: 
37674
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522 
assumes "2 \<le> DIM('a::euclidean_space)" 
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523 
shows "path_connected((UNIV::'a::euclidean_space set)  {a})" 
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524 
proof 
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525 
let ?A = "{x::'a. \<exists>i\<in>{..<DIM('a)}. x $$ i < a $$ i}" 
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526 
let ?B = "{x::'a. \<exists>i\<in>{..<DIM('a)}. a $$ i < x $$ i}" 
36583  527 

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528 
have A: "path_connected ?A" unfolding Collect_bex_eq 
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529 
proof (rule path_connected_UNION) 
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530 
fix i assume "i \<in> {..<DIM('a)}" 
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531 
thus "(\<chi>\<chi> i. a $$ i  1) \<in> {x::'a. x $$ i < a $$ i}" by simp 
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532 
show "path_connected {x. x $$ i < a $$ i}" unfolding euclidean_component_def 
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533 
by (rule convex_imp_path_connected [OF convex_halfspace_lt]) 
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534 
qed 
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changeset

535 
have B: "path_connected ?B" unfolding Collect_bex_eq 
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536 
proof (rule path_connected_UNION) 
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537 
fix i assume "i \<in> {..<DIM('a)}" 
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changeset

538 
thus "(\<chi>\<chi> i. a $$ i + 1) \<in> {x::'a. a $$ i < x $$ i}" by simp 
f86de9c00c47
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huffman
parents:
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diff
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539 
show "path_connected {x. a $$ i < x $$ i}" unfolding euclidean_component_def 
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

540 
by (rule convex_imp_path_connected [OF convex_halfspace_gt]) 
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convert theorem path_connected_sphere to euclidean_space class
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541 
qed 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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542 
from assms have "1 < DIM('a)" by auto 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

543 
hence "a + basis 0  basis 1 \<in> ?A \<inter> ?B" by auto 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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544 
hence "?A \<inter> ?B \<noteq> {}" by fast 
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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changeset

545 
with A B have "path_connected (?A \<union> ?B)" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

546 
by (rule path_connected_Un) 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

547 
also have "?A \<union> ?B = {x. \<exists>i\<in>{..<DIM('a)}. x $$ i \<noteq> a $$ i}" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

548 
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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changeset

549 
also have "\<dots> = {x. x \<noteq> a}" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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diff
changeset

550 
unfolding Bex_def euclidean_eq [where 'a='a] by simp 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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551 
also have "\<dots> = UNIV  {a}" by auto 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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552 
finally show ?thesis . 
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parents:
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553 
qed 
36583  554 

37674
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parents:
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changeset

555 
lemma path_connected_sphere: 
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parents:
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556 
assumes "2 \<le> DIM('a::euclidean_space)" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

557 
shows "path_connected {x::'a::euclidean_space. norm(x  a) = r}" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

558 
proof (rule linorder_cases [of r 0]) 
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parents:
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changeset

559 
assume "r < 0" hence "{x::'a. norm(x  a) = r} = {}" by auto 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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diff
changeset

560 
thus ?thesis using path_connected_empty by simp 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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diff
changeset

561 
next 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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changeset

562 
assume "r = 0" 
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convert theorem path_connected_sphere to euclidean_space class
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parents:
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diff
changeset

563 
thus ?thesis using path_connected_singleton by simp 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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diff
changeset

564 
next 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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diff
changeset

565 
assume r: "0 < r" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
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parents:
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diff
changeset

566 
hence *:"{x::'a. norm(x  a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" apply apply(rule set_eqI,rule) 
36583  567 
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) 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
37674
diff
changeset

568 
have **:"{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV  {0})" apply(rule set_eqI,rule) 
36583  569 
unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto split:split_if_asm) 
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
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parents:
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diff
changeset

570 
have "continuous_on (UNIV  {0}) (\<lambda>x::'a. 1 / norm x)" 
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modernize lemmas about 'continuous' and 'continuous_on';
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parents:
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diff
changeset

571 
unfolding field_divide_inverse by (simp add: continuous_on_intros) 
36583  572 
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms] 
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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diff
changeset

573 
by(auto intro!: path_connected_continuous_image continuous_on_intros) 
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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diff
changeset

574 
qed 
36583  575 

37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset

576 
lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x  a) = r}" 
36583  577 
using path_connected_sphere path_connected_imp_connected by auto 
578 

579 
end 