--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy Fri Sep 28 23:02:49 2012 +0200
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy Fri Sep 28 23:40:48 2012 +0200
@@ -10,44 +10,36 @@
subsection {* Paths. *}
-definition
- path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
+definition path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "path g \<longleftrightarrow> continuous_on {0 .. 1} g"
-definition
- pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
+definition pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathstart g = g 0"
-definition
- pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
+definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathfinish g = g 1"
-definition
- path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
+definition path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
where "path_image g = g ` {0 .. 1}"
-definition
- reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)"
+definition reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)"
where "reversepath g = (\<lambda>x. g(1 - x))"
-definition
- joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)"
+definition joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)"
(infixr "+++" 75)
where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
-definition
- simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
+definition simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "simple_path g \<longleftrightarrow>
- (\<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)"
+ (\<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)"
-definition
- injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
+definition injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "injective_path g \<longleftrightarrow> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"
+
subsection {* Some lemmas about these concepts. *}
-lemma injective_imp_simple_path:
- "injective_path g \<Longrightarrow> simple_path g"
+lemma injective_imp_simple_path: "injective_path g \<Longrightarrow> simple_path g"
unfolding injective_path_def simple_path_def by auto
lemma path_image_nonempty: "path_image g \<noteq> {}"
@@ -62,7 +54,8 @@
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
unfolding path_def path_image_def
apply (erule connected_continuous_image)
- by(rule convex_connected, rule convex_real_interval)
+ apply (rule convex_connected, rule convex_real_interval)
+ done
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
unfolding path_def path_image_def
@@ -77,177 +70,311 @@
lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
unfolding pathstart_def reversepath_def pathfinish_def by auto
-lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1"
+lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
unfolding pathstart_def joinpaths_def pathfinish_def by auto
-lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2"
+lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
unfolding pathstart_def joinpaths_def pathfinish_def by auto
-lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof-
- have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g"
- unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE)
- apply(rule_tac x="1 - xa" in bexI) by auto
- show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
-
-lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof-
- have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def
- apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
- apply(intro continuous_on_intros)
- apply(rule continuous_on_subset[of "{0..1}"], assumption) by auto
- show ?thesis using *[of "reversepath g"] *[of g] unfolding reversepath_reversepath by (rule iffI) qed
+lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g"
+proof -
+ have *: "\<And>g. path_image(reversepath g) \<subseteq> path_image g"
+ unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
+ apply(rule,rule,erule bexE)
+ apply(rule_tac x="1 - xa" in bexI)
+ apply auto
+ done
+ show ?thesis
+ using *[of g] *[of "reversepath g"]
+ unfolding reversepath_reversepath by auto
+qed
-lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
+lemma path_reversepath[simp]: "path (reversepath g) \<longleftrightarrow> path g"
+proof -
+ have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
+ unfolding path_def reversepath_def
+ apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
+ apply (intro continuous_on_intros)
+ apply (rule continuous_on_subset[of "{0..1}"], assumption)
+ apply auto
+ done
+ show ?thesis
+ using *[of "reversepath g"] *[of g]
+ unfolding reversepath_reversepath
+ by (rule iffI)
+qed
+
+lemmas reversepath_simps =
+ path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
-lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
- unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-
- assume as:"continuous_on {0..1} (g1 +++ g2)"
- have *:"g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)"
- "g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))"
+lemma path_join[simp]:
+ assumes "pathfinish g1 = pathstart g2"
+ shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
+ unfolding path_def pathfinish_def pathstart_def
+ apply rule defer
+ apply(erule conjE)
+proof -
+ assume as: "continuous_on {0..1} (g1 +++ g2)"
+ have *: "g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)"
+ "g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))"
unfolding o_def by (auto simp add: add_divide_distrib)
- 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}"
+ 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}"
by auto
- thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule
- apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)
+ then show "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2"
+ apply -
+ apply rule
+ apply (subst *) defer
+ apply (subst *)
+ apply (rule_tac[!] continuous_on_compose)
apply (intro continuous_on_intros) defer
apply (intro continuous_on_intros)
- apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
- apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)
- apply(rule) defer apply rule proof-
- fix x assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real..1}"
+ apply (rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
+ apply (rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"])
+ apply (rule as, assumption, rule as, assumption)
+ apply rule defer
+ apply rule
+ proof -
+ fix x
+ assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real..1}"
hence "x \<le> 1 / 2" unfolding image_iff by auto
- thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto next
- fix x assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real..1}"
+ thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto
+ next
+ fix x
+ assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real..1}"
hence "x \<ge> 1 / 2" unfolding image_iff by auto
- thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" proof(cases "x = 1 / 2")
- case True hence "x = (1/2) *\<^sub>R 1" by auto
- thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by (auto simp add: mult_ac)
- qed (auto simp add:le_less joinpaths_def) qed
-next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
- have *:"{0 .. 1::real} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by auto
- have **:"op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real}" apply(rule set_eqI, rule) unfolding image_iff
- defer apply(rule_tac x="(1/2)*\<^sub>R x" in bexI) by auto
- have ***:"(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real}"
+ thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)"
+ proof (cases "x = 1 / 2")
+ case True
+ hence "x = (1/2) *\<^sub>R 1" by auto
+ thus ?thesis
+ unfolding joinpaths_def
+ using assms[unfolded pathstart_def pathfinish_def]
+ by (auto simp add: mult_ac)
+ qed (auto simp add:le_less joinpaths_def)
+ qed
+next
+ assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
+ have *: "{0 .. 1::real} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by auto
+ have **: "op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real}"
+ apply (rule set_eqI, rule)
+ unfolding image_iff
+ defer
+ apply (rule_tac x="(1/2)*\<^sub>R x" in bexI)
+ apply auto
+ done
+ have ***: "(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real}"
apply (auto simp add: image_def)
apply (rule_tac x="(x + 1) / 2" in bexI)
apply (auto simp add: add_divide_distrib)
done
- show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply (rule closed_real_atLeastAtMost)+ proof-
- 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
- unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply (intro continuous_on_intros)
- unfolding ** apply(rule as(1)) unfolding joinpaths_def by auto next
- 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
- apply(rule continuous_on_compose) apply (intro continuous_on_intros)
- unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]
- by (auto simp add: mult_ac) qed qed
+ show "continuous_on {0..1} (g1 +++ g2)"
+ unfolding *
+ apply (rule continuous_on_union)
+ apply (rule closed_real_atLeastAtMost)+
+ proof -
+ 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
+ unfolding o_def[THEN sym]
+ apply (rule continuous_on_compose)
+ apply (intro continuous_on_intros)
+ unfolding **
+ apply (rule as(1))
+ unfolding joinpaths_def
+ apply auto
+ done
+ next
+ 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
+ apply (rule continuous_on_compose)
+ apply (intro continuous_on_intros)
+ unfolding *** o_def joinpaths_def
+ apply (rule as(2))
+ using assms[unfolded pathstart_def pathfinish_def]
+ apply (auto simp add: mult_ac)
+ done
+ qed
+qed
-lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof
- fix x assume "x \<in> path_image (g1 +++ g2)"
+lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)"
+proof
+ fix x
+ assume "x \<in> path_image (g1 +++ g2)"
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))"
unfolding path_image_def image_iff joinpaths_def by auto
- thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "y \<le> 1/2")
- apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)
- by(auto intro!: imageI) qed
+ thus "x \<in> path_image g1 \<union> path_image g2"
+ apply (cases "y \<le> 1/2")
+ apply (rule_tac UnI1) defer
+ apply (rule_tac UnI2)
+ unfolding y(2) path_image_def
+ using y(1)
+ apply (auto intro!: imageI)
+ done
+qed
lemma subset_path_image_join:
- assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s"
+ assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s"
+ shows "path_image(g1 +++ g2) \<subseteq> s"
using path_image_join_subset[of g1 g2] and assms by auto
lemma path_image_join:
assumes "path g1" "path g2" "pathfinish g1 = pathstart g2"
shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
-apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE)
- fix x assume "x \<in> path_image g1"
- then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto
- thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
- apply(rule_tac x="(1/2) *\<^sub>R y" in bexI) by auto next
- fix x assume "x \<in> path_image g2"
- then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto
- then show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
- apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
- by (auto simp add: add_divide_distrib) qed
+ apply (rule, rule path_image_join_subset, rule)
+ unfolding Un_iff
+proof (erule disjE)
+ fix x
+ assume "x \<in> path_image g1"
+ then obtain y where y: "y\<in>{0..1}" "x = g1 y"
+ unfolding path_image_def image_iff by auto
+ thus "x \<in> path_image (g1 +++ g2)"
+ unfolding joinpaths_def path_image_def image_iff
+ apply (rule_tac x="(1/2) *\<^sub>R y" in bexI)
+ apply auto
+ done
+next
+ fix x
+ assume "x \<in> path_image g2"
+ then obtain y where y: "y\<in>{0..1}" "x = g2 y"
+ unfolding path_image_def image_iff by auto
+ then show "x \<in> path_image (g1 +++ g2)"
+ unfolding joinpaths_def path_image_def image_iff
+ apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI)
+ using assms(3)[unfolded pathfinish_def pathstart_def]
+ apply (auto simp add: add_divide_distrib)
+ done
+qed
lemma not_in_path_image_join:
- assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)"
+ assumes "x \<notin> path_image g1" "x \<notin> path_image g2"
+ shows "x \<notin> path_image(g1 +++ g2)"
using assms and path_image_join_subset[of g1 g2] by auto
-lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)"
- using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+
- apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
- by auto
+lemma simple_path_reversepath:
+ assumes "simple_path g"
+ shows "simple_path (reversepath g)"
+ using assms
+ unfolding simple_path_def reversepath_def
+ apply -
+ apply (rule ballI)+
+ apply (erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
+ apply auto
+ done
lemma simple_path_join_loop:
assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
- "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
+ "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
shows "simple_path(g1 +++ g2)"
-unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2"
+ unfolding simple_path_def
+proof ((rule ballI)+, rule impI)
+ let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
- fix x y::"real" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
- 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)
- assume as:"x \<le> 1 / 2" "y \<le> 1 / 2"
- hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" using xy(3) unfolding joinpaths_def by auto
- moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
+ fix x y :: real
+ assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
+ 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)
+ assume as: "x \<le> 1 / 2" "y \<le> 1 / 2"
+ hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)"
+ using xy(3) unfolding joinpaths_def by auto
+ moreover
+ have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
by auto
- ultimately show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
- next assume as:"x > 1 / 2" "y > 1 / 2"
- hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" using xy(3) unfolding joinpaths_def by auto
- 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
- ultimately show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
- next assume as:"x \<le> 1 / 2" "y > 1 / 2"
- hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
+ ultimately
+ show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
+ next
+ assume as:"x > 1 / 2" "y > 1 / 2"
+ hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)"
+ using xy(3) unfolding joinpaths_def by auto
+ 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
+ ultimately
+ show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
+ next
+ assume as:"x \<le> 1 / 2" "y > 1 / 2"
+ hence "?g x \<in> path_image g1" "?g y \<in> path_image g2"
+ unfolding path_image_def joinpaths_def
using xy(1,2) by auto
- moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
+ moreover
+ have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)
by (auto simp add: field_simps)
- ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
+ ultimately
+ have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)
using inj(1)[of "2 *\<^sub>R x" 0] by auto
- moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
+ moreover
+ have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(2) and xy(2)
using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto
ultimately show ?thesis by auto
- next assume as:"x > 1 / 2" "y \<le> 1 / 2"
- hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
+ next
+ assume as: "x > 1 / 2" "y \<le> 1 / 2"
+ hence "?g x \<in> path_image g2" "?g y \<in> path_image g1"
+ unfolding path_image_def joinpaths_def
using xy(1,2) by auto
- moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
+ moreover
+ have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)
by (auto simp add: field_simps)
- ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
+ ultimately
+ have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)
using inj(1)[of "2 *\<^sub>R y" 0] by auto
- moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
+ moreover
+ have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(1) and xy(1)
using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto
- ultimately show ?thesis by auto qed qed
+ ultimately show ?thesis by auto
+ qed
+qed
lemma injective_path_join:
assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
- "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
+ "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
shows "injective_path(g1 +++ g2)"
- unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2"
+ unfolding injective_path_def
+proof (rule, rule, rule)
+ let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
- fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
- show "x = y" proof(cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
- 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
+ fix x y
+ assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
+ show "x = y"
+ proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
+ 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
unfolding joinpaths_def by auto
- 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
+ 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
unfolding joinpaths_def by auto
- next assume as:"x \<le> 1 / 2" "y > 1 / 2"
- hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
+ next
+ assume as: "x \<le> 1 / 2" "y > 1 / 2"
+ hence "?g x \<in> path_image g1" "?g y \<in> path_image g2"
+ unfolding path_image_def joinpaths_def
using xy(1,2) by auto
- hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto
- 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)
+ hence "?g x = pathfinish g1" "?g y = pathstart g2"
+ using assms(4) unfolding assms(3) xy(3) by auto
+ 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)
unfolding pathstart_def pathfinish_def joinpaths_def
by auto
- next assume as:"x > 1 / 2" "y \<le> 1 / 2"
- hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
+ next
+ assume as:"x > 1 / 2" "y \<le> 1 / 2"
+ hence "?g x \<in> path_image g2" "?g y \<in> path_image g1"
+ unfolding path_image_def joinpaths_def
using xy(1,2) by auto
- hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto
+ hence "?g x = pathstart g2" "?g y = pathfinish g1"
+ using assms(4) unfolding assms(3) xy(3) by auto
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)
unfolding pathstart_def pathfinish_def joinpaths_def
- by auto qed qed
+ by auto
+ qed
+qed
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
+
subsection {* Reparametrizing a closed curve to start at some chosen point. *}
definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) =
@@ -256,7 +383,8 @@
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
-lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g"
+lemma pathfinish_shiftpath:
+ assumes "0 \<le> a" "pathfinish g = pathstart g"
shows "pathfinish(shiftpath a g) = g a"
using assms unfolding pathstart_def pathfinish_def shiftpath_def
by auto
@@ -273,39 +401,60 @@
lemma path_shiftpath:
assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "path(shiftpath a g)" proof-
- have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto
- have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
+ shows "path(shiftpath a g)"
+proof -
+ have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto
+ have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
using assms(2)[unfolded pathfinish_def pathstart_def] by auto
- show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union)
- apply(rule closed_real_atLeastAtMost)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
- apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
- apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+
- apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
- using assms(3) and ** by(auto, auto simp add: field_simps) qed
+ show ?thesis
+ unfolding path_def shiftpath_def *
+ apply (rule continuous_on_union)
+ apply (rule closed_real_atLeastAtMost)+
+ apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
+ apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
+ apply (rule continuous_on_intros)+ prefer 2
+ apply (rule continuous_on_intros)+
+ apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
+ using assms(3) and **
+ apply (auto, auto simp add: field_simps)
+ done
+qed
-lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}"
+lemma shiftpath_shiftpath:
+ assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}"
shows "shiftpath (1 - a) (shiftpath a g) x = g x"
using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto
lemma path_image_shiftpath:
assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
- shows "path_image(shiftpath a g) = path_image g" proof-
- { 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)"
- hence "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x")
- case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI)
+ shows "path_image(shiftpath a g) = path_image g"
+proof -
+ { 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)"
+ hence "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
+ proof (cases "a \<le> x")
+ case False
+ thus ?thesis
+ apply (rule_tac x="1 + x - a" in bexI)
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
- by(auto simp add: field_simps atomize_not) next
- case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
- by(auto simp add: field_simps) qed }
- thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
- by(auto simp add: image_iff) qed
+ apply (auto simp add: field_simps atomize_not)
+ done
+ next
+ case True
+ thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
+ by(auto simp add: field_simps)
+ qed
+ }
+ thus ?thesis
+ using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
+ by(auto simp add: image_iff)
+qed
+
subsection {* Special case of straight-line paths. *}
-definition
- linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" where
- "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
+definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
+ where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
unfolding pathstart_def linepath_def by auto
@@ -323,166 +472,303 @@
unfolding path_def by(rule continuous_on_linepath)
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
- unfolding path_image_def segment linepath_def apply (rule set_eqI, rule) defer
- unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI)
+ unfolding path_image_def segment linepath_def
+ apply (rule set_eqI, rule) defer
+ unfolding mem_Collect_eq image_iff
+ apply(erule exE)
+ apply(rule_tac x="u *\<^sub>R 1" in bexI)
+ apply auto
+ done
+
+lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
+ unfolding reversepath_def linepath_def
by auto
-lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
- unfolding reversepath_def linepath_def by(rule ext, auto)
-
lemma injective_path_linepath:
- assumes "a \<noteq> b" shows "injective_path(linepath a b)"
+ assumes "a \<noteq> b"
+ shows "injective_path (linepath a b)"
proof -
{ fix x y :: "real"
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
hence "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps)
with assms have "x = y" by simp }
- thus ?thesis unfolding injective_path_def linepath_def by(auto simp add: algebra_simps) qed
+ thus ?thesis
+ unfolding injective_path_def linepath_def
+ by (auto simp add: algebra_simps)
+qed
-lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)
+lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)"
+ by(auto intro!: injective_imp_simple_path injective_path_linepath)
+
subsection {* Bounding a point away from a path. *}
lemma not_on_path_ball:
fixes g :: "real \<Rightarrow> 'a::heine_borel"
assumes "path g" "z \<notin> path_image g"
- shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof-
- obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y"
+ shows "\<exists>e > 0. ball z e \<inter> (path_image g) = {}"
+proof -
+ obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
using distance_attains_inf[OF _ path_image_nonempty, of g z]
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
- thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed
+ thus ?thesis
+ apply (rule_tac x="dist z a" in exI)
+ using assms(2)
+ apply (auto intro!: dist_pos_lt)
+ done
+qed
lemma not_on_path_cball:
fixes g :: "real \<Rightarrow> 'a::heine_borel"
assumes "path g" "z \<notin> path_image g"
- shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof-
- obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto
+ shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
+proof -
+ obtain e where "ball z e \<inter> path_image g = {}" "e>0"
+ using not_on_path_ball[OF assms] by auto
moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
- ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed
+ ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto
+qed
+
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
-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)"
+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)"
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
-lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s"
+lemma path_component_mem:
+ assumes "path_component s x y"
+ shows "x \<in> s" "y \<in> s"
using assms unfolding path_defs by auto
-lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x"
- unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms
- by(auto intro!:continuous_on_intros)
+lemma path_component_refl:
+ assumes "x \<in> s"
+ shows "path_component s x x"
+ unfolding path_defs
+ apply (rule_tac x="\<lambda>u. x" in exI)
+ using assms apply (auto intro!:continuous_on_intros) done
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
- by(auto intro!: path_component_mem path_component_refl)
+ by (auto intro!: path_component_mem path_component_refl)
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
- using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI)
- by auto
+ using assms
+ unfolding path_component_def
+ apply (erule exE)
+ apply (rule_tac x="reversepath g" in exI)
+ apply auto
+ done
-lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z"
- 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)
+lemma path_component_trans:
+ assumes "path_component s x y" "path_component s y z"
+ shows "path_component s x z"
+ using assms
+ unfolding path_component_def
+ apply -
+ apply (erule exE)+
+ apply (rule_tac x="g +++ ga" in exI)
+ apply (auto simp add: path_image_join)
+ done
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
unfolding path_component_def by auto
+
subsection {* Can also consider it as a set, as the name suggests. *}
-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 )}"
- apply(rule set_eqI) unfolding mem_Collect_eq unfolding path_component_def by auto
+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)}"
+ apply (rule set_eqI)
+ unfolding mem_Collect_eq
+ unfolding path_component_def
+ apply auto
+ done
lemma path_component_subset: "{y. path_component s x y} \<subseteq> s"
- apply(rule, rule path_component_mem(2)) by auto
+ apply (rule, rule path_component_mem(2))
+ apply auto
+ done
lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s"
- apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_Collect_eq
- apply(drule path_component_mem(1)) using path_component_refl by auto
+ apply rule
+ apply (drule equals0D[of _ x]) defer
+ apply (rule equals0I)
+ unfolding mem_Collect_eq
+ apply (drule path_component_mem(1))
+ using path_component_refl
+ apply auto
+ done
+
subsection {* Path connectedness of a space. *}
-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)"
+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)"
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
unfolding path_connected_def path_component_def by auto
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)"
- unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset)
- unfolding subset_eq mem_Collect_eq Ball_def by auto
+ unfolding path_connected_component
+ apply (rule, rule, rule, rule path_component_subset)
+ unfolding subset_eq mem_Collect_eq Ball_def
+ apply auto
+ done
+
subsection {* Some useful lemmas about path-connectedness. *}
lemma convex_imp_path_connected:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "path_connected s"
- unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI)
- unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto
+ unfolding path_connected_def
+ apply (rule, rule, rule_tac x = "linepath x y" in exI)
+ unfolding path_image_linepath
+ using assms [unfolded convex_contains_segment]
+ apply auto
+ done
-lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s"
- unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof-
- 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> {}"
+lemma path_connected_imp_connected:
+ assumes "path_connected s"
+ shows "connected s"
+ unfolding connected_def not_ex
+ apply (rule, rule, rule ccontr)
+ unfolding not_not
+ apply (erule conjE)+
+proof -
+ 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> {}"
then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
- have *:"connected {0..1::real}" by(auto intro!: convex_connected convex_real_interval)
- have "{0..1} \<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
- 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
- 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
+ have *: "connected {0..1::real}"
+ by (auto intro!: convex_connected convex_real_interval)
+ 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
+ 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
+ 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
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
- ultimately show False using *[unfolded connected_local not_ex,rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
+ 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}"]
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
- using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed
+ using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)]
+ by auto
+qed
lemma open_path_component:
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
- assumes "open s" shows "open {y. path_component s x y}"
- unfolding open_contains_ball proof
- fix y assume as:"y \<in> {y. path_component s x y}"
- hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_Collect_eq by auto
- then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
- 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-
- fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer
- apply(rule path_component_of_subset[OF e(2)]) apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using `e>0`
- using as by auto qed qed
+ assumes "open s"
+ shows "open {y. path_component s x y}"
+ unfolding open_contains_ball
+proof
+ fix y
+ assume as: "y \<in> {y. path_component s x y}"
+ hence "y \<in> s"
+ apply -
+ apply (rule path_component_mem(2))
+ unfolding mem_Collect_eq
+ apply auto
+ done
+ then obtain e where e:"e>0" "ball y e \<subseteq> s"
+ using assms[unfolded open_contains_ball] by auto
+ 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 -
+ fix z
+ assume "dist y z < e"
+ thus "path_component s x z"
+ apply (rule_tac path_component_trans[of _ _ y]) defer
+ apply (rule path_component_of_subset[OF e(2)])
+ apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
+ using `e>0` as
+ apply auto
+ done
+ qed
+qed
lemma open_non_path_component:
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
- assumes "open s" shows "open(s - {y. path_component s x y})"
- unfolding open_contains_ball proof
- fix y assume as:"y\<in>s - {y. path_component s x y}"
- then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
- 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)
- fix z assume "z\<in>ball y e" "\<not> z \<notin> {y. path_component s x y}"
- hence "y \<in> {y. path_component s x y}" unfolding not_not mem_Collect_eq using `e>0`
- apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)])
- apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto
- thus False using as by auto qed(insert e(2), auto) qed
+ assumes "open s"
+ shows "open(s - {y. path_component s x y})"
+ unfolding open_contains_ball
+proof
+ fix y
+ assume as: "y\<in>s - {y. path_component s x y}"
+ then obtain e where e:"e>0" "ball y e \<subseteq> s"
+ using assms [unfolded open_contains_ball] by auto
+ 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)
+ fix z
+ assume "z \<in> ball y e" "\<not> z \<notin> {y. path_component s x y}"
+ hence "y \<in> {y. path_component s x y}"
+ unfolding not_not mem_Collect_eq using `e>0`
+ apply -
+ apply (rule path_component_trans, assumption)
+ apply (rule path_component_of_subset[OF e(2)])
+ apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
+ apply auto
+ done
+ thus False using as by auto
+ qed (insert e(2), auto)
+qed
lemma connected_open_path_connected:
fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
- assumes "open s" "connected s" shows "path_connected s"
- unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule)
- fix x y assume "x \<in> s" "y \<in> s" show "y \<in> {y. path_component s x y}" proof(rule ccontr)
- assume "y \<notin> {y. path_component s x y}" moreover
- 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
- ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
- using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"] by auto
-qed qed
+ assumes "open s" "connected s"
+ shows "path_connected s"
+ unfolding path_connected_component_set
+proof (rule, rule, rule path_component_subset, rule)
+ fix x y
+ assume "x \<in> s" "y \<in> s"
+ show "y \<in> {y. path_component s x y}"
+ proof (rule ccontr)
+ assume "y \<notin> {y. path_component s x y}"
+ moreover
+ 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
+ ultimately
+ show False
+ using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
+ using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"]
+ by auto
+ qed
+qed
lemma path_connected_continuous_image:
- assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)"
- unfolding path_connected_def proof(rule,rule)
- fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s"
- then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
- guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] ..
+ assumes "continuous_on s f" "path_connected s"
+ shows "path_connected (f ` s)"
+ unfolding path_connected_def
+proof (rule, rule)
+ fix x' y'
+ assume "x' \<in> f ` s" "y' \<in> f ` s"
+ then obtain x y where xy: "x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
+ guess g using assms(2)[unfolded path_connected_def, rule_format, OF xy(1,2)] ..
thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
- unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs
- using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed
+ unfolding xy
+ apply (rule_tac x="f \<circ> g" in exI)
+ unfolding path_defs
+ using assms(1)
+ apply (auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"])
+ done
+qed
lemma homeomorphic_path_connectedness:
"s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
- unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule
- apply(drule_tac f=f in path_connected_continuous_image) prefer 3
- apply(drule_tac f=g in path_connected_continuous_image) by auto
+ unfolding homeomorphic_def homeomorphism_def
+ apply (erule exE|erule conjE)+
+ apply rule
+ apply (drule_tac f=f in path_connected_continuous_image) prefer 3
+ apply (drule_tac f=g in path_connected_continuous_image)
+ apply auto
+ done
lemma path_connected_empty: "path_connected {}"
unfolding path_connected_def by auto
@@ -493,19 +779,29 @@
apply (simp add: path_def continuous_on_const)
done
-lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
- shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)
- fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t"
+lemma path_connected_Un:
+ assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
+ shows "path_connected (s \<union> t)"
+ unfolding path_connected_component
+proof (rule, rule)
+ fix x y
+ assume as: "x \<in> s \<union> t" "y \<in> s \<union> t"
from assms(3) obtain z where "z \<in> s \<inter> t" by auto
- thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply-
- apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z])
- by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed
+ thus "path_component (s \<union> t) x y"
+ using as and assms(1-2)[unfolded path_connected_component]
+ apply -
+ apply (erule_tac[!] UnE)+
+ apply (rule_tac[2-3] path_component_trans[of _ _ z])
+ apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2])
+ done
+qed
lemma path_connected_UNION:
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
- assumes "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
+ and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
shows "path_connected (\<Union>i\<in>A. S i)"
-unfolding path_connected_component proof(clarify)
+ unfolding path_connected_component
+proof clarify
fix x i y j
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
hence "path_component (S i) x z" and "path_component (S j) z y"
@@ -516,25 +812,29 @@
by (rule path_component_trans)
qed
+
subsection {* sphere is path-connected. *}
lemma path_connected_punctured_universe:
assumes "2 \<le> DIM('a::euclidean_space)"
shows "path_connected((UNIV::'a::euclidean_space set) - {a})"
-proof-
+proof -
let ?A = "{x::'a. \<exists>i\<in>{..<DIM('a)}. x $$ i < a $$ i}"
let ?B = "{x::'a. \<exists>i\<in>{..<DIM('a)}. a $$ i < x $$ i}"
- have A: "path_connected ?A" unfolding Collect_bex_eq
+ have A: "path_connected ?A"
+ unfolding Collect_bex_eq
proof (rule path_connected_UNION)
- fix i assume "i \<in> {..<DIM('a)}"
+ fix i
+ assume "i \<in> {..<DIM('a)}"
thus "(\<chi>\<chi> i. a $$ i - 1) \<in> {x::'a. x $$ i < a $$ i}" by simp
show "path_connected {x. x $$ i < a $$ i}" unfolding euclidean_component_def
by (rule convex_imp_path_connected [OF convex_halfspace_lt])
qed
have B: "path_connected ?B" unfolding Collect_bex_eq
proof (rule path_connected_UNION)
- fix i assume "i \<in> {..<DIM('a)}"
+ fix i
+ assume "i \<in> {..<DIM('a)}"
thus "(\<chi>\<chi> i. a $$ i + 1) \<in> {x::'a. a $$ i < x $$ i}" by simp
show "path_connected {x. a $$ i < x $$ i}" unfolding euclidean_component_def
by (rule convex_imp_path_connected [OF convex_halfspace_gt])
@@ -556,21 +856,33 @@
assumes "2 \<le> DIM('a::euclidean_space)"
shows "path_connected {x::'a::euclidean_space. norm(x - a) = r}"
proof (rule linorder_cases [of r 0])
- assume "r < 0" hence "{x::'a. norm(x - a) = r} = {}" by auto
+ assume "r < 0"
+ hence "{x::'a. norm(x - a) = r} = {}" by auto
thus ?thesis using path_connected_empty by simp
next
assume "r = 0"
thus ?thesis using path_connected_singleton by simp
next
assume r: "0 < r"
- hence *:"{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" apply -apply(rule set_eqI,rule)
- 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)
- have **:"{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" apply(rule set_eqI,rule)
- unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto split:split_if_asm)
+ hence *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}"
+ apply -
+ apply (rule set_eqI, rule)
+ unfolding image_iff
+ apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI)
+ unfolding mem_Collect_eq norm_scaleR
+ apply (auto simp add: scaleR_right_diff_distrib)
+ done
+ have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})"
+ apply (rule set_eqI,rule)
+ unfolding image_iff
+ apply (rule_tac x=x in bexI)
+ unfolding mem_Collect_eq
+ apply (auto split:split_if_asm)
+ done
have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)"
unfolding field_divide_inverse by (simp add: continuous_on_intros)
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
- by(auto intro!: path_connected_continuous_image continuous_on_intros)
+ by (auto intro!: path_connected_continuous_image continuous_on_intros)
qed
lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x - a) = r}"