author | blanchet |
Fri, 01 Aug 2014 14:43:57 +0200 | |
changeset 57743 | 0af2d5dfb0ac |
parent 56371 | fb9ae0727548 |
child 58877 | 262572d90bc6 |
permissions | -rw-r--r-- |
41959 | 1 |
(* 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 path-connected sets *} |
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theory Path_Connected |
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f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
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imports Convex_Euclidean_Space |
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begin |
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subsection {* Paths. *} |
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definition path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "path g \<longleftrightarrow> continuous_on {0..1} g" |
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|
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definition pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathstart g = g 0" |
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definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathfinish g = g 1" |
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definition path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set" |
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where "path_image g = g ` {0 .. 1}" |
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definition reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
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where "reversepath g = (\<lambda>x. g(1 - x))" |
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definition joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a" |
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(infixr "+++" 75) |
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where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))" |
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definition simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "simple_path g \<longleftrightarrow> |
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(\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)" |
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|
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definition injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
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where "injective_path g \<longleftrightarrow> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)" |
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subsection {* Some lemmas about these concepts. *} |
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lemma injective_imp_simple_path: "injective_path g \<Longrightarrow> simple_path g" |
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unfolding injective_path_def simple_path_def |
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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 box_eq_empty |
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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 |
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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 |
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by auto |
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lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)" |
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unfolding path_def path_image_def |
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apply (erule connected_continuous_image) |
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apply (rule convex_connected, rule convex_real_interval) |
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done |
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lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)" |
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unfolding path_def path_image_def |
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apply (erule compact_continuous_image) |
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apply (rule compact_Icc) |
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done |
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lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g" |
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unfolding reversepath_def |
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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 |
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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 |
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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 |
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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 |
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by auto |
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lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g" |
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proof - |
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have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g" |
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unfolding path_image_def subset_eq reversepath_def Ball_def image_iff |
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apply rule |
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apply rule |
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apply (erule bexE) |
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apply (rule_tac x="1 - xa" in bexI) |
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apply auto |
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done |
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show ?thesis |
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using *[of g] *[of "reversepath g"] |
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unfolding reversepath_reversepath |
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by auto |
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qed |
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lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g" |
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proof - |
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have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)" |
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unfolding path_def reversepath_def |
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apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"]) |
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extend continuous_intros; remove continuous_on_intros and isCont_intros
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apply (intro continuous_intros) |
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apply (rule continuous_on_subset[of "{0..1}"]) |
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apply assumption |
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apply auto |
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done |
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show ?thesis |
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using *[of "reversepath g"] *[of g] |
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unfolding reversepath_reversepath |
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by (rule iffI) |
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qed |
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lemmas reversepath_simps = |
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path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
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lemma path_join[simp]: |
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assumes "pathfinish g1 = pathstart g2" |
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shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
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unfolding path_def pathfinish_def pathstart_def |
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move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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proof safe |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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assume cont: "continuous_on {0..1} (g1 +++ g2)" |
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move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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by (intro continuous_on_cong refl) (auto simp: joinpaths_def) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))" |
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using assms |
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by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def) |
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show "continuous_on {0..1} g1" and "continuous_on {0..1} g2" |
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51481
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move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
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unfolding g1 g2 |
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extend continuous_intros; remove continuous_on_intros and isCont_intros
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by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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parents:
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next |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2" |
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move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}" |
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by auto |
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{ |
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fix x :: real |
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assume "0 \<le> x" and "x \<le> 1" |
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then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}" |
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by (intro image_eqI[where x="x/2"]) auto |
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} |
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270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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note 1 = this |
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{ |
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fix x :: real |
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assume "0 \<le> x" and "x \<le> 1" |
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then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}" |
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by (intro image_eqI[where x="x/2 + 1/2"]) auto |
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} |
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51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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note 2 = this |
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show "continuous_on {0..1} (g1 +++ g2)" |
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using assms |
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unfolding joinpaths_def 01 |
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fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
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apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros) |
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apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2) |
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done |
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qed |
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lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2" |
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unfolding path_image_def joinpaths_def |
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by auto |
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lemma subset_path_image_join: |
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assumes "path_image g1 \<subseteq> s" |
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and "path_image g2 \<subseteq> s" |
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shows "path_image (g1 +++ g2) \<subseteq> s" |
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using path_image_join_subset[of g1 g2] and assms |
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by auto |
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lemma path_image_join: |
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51478
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move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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177 |
assumes "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 |
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apply (rule path_image_join_subset) |
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apply rule |
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unfolding Un_iff |
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proof (erule disjE) |
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fix x |
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assume "x \<in> path_image g1" |
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then obtain y where y: "y \<in> {0..1}" "x = g1 y" |
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unfolding path_image_def image_iff by auto |
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then show "x \<in> path_image (g1 +++ g2)" |
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unfolding joinpaths_def path_image_def image_iff |
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apply (rule_tac x="(1/2) *\<^sub>R y" in bexI) |
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apply auto |
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done |
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next |
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fix x |
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assume "x \<in> path_image g2" |
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then obtain y where y: "y \<in> {0..1}" "x = g2 y" |
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unfolding path_image_def image_iff by auto |
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then show "x \<in> path_image (g1 +++ g2)" |
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unfolding joinpaths_def path_image_def image_iff |
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53640 | 200 |
apply (rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
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using assms(1)[unfolded pathfinish_def pathstart_def] |
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apply (auto simp add: add_divide_distrib) |
49653 | 203 |
done |
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qed |
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lemma not_in_path_image_join: |
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assumes "x \<notin> path_image g1" |
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and "x \<notin> path_image g2" |
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shows "x \<notin> path_image (g1 +++ g2)" |
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using assms and path_image_join_subset[of g1 g2] |
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by auto |
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lemma simple_path_reversepath: |
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assumes "simple_path g" |
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shows "simple_path (reversepath g)" |
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using assms |
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unfolding simple_path_def reversepath_def |
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apply - |
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apply (rule ballI)+ |
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apply (erule_tac x="1-x" in ballE) |
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apply (erule_tac x="1-y" in ballE) |
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apply auto |
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done |
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lemma simple_path_join_loop: |
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53640 | 226 |
assumes "injective_path g1" |
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and "injective_path g2" |
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and "pathfinish g2 = pathstart g1" |
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and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
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shows "simple_path (g1 +++ g2)" |
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unfolding simple_path_def |
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proof (intro ballI impI) |
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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 |
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assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y" |
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show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" |
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proof (cases "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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then have "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" |
53640 | 241 |
using xy(3) |
242 |
unfolding joinpaths_def |
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by auto |
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moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" |
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using xy(1,2) as |
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36583 | 246 |
by auto |
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ultimately show ?thesis |
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using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] |
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by auto |
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next |
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assume as: "x > 1 / 2" "y > 1 / 2" |
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then have "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" |
53640 | 253 |
using xy(3) |
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unfolding joinpaths_def |
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255 |
by auto |
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moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" |
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using xy(1,2) as |
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258 |
by auto |
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259 |
ultimately show ?thesis |
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using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto |
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49653 | 261 |
next |
53640 | 262 |
assume as: "x \<le> 1 / 2" "y > 1 / 2" |
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then have "?g x \<in> path_image g1" "?g y \<in> path_image g2" |
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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" |
267 |
using as(2) |
|
268 |
unfolding pathstart_def joinpaths_def |
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36583 | 269 |
using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2) |
270 |
by (auto simp add: field_simps) |
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53640 | 271 |
ultimately have *: "?g x = pathstart g1" |
272 |
using assms(4) |
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273 |
unfolding xy(3) |
|
274 |
by auto |
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then have "x = 0" |
|
276 |
unfolding pathstart_def joinpaths_def |
|
277 |
using as(1) and xy(1) |
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278 |
using inj(1)[of "2 *\<^sub>R x" 0] |
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279 |
by auto |
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280 |
moreover have "y = 1" |
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281 |
using * |
|
282 |
unfolding xy(3) assms(3)[symmetric] |
|
283 |
unfolding joinpaths_def pathfinish_def |
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284 |
using as(2) and xy(2) |
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285 |
using inj(2)[of "2 *\<^sub>R y - 1" 1] |
|
286 |
by auto |
|
287 |
ultimately show ?thesis |
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288 |
by auto |
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49653 | 289 |
next |
290 |
assume as: "x > 1 / 2" "y \<le> 1 / 2" |
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53640 | 291 |
then have "?g x \<in> path_image g2" and "?g y \<in> path_image g1" |
49653 | 292 |
unfolding path_image_def joinpaths_def |
36583 | 293 |
using xy(1,2) by auto |
53640 | 294 |
moreover have "?g x \<noteq> pathstart g2" |
295 |
using as(1) |
|
296 |
unfolding pathstart_def joinpaths_def |
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36583 | 297 |
using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1) |
298 |
by (auto simp add: field_simps) |
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53640 | 299 |
ultimately have *: "?g y = pathstart g1" |
300 |
using assms(4) |
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301 |
unfolding xy(3) |
|
302 |
by auto |
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303 |
then have "y = 0" |
|
304 |
unfolding pathstart_def joinpaths_def |
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305 |
using as(2) and xy(2) |
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306 |
using inj(1)[of "2 *\<^sub>R y" 0] |
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307 |
by auto |
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308 |
moreover have "x = 1" |
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309 |
using * |
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310 |
unfolding xy(3)[symmetric] assms(3)[symmetric] |
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36583 | 311 |
unfolding joinpaths_def pathfinish_def using as(1) and xy(1) |
53640 | 312 |
using inj(2)[of "2 *\<^sub>R x - 1" 1] |
313 |
by auto |
|
314 |
ultimately show ?thesis |
|
315 |
by auto |
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49653 | 316 |
qed |
317 |
qed |
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36583 | 318 |
|
319 |
lemma injective_path_join: |
|
53640 | 320 |
assumes "injective_path g1" |
321 |
and "injective_path g2" |
|
322 |
and "pathfinish g1 = pathstart g2" |
|
323 |
and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" |
|
324 |
shows "injective_path (g1 +++ g2)" |
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49653 | 325 |
unfolding injective_path_def |
326 |
proof (rule, rule, rule) |
|
327 |
let ?g = "g1 +++ g2" |
|
36583 | 328 |
note inj = assms(1,2)[unfolded injective_path_def, rule_format] |
49653 | 329 |
fix x y |
330 |
assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y" |
|
331 |
show "x = y" |
|
332 |
proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le) |
|
53640 | 333 |
assume "x \<le> 1 / 2" and "y \<le> 1 / 2" |
334 |
then show ?thesis |
|
335 |
using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy |
|
36583 | 336 |
unfolding joinpaths_def by auto |
49653 | 337 |
next |
53640 | 338 |
assume "x > 1 / 2" and "y > 1 / 2" |
339 |
then show ?thesis |
|
340 |
using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy |
|
36583 | 341 |
unfolding joinpaths_def by auto |
49653 | 342 |
next |
343 |
assume as: "x \<le> 1 / 2" "y > 1 / 2" |
|
53640 | 344 |
then have "?g x \<in> path_image g1" and "?g y \<in> path_image g2" |
49653 | 345 |
unfolding path_image_def joinpaths_def |
53640 | 346 |
using xy(1,2) |
347 |
by auto |
|
348 |
then have "?g x = pathfinish g1" and "?g y = pathstart g2" |
|
349 |
using assms(4) |
|
350 |
unfolding assms(3) xy(3) |
|
351 |
by auto |
|
49654 | 352 |
then show ?thesis |
49653 | 353 |
using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2) |
36583 | 354 |
unfolding pathstart_def pathfinish_def joinpaths_def |
355 |
by auto |
|
49653 | 356 |
next |
53640 | 357 |
assume as:"x > 1 / 2" "y \<le> 1 / 2" |
358 |
then have "?g x \<in> path_image g2" and "?g y \<in> path_image g1" |
|
49653 | 359 |
unfolding path_image_def joinpaths_def |
53640 | 360 |
using xy(1,2) |
361 |
by auto |
|
362 |
then have "?g x = pathstart g2" and "?g y = pathfinish g1" |
|
363 |
using assms(4) |
|
364 |
unfolding assms(3) xy(3) |
|
365 |
by auto |
|
49654 | 366 |
then show ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2) |
36583 | 367 |
unfolding pathstart_def pathfinish_def joinpaths_def |
49653 | 368 |
by auto |
369 |
qed |
|
370 |
qed |
|
36583 | 371 |
|
372 |
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join |
|
53640 | 373 |
|
49653 | 374 |
|
53640 | 375 |
subsection {* Reparametrizing a closed curve to start at some chosen point *} |
36583 | 376 |
|
53640 | 377 |
definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
378 |
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))" |
|
36583 | 379 |
|
53640 | 380 |
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a" |
36583 | 381 |
unfolding pathstart_def shiftpath_def by auto |
382 |
||
49653 | 383 |
lemma pathfinish_shiftpath: |
53640 | 384 |
assumes "0 \<le> a" |
385 |
and "pathfinish g = pathstart g" |
|
386 |
shows "pathfinish (shiftpath a g) = g a" |
|
387 |
using assms |
|
388 |
unfolding pathstart_def pathfinish_def shiftpath_def |
|
36583 | 389 |
by auto |
390 |
||
391 |
lemma endpoints_shiftpath: |
|
53640 | 392 |
assumes "pathfinish g = pathstart g" |
393 |
and "a \<in> {0 .. 1}" |
|
394 |
shows "pathfinish (shiftpath a g) = g a" |
|
395 |
and "pathstart (shiftpath a g) = g a" |
|
396 |
using assms |
|
397 |
by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) |
|
36583 | 398 |
|
399 |
lemma closed_shiftpath: |
|
53640 | 400 |
assumes "pathfinish g = pathstart g" |
401 |
and "a \<in> {0..1}" |
|
402 |
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" |
|
403 |
using endpoints_shiftpath[OF assms] |
|
404 |
by auto |
|
36583 | 405 |
|
406 |
lemma path_shiftpath: |
|
53640 | 407 |
assumes "path g" |
408 |
and "pathfinish g = pathstart g" |
|
409 |
and "a \<in> {0..1}" |
|
410 |
shows "path (shiftpath a g)" |
|
49653 | 411 |
proof - |
53640 | 412 |
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" |
413 |
using assms(3) by auto |
|
49653 | 414 |
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
53640 | 415 |
using assms(2)[unfolded pathfinish_def pathstart_def] |
416 |
by auto |
|
49653 | 417 |
show ?thesis |
418 |
unfolding path_def shiftpath_def * |
|
419 |
apply (rule continuous_on_union) |
|
420 |
apply (rule closed_real_atLeastAtMost)+ |
|
53640 | 421 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) |
422 |
prefer 3 |
|
423 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) |
|
424 |
defer |
|
425 |
prefer 3 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
426 |
apply (rule continuous_intros)+ |
53640 | 427 |
prefer 2 |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
428 |
apply (rule continuous_intros)+ |
49653 | 429 |
apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) |
430 |
using assms(3) and ** |
|
53640 | 431 |
apply auto |
432 |
apply (auto simp add: field_simps) |
|
49653 | 433 |
done |
434 |
qed |
|
36583 | 435 |
|
49653 | 436 |
lemma shiftpath_shiftpath: |
53640 | 437 |
assumes "pathfinish g = pathstart g" |
438 |
and "a \<in> {0..1}" |
|
439 |
and "x \<in> {0..1}" |
|
36583 | 440 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
53640 | 441 |
using assms |
442 |
unfolding pathfinish_def pathstart_def shiftpath_def |
|
443 |
by auto |
|
36583 | 444 |
|
445 |
lemma path_image_shiftpath: |
|
53640 | 446 |
assumes "a \<in> {0..1}" |
447 |
and "pathfinish g = pathstart g" |
|
448 |
shows "path_image (shiftpath a g) = path_image g" |
|
49653 | 449 |
proof - |
450 |
{ fix x |
|
53640 | 451 |
assume as: "g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)" |
49654 | 452 |
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" |
49653 | 453 |
proof (cases "a \<le> x") |
454 |
case False |
|
49654 | 455 |
then show ?thesis |
49653 | 456 |
apply (rule_tac x="1 + x - a" in bexI) |
36583 | 457 |
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) |
49653 | 458 |
apply (auto simp add: field_simps atomize_not) |
459 |
done |
|
460 |
next |
|
461 |
case True |
|
53640 | 462 |
then show ?thesis |
463 |
using as(1-2) and assms(1) |
|
464 |
apply (rule_tac x="x - a" in bexI) |
|
465 |
apply (auto simp add: field_simps) |
|
466 |
done |
|
49653 | 467 |
qed |
468 |
} |
|
49654 | 469 |
then show ?thesis |
53640 | 470 |
using assms |
471 |
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
|
472 |
by (auto simp add: image_iff) |
|
49653 | 473 |
qed |
474 |
||
36583 | 475 |
|
53640 | 476 |
subsection {* Special case of straight-line paths *} |
36583 | 477 |
|
49653 | 478 |
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" |
479 |
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
|
36583 | 480 |
|
53640 | 481 |
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" |
482 |
unfolding pathstart_def linepath_def |
|
483 |
by auto |
|
36583 | 484 |
|
53640 | 485 |
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" |
486 |
unfolding pathfinish_def linepath_def |
|
487 |
by auto |
|
36583 | 488 |
|
489 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
|
53640 | 490 |
unfolding linepath_def |
491 |
by (intro continuous_intros) |
|
36583 | 492 |
|
493 |
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" |
|
53640 | 494 |
using continuous_linepath_at |
495 |
by (auto intro!: continuous_at_imp_continuous_on) |
|
36583 | 496 |
|
53640 | 497 |
lemma path_linepath[intro]: "path (linepath a b)" |
498 |
unfolding path_def |
|
499 |
by (rule continuous_on_linepath) |
|
36583 | 500 |
|
53640 | 501 |
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" |
49653 | 502 |
unfolding path_image_def segment linepath_def |
53640 | 503 |
apply (rule set_eqI) |
504 |
apply rule |
|
505 |
defer |
|
49653 | 506 |
unfolding mem_Collect_eq image_iff |
53640 | 507 |
apply (erule exE) |
508 |
apply (rule_tac x="u *\<^sub>R 1" in bexI) |
|
49653 | 509 |
apply auto |
510 |
done |
|
511 |
||
53640 | 512 |
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" |
49653 | 513 |
unfolding reversepath_def linepath_def |
36583 | 514 |
by auto |
515 |
||
516 |
lemma injective_path_linepath: |
|
49653 | 517 |
assumes "a \<noteq> b" |
518 |
shows "injective_path (linepath a b)" |
|
36583 | 519 |
proof - |
53640 | 520 |
{ |
521 |
fix x y :: "real" |
|
36583 | 522 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
53640 | 523 |
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" |
524 |
by (simp add: algebra_simps) |
|
525 |
with assms have "x = y" |
|
526 |
by simp |
|
527 |
} |
|
49654 | 528 |
then show ?thesis |
49653 | 529 |
unfolding injective_path_def linepath_def |
530 |
by (auto simp add: algebra_simps) |
|
531 |
qed |
|
36583 | 532 |
|
53640 | 533 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)" |
534 |
by (auto intro!: injective_imp_simple_path injective_path_linepath) |
|
49653 | 535 |
|
36583 | 536 |
|
53640 | 537 |
subsection {* Bounding a point away from a path *} |
36583 | 538 |
|
539 |
lemma not_on_path_ball: |
|
540 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 541 |
assumes "path g" |
542 |
and "z \<notin> path_image g" |
|
543 |
shows "\<exists>e > 0. ball z e \<inter> path_image g = {}" |
|
49653 | 544 |
proof - |
545 |
obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y" |
|
36583 | 546 |
using distance_attains_inf[OF _ path_image_nonempty, of g z] |
547 |
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto |
|
49654 | 548 |
then show ?thesis |
49653 | 549 |
apply (rule_tac x="dist z a" in exI) |
550 |
using assms(2) |
|
551 |
apply (auto intro!: dist_pos_lt) |
|
552 |
done |
|
553 |
qed |
|
36583 | 554 |
|
555 |
lemma not_on_path_cball: |
|
556 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 557 |
assumes "path g" |
558 |
and "z \<notin> path_image g" |
|
49653 | 559 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" |
560 |
proof - |
|
53640 | 561 |
obtain e where "ball z e \<inter> path_image g = {}" "e > 0" |
49653 | 562 |
using not_on_path_ball[OF assms] by auto |
53640 | 563 |
moreover have "cball z (e/2) \<subseteq> ball z e" |
564 |
using `e > 0` by auto |
|
565 |
ultimately show ?thesis |
|
566 |
apply (rule_tac x="e/2" in exI) |
|
567 |
apply auto |
|
568 |
done |
|
49653 | 569 |
qed |
570 |
||
36583 | 571 |
|
53640 | 572 |
subsection {* Path component, considered as a "joinability" relation (from Tom Hales) *} |
36583 | 573 |
|
49653 | 574 |
definition "path_component s x y \<longleftrightarrow> |
575 |
(\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
36583 | 576 |
|
53640 | 577 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
36583 | 578 |
|
49653 | 579 |
lemma path_component_mem: |
580 |
assumes "path_component s x y" |
|
53640 | 581 |
shows "x \<in> s" and "y \<in> s" |
582 |
using assms |
|
583 |
unfolding path_defs |
|
584 |
by auto |
|
36583 | 585 |
|
49653 | 586 |
lemma path_component_refl: |
587 |
assumes "x \<in> s" |
|
588 |
shows "path_component s x x" |
|
589 |
unfolding path_defs |
|
590 |
apply (rule_tac x="\<lambda>u. x" in exI) |
|
53640 | 591 |
using assms |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
592 |
apply (auto intro!: continuous_intros) |
53640 | 593 |
done |
36583 | 594 |
|
595 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
49653 | 596 |
by (auto intro!: path_component_mem path_component_refl) |
36583 | 597 |
|
598 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
49653 | 599 |
using assms |
600 |
unfolding path_component_def |
|
601 |
apply (erule exE) |
|
602 |
apply (rule_tac x="reversepath g" in exI) |
|
603 |
apply auto |
|
604 |
done |
|
36583 | 605 |
|
49653 | 606 |
lemma path_component_trans: |
53640 | 607 |
assumes "path_component s x y" |
608 |
and "path_component s y z" |
|
49653 | 609 |
shows "path_component s x z" |
610 |
using assms |
|
611 |
unfolding path_component_def |
|
53640 | 612 |
apply (elim exE) |
49653 | 613 |
apply (rule_tac x="g +++ ga" in exI) |
614 |
apply (auto simp add: path_image_join) |
|
615 |
done |
|
36583 | 616 |
|
53640 | 617 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
36583 | 618 |
unfolding path_component_def by auto |
619 |
||
49653 | 620 |
|
53640 | 621 |
text {* Can also consider it as a set, as the name suggests. *} |
36583 | 622 |
|
49653 | 623 |
lemma path_component_set: |
624 |
"{y. path_component s x y} = |
|
625 |
{y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}" |
|
626 |
apply (rule set_eqI) |
|
627 |
unfolding mem_Collect_eq |
|
628 |
unfolding path_component_def |
|
629 |
apply auto |
|
630 |
done |
|
36583 | 631 |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
41959
diff
changeset
|
632 |
lemma path_component_subset: "{y. path_component s x y} \<subseteq> s" |
53640 | 633 |
apply rule |
634 |
apply (rule path_component_mem(2)) |
|
49653 | 635 |
apply auto |
636 |
done |
|
36583 | 637 |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
41959
diff
changeset
|
638 |
lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s" |
49653 | 639 |
apply rule |
53640 | 640 |
apply (drule equals0D[of _ x]) |
641 |
defer |
|
49653 | 642 |
apply (rule equals0I) |
643 |
unfolding mem_Collect_eq |
|
644 |
apply (drule path_component_mem(1)) |
|
645 |
using path_component_refl |
|
646 |
apply auto |
|
647 |
done |
|
648 |
||
36583 | 649 |
|
53640 | 650 |
subsection {* Path connectedness of a space *} |
36583 | 651 |
|
49653 | 652 |
definition "path_connected s \<longleftrightarrow> |
53640 | 653 |
(\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
36583 | 654 |
|
655 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
|
656 |
unfolding path_connected_def path_component_def by auto |
|
657 |
||
53640 | 658 |
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)" |
49653 | 659 |
unfolding path_connected_component |
53640 | 660 |
apply rule |
661 |
apply rule |
|
662 |
apply rule |
|
663 |
apply (rule path_component_subset) |
|
49653 | 664 |
unfolding subset_eq mem_Collect_eq Ball_def |
665 |
apply auto |
|
666 |
done |
|
667 |
||
36583 | 668 |
|
53640 | 669 |
subsection {* Some useful lemmas about path-connectedness *} |
36583 | 670 |
|
671 |
lemma convex_imp_path_connected: |
|
672 |
fixes s :: "'a::real_normed_vector set" |
|
53640 | 673 |
assumes "convex s" |
674 |
shows "path_connected s" |
|
49653 | 675 |
unfolding path_connected_def |
53640 | 676 |
apply rule |
677 |
apply rule |
|
678 |
apply (rule_tac x = "linepath x y" in exI) |
|
49653 | 679 |
unfolding path_image_linepath |
680 |
using assms [unfolded convex_contains_segment] |
|
681 |
apply auto |
|
682 |
done |
|
36583 | 683 |
|
49653 | 684 |
lemma path_connected_imp_connected: |
685 |
assumes "path_connected s" |
|
686 |
shows "connected s" |
|
687 |
unfolding connected_def not_ex |
|
53640 | 688 |
apply rule |
689 |
apply rule |
|
690 |
apply (rule ccontr) |
|
49653 | 691 |
unfolding not_not |
53640 | 692 |
apply (elim conjE) |
49653 | 693 |
proof - |
694 |
fix e1 e2 |
|
695 |
assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}" |
|
53640 | 696 |
then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s" |
697 |
by auto |
|
698 |
then obtain g where g: "path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" |
|
36583 | 699 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
49653 | 700 |
have *: "connected {0..1::real}" |
701 |
by (auto intro!: convex_connected convex_real_interval) |
|
702 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" |
|
703 |
using as(3) g(2)[unfolded path_defs] by blast |
|
704 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" |
|
53640 | 705 |
using as(4) g(2)[unfolded path_defs] |
706 |
unfolding subset_eq |
|
707 |
by auto |
|
49653 | 708 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" |
53640 | 709 |
using g(3,4)[unfolded path_defs] |
710 |
using obt |
|
36583 | 711 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
49653 | 712 |
ultimately show False |
53640 | 713 |
using *[unfolded connected_local not_ex, rule_format, |
714 |
of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"] |
|
36583 | 715 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)] |
49653 | 716 |
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] |
717 |
by auto |
|
718 |
qed |
|
36583 | 719 |
|
720 |
lemma open_path_component: |
|
53593 | 721 |
fixes s :: "'a::real_normed_vector set" |
49653 | 722 |
assumes "open s" |
723 |
shows "open {y. path_component s x y}" |
|
724 |
unfolding open_contains_ball |
|
725 |
proof |
|
726 |
fix y |
|
727 |
assume as: "y \<in> {y. path_component s x y}" |
|
49654 | 728 |
then have "y \<in> s" |
49653 | 729 |
apply - |
730 |
apply (rule path_component_mem(2)) |
|
731 |
unfolding mem_Collect_eq |
|
732 |
apply auto |
|
733 |
done |
|
53640 | 734 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
735 |
using assms[unfolded open_contains_ball] |
|
736 |
by auto |
|
49653 | 737 |
show "\<exists>e > 0. ball y e \<subseteq> {y. path_component s x y}" |
738 |
apply (rule_tac x=e in exI) |
|
53640 | 739 |
apply (rule,rule `e>0`) |
740 |
apply rule |
|
49653 | 741 |
unfolding mem_ball mem_Collect_eq |
742 |
proof - |
|
743 |
fix z |
|
744 |
assume "dist y z < e" |
|
49654 | 745 |
then show "path_component s x z" |
53640 | 746 |
apply (rule_tac path_component_trans[of _ _ y]) |
747 |
defer |
|
49653 | 748 |
apply (rule path_component_of_subset[OF e(2)]) |
749 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
53640 | 750 |
using `e > 0` as |
49653 | 751 |
apply auto |
752 |
done |
|
753 |
qed |
|
754 |
qed |
|
36583 | 755 |
|
756 |
lemma open_non_path_component: |
|
53593 | 757 |
fixes s :: "'a::real_normed_vector set" |
49653 | 758 |
assumes "open s" |
53640 | 759 |
shows "open (s - {y. path_component s x y})" |
49653 | 760 |
unfolding open_contains_ball |
761 |
proof |
|
762 |
fix y |
|
53640 | 763 |
assume as: "y \<in> s - {y. path_component s x y}" |
764 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
|
765 |
using assms [unfolded open_contains_ball] |
|
766 |
by auto |
|
49653 | 767 |
show "\<exists>e>0. ball y e \<subseteq> s - {y. path_component s x y}" |
768 |
apply (rule_tac x=e in exI) |
|
53640 | 769 |
apply rule |
770 |
apply (rule `e>0`) |
|
771 |
apply rule |
|
772 |
apply rule |
|
773 |
defer |
|
49653 | 774 |
proof (rule ccontr) |
775 |
fix z |
|
776 |
assume "z \<in> ball y e" "\<not> z \<notin> {y. path_component s x y}" |
|
49654 | 777 |
then have "y \<in> {y. path_component s x y}" |
49653 | 778 |
unfolding not_not mem_Collect_eq using `e>0` |
779 |
apply - |
|
780 |
apply (rule path_component_trans, assumption) |
|
781 |
apply (rule path_component_of_subset[OF e(2)]) |
|
782 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
783 |
apply auto |
|
784 |
done |
|
53640 | 785 |
then show False |
786 |
using as by auto |
|
49653 | 787 |
qed (insert e(2), auto) |
788 |
qed |
|
36583 | 789 |
|
790 |
lemma connected_open_path_connected: |
|
53593 | 791 |
fixes s :: "'a::real_normed_vector set" |
53640 | 792 |
assumes "open s" |
793 |
and "connected s" |
|
49653 | 794 |
shows "path_connected s" |
795 |
unfolding path_connected_component_set |
|
796 |
proof (rule, rule, rule path_component_subset, rule) |
|
797 |
fix x y |
|
53640 | 798 |
assume "x \<in> s" and "y \<in> s" |
49653 | 799 |
show "y \<in> {y. path_component s x y}" |
800 |
proof (rule ccontr) |
|
53640 | 801 |
assume "\<not> ?thesis" |
802 |
moreover have "{y. path_component s x y} \<inter> s \<noteq> {}" |
|
803 |
using `x \<in> s` path_component_eq_empty path_component_subset[of s x] |
|
804 |
by auto |
|
49653 | 805 |
ultimately |
806 |
show False |
|
53640 | 807 |
using `y \<in> s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
808 |
using assms(2)[unfolded connected_def not_ex, rule_format, |
|
809 |
of"{y. path_component s x y}" "s - {y. path_component s x y}"] |
|
49653 | 810 |
by auto |
811 |
qed |
|
812 |
qed |
|
36583 | 813 |
|
814 |
lemma path_connected_continuous_image: |
|
53640 | 815 |
assumes "continuous_on s f" |
816 |
and "path_connected s" |
|
49653 | 817 |
shows "path_connected (f ` s)" |
818 |
unfolding path_connected_def |
|
819 |
proof (rule, rule) |
|
820 |
fix x' y' |
|
821 |
assume "x' \<in> f ` s" "y' \<in> f ` s" |
|
53640 | 822 |
then obtain x y where x: "x \<in> s" and y: "y \<in> s" and x': "x' = f x" and y': "y' = f y" |
823 |
by auto |
|
824 |
from x y obtain g where "path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y" |
|
825 |
using assms(2)[unfolded path_connected_def] by fast |
|
49654 | 826 |
then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" |
53640 | 827 |
unfolding x' y' |
49653 | 828 |
apply (rule_tac x="f \<circ> g" in exI) |
829 |
unfolding path_defs |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
830 |
apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
831 |
apply auto |
49653 | 832 |
done |
833 |
qed |
|
36583 | 834 |
|
835 |
lemma homeomorphic_path_connectedness: |
|
53640 | 836 |
"s homeomorphic t \<Longrightarrow> path_connected s \<longleftrightarrow> path_connected t" |
49653 | 837 |
unfolding homeomorphic_def homeomorphism_def |
53640 | 838 |
apply (erule exE|erule conjE)+ |
49653 | 839 |
apply rule |
53640 | 840 |
apply (drule_tac f=f in path_connected_continuous_image) |
841 |
prefer 3 |
|
49653 | 842 |
apply (drule_tac f=g in path_connected_continuous_image) |
843 |
apply auto |
|
844 |
done |
|
36583 | 845 |
|
846 |
lemma path_connected_empty: "path_connected {}" |
|
847 |
unfolding path_connected_def by auto |
|
848 |
||
849 |
lemma path_connected_singleton: "path_connected {a}" |
|
850 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
|
53640 | 851 |
apply clarify |
852 |
apply (rule_tac x="\<lambda>x. a" in exI) |
|
853 |
apply (simp add: image_constant_conv) |
|
36583 | 854 |
apply (simp add: path_def continuous_on_const) |
855 |
done |
|
856 |
||
49653 | 857 |
lemma path_connected_Un: |
53640 | 858 |
assumes "path_connected s" |
859 |
and "path_connected t" |
|
860 |
and "s \<inter> t \<noteq> {}" |
|
49653 | 861 |
shows "path_connected (s \<union> t)" |
862 |
unfolding path_connected_component |
|
863 |
proof (rule, rule) |
|
864 |
fix x y |
|
865 |
assume as: "x \<in> s \<union> t" "y \<in> s \<union> t" |
|
53640 | 866 |
from assms(3) obtain z where "z \<in> s \<inter> t" |
867 |
by auto |
|
49654 | 868 |
then show "path_component (s \<union> t) x y" |
49653 | 869 |
using as and assms(1-2)[unfolded path_connected_component] |
53640 | 870 |
apply - |
49653 | 871 |
apply (erule_tac[!] UnE)+ |
872 |
apply (rule_tac[2-3] path_component_trans[of _ _ z]) |
|
873 |
apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) |
|
874 |
done |
|
875 |
qed |
|
36583 | 876 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
877 |
lemma path_connected_UNION: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
878 |
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" |
49653 | 879 |
and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
880 |
shows "path_connected (\<Union>i\<in>A. S i)" |
49653 | 881 |
unfolding path_connected_component |
882 |
proof clarify |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
883 |
fix x i y j |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
884 |
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" |
49654 | 885 |
then have "path_component (S i) x z" and "path_component (S j) z y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
886 |
using assms by (simp_all add: path_connected_component) |
49654 | 887 |
then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y" |
48125
602dc0215954
tuned proofs -- prefer direct "rotated" instead of old-style COMP;
wenzelm
parents:
44647
diff
changeset
|
888 |
using *(1,3) by (auto elim!: path_component_of_subset [rotated]) |
49654 | 889 |
then show "path_component (\<Union>i\<in>A. S i) x y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
890 |
by (rule path_component_trans) |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
891 |
qed |
36583 | 892 |
|
49653 | 893 |
|
53640 | 894 |
subsection {* Sphere is path-connected *} |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
895 |
|
36583 | 896 |
lemma path_connected_punctured_universe: |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
897 |
assumes "2 \<le> DIM('a::euclidean_space)" |
53640 | 898 |
shows "path_connected ((UNIV::'a set) - {a})" |
49653 | 899 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
900 |
let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
901 |
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}" |
36583 | 902 |
|
49653 | 903 |
have A: "path_connected ?A" |
904 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
905 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
906 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
907 |
assume "i \<in> Basis" |
53640 | 908 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" |
909 |
by simp |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
910 |
show "path_connected {x. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
911 |
using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
912 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
913 |
qed |
53640 | 914 |
have B: "path_connected ?B" |
915 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
916 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
917 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
918 |
assume "i \<in> Basis" |
53640 | 919 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" |
920 |
by simp |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
921 |
show "path_connected {x. a \<bullet> i < x \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
922 |
using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
923 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
924 |
qed |
53640 | 925 |
obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)" |
926 |
using ex_card[OF assms] |
|
927 |
by auto |
|
928 |
then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
929 |
unfolding card_Suc_eq by auto |
53640 | 930 |
then have "a + b0 - b1 \<in> ?A \<inter> ?B" |
931 |
by (auto simp: inner_simps inner_Basis) |
|
932 |
then have "?A \<inter> ?B \<noteq> {}" |
|
933 |
by fast |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
934 |
with A B have "path_connected (?A \<union> ?B)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
935 |
by (rule path_connected_Un) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
936 |
also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
937 |
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
938 |
also have "\<dots> = {x. x \<noteq> a}" |
53640 | 939 |
unfolding euclidean_eq_iff [where 'a='a] |
940 |
by (simp add: Bex_def) |
|
941 |
also have "\<dots> = UNIV - {a}" |
|
942 |
by auto |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
943 |
finally show ?thesis . |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
944 |
qed |
36583 | 945 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
946 |
lemma path_connected_sphere: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
947 |
assumes "2 \<le> DIM('a::euclidean_space)" |
53640 | 948 |
shows "path_connected {x::'a. norm (x - a) = r}" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
949 |
proof (rule linorder_cases [of r 0]) |
49653 | 950 |
assume "r < 0" |
53640 | 951 |
then have "{x::'a. norm(x - a) = r} = {}" |
952 |
by auto |
|
953 |
then show ?thesis |
|
954 |
using path_connected_empty by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
955 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
956 |
assume "r = 0" |
53640 | 957 |
then show ?thesis |
958 |
using path_connected_singleton by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
959 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
960 |
assume r: "0 < r" |
53640 | 961 |
have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" |
962 |
apply (rule set_eqI) |
|
963 |
apply rule |
|
49653 | 964 |
unfolding image_iff |
965 |
apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) |
|
966 |
unfolding mem_Collect_eq norm_scaleR |
|
53640 | 967 |
using r |
49653 | 968 |
apply (auto simp add: scaleR_right_diff_distrib) |
969 |
done |
|
970 |
have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" |
|
53640 | 971 |
apply (rule set_eqI) |
972 |
apply rule |
|
49653 | 973 |
unfolding image_iff |
974 |
apply (rule_tac x=x in bexI) |
|
975 |
unfolding mem_Collect_eq |
|
53640 | 976 |
apply (auto split: split_if_asm) |
49653 | 977 |
done |
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44531
diff
changeset
|
978 |
have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)" |
53640 | 979 |
unfolding field_divide_inverse |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
980 |
by (simp add: continuous_intros) |
53640 | 981 |
then show ?thesis |
982 |
unfolding * ** |
|
983 |
using path_connected_punctured_universe[OF assms] |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
984 |
by (auto intro!: path_connected_continuous_image continuous_intros) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
985 |
qed |
36583 | 986 |
|
53640 | 987 |
lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm (x - a) = r}" |
988 |
using path_connected_sphere path_connected_imp_connected |
|
989 |
by auto |
|
36583 | 990 |
|
991 |
end |