--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy Mon Jan 07 14:57:45 2019 +0100
@@ -15,7 +15,10 @@
section \<open>Brouwer's Fixed Point Theorem\<close>
theory Brouwer_Fixpoint
-imports Path_Connected Homeomorphism
+ imports
+ Path_Connected
+ Homeomorphism
+ Continuous_Extension
begin
(* FIXME mv topology euclidean space *)
--- a/src/HOL/Analysis/Homeomorphism.thy Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Homeomorphism.thy Mon Jan 07 14:57:45 2019 +0100
@@ -5,7 +5,7 @@
section%important \<open>Homeomorphism Theorems\<close>
theory Homeomorphism
-imports Path_Connected
+imports Homotopy
begin
lemma%unimportant homeomorphic_spheres':
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Homotopy.thy Mon Jan 07 14:57:45 2019 +0100
@@ -0,0 +1,5159 @@
+(* Title: HOL/Analysis/Path_Connected.thy
+ Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
+*)
+
+section \<open>Homotopy of Maps\<close>
+
+theory Homotopy
+ imports Path_Connected Continuum_Not_Denumerable
+begin
+
+definition%important homotopic_with ::
+ "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
+where
+ "homotopic_with P X Y p q \<equiv>
+ (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
+ continuous_on ({0..1} \<times> X) h \<and>
+ h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+ (\<forall>x. h(0, x) = p x) \<and>
+ (\<forall>x. h(1, x) = q x) \<and>
+ (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+
+text\<open>\<open>p\<close>, \<open>q\<close> are functions \<open>X \<rightarrow> Y\<close>, and the property \<open>P\<close> restricts all intermediate maps.
+We often just want to require that \<open>P\<close> fixes some subset, but to include the case of a loop homotopy,
+it is convenient to have a general property \<open>P\<close>.\<close>
+
+text \<open>We often want to just localize the ending function equality or whatever.\<close>
+text%important \<open>%whitespace\<close>
+proposition homotopic_with:
+ fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
+ assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
+ shows "homotopic_with P X Y p q \<longleftrightarrow>
+ (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
+ continuous_on ({0..1} \<times> X) h \<and>
+ h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+ (\<forall>x \<in> X. h(0,x) = p x) \<and>
+ (\<forall>x \<in> X. h(1,x) = q x) \<and>
+ (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+ unfolding homotopic_with_def
+ apply (rule iffI, blast, clarify)
+ apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
+ apply auto
+ apply (force elim: continuous_on_eq)
+ apply (drule_tac x=t in bspec, force)
+ apply (subst assms; simp)
+ done
+
+proposition homotopic_with_eq:
+ assumes h: "homotopic_with P X Y f g"
+ and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
+ and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
+ and P: "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
+ shows "homotopic_with P X Y f' g'"
+ using h unfolding homotopic_with_def
+ apply safe
+ apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
+ apply (simp add: f' g', safe)
+ apply (fastforce intro: continuous_on_eq, fastforce)
+ apply (subst P; fastforce)
+ done
+
+proposition homotopic_with_equal:
+ assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
+ and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
+ and P: "P f" "P g"
+ shows "homotopic_with P X Y f g"
+ unfolding homotopic_with_def
+ apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
+ using assms
+ apply (intro conjI)
+ apply (rule continuous_on_eq [where f = "f \<circ> snd"])
+ apply (rule continuous_intros | force)+
+ apply clarify
+ apply (case_tac "t=1"; force)
+ done
+
+
+lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
+ by auto
+
+lemma homotopic_constant_maps:
+ "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
+proof (cases "s = {} \<or> t = {}")
+ case True with continuous_on_const show ?thesis
+ by (auto simp: homotopic_with path_component_def)
+next
+ case False
+ then obtain c where "c \<in> s" by blast
+ show ?thesis
+ proof
+ assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+ then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+ where conth: "continuous_on ({0..1} \<times> s) h"
+ and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
+ by (auto simp: homotopic_with)
+ have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
+ apply (rule continuous_intros conth | simp add: image_Pair_const)+
+ apply (blast intro: \<open>c \<in> s\<close> continuous_on_subset [OF conth])
+ done
+ with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
+ apply (simp_all add: homotopic_with path_component_def, auto)
+ apply (drule_tac x="h \<circ> (\<lambda>t. (t, c))" in spec)
+ apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
+ done
+ next
+ assume "s = {} \<or> path_component t a b"
+ with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+ apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
+ apply (rule_tac x="g \<circ> fst" in exI)
+ apply (rule conjI continuous_intros | force)+
+ done
+ qed
+qed
+
+
+subsection%unimportant\<open>Trivial properties\<close>
+
+lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
+ unfolding homotopic_with_def Ball_def
+ apply clarify
+ apply (frule_tac x=0 in spec)
+ apply (drule_tac x=1 in spec, auto)
+ done
+
+lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h \<circ> Pair t)"
+ by (fast intro: continuous_intros elim!: continuous_on_subset)
+
+lemma homotopic_with_imp_continuous:
+ assumes "homotopic_with P X Y f g"
+ shows "continuous_on X f \<and> continuous_on X g"
+proof -
+ obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+ where conth: "continuous_on ({0..1} \<times> X) h"
+ and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
+ using assms by (auto simp: homotopic_with_def)
+ have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h \<circ> (\<lambda>x. (t,x)))" for t
+ by (rule continuous_intros continuous_on_subset [OF conth] | force)+
+ show ?thesis
+ using h *[of 0] *[of 1] by auto
+qed
+
+proposition homotopic_with_imp_subset1:
+ "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
+ by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_imp_subset2:
+ "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
+ by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_mono:
+ assumes hom: "homotopic_with P X Y f g"
+ and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
+ shows "homotopic_with Q X Y f g"
+ using hom
+ apply (simp add: homotopic_with_def)
+ apply (erule ex_forward)
+ apply (force simp: intro!: Q dest: continuous_on_o_Pair)
+ done
+
+proposition homotopic_with_subset_left:
+ "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
+ apply (simp add: homotopic_with_def)
+ apply (fast elim!: continuous_on_subset ex_forward)
+ done
+
+proposition homotopic_with_subset_right:
+ "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
+ apply (simp add: homotopic_with_def)
+ apply (fast elim!: continuous_on_subset ex_forward)
+ done
+
+proposition homotopic_with_compose_continuous_right:
+ "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+ \<Longrightarrow> homotopic_with p W Y (f \<circ> h) (g \<circ> h)"
+ apply (clarsimp simp add: homotopic_with_def)
+ apply (rename_tac k)
+ apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
+ apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+ apply (erule continuous_on_subset)
+ apply (fastforce simp: o_def)+
+ done
+
+proposition homotopic_compose_continuous_right:
+ "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+ \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
+ using homotopic_with_compose_continuous_right by fastforce
+
+proposition homotopic_with_compose_continuous_left:
+ "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+ \<Longrightarrow> homotopic_with p X Z (h \<circ> f) (h \<circ> g)"
+ apply (clarsimp simp add: homotopic_with_def)
+ apply (rename_tac k)
+ apply (rule_tac x="h \<circ> k" in exI)
+ apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+ apply (erule continuous_on_subset)
+ apply (fastforce simp: o_def)+
+ done
+
+proposition homotopic_compose_continuous_left:
+ "\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g;
+ continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+ \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
+ using homotopic_with_compose_continuous_left by fastforce
+
+proposition homotopic_with_Pair:
+ assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
+ and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
+ shows "homotopic_with q (s \<times> s') (t \<times> t')
+ (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
+ using hom
+ apply (clarsimp simp add: homotopic_with_def)
+ apply (rename_tac k k')
+ apply (rule_tac x="\<lambda>z. ((k \<circ> (\<lambda>x. (fst x, fst (snd x)))) z, (k' \<circ> (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
+ apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
+ apply (auto intro!: q [unfolded case_prod_unfold])
+ done
+
+lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g"
+ by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
+
+
+text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
+ though this only affects reflexivity.\<close>
+
+
+proposition homotopic_with_refl:
+ "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
+ apply (rule iffI)
+ using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
+ apply (simp add: homotopic_with_def)
+ apply (rule_tac x="f \<circ> snd" in exI)
+ apply (rule conjI continuous_intros | force)+
+ done
+
+lemma homotopic_with_symD:
+ fixes X :: "'a::real_normed_vector set"
+ assumes "homotopic_with P X Y f g"
+ shows "homotopic_with P X Y g f"
+ using assms
+ apply (clarsimp simp add: homotopic_with_def)
+ apply (rename_tac h)
+ apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
+ apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
+ done
+
+proposition homotopic_with_sym:
+ fixes X :: "'a::real_normed_vector set"
+ shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
+ using homotopic_with_symD by blast
+
+lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
+ by force
+
+lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
+ by force
+
+proposition homotopic_with_trans:
+ fixes X :: "'a::real_normed_vector set"
+ assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
+ shows "homotopic_with P X Y f h"
+proof -
+ have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
+ apply (simp add: closedin_closed split_01_prod [symmetric])
+ apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
+ apply (force simp: closed_Times)
+ done
+ have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
+ apply (simp add: closedin_closed split_01_prod [symmetric])
+ apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
+ apply (force simp: closed_Times)
+ done
+ { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
+ assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
+ and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
+ and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
+ and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
+ and P: "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
+ define k where "k y =
+ (if fst y \<le> 1 / 2
+ then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
+ else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y
+ have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2" for u v
+ by (simp add: geq that)
+ have "continuous_on ({0..1} \<times> X) k"
+ using cont
+ apply (simp add: split_01_prod k_def)
+ apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
+ apply (force simp: keq)
+ done
+ moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
+ using Y by (force simp: k_def)
+ moreover have "\<forall>x. k (0, x) = f x"
+ by (simp add: k_def k12)
+ moreover have "(\<forall>x. k (1, x) = h x)"
+ by (simp add: k_def k12)
+ moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
+ using P
+ apply (clarsimp simp add: k_def)
+ apply (case_tac "t \<le> 1/2", auto)
+ done
+ ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
+ continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
+ (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
+ by blast
+ } note * = this
+ show ?thesis
+ using assms by (auto intro: * simp add: homotopic_with_def)
+qed
+
+proposition homotopic_compose:
+ fixes s :: "'a::real_normed_vector set"
+ shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
+ \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g \<circ> f) (g' \<circ> f')"
+ apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
+ apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
+ by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
+
+
+text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
+lemma homotopic_triviality:
+ fixes S :: "'a::real_normed_vector set"
+ shows "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
+ continuous_on S g \<and> g ` S \<subseteq> T
+ \<longrightarrow> homotopic_with (\<lambda>x. True) S T f g) \<longleftrightarrow>
+ (S = {} \<or> path_connected T) \<and>
+ (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)))"
+ (is "?lhs = ?rhs")
+proof (cases "S = {} \<or> T = {}")
+ case True then show ?thesis by auto
+next
+ case False show ?thesis
+ proof
+ assume LHS [rule_format]: ?lhs
+ have pab: "path_component T a b" if "a \<in> T" "b \<in> T" for a b
+ proof -
+ have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
+ by (simp add: LHS continuous_on_const image_subset_iff that)
+ then show ?thesis
+ using False homotopic_constant_maps by blast
+ qed
+ moreover
+ have "\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
+ by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
+ ultimately show ?rhs
+ by (simp add: path_connected_component)
+ next
+ assume RHS: ?rhs
+ with False have T: "path_connected T"
+ by blast
+ show ?lhs
+ proof clarify
+ fix f g
+ assume "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
+ obtain c d where c: "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with (\<lambda>x. True) S T g (\<lambda>x. d)"
+ using False \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> T\<close> RHS \<open>continuous_on S g\<close> \<open>g ` S \<subseteq> T\<close> by blast
+ then have "c \<in> T" "d \<in> T"
+ using False homotopic_with_imp_subset2 by fastforce+
+ with T have "path_component T c d"
+ using path_connected_component by blast
+ then have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
+ by (simp add: homotopic_constant_maps)
+ with c d show "homotopic_with (\<lambda>x. True) S T f g"
+ by (meson homotopic_with_symD homotopic_with_trans)
+ qed
+ qed
+qed
+
+
+subsection\<open>Homotopy of paths, maintaining the same endpoints\<close>
+
+
+definition%important homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
+ where
+ "homotopic_paths s p q \<equiv>
+ homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
+
+lemma homotopic_paths:
+ "homotopic_paths s p q \<longleftrightarrow>
+ (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
+ h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
+ (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
+ (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
+ (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
+ pathfinish(h \<circ> Pair t) = pathfinish p))"
+ by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
+
+proposition homotopic_paths_imp_pathstart:
+ "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
+ by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
+
+proposition homotopic_paths_imp_pathfinish:
+ "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
+ by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
+
+lemma homotopic_paths_imp_path:
+ "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
+ using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
+
+lemma homotopic_paths_imp_subset:
+ "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+ by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
+
+proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
+by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
+
+proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p"
+ by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
+
+proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
+ by (metis homotopic_paths_sym)
+
+proposition homotopic_paths_trans [trans]:
+ "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
+ apply (simp add: homotopic_paths_def)
+ apply (rule homotopic_with_trans, assumption)
+ by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
+
+proposition homotopic_paths_eq:
+ "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
+ apply (simp add: homotopic_paths_def)
+ apply (rule homotopic_with_eq)
+ apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
+ done
+
+proposition homotopic_paths_reparametrize:
+ assumes "path p"
+ and pips: "path_image p \<subseteq> s"
+ and contf: "continuous_on {0..1} f"
+ and f01:"f ` {0..1} \<subseteq> {0..1}"
+ and [simp]: "f(0) = 0" "f(1) = 1"
+ and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
+ shows "homotopic_paths s p q"
+proof -
+ have contp: "continuous_on {0..1} p"
+ by (metis \<open>path p\<close> path_def)
+ then have "continuous_on {0..1} (p \<circ> f)"
+ using contf continuous_on_compose continuous_on_subset f01 by blast
+ then have "path q"
+ by (simp add: path_def) (metis q continuous_on_cong)
+ have piqs: "path_image q \<subseteq> s"
+ by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
+ have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
+ using f01 by force
+ have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
+ using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
+ have "homotopic_paths s q p"
+ proof (rule homotopic_paths_trans)
+ show "homotopic_paths s q (p \<circ> f)"
+ using q by (force intro: homotopic_paths_eq [OF \<open>path q\<close> piqs])
+ next
+ show "homotopic_paths s (p \<circ> f) p"
+ apply (simp add: homotopic_paths_def homotopic_with_def)
+ apply (rule_tac x="p \<circ> (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f \<circ> snd) y) + (fst y) *\<^sub>R snd y)" in exI)
+ apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
+ using pips [unfolded path_image_def]
+ apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
+ done
+ qed
+ then show ?thesis
+ by (simp add: homotopic_paths_sym)
+qed
+
+lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
+ using homotopic_paths_def homotopic_with_subset_right by blast
+
+
+text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
+lemma homotopic_join_lemma:
+ fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
+ assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
+ and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
+ and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
+ shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
+proof -
+ have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
+ by (rule ext) (simp)
+ have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
+ by (rule ext) (simp)
+ show ?thesis
+ apply (simp add: joinpaths_def)
+ apply (rule continuous_on_cases_le)
+ apply (simp_all only: 1 2)
+ apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
+ using pf
+ apply (auto simp: mult.commute pathstart_def pathfinish_def)
+ done
+qed
+
+text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
+
+lemma homotopic_paths_reversepath_D:
+ assumes "homotopic_paths s p q"
+ shows "homotopic_paths s (reversepath p) (reversepath q)"
+ using assms
+ apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
+ apply (rule_tac x="h \<circ> (\<lambda>x. (fst x, 1 - snd x))" in exI)
+ apply (rule conjI continuous_intros)+
+ apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
+ done
+
+proposition homotopic_paths_reversepath:
+ "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
+ using homotopic_paths_reversepath_D by force
+
+
+proposition homotopic_paths_join:
+ "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
+ apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
+ apply (rename_tac k1 k2)
+ apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
+ apply (rule conjI continuous_intros homotopic_join_lemma)+
+ apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
+ done
+
+proposition homotopic_paths_continuous_image:
+ "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
+ unfolding homotopic_paths_def
+ apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
+ apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
+ done
+
+
+subsection\<open>Group properties for homotopy of paths\<close>
+
+text%important\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
+
+proposition homotopic_paths_rid:
+ "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
+ apply (subst homotopic_paths_sym)
+ apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
+ apply (simp_all del: le_divide_eq_numeral1)
+ apply (subst split_01)
+ apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
+ done
+
+proposition homotopic_paths_lid:
+ "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
+ using homotopic_paths_rid [of "reversepath p" s]
+ by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
+ pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
+
+proposition homotopic_paths_assoc:
+ "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
+ pathfinish q = pathstart r\<rbrakk>
+ \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
+ apply (subst homotopic_paths_sym)
+ apply (rule homotopic_paths_reparametrize
+ [where f = "\<lambda>t. if t \<le> 1 / 2 then inverse 2 *\<^sub>R t
+ else if t \<le> 3 / 4 then t - (1 / 4)
+ else 2 *\<^sub>R t - 1"])
+ apply (simp_all del: le_divide_eq_numeral1)
+ apply (simp add: subset_path_image_join)
+ apply (rule continuous_on_cases_1 continuous_intros)+
+ apply (auto simp: joinpaths_def)
+ done
+
+proposition homotopic_paths_rinv:
+ assumes "path p" "path_image p \<subseteq> s"
+ shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
+proof -
+ have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
+ using assms
+ apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
+ apply (rule continuous_on_cases_le)
+ apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
+ apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+ apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
+ apply (force elim!: continuous_on_subset simp add: mult_le_one)+
+ done
+ then show ?thesis
+ using assms
+ apply (subst homotopic_paths_sym_eq)
+ unfolding homotopic_paths_def homotopic_with_def
+ apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
+ apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
+ apply (force simp: mult_le_one)
+ done
+qed
+
+proposition homotopic_paths_linv:
+ assumes "path p" "path_image p \<subseteq> s"
+ shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
+ using homotopic_paths_rinv [of "reversepath p" s] assms by simp
+
+
+subsection\<open>Homotopy of loops without requiring preservation of endpoints\<close>
+
+definition%important homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "homotopic_loops s p q \<equiv>
+ homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
+
+lemma homotopic_loops:
+ "homotopic_loops s p q \<longleftrightarrow>
+ (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
+ image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
+ (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
+ (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
+ (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
+ by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
+
+proposition homotopic_loops_imp_loop:
+ "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
+using homotopic_with_imp_property homotopic_loops_def by blast
+
+proposition homotopic_loops_imp_path:
+ "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
+ unfolding homotopic_loops_def path_def
+ using homotopic_with_imp_continuous by blast
+
+proposition homotopic_loops_imp_subset:
+ "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+ unfolding homotopic_loops_def path_image_def
+ by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
+
+proposition homotopic_loops_refl:
+ "homotopic_loops s p p \<longleftrightarrow>
+ path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
+ by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
+
+proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
+ by (simp add: homotopic_loops_def homotopic_with_sym)
+
+proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
+ by (metis homotopic_loops_sym)
+
+proposition homotopic_loops_trans:
+ "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
+ unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
+
+proposition homotopic_loops_subset:
+ "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
+ by (simp add: homotopic_loops_def homotopic_with_subset_right)
+
+proposition homotopic_loops_eq:
+ "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
+ \<Longrightarrow> homotopic_loops s p q"
+ unfolding homotopic_loops_def
+ apply (rule homotopic_with_eq)
+ apply (rule homotopic_with_refl [where f = p, THEN iffD2])
+ apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
+ done
+
+proposition homotopic_loops_continuous_image:
+ "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
+ unfolding homotopic_loops_def
+ apply (rule homotopic_with_compose_continuous_left)
+ apply (erule homotopic_with_mono)
+ by (simp add: pathfinish_def pathstart_def)
+
+
+subsection\<open>Relations between the two variants of homotopy\<close>
+
+proposition homotopic_paths_imp_homotopic_loops:
+ "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
+ by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
+
+proposition homotopic_loops_imp_homotopic_paths_null:
+ assumes "homotopic_loops s p (linepath a a)"
+ shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
+proof -
+ have "path p" by (metis assms homotopic_loops_imp_path)
+ have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
+ have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
+ obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
+ and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
+ and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
+ and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
+ and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
+ using assms by (auto simp: homotopic_loops homotopic_with)
+ have conth0: "path (\<lambda>u. h (u, 0))"
+ unfolding path_def
+ apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+ apply (force intro: continuous_intros continuous_on_subset [OF conth])+
+ done
+ have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
+ using hs by (force simp: path_image_def)
+ have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
+ apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+ apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
+ done
+ have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
+ apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+ apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
+ apply (rule continuous_on_subset [OF conth])
+ apply (auto simp: algebra_simps add_increasing2 mult_left_le)
+ done
+ have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
+ using ends by (simp add: pathfinish_def pathstart_def)
+ have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
+ proof -
+ have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
+ with \<open>c \<le> 1\<close> show ?thesis by fastforce
+ qed
+ have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
+ (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
+ (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
+ pathstart(reversepath p) = a) \<and> pathstart p = x
+ \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
+ by (metis homotopic_paths_lid homotopic_paths_join
+ homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
+ have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
+ using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast
+ moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
+ (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
+ apply (rule homotopic_paths_sym)
+ using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
+ by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
+ moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
+ ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
+ apply (simp add: homotopic_paths_def homotopic_with_def)
+ apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
+ apply (simp add: subpath_reversepath)
+ apply (intro conjI homotopic_join_lemma)
+ using ploop
+ apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
+ apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
+ done
+ moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
+ (linepath (pathstart p) (pathstart p))"
+ apply (rule *)
+ apply (simp add: pih0 pathstart_def pathfinish_def conth0)
+ apply (simp add: reversepath_def joinpaths_def)
+ done
+ ultimately show ?thesis
+ by (blast intro: homotopic_paths_trans)
+qed
+
+proposition homotopic_loops_conjugate:
+ fixes s :: "'a::real_normed_vector set"
+ assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
+ and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
+ shows "homotopic_loops s (p +++ q +++ reversepath p) q"
+proof -
+ have contp: "continuous_on {0..1} p" using \<open>path p\<close> [unfolded path_def] by blast
+ have contq: "continuous_on {0..1} q" using \<open>path q\<close> [unfolded path_def] by blast
+ have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
+ apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+ apply (force simp: mult_le_one intro!: continuous_intros)
+ apply (rule continuous_on_subset [OF contp])
+ apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
+ done
+ have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
+ apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+ apply (force simp: mult_le_one intro!: continuous_intros)
+ apply (rule continuous_on_subset [OF contp])
+ apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
+ done
+ have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s"
+ using sum_le_prod1
+ by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
+ have ps2: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
+ apply (rule pip [unfolded path_image_def, THEN subsetD])
+ apply (rule image_eqI, blast)
+ apply (simp add: algebra_simps)
+ by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
+ add.commute zero_le_numeral)
+ have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s"
+ using path_image_def piq by fastforce
+ have "homotopic_loops s (p +++ q +++ reversepath p)
+ (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
+ apply (simp add: homotopic_loops_def homotopic_with_def)
+ apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
+ apply (simp add: subpath_refl subpath_reversepath)
+ apply (intro conjI homotopic_join_lemma)
+ using papp qloop
+ apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
+ apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
+ apply (auto simp: ps1 ps2 qs)
+ done
+ moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
+ proof -
+ have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
+ using \<open>path q\<close> homotopic_paths_lid qloop piq by auto
+ hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
+ using homotopic_paths_trans by blast
+ hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
+ proof -
+ have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
+ by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
+ thus ?thesis
+ by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
+ homotopic_paths_trans qloop pathfinish_linepath piq)
+ qed
+ thus ?thesis
+ by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
+ qed
+ ultimately show ?thesis
+ by (blast intro: homotopic_loops_trans)
+qed
+
+lemma homotopic_paths_loop_parts:
+ assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
+ shows "homotopic_paths S p q"
+proof -
+ have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
+ using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
+ then have "path p"
+ using \<open>path q\<close> homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
+ show ?thesis
+ proof (cases "pathfinish p = pathfinish q")
+ case True
+ have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
+ by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> homotopic_loops_imp_subset homotopic_paths_imp_path loops
+ path_image_join path_image_reversepath path_imp_reversepath path_join_eq)+
+ have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
+ using \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> homotopic_paths_rid homotopic_paths_sym by blast
+ moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
+ by (simp add: True \<open>path p\<close> \<open>path q\<close> pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
+ moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
+ by (simp add: True \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
+ moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
+ by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
+ moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
+ by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
+ ultimately show ?thesis
+ using homotopic_paths_trans by metis
+ next
+ case False
+ then show ?thesis
+ using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
+ qed
+qed
+
+
+subsection%unimportant\<open>Homotopy of "nearby" function, paths and loops\<close>
+
+lemma homotopic_with_linear:
+ fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
+ assumes contf: "continuous_on s f"
+ and contg:"continuous_on s g"
+ and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
+ shows "homotopic_with (\<lambda>z. True) s t f g"
+ apply (simp add: homotopic_with_def)
+ apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
+ apply (intro conjI)
+ apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
+ continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
+ using sub closed_segment_def apply fastforce+
+ done
+
+lemma homotopic_paths_linear:
+ fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
+ assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
+ "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
+ shows "homotopic_paths s g h"
+ using assms
+ unfolding path_def
+ apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
+ apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g \<circ> snd) y + (fst y) *\<^sub>R (h \<circ> snd) y)" in exI)
+ apply (intro conjI subsetI continuous_intros; force)
+ done
+
+lemma homotopic_loops_linear:
+ fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
+ assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
+ "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
+ shows "homotopic_loops s g h"
+ using assms
+ unfolding path_def
+ apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
+ apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
+ apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
+ apply (force simp: closed_segment_def)
+ done
+
+lemma homotopic_paths_nearby_explicit:
+ assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
+ and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
+ shows "homotopic_paths s g h"
+ apply (rule homotopic_paths_linear [OF assms(1-4)])
+ by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
+
+lemma homotopic_loops_nearby_explicit:
+ assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
+ and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
+ shows "homotopic_loops s g h"
+ apply (rule homotopic_loops_linear [OF assms(1-4)])
+ by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
+
+lemma homotopic_nearby_paths:
+ fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "path g" "open s" "path_image g \<subseteq> s"
+ shows "\<exists>e. 0 < e \<and>
+ (\<forall>h. path h \<and>
+ pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
+ (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
+proof -
+ obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
+ using separate_compact_closed [of "path_image g" "-s"] assms by force
+ show ?thesis
+ apply (intro exI conjI)
+ using e [unfolded dist_norm]
+ apply (auto simp: intro!: homotopic_paths_nearby_explicit assms \<open>e > 0\<close>)
+ by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
+qed
+
+lemma homotopic_nearby_loops:
+ fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
+ shows "\<exists>e. 0 < e \<and>
+ (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
+ (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
+proof -
+ obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
+ using separate_compact_closed [of "path_image g" "-s"] assms by force
+ show ?thesis
+ apply (intro exI conjI)
+ using e [unfolded dist_norm]
+ apply (auto simp: intro!: homotopic_loops_nearby_explicit assms \<open>e > 0\<close>)
+ by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
+qed
+
+
+subsection\<open> Homotopy and subpaths\<close>
+
+lemma homotopic_join_subpaths1:
+ assumes "path g" and pag: "path_image g \<subseteq> s"
+ and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
+ shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+proof -
+ have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
+ using affine_ineq \<open>u \<le> v\<close> by fastforce
+ have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
+ by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
+ have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
+ show ?thesis
+ apply (rule homotopic_paths_subset [OF _ pag])
+ using assms
+ apply (cases "w = u")
+ using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
+ apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
+ apply (rule homotopic_paths_sym)
+ apply (rule homotopic_paths_reparametrize
+ [where f = "\<lambda>t. if t \<le> 1 / 2
+ then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
+ else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
+ using \<open>path g\<close> path_subpath u w apply blast
+ using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
+ apply simp_all
+ apply (subst split_01)
+ apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
+ apply (simp_all add: field_simps not_le)
+ apply (force dest!: t2)
+ apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
+ apply (simp add: joinpaths_def subpath_def)
+ apply (force simp: algebra_simps)
+ done
+qed
+
+lemma homotopic_join_subpaths2:
+ assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+ shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
+by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
+
+lemma homotopic_join_subpaths3:
+ assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+ and "path g" and pag: "path_image g \<subseteq> s"
+ and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
+ shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
+proof -
+ have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
+ apply (rule homotopic_paths_join)
+ using hom homotopic_paths_sym_eq apply blast
+ apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
+ done
+ also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
+ apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
+ using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
+ also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
+ (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
+ apply (rule homotopic_paths_join)
+ apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
+ apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
+ apply simp
+ done
+ also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
+ apply (rule homotopic_paths_rid)
+ using \<open>path g\<close> path_subpath u v apply blast
+ apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
+ done
+ finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
+ then show ?thesis
+ using homotopic_join_subpaths2 by blast
+qed
+
+proposition homotopic_join_subpaths:
+ "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
+ \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+ apply (rule le_cases3 [of u v w])
+using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
+
+text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
+
+lemma path_component_imp_homotopic_points:
+ "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
+apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
+ pathstart_def pathfinish_def path_image_def path_def, clarify)
+apply (rule_tac x="g \<circ> fst" in exI)
+apply (intro conjI continuous_intros continuous_on_compose)+
+apply (auto elim!: continuous_on_subset)
+done
+
+lemma homotopic_loops_imp_path_component_value:
+ "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
+ \<Longrightarrow> path_component S (p t) (q t)"
+apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
+ pathstart_def pathfinish_def path_image_def path_def, clarify)
+apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
+apply (intro conjI continuous_intros continuous_on_compose)+
+apply (auto elim!: continuous_on_subset)
+done
+
+lemma homotopic_points_eq_path_component:
+ "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
+ path_component S a b"
+by (auto simp: path_component_imp_homotopic_points
+ dest: homotopic_loops_imp_path_component_value [where t=1])
+
+lemma path_connected_eq_homotopic_points:
+ "path_connected S \<longleftrightarrow>
+ (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
+by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
+
+
+subsection\<open>Simply connected sets\<close>
+
+text%important\<open>defined as "all loops are homotopic (as loops)\<close>
+
+definition%important simply_connected where
+ "simply_connected S \<equiv>
+ \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and>
+ path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S
+ \<longrightarrow> homotopic_loops S p q"
+
+lemma simply_connected_empty [iff]: "simply_connected {}"
+ by (simp add: simply_connected_def)
+
+lemma simply_connected_imp_path_connected:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<Longrightarrow> path_connected S"
+by (simp add: simply_connected_def path_connected_eq_homotopic_points)
+
+lemma simply_connected_imp_connected:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<Longrightarrow> connected S"
+by (simp add: path_connected_imp_connected simply_connected_imp_path_connected)
+
+lemma simply_connected_eq_contractible_loop_any:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<longleftrightarrow>
+ (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and>
+ pathfinish p = pathstart p \<and> a \<in> S
+ \<longrightarrow> homotopic_loops S p (linepath a a))"
+apply (simp add: simply_connected_def)
+apply (rule iffI, force, clarify)
+apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
+apply (fastforce simp add:)
+using homotopic_loops_sym apply blast
+done
+
+lemma simply_connected_eq_contractible_loop_some:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<longleftrightarrow>
+ path_connected S \<and>
+ (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+ \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
+apply (rule iffI)
+ apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
+apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
+apply (drule_tac x=p in spec)
+using homotopic_loops_trans path_connected_eq_homotopic_points
+ apply blast
+done
+
+lemma simply_connected_eq_contractible_loop_all:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<longleftrightarrow>
+ S = {} \<or>
+ (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+ \<longrightarrow> homotopic_loops S p (linepath a a))"
+ (is "?lhs = ?rhs")
+proof (cases "S = {}")
+ case True then show ?thesis by force
+next
+ case False
+ then obtain a where "a \<in> S" by blast
+ show ?thesis
+ proof
+ assume "simply_connected S"
+ then show ?rhs
+ using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
+ by blast
+ next
+ assume ?rhs
+ then show "simply_connected S"
+ apply (simp add: simply_connected_eq_contractible_loop_any False)
+ by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
+ path_component_imp_homotopic_points path_component_refl)
+ qed
+qed
+
+lemma simply_connected_eq_contractible_path:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<longleftrightarrow>
+ path_connected S \<and>
+ (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+ \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
+apply (rule iffI)
+ apply (simp add: simply_connected_imp_path_connected)
+ apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
+by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
+ simply_connected_eq_contractible_loop_some subset_iff)
+
+lemma simply_connected_eq_homotopic_paths:
+ fixes S :: "_::real_normed_vector set"
+ shows "simply_connected S \<longleftrightarrow>
+ path_connected S \<and>
+ (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
+ path q \<and> path_image q \<subseteq> S \<and>
+ pathstart q = pathstart p \<and> pathfinish q = pathfinish p
+ \<longrightarrow> homotopic_paths S p q)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then have pc: "path_connected S"
+ and *: "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
+ pathfinish p = pathstart p\<rbrakk>
+ \<Longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))"
+ by (auto simp: simply_connected_eq_contractible_path)
+ have "homotopic_paths S p q"
+ if "path p" "path_image p \<subseteq> S" "path q"
+ "path_image q \<subseteq> S" "pathstart q = pathstart p"
+ "pathfinish q = pathfinish p" for p q
+ proof -
+ have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
+ by (simp add: homotopic_paths_rid homotopic_paths_sym that)
+ also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
+ (p +++ reversepath q +++ q)"
+ using that
+ by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
+ also have "homotopic_paths S (p +++ reversepath q +++ q)
+ ((p +++ reversepath q) +++ q)"
+ by (simp add: that homotopic_paths_assoc)
+ also have "homotopic_paths S ((p +++ reversepath q) +++ q)
+ (linepath (pathstart q) (pathstart q) +++ q)"
+ using * [of "p +++ reversepath q"] that
+ by (simp add: homotopic_paths_join path_image_join)
+ also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q"
+ using that homotopic_paths_lid by blast
+ finally show ?thesis .
+ qed
+ then show ?rhs
+ by (blast intro: pc *)
+next
+ assume ?rhs
+ then show ?lhs
+ by (force simp: simply_connected_eq_contractible_path)
+qed
+
+proposition simply_connected_Times:
+ fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+ assumes S: "simply_connected S" and T: "simply_connected T"
+ shows "simply_connected(S \<times> T)"
+proof -
+ have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
+ if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
+ for p a b
+ proof -
+ have "path (fst \<circ> p)"
+ apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
+ apply (rule continuous_intros)+
+ done
+ moreover have "path_image (fst \<circ> p) \<subseteq> S"
+ using that apply (simp add: path_image_def) by force
+ ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
+ using S that
+ apply (simp add: simply_connected_eq_contractible_loop_any)
+ apply (drule_tac x="fst \<circ> p" in spec)
+ apply (drule_tac x=a in spec)
+ apply (auto simp: pathstart_def pathfinish_def)
+ done
+ have "path (snd \<circ> p)"
+ apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
+ apply (rule continuous_intros)+
+ done
+ moreover have "path_image (snd \<circ> p) \<subseteq> T"
+ using that apply (simp add: path_image_def) by force
+ ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
+ using T that
+ apply (simp add: simply_connected_eq_contractible_loop_any)
+ apply (drule_tac x="snd \<circ> p" in spec)
+ apply (drule_tac x=b in spec)
+ apply (auto simp: pathstart_def pathfinish_def)
+ done
+ show ?thesis
+ using p1 p2
+ apply (simp add: homotopic_loops, clarify)
+ apply (rename_tac h k)
+ apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
+ apply (intro conjI continuous_intros | assumption)+
+ apply (auto simp: pathstart_def pathfinish_def)
+ done
+ qed
+ with assms show ?thesis
+ by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
+qed
+
+
+subsection\<open>Contractible sets\<close>
+
+definition%important contractible where
+ "contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
+
+proposition contractible_imp_simply_connected:
+ fixes S :: "_::real_normed_vector set"
+ assumes "contractible S" shows "simply_connected S"
+proof (cases "S = {}")
+ case True then show ?thesis by force
+next
+ case False
+ obtain a where a: "homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
+ using assms by (force simp: contractible_def)
+ then have "a \<in> S"
+ by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
+ show ?thesis
+ apply (simp add: simply_connected_eq_contractible_loop_all False)
+ apply (rule bexI [OF _ \<open>a \<in> S\<close>])
+ using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
+ apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
+ apply (intro conjI continuous_on_compose continuous_intros)
+ apply (erule continuous_on_subset | force)+
+ done
+qed
+
+corollary contractible_imp_connected:
+ fixes S :: "_::real_normed_vector set"
+ shows "contractible S \<Longrightarrow> connected S"
+by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)
+
+lemma contractible_imp_path_connected:
+ fixes S :: "_::real_normed_vector set"
+ shows "contractible S \<Longrightarrow> path_connected S"
+by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)
+
+lemma nullhomotopic_through_contractible:
+ fixes S :: "_::topological_space set"
+ assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+ and g: "continuous_on T g" "g ` T \<subseteq> U"
+ and T: "contractible T"
+ obtains c where "homotopic_with (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
+proof -
+ obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)"
+ using assms by (force simp: contractible_def)
+ have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
+ by (rule homotopic_compose_continuous_left [OF b g])
+ then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
+ by (rule homotopic_compose_continuous_right [OF _ f])
+ then show ?thesis
+ by (simp add: comp_def that)
+qed
+
+lemma nullhomotopic_into_contractible:
+ assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+ and T: "contractible T"
+ obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
+apply (rule nullhomotopic_through_contractible [OF f, of id T])
+using assms
+apply (auto simp: continuous_on_id)
+done
+
+lemma nullhomotopic_from_contractible:
+ assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+ and S: "contractible S"
+ obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
+apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
+using assms
+apply (auto simp: comp_def)
+done
+
+lemma homotopic_through_contractible:
+ fixes S :: "_::real_normed_vector set"
+ assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
+ "continuous_on T g1" "g1 ` T \<subseteq> U"
+ "continuous_on S f2" "f2 ` S \<subseteq> T"
+ "continuous_on T g2" "g2 ` T \<subseteq> U"
+ "contractible T" "path_connected U"
+ shows "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
+proof -
+ obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
+ apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
+ using assms apply auto
+ done
+ obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
+ apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
+ using assms apply auto
+ done
+ have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
+ proof (cases "S = {}")
+ case True then show ?thesis by force
+ next
+ case False
+ with c1 c2 have "c1 \<in> U" "c2 \<in> U"
+ using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
+ with \<open>path_connected U\<close> show ?thesis by blast
+ qed
+ show ?thesis
+ apply (rule homotopic_with_trans [OF c1])
+ apply (rule homotopic_with_symD)
+ apply (rule homotopic_with_trans [OF c2])
+ apply (simp add: path_component homotopic_constant_maps *)
+ done
+qed
+
+lemma homotopic_into_contractible:
+ fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
+ assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+ and g: "continuous_on S g" "g ` S \<subseteq> T"
+ and T: "contractible T"
+ shows "homotopic_with (\<lambda>h. True) S T f g"
+using homotopic_through_contractible [of S f T id T g id]
+by (simp add: assms contractible_imp_path_connected continuous_on_id)
+
+lemma homotopic_from_contractible:
+ fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
+ assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+ and g: "continuous_on S g" "g ` S \<subseteq> T"
+ and "contractible S" "path_connected T"
+ shows "homotopic_with (\<lambda>h. True) S T f g"
+using homotopic_through_contractible [of S id S f T id g]
+by (simp add: assms contractible_imp_path_connected continuous_on_id)
+
+lemma starlike_imp_contractible_gen:
+ fixes S :: "'a::real_normed_vector set"
+ assumes S: "starlike S"
+ and P: "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P(\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)"
+ obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)"
+proof -
+ obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
+ using S by (auto simp: starlike_def)
+ have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
+ apply clarify
+ apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
+ apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
+ done
+ then show ?thesis
+ apply (rule_tac a=a in that)
+ using \<open>a \<in> S\<close>
+ apply (simp add: homotopic_with_def)
+ apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
+ apply (intro conjI ballI continuous_on_compose continuous_intros)
+ apply (simp_all add: P)
+ done
+qed
+
+lemma starlike_imp_contractible:
+ fixes S :: "'a::real_normed_vector set"
+ shows "starlike S \<Longrightarrow> contractible S"
+using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)
+
+lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
+ by (simp add: starlike_imp_contractible)
+
+lemma starlike_imp_simply_connected:
+ fixes S :: "'a::real_normed_vector set"
+ shows "starlike S \<Longrightarrow> simply_connected S"
+by (simp add: contractible_imp_simply_connected starlike_imp_contractible)
+
+lemma convex_imp_simply_connected:
+ fixes S :: "'a::real_normed_vector set"
+ shows "convex S \<Longrightarrow> simply_connected S"
+using convex_imp_starlike starlike_imp_simply_connected by blast
+
+lemma starlike_imp_path_connected:
+ fixes S :: "'a::real_normed_vector set"
+ shows "starlike S \<Longrightarrow> path_connected S"
+by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)
+
+lemma starlike_imp_connected:
+ fixes S :: "'a::real_normed_vector set"
+ shows "starlike S \<Longrightarrow> connected S"
+by (simp add: path_connected_imp_connected starlike_imp_path_connected)
+
+lemma is_interval_simply_connected_1:
+ fixes S :: "real set"
+ shows "is_interval S \<longleftrightarrow> simply_connected S"
+using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
+
+lemma contractible_empty [simp]: "contractible {}"
+ by (simp add: contractible_def homotopic_with)
+
+lemma contractible_convex_tweak_boundary_points:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S"
+ shows "contractible T"
+proof (cases "S = {}")
+ case True
+ with assms show ?thesis
+ by (simp add: subsetCE)
+next
+ case False
+ show ?thesis
+ apply (rule starlike_imp_contractible)
+ apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
+ done
+qed
+
+lemma convex_imp_contractible:
+ fixes S :: "'a::real_normed_vector set"
+ shows "convex S \<Longrightarrow> contractible S"
+ using contractible_empty convex_imp_starlike starlike_imp_contractible by blast
+
+lemma contractible_sing [simp]:
+ fixes a :: "'a::real_normed_vector"
+ shows "contractible {a}"
+by (rule convex_imp_contractible [OF convex_singleton])
+
+lemma is_interval_contractible_1:
+ fixes S :: "real set"
+ shows "is_interval S \<longleftrightarrow> contractible S"
+using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
+ is_interval_simply_connected_1 by auto
+
+lemma contractible_Times:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes S: "contractible S" and T: "contractible T"
+ shows "contractible (S \<times> T)"
+proof -
+ obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
+ and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
+ and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
+ and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (1::real, x) = a"
+ using S by (auto simp: contractible_def homotopic_with)
+ obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
+ and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
+ and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
+ and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (1::real, x) = b"
+ using T by (auto simp: contractible_def homotopic_with)
+ show ?thesis
+ apply (simp add: contractible_def homotopic_with)
+ apply (rule exI [where x=a])
+ apply (rule exI [where x=b])
+ apply (rule exI [where x = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
+ apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
+ using hsub ksub
+ apply auto
+ done
+qed
+
+lemma homotopy_dominated_contractibility:
+ fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+ assumes S: "contractible S"
+ and f: "continuous_on S f" "image f S \<subseteq> T"
+ and g: "continuous_on T g" "image g T \<subseteq> S"
+ and hom: "homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
+ shows "contractible T"
+proof -
+ obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)"
+ using nullhomotopic_from_contractible [OF f S] .
+ then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)"
+ by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
+ show ?thesis
+ apply (simp add: contractible_def)
+ apply (rule exI [where x = b])
+ apply (rule homotopic_with_symD)
+ apply (rule homotopic_with_trans [OF _ hom])
+ using homg apply (simp add: o_def)
+ done
+qed
+
+
+subsection\<open>Local versions of topological properties in general\<close>
+
+definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+where
+ "locally P S \<equiv>
+ \<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w
+ \<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+ x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
+
+lemma locallyI:
+ assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk>
+ \<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+ x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
+ shows "locally P S"
+using assms by (force simp: locally_def)
+
+lemma locallyE:
+ assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w"
+ obtains u v where "openin (subtopology euclidean S) u"
+ "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
+ using assms unfolding locally_def by meson
+
+lemma locally_mono:
+ assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t"
+ shows "locally Q S"
+by (metis assms locally_def)
+
+lemma locally_open_subset:
+ assumes "locally P S" "openin (subtopology euclidean S) t"
+ shows "locally P t"
+using assms
+apply (simp add: locally_def)
+apply (erule all_forward)+
+apply (rule impI)
+apply (erule impCE)
+ using openin_trans apply blast
+apply (erule ex_forward)
+by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
+
+lemma locally_diff_closed:
+ "\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> locally P (S - t)"
+ using locally_open_subset closedin_def by fastforce
+
+lemma locally_empty [iff]: "locally P {}"
+ by (simp add: locally_def openin_subtopology)
+
+lemma locally_singleton [iff]:
+ fixes a :: "'a::metric_space"
+ shows "locally P {a} \<longleftrightarrow> P {a}"
+apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
+using zero_less_one by blast
+
+lemma locally_iff:
+ "locally P S \<longleftrightarrow>
+ (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))"
+apply (simp add: le_inf_iff locally_def openin_open, safe)
+apply (metis IntE IntI le_inf_iff)
+apply (metis IntI Int_subset_iff)
+done
+
+lemma locally_Int:
+ assumes S: "locally P S" and t: "locally P t"
+ and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
+ shows "locally P (S \<inter> t)"
+using S t unfolding locally_iff
+apply clarify
+apply (drule_tac x=T in spec)+
+apply (drule_tac x=x in spec)+
+apply clarsimp
+apply (rename_tac U1 U2 V1 V2)
+apply (rule_tac x="U1 \<inter> U2" in exI)
+apply (simp add: open_Int)
+apply (rule_tac x="V1 \<inter> V2" in exI)
+apply (auto intro: P)
+done
+
+lemma locally_Times:
+ fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
+ assumes PS: "locally P S" and QT: "locally Q T" and R: "\<And>S T. P S \<and> Q T \<Longrightarrow> R(S \<times> T)"
+ shows "locally R (S \<times> T)"
+ unfolding locally_def
+proof (clarify)
+ fix W x y
+ assume W: "openin (subtopology euclidean (S \<times> T)) W" and xy: "(x, y) \<in> W"
+ then obtain U V where "openin (subtopology euclidean S) U" "x \<in> U"
+ "openin (subtopology euclidean T) V" "y \<in> V" "U \<times> V \<subseteq> W"
+ using Times_in_interior_subtopology by metis
+ then obtain U1 U2 V1 V2
+ where opeS: "openin (subtopology euclidean S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
+ and opeT: "openin (subtopology euclidean T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
+ by (meson PS QT locallyE)
+ with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (subtopology euclidean (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
+ apply (rule_tac x="U1 \<times> V1" in exI)
+ apply (rule_tac x="U2 \<times> V2" in exI)
+ apply (auto simp: openin_Times R)
+ done
+qed
+
+
+proposition homeomorphism_locally_imp:
+ fixes S :: "'a::metric_space set" and t :: "'b::t2_space set"
+ assumes S: "locally P S" and hom: "homeomorphism S t f g"
+ and Q: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
+ shows "locally Q t"
+proof (clarsimp simp: locally_def)
+ fix W y
+ assume "y \<in> W" and "openin (subtopology euclidean t) W"
+ then obtain T where T: "open T" "W = t \<inter> T"
+ by (force simp: openin_open)
+ then have "W \<subseteq> t" by auto
+ have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
+ and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
+ using hom by (auto simp: homeomorphism_def)
+ have gw: "g ` W = S \<inter> f -` W"
+ using \<open>W \<subseteq> t\<close>
+ apply auto
+ using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
+ using g \<open>W \<subseteq> t\<close> apply auto[1]
+ by (simp add: f rev_image_eqI)
+ have \<circ>: "openin (subtopology euclidean S) (g ` W)"
+ proof -
+ have "continuous_on S f"
+ using f(3) by blast
+ then show "openin (subtopology euclidean S) (g ` W)"
+ by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) W\<close> continuous_on_open f(2))
+ qed
+ then obtain u v
+ where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
+ using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
+ have "v \<subseteq> S" using uv by (simp add: gw)
+ have fv: "f ` v = t \<inter> {x. g x \<in> v}"
+ using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto
+ have "f ` v \<subseteq> W"
+ using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
+ have contvf: "continuous_on v f"
+ using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast
+ have contvg: "continuous_on (f ` v) g"
+ using \<open>f ` v \<subseteq> W\<close> \<open>W \<subseteq> t\<close> continuous_on_subset [OF g(3)] by blast
+ have homv: "homeomorphism v (f ` v) f g"
+ using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
+ apply (simp add: homeomorphism_def contvf contvg, auto)
+ by (metis f(1) rev_image_eqI rev_subsetD)
+ have 1: "openin (subtopology euclidean t) (t \<inter> g -` u)"
+ apply (rule continuous_on_open [THEN iffD1, rule_format])
+ apply (rule \<open>continuous_on t g\<close>)
+ using \<open>g ` t = S\<close> apply (simp add: osu)
+ done
+ have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
+ apply (rule_tac x="f ` v" in exI)
+ apply (intro conjI Q [OF \<open>P v\<close> homv])
+ using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close> \<open>f ` v \<subseteq> W\<close> uv apply (auto simp: fv)
+ done
+ show "\<exists>U. openin (subtopology euclidean t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
+ by (meson 1 2)
+qed
+
+lemma homeomorphism_locally:
+ fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes hom: "homeomorphism S t f g"
+ and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)"
+ shows "locally P S \<longleftrightarrow> locally Q t"
+apply (rule iffI)
+apply (erule homeomorphism_locally_imp [OF _ hom])
+apply (simp add: eq)
+apply (erule homeomorphism_locally_imp)
+using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
+done
+
+lemma homeomorphic_locally:
+ fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
+ assumes hom: "S homeomorphic T"
+ and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
+ shows "locally P S \<longleftrightarrow> locally Q T"
+proof -
+ obtain f g where hom: "homeomorphism S T f g"
+ using assms by (force simp: homeomorphic_def)
+ then show ?thesis
+ using homeomorphic_def local.iff
+ by (blast intro!: homeomorphism_locally)
+qed
+
+lemma homeomorphic_local_compactness:
+ fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
+ shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
+by (simp add: homeomorphic_compactness homeomorphic_locally)
+
+lemma locally_translation:
+ fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
+ shows
+ "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
+ \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
+apply (rule homeomorphism_locally [OF homeomorphism_translation])
+apply (simp add: homeomorphism_def)
+by metis
+
+lemma locally_injective_linear_image:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
+ shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
+apply (rule linear_homeomorphism_image [OF f])
+apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
+by (metis iff homeomorphism_def)
+
+lemma locally_open_map_image:
+ fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+ assumes P: "locally P S"
+ and f: "continuous_on S f"
+ and oo: "\<And>t. openin (subtopology euclidean S) t
+ \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)"
+ and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
+ shows "locally Q (f ` S)"
+proof (clarsimp simp add: locally_def)
+ fix W y
+ assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y \<in> W"
+ then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
+ have oivf: "openin (subtopology euclidean S) (S \<inter> f -` W)"
+ by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
+ then obtain x where "x \<in> S" "f x = y"
+ using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
+ then obtain U V
+ where "openin (subtopology euclidean S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
+ using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
+ by auto
+ then show "\<exists>X. openin (subtopology euclidean (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
+ apply (rule_tac x="f ` U" in exI)
+ apply (rule conjI, blast intro!: oo)
+ apply (rule_tac x="f ` V" in exI)
+ apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
+ done
+qed
+
+
+subsection\<open>An induction principle for connected sets\<close>
+
+proposition connected_induction:
+ assumes "connected S"
+ and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+ and opI: "\<And>a. a \<in> S
+ \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+ (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
+ and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
+ shows "Q b"
+proof -
+ have 1: "openin (subtopology euclidean S)
+ {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+ b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
+ apply (subst openin_subopen, clarify)
+ apply (rule_tac x=T in exI, auto)
+ done
+ have 2: "openin (subtopology euclidean S)
+ {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+ b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
+ apply (subst openin_subopen, clarify)
+ apply (rule_tac x=T in exI, auto)
+ done
+ show ?thesis
+ using \<open>connected S\<close>
+ apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
+ apply (elim disjE allE)
+ apply (blast intro: 1)
+ apply (blast intro: 2, simp_all)
+ apply clarify apply (metis opI)
+ using opD apply (blast intro: etc elim: dest:)
+ using opI etc apply meson+
+ done
+qed
+
+lemma connected_equivalence_relation_gen:
+ assumes "connected S"
+ and etc: "a \<in> S" "b \<in> S" "P a" "P b"
+ and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
+ and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+ and opI: "\<And>a. a \<in> S
+ \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+ (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
+ shows "R a b"
+proof -
+ have "\<And>a b c. \<lbrakk>a \<in> S; P a; b \<in> S; c \<in> S; P b; P c; R a b\<rbrakk> \<Longrightarrow> R a c"
+ apply (rule connected_induction [OF \<open>connected S\<close> opD], simp_all)
+ by (meson trans opI)
+ then show ?thesis by (metis etc opI)
+qed
+
+lemma connected_induction_simple:
+ assumes "connected S"
+ and etc: "a \<in> S" "b \<in> S" "P a"
+ and opI: "\<And>a. a \<in> S
+ \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+ (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
+ shows "P b"
+apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
+apply (frule opI)
+using etc apply simp_all
+done
+
+lemma connected_equivalence_relation:
+ assumes "connected S"
+ and etc: "a \<in> S" "b \<in> S"
+ and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
+ and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
+ and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
+ shows "R a b"
+proof -
+ have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
+ apply (rule connected_induction_simple [OF \<open>connected S\<close>], simp_all)
+ by (meson local.sym local.trans opI openin_imp_subset subsetCE)
+ then show ?thesis by (metis etc opI)
+qed
+
+lemma locally_constant_imp_constant:
+ assumes "connected S"
+ and opI: "\<And>a. a \<in> S
+ \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
+ shows "f constant_on S"
+proof -
+ have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
+ apply (rule connected_equivalence_relation [OF \<open>connected S\<close>], simp_all)
+ by (metis opI)
+ then show ?thesis
+ by (metis constant_on_def)
+qed
+
+lemma locally_constant:
+ "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
+apply (simp add: locally_def)
+apply (rule iffI)
+ apply (rule locally_constant_imp_constant, assumption)
+ apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
+by (meson constant_on_subset openin_imp_subset order_refl)
+
+
+subsection\<open>Basic properties of local compactness\<close>
+
+proposition locally_compact:
+ fixes s :: "'a :: metric_space set"
+ shows
+ "locally compact s \<longleftrightarrow>
+ (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+ openin (subtopology euclidean s) u \<and> compact v)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply clarify
+ apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
+ by auto
+next
+ assume r [rule_format]: ?rhs
+ have *: "\<exists>u v.
+ openin (subtopology euclidean s) u \<and>
+ compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
+ if "open T" "x \<in> s" "x \<in> T" for x T
+ proof -
+ obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u"
+ using r [OF \<open>x \<in> s\<close>] by auto
+ obtain e where "e>0" and e: "cball x e \<subseteq> T"
+ using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
+ show ?thesis
+ apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
+ apply (rule_tac x="cball x e \<inter> v" in exI)
+ using that \<open>e > 0\<close> e uv
+ apply auto
+ done
+ qed
+ show ?lhs
+ apply (rule locallyI)
+ apply (subst (asm) openin_open)
+ apply (blast intro: *)
+ done
+qed
+
+lemma locally_compactE:
+ fixes s :: "'a :: metric_space set"
+ assumes "locally compact s"
+ obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
+ openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+using assms
+unfolding locally_compact by metis
+
+lemma locally_compact_alt:
+ fixes s :: "'a :: heine_borel set"
+ shows "locally compact s \<longleftrightarrow>
+ (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
+ openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
+apply (simp add: locally_compact)
+apply (intro ball_cong ex_cong refl iffI)
+apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
+by (meson closure_subset compact_closure)
+
+lemma locally_compact_Int_cball:
+ fixes s :: "'a :: heine_borel set"
+ shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (simp add: locally_compact openin_contains_cball)
+ apply (clarify | assumption | drule bspec)+
+ by (metis (no_types, lifting) compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
+next
+ assume ?rhs
+ then show ?lhs
+ apply (simp add: locally_compact openin_contains_cball)
+ apply (clarify | assumption | drule bspec)+
+ apply (rule_tac x="ball x e \<inter> s" in exI, simp)
+ apply (rule_tac x="cball x e \<inter> s" in exI)
+ using compact_eq_bounded_closed
+ apply auto
+ apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
+ done
+qed
+
+lemma locally_compact_compact:
+ fixes s :: "'a :: heine_borel set"
+ shows "locally compact s \<longleftrightarrow>
+ (\<forall>k. k \<subseteq> s \<and> compact k
+ \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+ openin (subtopology euclidean s) u \<and> compact v))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then obtain u v where
+ uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
+ openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+ by (metis locally_compactE)
+ have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+ if "k \<subseteq> s" "compact k" for k
+ proof -
+ have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
+ \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
+ using that by (simp add: compact_eq_openin_cover)
+ moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c"
+ using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
+ moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
+ using that by clarsimp (meson subsetCE uv)
+ ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D"
+ by metis
+ then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)"
+ by (metis finite_subset_image)
+ have Tuv: "\<Union>(u ` T) \<subseteq> \<Union>(v ` T)"
+ using T that by (force simp: dest!: uv)
+ show ?thesis
+ apply (rule_tac x="\<Union>(u ` T)" in exI)
+ apply (rule_tac x="\<Union>(v ` T)" in exI)
+ apply (simp add: Tuv)
+ using T that
+ apply (auto simp: dest!: uv)
+ done
+ qed
+ show ?rhs
+ by (blast intro: *)
+next
+ assume ?rhs
+ then show ?lhs
+ apply (clarsimp simp add: locally_compact)
+ apply (drule_tac x="{x}" in spec, simp)
+ done
+qed
+
+lemma open_imp_locally_compact:
+ fixes s :: "'a :: heine_borel set"
+ assumes "open s"
+ shows "locally compact s"
+proof -
+ have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+ if "x \<in> s" for x
+ proof -
+ obtain e where "e>0" and e: "cball x e \<subseteq> s"
+ using open_contains_cball assms \<open>x \<in> s\<close> by blast
+ have ope: "openin (subtopology euclidean s) (ball x e)"
+ by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
+ show ?thesis
+ apply (rule_tac x="ball x e" in exI)
+ apply (rule_tac x="cball x e" in exI)
+ using \<open>e > 0\<close> e apply (auto simp: ope)
+ done
+ qed
+ show ?thesis
+ unfolding locally_compact
+ by (blast intro: *)
+qed
+
+lemma closed_imp_locally_compact:
+ fixes s :: "'a :: heine_borel set"
+ assumes "closed s"
+ shows "locally compact s"
+proof -
+ have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+ openin (subtopology euclidean s) u \<and> compact v"
+ if "x \<in> s" for x
+ proof -
+ show ?thesis
+ apply (rule_tac x = "s \<inter> ball x 1" in exI)
+ apply (rule_tac x = "s \<inter> cball x 1" in exI)
+ using \<open>x \<in> s\<close> assms apply auto
+ done
+ qed
+ show ?thesis
+ unfolding locally_compact
+ by (blast intro: *)
+qed
+
+lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
+ by (simp add: closed_imp_locally_compact)
+
+lemma locally_compact_Int:
+ fixes s :: "'a :: t2_space set"
+ shows "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
+by (simp add: compact_Int locally_Int)
+
+lemma locally_compact_closedin:
+ fixes s :: "'a :: heine_borel set"
+ shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk>
+ \<Longrightarrow> locally compact t"
+unfolding closedin_closed
+using closed_imp_locally_compact locally_compact_Int by blast
+
+lemma locally_compact_delete:
+ fixes s :: "'a :: t1_space set"
+ shows "locally compact s \<Longrightarrow> locally compact (s - {a})"
+ by (auto simp: openin_delete locally_open_subset)
+
+lemma locally_closed:
+ fixes s :: "'a :: heine_borel set"
+ shows "locally closed s \<longleftrightarrow> locally compact s"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (simp only: locally_def)
+ apply (erule all_forward imp_forward asm_rl exE)+
+ apply (rule_tac x = "u \<inter> ball x 1" in exI)
+ apply (rule_tac x = "v \<inter> cball x 1" in exI)
+ apply (force intro: openin_trans)
+ done
+next
+ assume ?rhs then show ?lhs
+ using compact_eq_bounded_closed locally_mono by blast
+qed
+
+lemma locally_compact_openin_Un:
+ fixes S :: "'a::euclidean_space set"
+ assumes LCS: "locally compact S" and LCT:"locally compact T"
+ and opS: "openin (subtopology euclidean (S \<union> T)) S"
+ and opT: "openin (subtopology euclidean (S \<union> T)) T"
+ shows "locally compact (S \<union> T)"
+proof -
+ have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
+ proof -
+ obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+ using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+ moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> S"
+ by (meson \<open>x \<in> S\<close> opS openin_contains_cball)
+ then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> S"
+ by force
+ ultimately show ?thesis
+ apply (rule_tac x="min e1 e2" in exI)
+ apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
+ by (metis closed_Int closed_cball inf_left_commute)
+ qed
+ moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
+ proof -
+ obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
+ using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+ moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
+ by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
+ then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
+ by force
+ ultimately show ?thesis
+ apply (rule_tac x="min e1 e2" in exI)
+ apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
+ by (metis closed_Int closed_cball inf_left_commute)
+ qed
+ ultimately show ?thesis
+ by (force simp: locally_compact_Int_cball)
+qed
+
+lemma locally_compact_closedin_Un:
+ fixes S :: "'a::euclidean_space set"
+ assumes LCS: "locally compact S" and LCT:"locally compact T"
+ and clS: "closedin (subtopology euclidean (S \<union> T)) S"
+ and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+ shows "locally compact (S \<union> T)"
+proof -
+ have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
+ proof -
+ obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+ using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+ moreover
+ obtain e2 where "e2 > 0" and e2: "closed (cball x e2 \<inter> T)"
+ using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+ ultimately show ?thesis
+ apply (rule_tac x="min e1 e2" in exI)
+ apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+ by (metis closed_Int closed_Un closed_cball inf_left_commute)
+ qed
+ moreover
+ have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
+ proof -
+ obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+ using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+ moreover
+ obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
+ using clT x by (fastforce simp: openin_contains_cball closedin_def)
+ then have "closed (cball x e2 \<inter> T)"
+ proof -
+ have "{} = T - (T - cball x e2)"
+ using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
+ then show ?thesis
+ by (simp add: Diff_Diff_Int inf_commute)
+ qed
+ ultimately show ?thesis
+ apply (rule_tac x="min e1 e2" in exI)
+ apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+ by (metis closed_Int closed_Un closed_cball inf_left_commute)
+ qed
+ moreover
+ have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
+ proof -
+ obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
+ using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+ moreover
+ obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
+ using clS x by (fastforce simp: openin_contains_cball closedin_def)
+ then have "closed (cball x e2 \<inter> S)"
+ by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
+ ultimately show ?thesis
+ apply (rule_tac x="min e1 e2" in exI)
+ apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+ by (metis closed_Int closed_Un closed_cball inf_left_commute)
+ qed
+ ultimately show ?thesis
+ by (auto simp: locally_compact_Int_cball)
+qed
+
+lemma locally_compact_Times:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ shows "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
+ by (auto simp: compact_Times locally_Times)
+
+lemma locally_compact_compact_subopen:
+ fixes S :: "'a :: heine_borel set"
+ shows
+ "locally compact S \<longleftrightarrow>
+ (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
+ \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
+ openin (subtopology euclidean S) U \<and> compact V))"
+ (is "?lhs = ?rhs")
+proof
+ assume L: ?lhs
+ show ?rhs
+ proof clarify
+ fix K :: "'a set" and T :: "'a set"
+ assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
+ obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
+ and ope: "openin (subtopology euclidean S) U"
+ using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
+ show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
+ openin (subtopology euclidean S) U \<and> compact V"
+ proof (intro exI conjI)
+ show "K \<subseteq> U \<inter> T"
+ by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
+ show "U \<inter> T \<subseteq> closure(U \<inter> T)"
+ by (rule closure_subset)
+ show "closure (U \<inter> T) \<subseteq> S"
+ by (metis \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S\<close> \<open>compact V\<close> closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
+ show "openin (subtopology euclidean S) (U \<inter> T)"
+ by (simp add: \<open>open T\<close> ope openin_Int_open)
+ show "compact (closure (U \<inter> T))"
+ by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> bounded_subset compact_closure compact_eq_bounded_closed)
+ qed auto
+ qed
+next
+ assume ?rhs then show ?lhs
+ unfolding locally_compact_compact
+ by (metis open_openin openin_topspace subtopology_superset top.extremum topspace_euclidean_subtopology)
+qed
+
+
+subsection\<open>Sura-Bura's results about compact components of sets\<close>
+
+proposition Sura_Bura_compact:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S" and C: "C \<in> components S"
+ shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean S) T \<and>
+ closedin (subtopology euclidean S) T}"
+ (is "C = \<Inter>?\<T>")
+proof
+ obtain x where x: "C = connected_component_set S x" and "x \<in> S"
+ using C by (auto simp: components_def)
+ have "C \<subseteq> S"
+ by (simp add: C in_components_subset)
+ have "\<Inter>?\<T> \<subseteq> connected_component_set S x"
+ proof (rule connected_component_maximal)
+ have "x \<in> C"
+ by (simp add: \<open>x \<in> S\<close> x)
+ then show "x \<in> \<Inter>?\<T>"
+ by blast
+ have clo: "closed (\<Inter>?\<T>)"
+ by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
+ have False
+ if K1: "closedin (subtopology euclidean (\<Inter>?\<T>)) K1" and
+ K2: "closedin (subtopology euclidean (\<Inter>?\<T>)) K2" and
+ K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
+ for K1 K2
+ proof -
+ have "closed K1" "closed K2"
+ using closedin_closed_trans clo K1 K2 by blast+
+ then obtain V1 V2 where "open V1" "open V2" "K1 \<subseteq> V1" "K2 \<subseteq> V2" and V12: "V1 \<inter> V2 = {}"
+ using separation_normal \<open>K1 \<inter> K2 = {}\<close> by metis
+ have SV12_ne: "(S - (V1 \<union> V2)) \<inter> (\<Inter>?\<T>) \<noteq> {}"
+ proof (rule compact_imp_fip)
+ show "compact (S - (V1 \<union> V2))"
+ by (simp add: \<open>open V1\<close> \<open>open V2\<close> \<open>compact S\<close> compact_diff open_Un)
+ show clo\<T>: "closed T" if "T \<in> ?\<T>" for T
+ using that \<open>compact S\<close>
+ by (force intro: closedin_closed_trans simp add: compact_imp_closed)
+ show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
+ proof
+ assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
+ obtain D where opeD: "openin (subtopology euclidean S) D"
+ and cloD: "closedin (subtopology euclidean S) D"
+ and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
+ proof (cases "\<F> = {}")
+ case True
+ with \<open>C \<subseteq> S\<close> djo that show ?thesis
+ by force
+ next
+ case False show ?thesis
+ proof
+ show ope: "openin (subtopology euclidean S) (\<Inter>\<F>)"
+ using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
+ then show "closedin (subtopology euclidean S) (\<Inter>\<F>)"
+ by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
+ show "C \<subseteq> \<Inter>\<F>"
+ using \<F> by auto
+ show "\<Inter>\<F> \<subseteq> V1 \<union> V2"
+ using ope djo openin_imp_subset by fastforce
+ qed
+ qed
+ have "connected C"
+ by (simp add: x)
+ have "closed D"
+ using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
+ have cloV1: "closedin (subtopology euclidean D) (D \<inter> closure V1)"
+ and cloV2: "closedin (subtopology euclidean D) (D \<inter> closure V2)"
+ by (simp_all add: closedin_closed_Int)
+ moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
+ apply safe
+ using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
+ apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
+ done
+ ultimately have cloDV1: "closedin (subtopology euclidean D) (D \<inter> V1)"
+ and cloDV2: "closedin (subtopology euclidean D) (D \<inter> V2)"
+ by metis+
+ then obtain U1 U2 where "closed U1" "closed U2"
+ and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
+ by (auto simp: closedin_closed)
+ have "D \<inter> U1 \<inter> C \<noteq> {}"
+ proof
+ assume "D \<inter> U1 \<inter> C = {}"
+ then have *: "C \<subseteq> D \<inter> V2"
+ using D1 DV12 \<open>C \<subseteq> D\<close> by auto
+ have "\<Inter>?\<T> \<subseteq> D \<inter> V2"
+ apply (rule Inter_lower)
+ using * apply simp
+ by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
+ then show False
+ using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
+ qed
+ moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
+ proof
+ assume "D \<inter> U2 \<inter> C = {}"
+ then have *: "C \<subseteq> D \<inter> V1"
+ using D2 DV12 \<open>C \<subseteq> D\<close> by auto
+ have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
+ apply (rule Inter_lower)
+ using * apply simp
+ by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
+ then show False
+ using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
+ qed
+ ultimately show False
+ using \<open>connected C\<close> unfolding connected_closed
+ apply (simp only: not_ex)
+ apply (drule_tac x="D \<inter> U1" in spec)
+ apply (drule_tac x="D \<inter> U2" in spec)
+ using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
+ by blast
+ qed
+ qed
+ show False
+ by (metis (full_types) DiffE UnE Un_upper2 SV12_ne \<open>K1 \<subseteq> V1\<close> \<open>K2 \<subseteq> V2\<close> disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
+ qed
+ then show "connected (\<Inter>?\<T>)"
+ by (auto simp: connected_closedin_eq)
+ show "\<Inter>?\<T> \<subseteq> S"
+ by (fastforce simp: C in_components_subset)
+ qed
+ with x show "\<Inter>?\<T> \<subseteq> C" by simp
+qed auto
+
+
+corollary Sura_Bura_clopen_subset:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
+ and U: "open U" "C \<subseteq> U"
+ obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+proof (rule ccontr)
+ assume "\<not> thesis"
+ with that have neg: "\<nexists>K. openin (subtopology euclidean S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
+ by metis
+ obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
+ and opeSV: "openin (subtopology euclidean S) V"
+ using S U \<open>compact C\<close>
+ apply (simp add: locally_compact_compact_subopen)
+ by (meson C in_components_subset)
+ let ?\<T> = "{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> compact T \<and> T \<subseteq> K}"
+ have CK: "C \<in> components K"
+ by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
+ with \<open>compact K\<close>
+ have "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> closedin (subtopology euclidean K) T}"
+ by (simp add: Sura_Bura_compact)
+ then have Ceq: "C = \<Inter>?\<T>"
+ by (simp add: closedin_compact_eq \<open>compact K\<close>)
+ obtain W where "open W" and W: "V = S \<inter> W"
+ using opeSV by (auto simp: openin_open)
+ have "-(U \<inter> W) \<inter> \<Inter>?\<T> \<noteq> {}"
+ proof (rule closed_imp_fip_compact)
+ show "- (U \<inter> W) \<inter> \<Inter>\<F> \<noteq> {}"
+ if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
+ proof (cases "\<F> = {}")
+ case True
+ have False if "U = UNIV" "W = UNIV"
+ proof -
+ have "V = S"
+ by (simp add: W \<open>W = UNIV\<close>)
+ with neg show False
+ using \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> \<open>V \<subseteq> U\<close> \<open>compact K\<close> by auto
+ qed
+ with True show ?thesis
+ by auto
+ next
+ case False
+ show ?thesis
+ proof
+ assume "- (U \<inter> W) \<inter> \<Inter>\<F> = {}"
+ then have FUW: "\<Inter>\<F> \<subseteq> U \<inter> W"
+ by blast
+ have "C \<subseteq> \<Inter>\<F>"
+ using \<F> by auto
+ moreover have "compact (\<Inter>\<F>)"
+ by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
+ moreover have "\<Inter>\<F> \<subseteq> K"
+ using False that(2) by fastforce
+ moreover have opeKF: "openin (subtopology euclidean K) (\<Inter>\<F>)"
+ using False \<F> \<open>finite \<F>\<close> by blast
+ then have opeVF: "openin (subtopology euclidean V) (\<Inter>\<F>)"
+ using W \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> opeKF \<open>\<Inter>\<F> \<subseteq> K\<close> FUW openin_subset_trans by fastforce
+ then have "openin (subtopology euclidean S) (\<Inter>\<F>)"
+ by (metis opeSV openin_trans)
+ moreover have "\<Inter>\<F> \<subseteq> U"
+ by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
+ ultimately show False
+ using neg by blast
+ qed
+ qed
+ qed (use \<open>open W\<close> \<open>open U\<close> in auto)
+ with W Ceq \<open>C \<subseteq> V\<close> \<open>C \<subseteq> U\<close> show False
+ by auto
+qed
+
+
+corollary Sura_Bura_clopen_subset_alt:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
+ and opeSU: "openin (subtopology euclidean S) U" and "C \<subseteq> U"
+ obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+proof -
+ obtain V where "open V" "U = S \<inter> V"
+ using opeSU by (auto simp: openin_open)
+ with \<open>C \<subseteq> U\<close> have "C \<subseteq> V"
+ by auto
+ then show ?thesis
+ using Sura_Bura_clopen_subset [OF S C \<open>compact C\<close> \<open>open V\<close>]
+ by (metis \<open>U = S \<inter> V\<close> inf.bounded_iff openin_imp_subset that)
+qed
+
+corollary Sura_Bura:
+ fixes S :: "'a::euclidean_space set"
+ assumes "locally compact S" "C \<in> components S" "compact C"
+ shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (subtopology euclidean S) K}"
+ (is "C = ?rhs")
+proof
+ show "?rhs \<subseteq> C"
+ proof (clarsimp, rule ccontr)
+ fix x
+ assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (subtopology euclidean S) X \<longrightarrow> x \<in> X"
+ and "x \<notin> C"
+ obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
+ using separation_normal [of "{x}" C]
+ by (metis Int_empty_left \<open>x \<notin> C\<close> \<open>compact C\<close> closed_empty closed_insert compact_imp_closed insert_disjoint(1))
+ have "x \<notin> V"
+ using \<open>U \<inter> V = {}\<close> \<open>{x} \<subseteq> U\<close> by blast
+ then show False
+ by (meson "*" Sura_Bura_clopen_subset \<open>C \<subseteq> V\<close> \<open>open V\<close> assms(1) assms(2) assms(3) subsetCE)
+ qed
+qed blast
+
+
+subsection\<open>Special cases of local connectedness and path connectedness\<close>
+
+lemma locally_connected_1:
+ assumes
+ "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
+ \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and>
+ connected u \<and> x \<in> u \<and> u \<subseteq> v"
+ shows "locally connected S"
+apply (clarsimp simp add: locally_def)
+apply (drule assms; blast)
+done
+
+lemma locally_connected_2:
+ assumes "locally connected S"
+ "openin (subtopology euclidean S) t"
+ "x \<in> t"
+ shows "openin (subtopology euclidean S) (connected_component_set t x)"
+proof -
+ { fix y :: 'a
+ let ?SS = "subtopology euclidean S"
+ assume 1: "openin ?SS t"
+ "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
+ and "connected_component t x y"
+ then have "y \<in> t" and y: "y \<in> connected_component_set t x"
+ using connected_component_subset by blast+
+ obtain F where
+ "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
+ by moura
+ then obtain G where
+ "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
+ by moura
+ then have *: "openin ?SS (F y t) \<and> connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
+ using 1 \<open>y \<in> t\<close> by presburger
+ have "G y t \<subseteq> connected_component_set t y"
+ by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
+ then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> connected_component_set t x"
+ by (metis (no_types) * connected_component_eq dual_order.trans y)
+ }
+ then show ?thesis
+ using assms openin_subopen by (force simp: locally_def)
+qed
+
+lemma locally_connected_3:
+ assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean S)
+ (connected_component_set t x)"
+ "openin (subtopology euclidean S) v" "x \<in> v"
+ shows "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
+using assms connected_component_subset by fastforce
+
+lemma locally_connected:
+ "locally connected S \<longleftrightarrow>
+ (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+ \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
+by (metis locally_connected_1 locally_connected_2 locally_connected_3)
+
+lemma locally_connected_open_connected_component:
+ "locally connected S \<longleftrightarrow>
+ (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
+ \<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))"
+by (metis locally_connected_1 locally_connected_2 locally_connected_3)
+
+lemma locally_path_connected_1:
+ assumes
+ "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
+ \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+ shows "locally path_connected S"
+apply (clarsimp simp add: locally_def)
+apply (drule assms; blast)
+done
+
+lemma locally_path_connected_2:
+ assumes "locally path_connected S"
+ "openin (subtopology euclidean S) t"
+ "x \<in> t"
+ shows "openin (subtopology euclidean S) (path_component_set t x)"
+proof -
+ { fix y :: 'a
+ let ?SS = "subtopology euclidean S"
+ assume 1: "openin ?SS t"
+ "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
+ and "path_component t x y"
+ then have "y \<in> t" and y: "y \<in> path_component_set t x"
+ using path_component_mem(2) by blast+
+ obtain F where
+ "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. path_connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. path_connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
+ by moura
+ then obtain G where
+ "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. path_connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> path_connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
+ by moura
+ then have *: "openin ?SS (F y t) \<and> path_connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
+ using 1 \<open>y \<in> t\<close> by presburger
+ have "G y t \<subseteq> path_component_set t y"
+ using * path_component_maximal set_rev_mp by blast
+ then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x"
+ by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> dual_order.trans path_component_eq y)
+ }
+ then show ?thesis
+ using assms openin_subopen by (force simp: locally_def)
+qed
+
+lemma locally_path_connected_3:
+ assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)"
+ "openin (subtopology euclidean S) v" "x \<in> v"
+ shows "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+proof -
+ have "path_component v x x"
+ by (meson assms(3) path_component_refl)
+ then show ?thesis
+ by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
+qed
+
+proposition locally_path_connected:
+ "locally path_connected S \<longleftrightarrow>
+ (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+ \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))"
+ by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
+
+proposition locally_path_connected_open_path_component:
+ "locally path_connected S \<longleftrightarrow>
+ (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
+ \<longrightarrow> openin (subtopology euclidean S) (path_component_set t x))"
+ by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
+
+lemma locally_connected_open_component:
+ "locally connected S \<longleftrightarrow>
+ (\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t
+ \<longrightarrow> openin (subtopology euclidean S) c)"
+by (metis components_iff locally_connected_open_connected_component)
+
+proposition locally_connected_im_kleinen:
+ "locally connected S \<longleftrightarrow>
+ (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+ \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+ x \<in> u \<and> u \<subseteq> v \<and>
+ (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ by (fastforce simp add: locally_connected)
+next
+ assume ?rhs
+ have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c"
+ if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
+ proof -
+ from that \<open>?rhs\<close> [rule_format, of t x]
+ obtain u where u:
+ "openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
+ (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
+ using in_components_subset by auto
+ obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
+ "\<forall>x y. (\<exists>z. z \<in> x \<and> y = connected_component_set x z) = (F x y \<in> x \<and> y = connected_component_set x (F x y))"
+ by moura
+ then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)"
+ by (meson components_iff c)
+ obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
+ G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)"
+ by moura
+ have "G c u \<notin> u \<or> G c u \<in> c"
+ using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3))
+ then show ?thesis
+ using G u by auto
+ qed
+ show ?lhs
+ apply (clarsimp simp add: locally_connected_open_component)
+ apply (subst openin_subopen)
+ apply (blast intro: *)
+ done
+qed
+
+proposition locally_path_connected_im_kleinen:
+ "locally path_connected S \<longleftrightarrow>
+ (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+ \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+ x \<in> u \<and> u \<subseteq> v \<and>
+ (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
+ pathstart p = x \<and> pathfinish p = y))))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (simp add: locally_path_connected path_connected_def)
+ apply (erule all_forward ex_forward imp_forward conjE | simp)+
+ by (meson dual_order.trans)
+next
+ assume ?rhs
+ have *: "\<exists>T. openin (subtopology euclidean S) T \<and>
+ x \<in> T \<and> T \<subseteq> path_component_set u z"
+ if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x
+ proof -
+ have "x \<in> u"
+ by (meson c path_component_mem(2))
+ with that \<open>?rhs\<close> [rule_format, of u x]
+ obtain U where U:
+ "openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
+ (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
+ by blast
+ show ?thesis
+ apply (rule_tac x=U in exI)
+ apply (auto simp: U)
+ apply (metis U c path_component_trans path_component_def)
+ done
+ qed
+ show ?lhs
+ apply (clarsimp simp add: locally_path_connected_open_path_component)
+ apply (subst openin_subopen)
+ apply (blast intro: *)
+ done
+qed
+
+lemma locally_path_connected_imp_locally_connected:
+ "locally path_connected S \<Longrightarrow> locally connected S"
+using locally_mono path_connected_imp_connected by blast
+
+lemma locally_connected_components:
+ "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
+by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
+
+lemma locally_path_connected_components:
+ "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally path_connected c"
+by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)
+
+lemma locally_path_connected_connected_component:
+ "locally path_connected S \<Longrightarrow> locally path_connected (connected_component_set S x)"
+by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)
+
+lemma open_imp_locally_path_connected:
+ fixes S :: "'a :: real_normed_vector set"
+ shows "open S \<Longrightarrow> locally path_connected S"
+apply (rule locally_mono [of convex])
+apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
+apply (meson open_ball centre_in_ball convex_ball openE order_trans)
+done
+
+lemma open_imp_locally_connected:
+ fixes S :: "'a :: real_normed_vector set"
+ shows "open S \<Longrightarrow> locally connected S"
+by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)
+
+lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
+ by (simp add: open_imp_locally_path_connected)
+
+lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
+ by (simp add: open_imp_locally_connected)
+
+lemma openin_connected_component_locally_connected:
+ "locally connected S
+ \<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)"
+apply (simp add: locally_connected_open_connected_component)
+by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
+
+lemma openin_components_locally_connected:
+ "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c"
+ using locally_connected_open_component openin_subtopology_self by blast
+
+lemma openin_path_component_locally_path_connected:
+ "locally path_connected S
+ \<Longrightarrow> openin (subtopology euclidean S) (path_component_set S x)"
+by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
+
+lemma closedin_path_component_locally_path_connected:
+ "locally path_connected S
+ \<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)"
+apply (simp add: closedin_def path_component_subset complement_path_component_Union)
+apply (rule openin_Union)
+using openin_path_component_locally_path_connected by auto
+
+lemma convex_imp_locally_path_connected:
+ fixes S :: "'a:: real_normed_vector set"
+ shows "convex S \<Longrightarrow> locally path_connected S"
+apply (clarsimp simp add: locally_path_connected)
+apply (subst (asm) openin_open)
+apply clarify
+apply (erule (1) openE)
+apply (rule_tac x = "S \<inter> ball x e" in exI)
+apply (force simp: convex_Int convex_imp_path_connected)
+done
+
+lemma convex_imp_locally_connected:
+ fixes S :: "'a:: real_normed_vector set"
+ shows "convex S \<Longrightarrow> locally connected S"
+ by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
+
+
+subsection\<open>Relations between components and path components\<close>
+
+lemma path_component_eq_connected_component:
+ assumes "locally path_connected S"
+ shows "(path_component S x = connected_component S x)"
+proof (cases "x \<in> S")
+ case True
+ have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+ apply (rule openin_subset_trans [of S])
+ apply (intro conjI openin_path_component_locally_path_connected [OF assms])
+ using path_component_subset_connected_component apply (auto simp: connected_component_subset)
+ done
+ moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+ apply (rule closedin_subset_trans [of S])
+ apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
+ using path_component_subset_connected_component apply (auto simp: connected_component_subset)
+ done
+ ultimately have *: "path_component_set S x = connected_component_set S x"
+ by (metis connected_connected_component connected_clopen True path_component_eq_empty)
+ then show ?thesis
+ by blast
+next
+ case False then show ?thesis
+ by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
+qed
+
+lemma path_component_eq_connected_component_set:
+ "locally path_connected S \<Longrightarrow> (path_component_set S x = connected_component_set S x)"
+by (simp add: path_component_eq_connected_component)
+
+lemma locally_path_connected_path_component:
+ "locally path_connected S \<Longrightarrow> locally path_connected (path_component_set S x)"
+using locally_path_connected_connected_component path_component_eq_connected_component by fastforce
+
+lemma open_path_connected_component:
+ fixes S :: "'a :: real_normed_vector set"
+ shows "open S \<Longrightarrow> path_component S x = connected_component S x"
+by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)
+
+lemma open_path_connected_component_set:
+ fixes S :: "'a :: real_normed_vector set"
+ shows "open S \<Longrightarrow> path_component_set S x = connected_component_set S x"
+by (simp add: open_path_connected_component)
+
+proposition locally_connected_quotient_image:
+ assumes lcS: "locally connected S"
+ and oo: "\<And>T. T \<subseteq> f ` S
+ \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
+ openin (subtopology euclidean (f ` S)) T"
+ shows "locally connected (f ` S)"
+proof (clarsimp simp: locally_connected_open_component)
+ fix U C
+ assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C \<in> components U"
+ then have "C \<subseteq> U" "U \<subseteq> f ` S"
+ by (meson in_components_subset openin_imp_subset)+
+ then have "openin (subtopology euclidean (f ` S)) C \<longleftrightarrow>
+ openin (subtopology euclidean S) (S \<inter> f -` C)"
+ by (auto simp: oo)
+ moreover have "openin (subtopology euclidean S) (S \<inter> f -` C)"
+ proof (subst openin_subopen, clarify)
+ fix x
+ assume "x \<in> S" "f x \<in> C"
+ show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
+ proof (intro conjI exI)
+ show "openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+ proof (rule ccontr)
+ assume **: "\<not> openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+ then have "x \<notin> (S \<inter> f -` U)"
+ using \<open>U \<subseteq> f ` S\<close> opefSU lcS locally_connected_2 oo by blast
+ with ** show False
+ by (metis (no_types) connected_component_eq_empty empty_iff openin_subopen)
+ qed
+ next
+ show "x \<in> connected_component_set (S \<inter> f -` U) x"
+ using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by auto
+ next
+ have contf: "continuous_on S f"
+ by (simp add: continuous_on_open oo openin_imp_subset)
+ then have "continuous_on (connected_component_set (S \<inter> f -` U) x) f"
+ apply (rule continuous_on_subset)
+ using connected_component_subset apply blast
+ done
+ then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
+ by (rule connected_continuous_image [OF _ connected_connected_component])
+ moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
+ using connected_component_in by blast
+ moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
+ using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
+ ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
+ by (rule components_maximal [OF \<open>C \<in> components U\<close>])
+ have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
+ using connected_component_subset fC by blast
+ have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
+ proof -
+ { assume "x \<in> connected_component_set (S \<inter> f -` U) x"
+ then have ?thesis
+ using cUC connected_component_idemp connected_component_mono by blast }
+ then show ?thesis
+ using connected_component_eq_empty by auto
+ qed
+ also have "\<dots> \<subseteq> (S \<inter> f -` C)"
+ by (rule connected_component_subset)
+ finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
+ qed
+ qed
+ ultimately show "openin (subtopology euclidean (f ` S)) C"
+ by metis
+qed
+
+text\<open>The proof resembles that above but is not identical!\<close>
+proposition locally_path_connected_quotient_image:
+ assumes lcS: "locally path_connected S"
+ and oo: "\<And>T. T \<subseteq> f ` S
+ \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow> openin (subtopology euclidean (f ` S)) T"
+ shows "locally path_connected (f ` S)"
+proof (clarsimp simp: locally_path_connected_open_path_component)
+ fix U y
+ assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "y \<in> U"
+ then have "path_component_set U y \<subseteq> U" "U \<subseteq> f ` S"
+ by (meson path_component_subset openin_imp_subset)+
+ then have "openin (subtopology euclidean (f ` S)) (path_component_set U y) \<longleftrightarrow>
+ openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+ proof -
+ have "path_component_set U y \<subseteq> f ` S"
+ using \<open>U \<subseteq> f ` S\<close> \<open>path_component_set U y \<subseteq> U\<close> by blast
+ then show ?thesis
+ using oo by blast
+ qed
+ moreover have "openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+ proof (subst openin_subopen, clarify)
+ fix x
+ assume "x \<in> S" and Uyfx: "path_component U y (f x)"
+ then have "f x \<in> U"
+ using path_component_mem by blast
+ show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
+ proof (intro conjI exI)
+ show "openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+ proof (rule ccontr)
+ assume **: "\<not> openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+ then have "x \<notin> (S \<inter> f -` U)"
+ by (metis (no_types, lifting) \<open>U \<subseteq> f ` S\<close> opefSU lcS oo locally_path_connected_open_path_component)
+ then show False
+ using ** \<open>path_component_set U y \<subseteq> U\<close> \<open>x \<in> S\<close> \<open>path_component U y (f x)\<close> by blast
+ qed
+ next
+ show "x \<in> path_component_set (S \<inter> f -` U) x"
+ by (simp add: \<open>f x \<in> U\<close> \<open>x \<in> S\<close> path_component_refl)
+ next
+ have contf: "continuous_on S f"
+ by (simp add: continuous_on_open oo openin_imp_subset)
+ then have "continuous_on (path_component_set (S \<inter> f -` U) x) f"
+ apply (rule continuous_on_subset)
+ using path_component_subset apply blast
+ done
+ then have "path_connected (f ` path_component_set (S \<inter> f -` U) x)"
+ by (simp add: path_connected_continuous_image)
+ moreover have "f ` path_component_set (S \<inter> f -` U) x \<subseteq> U"
+ using path_component_mem by fastforce
+ moreover have "f x \<in> f ` path_component_set (S \<inter> f -` U) x"
+ by (force simp: \<open>x \<in> S\<close> \<open>f x \<in> U\<close> path_component_refl_eq)
+ ultimately have "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U (f x)"
+ by (meson path_component_maximal)
+ also have "\<dots> \<subseteq> path_component_set U y"
+ by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
+ finally have fC: "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U y" .
+ have cUC: "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)"
+ using path_component_subset fC by blast
+ have "path_component_set (S \<inter> f -` U) x \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) x"
+ proof -
+ have "\<And>a. path_component_set (path_component_set (S \<inter> f -` U) x) a \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) a"
+ using cUC path_component_mono by blast
+ then show ?thesis
+ using path_component_path_component by blast
+ qed
+ also have "\<dots> \<subseteq> (S \<inter> f -` path_component_set U y)"
+ by (rule path_component_subset)
+ finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
+ qed
+ qed
+ ultimately show "openin (subtopology euclidean (f ` S)) (path_component_set U y)"
+ by metis
+qed
+
+subsection%unimportant\<open>Components, continuity, openin, closedin\<close>
+
+lemma continuous_on_components_gen:
+ fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
+ assumes "\<And>c. c \<in> components S \<Longrightarrow>
+ openin (subtopology euclidean S) c \<and> continuous_on c f"
+ shows "continuous_on S f"
+proof (clarsimp simp: continuous_openin_preimage_eq)
+ fix t :: "'b set"
+ assume "open t"
+ have *: "S \<inter> f -` t = (\<Union>c \<in> components S. c \<inter> f -` t)"
+ by auto
+ show "openin (subtopology euclidean S) (S \<inter> f -` t)"
+ unfolding * using \<open>open t\<close> assms continuous_openin_preimage_gen openin_trans openin_Union by blast
+qed
+
+lemma continuous_on_components:
+ fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
+ assumes "locally connected S "
+ "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
+ shows "continuous_on S f"
+apply (rule continuous_on_components_gen)
+apply (auto simp: assms intro: openin_components_locally_connected)
+done
+
+lemma continuous_on_components_eq:
+ "locally connected S
+ \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
+by (meson continuous_on_components continuous_on_subset in_components_subset)
+
+lemma continuous_on_components_open:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "open S "
+ "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
+ shows "continuous_on S f"
+using continuous_on_components open_imp_locally_connected assms by blast
+
+lemma continuous_on_components_open_eq:
+ fixes S :: "'a::real_normed_vector set"
+ shows "open S \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
+using continuous_on_subset in_components_subset
+by (blast intro: continuous_on_components_open)
+
+lemma closedin_union_complement_components:
+ assumes u: "locally connected u"
+ and S: "closedin (subtopology euclidean u) S"
+ and cuS: "c \<subseteq> components(u - S)"
+ shows "closedin (subtopology euclidean u) (S \<union> \<Union>c)"
+proof -
+ have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c'
+ by (simp add: disjnt_def) blast
+ have "S \<subseteq> u"
+ using S closedin_imp_subset by blast
+ moreover have "u - S = \<Union>c \<union> \<Union>(components (u - S) - c)"
+ by (metis Diff_partition Union_components Union_Un_distrib assms(3))
+ moreover have "disjnt (\<Union>c) (\<Union>(components (u - S) - c))"
+ apply (rule di)
+ by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
+ ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))"
+ by (auto simp: disjnt_def)
+ have *: "openin (subtopology euclidean u) (\<Union>(components (u - S) - c))"
+ apply (rule openin_Union)
+ apply (rule openin_trans [of "u - S"])
+ apply (simp add: u S locally_diff_closed openin_components_locally_connected)
+ apply (simp add: openin_diff S)
+ done
+ have "openin (subtopology euclidean u) (u - (u - \<Union>(components (u - S) - c)))"
+ apply (rule openin_diff, simp)
+ apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
+ done
+ then show ?thesis
+ by (force simp: eq closedin_def)
+qed
+
+lemma closed_union_complement_components:
+ fixes S :: "'a::real_normed_vector set"
+ assumes S: "closed S" and c: "c \<subseteq> components(- S)"
+ shows "closed(S \<union> \<Union> c)"
+proof -
+ have "closedin (subtopology euclidean UNIV) (S \<union> \<Union>c)"
+ apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
+ using S c apply (simp_all add: Compl_eq_Diff_UNIV)
+ done
+ then show ?thesis by simp
+qed
+
+lemma closedin_Un_complement_component:
+ fixes S :: "'a::real_normed_vector set"
+ assumes u: "locally connected u"
+ and S: "closedin (subtopology euclidean u) S"
+ and c: " c \<in> components(u - S)"
+ shows "closedin (subtopology euclidean u) (S \<union> c)"
+proof -
+ have "closedin (subtopology euclidean u) (S \<union> \<Union>{c})"
+ using c by (blast intro: closedin_union_complement_components [OF u S])
+ then show ?thesis
+ by simp
+qed
+
+lemma closed_Un_complement_component:
+ fixes S :: "'a::real_normed_vector set"
+ assumes S: "closed S" and c: " c \<in> components(-S)"
+ shows "closed (S \<union> c)"
+ by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
+ locally_connected_UNIV subtopology_UNIV)
+
+
+subsection\<open>Existence of isometry between subspaces of same dimension\<close>
+
+lemma isometry_subset_subspace:
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes S: "subspace S"
+ and T: "subspace T"
+ and d: "dim S \<le> dim T"
+ obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+proof -
+ obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
+ and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+ and "independent B" "finite B" "card B = dim S" "span B = S"
+ by (metis orthonormal_basis_subspace [OF S] independent_finite)
+ obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
+ and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
+ and "independent C" "finite C" "card C = dim T" "span C = T"
+ by (metis orthonormal_basis_subspace [OF T] independent_finite)
+ obtain fb where "fb ` B \<subseteq> C" "inj_on fb B"
+ by (metis \<open>card B = dim S\<close> \<open>card C = dim T\<close> \<open>finite B\<close> \<open>finite C\<close> card_le_inj d)
+ then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
+ using Corth
+ apply (auto simp: pairwise_def orthogonal_clauses)
+ by (meson subsetD image_eqI inj_on_def)
+ obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
+ using linear_independent_extend \<open>independent B\<close> by fastforce
+ have "span (f ` B) \<subseteq> span C"
+ by (metis \<open>fb ` B \<subseteq> C\<close> ffb image_cong span_mono)
+ then have "f ` S \<subseteq> T"
+ unfolding \<open>span B = S\<close> \<open>span C = T\<close> span_linear_image[OF \<open>linear f\<close>] .
+ have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
+ using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto
+ have "norm (f x) = norm x" if "x \<in> S" for x
+ proof -
+ interpret linear f by fact
+ obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
+ using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
+ have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by (simp add: sum scale ffb x)
+ also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
+ apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
+ apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
+ done
+ also have "\<dots> = norm x ^2"
+ by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
+ finally show ?thesis
+ by (simp add: norm_eq_sqrt_inner)
+ qed
+ then show ?thesis
+ by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
+qed
+
+proposition isometries_subspaces:
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes S: "subspace S"
+ and T: "subspace T"
+ and d: "dim S = dim T"
+ obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
+ "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+ "\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x"
+ "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
+ "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
+proof -
+ obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
+ and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+ and "independent B" "finite B" "card B = dim S" "span B = S"
+ by (metis orthonormal_basis_subspace [OF S] independent_finite)
+ obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
+ and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
+ and "independent C" "finite C" "card C = dim T" "span C = T"
+ by (metis orthonormal_basis_subspace [OF T] independent_finite)
+ obtain fb where "bij_betw fb B C"
+ by (metis \<open>finite B\<close> \<open>finite C\<close> bij_betw_iff_card \<open>card B = dim S\<close> \<open>card C = dim T\<close> d)
+ then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
+ using Corth
+ apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
+ by (meson subsetD image_eqI inj_on_def)
+ obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
+ using linear_independent_extend \<open>independent B\<close> by fastforce
+ interpret f: linear f by fact
+ define gb where "gb \<equiv> inv_into B fb"
+ then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C"
+ using Borth
+ apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
+ by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
+ obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x"
+ using linear_independent_extend \<open>independent C\<close> by fastforce
+ interpret g: linear g by fact
+ have "span (f ` B) \<subseteq> span C"
+ by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb image_cong)
+ then have "f ` S \<subseteq> T"
+ unfolding \<open>span B = S\<close> \<open>span C = T\<close>
+ span_linear_image[OF \<open>linear f\<close>] .
+ have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
+ using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce
+ have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x
+ proof -
+ obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
+ using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
+ have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
+ using linear_sum [OF \<open>linear f\<close>] x by auto
+ also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
+ by (simp add: f.sum f.scale)
+ also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
+ by (simp add: ffb cong: sum.cong)
+ finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
+ then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
+ also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
+ apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
+ apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
+ done
+ also have "\<dots> = (norm x)\<^sup>2"
+ by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
+ finally show "norm (f x) = norm x"
+ by (simp add: norm_eq_sqrt_inner)
+ have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
+ also have "\<dots> = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
+ by (simp add: g.sum g.scale)
+ also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
+ by (simp add: g.scale)
+ also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R v)"
+ apply (rule sum.cong [OF refl])
+ using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
+ also have "\<dots> = x"
+ using x by blast
+ finally show "g (f x) = x" .
+ qed
+ have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x"
+ by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into)
+ have g [simp]: "f (g x) = x" if "x \<in> T" for x
+ proof -
+ obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
+ using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
+ have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
+ by (simp add: x g.sum)
+ also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
+ by (simp add: g.scale)
+ also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
+ by (simp add: ggb cong: sum.cong)
+ finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
+ also have "\<dots> = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
+ by (simp add: f.scale f.sum)
+ also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
+ by (simp add: f.scale f.sum)
+ also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R v)"
+ using \<open>bij_betw fb B C\<close>
+ by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
+ also have "\<dots> = x"
+ using x by blast
+ finally show "f (g x) = x" .
+ qed
+ have gim: "g ` T = S"
+ by (metis (full_types) S T \<open>f ` S \<subseteq> T\<close> d dim_eq_span dim_image_le f(2) g.linear_axioms
+ image_iff linear_subspace_image span_eq_iff subset_iff)
+ have fim: "f ` S = T"
+ using \<open>g ` T = S\<close> image_iff by fastforce
+ have [simp]: "norm (g x) = norm x" if "x \<in> T" for x
+ using fim that by auto
+ show ?thesis
+ apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>])
+ apply (simp_all add: fim gim)
+ done
+qed
+
+corollary isometry_subspaces:
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes S: "subspace S"
+ and T: "subspace T"
+ and d: "dim S = dim T"
+ obtains f where "linear f" "f ` S = T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+using isometries_subspaces [OF assms]
+by metis
+
+corollary isomorphisms_UNIV_UNIV:
+ assumes "DIM('M) = DIM('N)"
+ obtains f::"'M::euclidean_space \<Rightarrow>'N::euclidean_space" and g
+ where "linear f" "linear g"
+ "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y"
+ "\<And>x. g (f x) = x" "\<And>y. f(g y) = y"
+ using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])
+
+lemma homeomorphic_subspaces:
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes S: "subspace S"
+ and T: "subspace T"
+ and d: "dim S = dim T"
+ shows "S homeomorphic T"
+proof -
+ obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
+ "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
+ by (blast intro: isometries_subspaces [OF assms])
+ then show ?thesis
+ apply (simp add: homeomorphic_def homeomorphism_def)
+ apply (rule_tac x=f in exI)
+ apply (rule_tac x=g in exI)
+ apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
+ done
+qed
+
+lemma homeomorphic_affine_sets:
+ assumes "affine S" "affine T" "aff_dim S = aff_dim T"
+ shows "S homeomorphic T"
+proof (cases "S = {} \<or> T = {}")
+ case True with assms aff_dim_empty homeomorphic_empty show ?thesis
+ by metis
+next
+ case False
+ then obtain a b where ab: "a \<in> S" "b \<in> T" by auto
+ then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
+ using affine_diffs_subspace assms by blast+
+ have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
+ using assms ab by (simp add: aff_dim_eq_dim [OF hull_inc] image_def)
+ have "S homeomorphic ((+) (- a) ` S)"
+ by (simp add: homeomorphic_translation)
+ also have "\<dots> homeomorphic ((+) (- b) ` T)"
+ by (rule homeomorphic_subspaces [OF ss dd])
+ also have "\<dots> homeomorphic T"
+ using homeomorphic_sym homeomorphic_translation by auto
+ finally show ?thesis .
+qed
+
+
+subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
+
+locale%important Retracts =
+ fixes s h t k
+ assumes conth: "continuous_on s h"
+ and imh: "h ` s = t"
+ and contk: "continuous_on t k"
+ and imk: "k ` t \<subseteq> s"
+ and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
+
+begin
+
+lemma homotopically_trivial_retraction_gen:
+ assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+ and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+ and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+ and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f;
+ continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk>
+ \<Longrightarrow> homotopic_with P u s f g"
+ and contf: "continuous_on u f" and imf: "f ` u \<subseteq> t" and Qf: "Q f"
+ and contg: "continuous_on u g" and img: "g ` u \<subseteq> t" and Qg: "Q g"
+ shows "homotopic_with Q u t f g"
+proof -
+ have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
+ have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto
+ have "continuous_on u (k \<circ> f)"
+ using contf continuous_on_compose continuous_on_subset contk imf by blast
+ moreover have "(k \<circ> f) ` u \<subseteq> s"
+ using imf imk by fastforce
+ moreover have "P (k \<circ> f)"
+ by (simp add: P Qf contf imf)
+ moreover have "continuous_on u (k \<circ> g)"
+ using contg continuous_on_compose continuous_on_subset contk img by blast
+ moreover have "(k \<circ> g) ` u \<subseteq> s"
+ using img imk by fastforce
+ moreover have "P (k \<circ> g)"
+ by (simp add: P Qg contg img)
+ ultimately have "homotopic_with P u s (k \<circ> f) (k \<circ> g)"
+ by (rule hom)
+ then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
+ apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
+ using Q by (auto simp: conth imh)
+ then show ?thesis
+ apply (rule homotopic_with_eq)
+ apply (metis feq)
+ apply (metis geq)
+ apply (metis Qeq)
+ done
+qed
+
+lemma homotopically_trivial_retraction_null_gen:
+ assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+ and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+ and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+ and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk>
+ \<Longrightarrow> \<exists>c. homotopic_with P u s f (\<lambda>x. c)"
+ and contf: "continuous_on u f" and imf:"f ` u \<subseteq> t" and Qf: "Q f"
+ obtains c where "homotopic_with Q u t f (\<lambda>x. c)"
+proof -
+ have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
+ have "continuous_on u (k \<circ> f)"
+ using contf continuous_on_compose continuous_on_subset contk imf by blast
+ moreover have "(k \<circ> f) ` u \<subseteq> s"
+ using imf imk by fastforce
+ moreover have "P (k \<circ> f)"
+ by (simp add: P Qf contf imf)
+ ultimately obtain c where "homotopic_with P u s (k \<circ> f) (\<lambda>x. c)"
+ by (metis hom)
+ then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+ apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
+ using Q by (auto simp: conth imh)
+ then show ?thesis
+ apply (rule_tac c = "h c" in that)
+ apply (erule homotopic_with_eq)
+ apply (metis feq, simp)
+ apply (metis Qeq)
+ done
+qed
+
+lemma cohomotopically_trivial_retraction_gen:
+ assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+ and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+ and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+ and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f;
+ continuous_on s g; g ` s \<subseteq> u; P g\<rbrakk>
+ \<Longrightarrow> homotopic_with P s u f g"
+ and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
+ and contg: "continuous_on t g" and img: "g ` t \<subseteq> u" and Qg: "Q g"
+ shows "homotopic_with Q t u f g"
+proof -
+ have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
+ have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto
+ have "continuous_on s (f \<circ> h)"
+ using contf conth continuous_on_compose imh by blast
+ moreover have "(f \<circ> h) ` s \<subseteq> u"
+ using imf imh by fastforce
+ moreover have "P (f \<circ> h)"
+ by (simp add: P Qf contf imf)
+ moreover have "continuous_on s (g \<circ> h)"
+ using contg continuous_on_compose continuous_on_subset conth imh by blast
+ moreover have "(g \<circ> h) ` s \<subseteq> u"
+ using img imh by fastforce
+ moreover have "P (g \<circ> h)"
+ by (simp add: P Qg contg img)
+ ultimately have "homotopic_with P s u (f \<circ> h) (g \<circ> h)"
+ by (rule hom)
+ then have "homotopic_with Q t u (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
+ apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
+ using Q by (auto simp: contk imk)
+ then show ?thesis
+ apply (rule homotopic_with_eq)
+ apply (metis feq)
+ apply (metis geq)
+ apply (metis Qeq)
+ done
+qed
+
+lemma cohomotopically_trivial_retraction_null_gen:
+ assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+ and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+ and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+ and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk>
+ \<Longrightarrow> \<exists>c. homotopic_with P s u f (\<lambda>x. c)"
+ and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
+ obtains c where "homotopic_with Q t u f (\<lambda>x. c)"
+proof -
+ have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
+ have "continuous_on s (f \<circ> h)"
+ using contf conth continuous_on_compose imh by blast
+ moreover have "(f \<circ> h) ` s \<subseteq> u"
+ using imf imh by fastforce
+ moreover have "P (f \<circ> h)"
+ by (simp add: P Qf contf imf)
+ ultimately obtain c where "homotopic_with P s u (f \<circ> h) (\<lambda>x. c)"
+ by (metis hom)
+ then have "homotopic_with Q t u (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
+ apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
+ using Q by (auto simp: contk imk)
+ then show ?thesis
+ apply (rule_tac c = c in that)
+ apply (erule homotopic_with_eq)
+ apply (metis feq, simp)
+ apply (metis Qeq)
+ done
+qed
+
+end
+
+lemma simply_connected_retraction_gen:
+ shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T;
+ continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
+ \<Longrightarrow> simply_connected T"
+apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
+apply (rule Retracts.homotopically_trivial_retraction_gen
+ [of S h _ k _ "\<lambda>p. pathfinish p = pathstart p" "\<lambda>p. pathfinish p = pathstart p"])
+apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
+done
+
+lemma homeomorphic_simply_connected:
+ "\<lbrakk>S homeomorphic T; simply_connected S\<rbrakk> \<Longrightarrow> simply_connected T"
+ by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)
+
+lemma homeomorphic_simply_connected_eq:
+ "S homeomorphic T \<Longrightarrow> (simply_connected S \<longleftrightarrow> simply_connected T)"
+ by (metis homeomorphic_simply_connected homeomorphic_sym)
+
+
+subsection\<open>Homotopy equivalence\<close>
+
+definition%important homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
+ (infix "homotopy'_eqv" 50)
+ where "S homotopy_eqv T \<equiv>
+ \<exists>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
+ continuous_on T g \<and> g ` T \<subseteq> S \<and>
+ homotopic_with (\<lambda>x. True) S S (g \<circ> f) id \<and>
+ homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
+
+lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
+ unfolding homeomorphic_def homotopy_eqv_def homeomorphism_def
+ by (fastforce intro!: homotopic_with_equal continuous_on_compose)
+
+lemma homotopy_eqv_refl: "S homotopy_eqv S"
+ by (rule homeomorphic_imp_homotopy_eqv homeomorphic_refl)+
+
+lemma homotopy_eqv_sym: "S homotopy_eqv T \<longleftrightarrow> T homotopy_eqv S"
+ by (auto simp: homotopy_eqv_def)
+
+lemma homotopy_eqv_trans [trans]:
+ fixes S :: "'a::real_normed_vector set" and U :: "'c::real_normed_vector set"
+ assumes ST: "S homotopy_eqv T" and TU: "T homotopy_eqv U"
+ shows "S homotopy_eqv U"
+proof -
+ obtain f1 g1 where f1: "continuous_on S f1" "f1 ` S \<subseteq> T"
+ and g1: "continuous_on T g1" "g1 ` T \<subseteq> S"
+ and hom1: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> f1) id"
+ "homotopic_with (\<lambda>x. True) T T (f1 \<circ> g1) id"
+ using ST by (auto simp: homotopy_eqv_def)
+ obtain f2 g2 where f2: "continuous_on T f2" "f2 ` T \<subseteq> U"
+ and g2: "continuous_on U g2" "g2 ` U \<subseteq> T"
+ and hom2: "homotopic_with (\<lambda>x. True) T T (g2 \<circ> f2) id"
+ "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
+ using TU by (auto simp: homotopy_eqv_def)
+ have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
+ by (rule homotopic_with_compose_continuous_right hom2 f1)+
+ then have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
+ by (simp add: o_assoc)
+ then have "homotopic_with (\<lambda>x. True) S S
+ (g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
+ by (simp add: g1 homotopic_with_compose_continuous_left)
+ moreover have "homotopic_with (\<lambda>x. True) S S (g1 \<circ> id \<circ> f1) id"
+ using hom1 by simp
+ ultimately have SS: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
+ apply (simp add: o_assoc)
+ apply (blast intro: homotopic_with_trans)
+ done
+ have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
+ by (rule homotopic_with_compose_continuous_right hom1 g2)+
+ then have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> (g1 \<circ> g2)) (id \<circ> g2)"
+ by (simp add: o_assoc)
+ then have "homotopic_with (\<lambda>x. True) U U
+ (f2 \<circ> (f1 \<circ> (g1 \<circ> g2))) (f2 \<circ> (id \<circ> g2))"
+ by (simp add: f2 homotopic_with_compose_continuous_left)
+ moreover have "homotopic_with (\<lambda>x. True) U U (f2 \<circ> id \<circ> g2) id"
+ using hom2 by simp
+ ultimately have UU: "homotopic_with (\<lambda>x. True) U U (f2 \<circ> f1 \<circ> (g1 \<circ> g2)) id"
+ apply (simp add: o_assoc)
+ apply (blast intro: homotopic_with_trans)
+ done
+ show ?thesis
+ unfolding homotopy_eqv_def
+ apply (rule_tac x = "f2 \<circ> f1" in exI)
+ apply (rule_tac x = "g1 \<circ> g2" in exI)
+ apply (intro conjI continuous_on_compose SS UU)
+ using f1 f2 g1 g2 apply (force simp: elim!: continuous_on_subset)+
+ done
+qed
+
+lemma homotopy_eqv_inj_linear_image:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "linear f" "inj f"
+ shows "(f ` S) homotopy_eqv S"
+apply (rule homeomorphic_imp_homotopy_eqv)
+using assms homeomorphic_sym linear_homeomorphic_image by auto
+
+lemma homotopy_eqv_translation:
+ fixes S :: "'a::real_normed_vector set"
+ shows "(+) a ` S homotopy_eqv S"
+ apply (rule homeomorphic_imp_homotopy_eqv)
+ using homeomorphic_translation homeomorphic_sym by blast
+
+lemma homotopy_eqv_homotopic_triviality_imp:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ and f: "continuous_on U f" "f ` U \<subseteq> T"
+ and g: "continuous_on U g" "g ` U \<subseteq> T"
+ and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
+ continuous_on U g; g ` U \<subseteq> S\<rbrakk>
+ \<Longrightarrow> homotopic_with (\<lambda>x. True) U S f g"
+ shows "homotopic_with (\<lambda>x. True) U T f g"
+proof -
+ obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+ and k: "continuous_on T k" "k ` T \<subseteq> S"
+ and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+ "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+ using assms by (auto simp: homotopy_eqv_def)
+ have "homotopic_with (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
+ apply (rule homUS)
+ using f g k
+ apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
+ apply (force simp: o_def)+
+ done
+ then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
+ apply (rule homotopic_with_compose_continuous_left)
+ apply (simp_all add: h)
+ done
+ moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
+ apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
+ apply (auto simp: hom f)
+ done
+ moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
+ apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
+ apply (auto simp: hom g)
+ done
+ ultimately show "homotopic_with (\<lambda>x. True) U T f g"
+ apply (simp add: o_assoc)
+ using homotopic_with_trans homotopic_with_sym by blast
+qed
+
+lemma homotopy_eqv_homotopic_triviality:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
+ continuous_on U g \<and> g ` U \<subseteq> S
+ \<longrightarrow> homotopic_with (\<lambda>x. True) U S f g) \<longleftrightarrow>
+ (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and>
+ continuous_on U g \<and> g ` U \<subseteq> T
+ \<longrightarrow> homotopic_with (\<lambda>x. True) U T f g)"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_imp)
+by (metis (no_types) assms homotopy_eqv_homotopic_triviality_imp homotopy_eqv_sym)
+
+lemma homotopy_eqv_cohomotopic_triviality_null_imp:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ and f: "continuous_on T f" "f ` T \<subseteq> U"
+ and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
+ \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c)"
+ obtains c where "homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)"
+proof -
+ obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+ and k: "continuous_on T k" "k ` T \<subseteq> S"
+ and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+ "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+ using assms by (auto simp: homotopy_eqv_def)
+ obtain c where "homotopic_with (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
+ apply (rule exE [OF homSU [of "f \<circ> h"]])
+ apply (intro continuous_on_compose h)
+ using h f apply (force elim!: continuous_on_subset)+
+ done
+ then have "homotopic_with (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
+ apply (rule homotopic_with_compose_continuous_right [where X=S])
+ using k by auto
+ moreover have "homotopic_with (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
+ apply (rule homotopic_with_compose_continuous_left [where Y=T])
+ apply (simp add: hom homotopic_with_symD)
+ using f apply auto
+ done
+ ultimately show ?thesis
+ apply (rule_tac c=c in that)
+ apply (simp add: o_def)
+ using homotopic_with_trans by blast
+qed
+
+lemma homotopy_eqv_cohomotopic_triviality_null:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c))) \<longleftrightarrow>
+ (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
+
+lemma homotopy_eqv_homotopic_triviality_null_imp:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ and f: "continuous_on U f" "f ` U \<subseteq> T"
+ and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+ \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
+ shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
+proof -
+ obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+ and k: "continuous_on T k" "k ` T \<subseteq> S"
+ and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+ "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+ using assms by (auto simp: homotopy_eqv_def)
+ obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
+ apply (rule exE [OF homSU [of "k \<circ> f"]])
+ apply (intro continuous_on_compose h)
+ using k f apply (force elim!: continuous_on_subset)+
+ done
+ then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+ apply (rule homotopic_with_compose_continuous_left [where Y=S])
+ using h by auto
+ moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
+ apply (rule homotopic_with_compose_continuous_right [where X=T])
+ apply (simp add: hom homotopic_with_symD)
+ using f apply auto
+ done
+ ultimately show ?thesis
+ using homotopic_with_trans by (fastforce simp add: o_def)
+qed
+
+lemma homotopy_eqv_homotopic_triviality_null:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
+ (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
+
+lemma homotopy_eqv_contractible_sets:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ assumes "contractible S" "contractible T" "S = {} \<longleftrightarrow> T = {}"
+ shows "S homotopy_eqv T"
+proof (cases "S = {}")
+ case True with assms show ?thesis
+ by (simp add: homeomorphic_imp_homotopy_eqv)
+next
+ case False
+ with assms obtain a b where "a \<in> S" "b \<in> T"
+ by auto
+ then show ?thesis
+ unfolding homotopy_eqv_def
+ apply (rule_tac x="\<lambda>x. b" in exI)
+ apply (rule_tac x="\<lambda>x. a" in exI)
+ apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
+ apply (auto simp: o_def continuous_on_const)
+ done
+qed
+
+lemma homotopy_eqv_empty1 [simp]:
+ fixes S :: "'a::real_normed_vector set"
+ shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}"
+apply (rule iffI)
+using homotopy_eqv_def apply fastforce
+by (simp add: homotopy_eqv_contractible_sets)
+
+lemma homotopy_eqv_empty2 [simp]:
+ fixes S :: "'a::real_normed_vector set"
+ shows "({}::'b::real_normed_vector set) homotopy_eqv S \<longleftrightarrow> S = {}"
+by (metis homotopy_eqv_empty1 homotopy_eqv_sym)
+
+lemma homotopy_eqv_contractibility:
+ fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+ shows "S homotopy_eqv T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
+unfolding homotopy_eqv_def
+by (blast intro: homotopy_dominated_contractibility)
+
+lemma homotopy_eqv_sing:
+ fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
+ shows "S homotopy_eqv {a} \<longleftrightarrow> S \<noteq> {} \<and> contractible S"
+proof (cases "S = {}")
+ case True then show ?thesis
+ by simp
+next
+ case False then show ?thesis
+ by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
+qed
+
+lemma homeomorphic_contractible_eq:
+ fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+ shows "S homeomorphic T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
+by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)
+
+lemma homeomorphic_contractible:
+ fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+ shows "\<lbrakk>contractible S; S homeomorphic T\<rbrakk> \<Longrightarrow> contractible T"
+ by (metis homeomorphic_contractible_eq)
+
+
+subsection%unimportant\<open>Misc other results\<close>
+
+lemma bounded_connected_Compl_real:
+ fixes S :: "real set"
+ assumes "bounded S" and conn: "connected(- S)"
+ shows "S = {}"
+proof -
+ obtain a b where "S \<subseteq> box a b"
+ by (meson assms bounded_subset_box_symmetric)
+ then have "a \<notin> S" "b \<notin> S"
+ by auto
+ then have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> - S"
+ by (meson Compl_iff conn connected_iff_interval)
+ then show ?thesis
+ using \<open>S \<subseteq> box a b\<close> by auto
+qed
+
+lemma bounded_connected_Compl_1:
+ fixes S :: "'a::{euclidean_space} set"
+ assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
+ shows "S = {}"
+proof -
+ have "DIM('a) = DIM(real)"
+ by (simp add: "1")
+ then obtain f::"'a \<Rightarrow> real" and g
+ where "linear f" "\<And>x. norm(f x) = norm x" "\<And>x. g(f x) = x" "\<And>y. f(g y) = y"
+ by (rule isomorphisms_UNIV_UNIV) blast
+ with \<open>bounded S\<close> have "bounded (f ` S)"
+ using bounded_linear_image linear_linear by blast
+ have "connected (f ` (-S))"
+ using connected_linear_image assms \<open>linear f\<close> by blast
+ moreover have "f ` (-S) = - (f ` S)"
+ apply (rule bij_image_Compl_eq)
+ apply (auto simp: bij_def)
+ apply (metis \<open>\<And>x. g (f x) = x\<close> injI)
+ by (metis UNIV_I \<open>\<And>y. f (g y) = y\<close> image_iff)
+ finally have "connected (- (f ` S))"
+ by simp
+ then have "f ` S = {}"
+ using \<open>bounded (f ` S)\<close> bounded_connected_Compl_real by blast
+ then show ?thesis
+ by blast
+qed
+
+
+subsection%unimportant\<open>Some Uncountable Sets\<close>
+
+lemma uncountable_closed_segment:
+ fixes a :: "'a::real_normed_vector"
+ assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
+unfolding path_image_linepath [symmetric] path_image_def
+ using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
+ countable_image_inj_on by auto
+
+lemma uncountable_open_segment:
+ fixes a :: "'a::real_normed_vector"
+ assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
+ by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
+
+lemma uncountable_convex:
+ fixes a :: "'a::real_normed_vector"
+ assumes "convex S" "a \<in> S" "b \<in> S" "a \<noteq> b"
+ shows "uncountable S"
+proof -
+ have "uncountable (closed_segment a b)"
+ by (simp add: uncountable_closed_segment assms)
+ then show ?thesis
+ by (meson assms convex_contains_segment countable_subset)
+qed
+
+lemma uncountable_ball:
+ fixes a :: "'a::euclidean_space"
+ assumes "r > 0"
+ shows "uncountable (ball a r)"
+proof -
+ have "uncountable (open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)))"
+ by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
+ moreover have "open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)) \<subseteq> ball a r"
+ using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
+ ultimately show ?thesis
+ by (metis countable_subset)
+qed
+
+lemma ball_minus_countable_nonempty:
+ assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
+ shows "ball z r - A \<noteq> {}"
+proof
+ assume *: "ball z r - A = {}"
+ have "uncountable (ball z r - A)"
+ by (intro uncountable_minus_countable assms uncountable_ball)
+ thus False by (subst (asm) *) auto
+qed
+
+lemma uncountable_cball:
+ fixes a :: "'a::euclidean_space"
+ assumes "r > 0"
+ shows "uncountable (cball a r)"
+ using assms countable_subset uncountable_ball by auto
+
+lemma pairwise_disjnt_countable:
+ fixes \<N> :: "nat set set"
+ assumes "pairwise disjnt \<N>"
+ shows "countable \<N>"
+proof -
+ have "inj_on (\<lambda>X. SOME n. n \<in> X) (\<N> - {{}})"
+ apply (clarsimp simp add: inj_on_def)
+ by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
+ then show ?thesis
+ by (metis countable_Diff_eq countable_def)
+qed
+
+lemma pairwise_disjnt_countable_Union:
+ assumes "countable (\<Union>\<N>)" and pwd: "pairwise disjnt \<N>"
+ shows "countable \<N>"
+proof -
+ obtain f :: "_ \<Rightarrow> nat" where f: "inj_on f (\<Union>\<N>)"
+ using assms by blast
+ then have "pairwise disjnt (\<Union> X \<in> \<N>. {f ` X})"
+ using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
+ then have "countable (\<Union> X \<in> \<N>. {f ` X})"
+ using pairwise_disjnt_countable by blast
+ then show ?thesis
+ by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
+qed
+
+lemma connected_uncountable:
+ fixes S :: "'a::metric_space set"
+ assumes "connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
+proof -
+ have "continuous_on S (dist a)"
+ by (intro continuous_intros)
+ then have "connected (dist a ` S)"
+ by (metis connected_continuous_image \<open>connected S\<close>)
+ then have "closed_segment 0 (dist a b) \<subseteq> (dist a ` S)"
+ by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
+ then have "uncountable (dist a ` S)"
+ by (metis \<open>a \<noteq> b\<close> countable_subset dist_eq_0_iff uncountable_closed_segment)
+ then show ?thesis
+ by blast
+qed
+
+lemma path_connected_uncountable:
+ fixes S :: "'a::metric_space set"
+ assumes "path_connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
+ using path_connected_imp_connected assms connected_uncountable by metis
+
+lemma connected_finite_iff_sing:
+ fixes S :: "'a::metric_space set"
+ assumes "connected S"
+ shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})" (is "_ = ?rhs")
+proof -
+ have "uncountable S" if "\<not> ?rhs"
+ using connected_uncountable assms that by blast
+ then show ?thesis
+ using uncountable_infinite by auto
+qed
+
+lemma connected_card_eq_iff_nontrivial:
+ fixes S :: "'a::metric_space set"
+ shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
+ apply (auto simp: countable_finite finite_subset)
+ by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
+
+lemma simple_path_image_uncountable:
+ fixes g :: "real \<Rightarrow> 'a::metric_space"
+ assumes "simple_path g"
+ shows "uncountable (path_image g)"
+proof -
+ have "g 0 \<in> path_image g" "g (1/2) \<in> path_image g"
+ by (simp_all add: path_defs)
+ moreover have "g 0 \<noteq> g (1/2)"
+ using assms by (fastforce simp add: simple_path_def)
+ ultimately show ?thesis
+ apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
+ by blast
+qed
+
+lemma arc_image_uncountable:
+ fixes g :: "real \<Rightarrow> 'a::metric_space"
+ assumes "arc g"
+ shows "uncountable (path_image g)"
+ by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
+
+
+subsection%unimportant\<open> Some simple positive connection theorems\<close>
+
+proposition path_connected_convex_diff_countable:
+ fixes U :: "'a::euclidean_space set"
+ assumes "convex U" "\<not> collinear U" "countable S"
+ shows "path_connected(U - S)"
+proof (clarsimp simp add: path_connected_def)
+ fix a b
+ assume "a \<in> U" "a \<notin> S" "b \<in> U" "b \<notin> S"
+ let ?m = "midpoint a b"
+ show "\<exists>g. path g \<and> path_image g \<subseteq> U - S \<and> pathstart g = a \<and> pathfinish g = b"
+ proof (cases "a = b")
+ case True
+ then show ?thesis
+ by (metis DiffI \<open>a \<in> U\<close> \<open>a \<notin> S\<close> path_component_def path_component_refl)
+ next
+ case False
+ then have "a \<noteq> ?m" "b \<noteq> ?m"
+ using midpoint_eq_endpoint by fastforce+
+ have "?m \<in> U"
+ using \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>convex U\<close> convex_contains_segment by force
+ obtain c where "c \<in> U" and nc_abc: "\<not> collinear {a,b,c}"
+ by (metis False \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>\<not> collinear U\<close> collinear_triples insert_absorb)
+ have ncoll_mca: "\<not> collinear {?m,c,a}"
+ by (metis (full_types) \<open>a \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
+ have ncoll_mcb: "\<not> collinear {?m,c,b}"
+ by (metis (full_types) \<open>b \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
+ have "c \<noteq> ?m"
+ by (metis collinear_midpoint insert_commute nc_abc)
+ then have "closed_segment ?m c \<subseteq> U"
+ by (simp add: \<open>c \<in> U\<close> \<open>?m \<in> U\<close> \<open>convex U\<close> closed_segment_subset)
+ then obtain z where z: "z \<in> closed_segment ?m c"
+ and disjS: "(closed_segment a z \<union> closed_segment z b) \<inter> S = {}"
+ proof -
+ have False if "closed_segment ?m c \<subseteq> {z. (closed_segment a z \<union> closed_segment z b) \<inter> S \<noteq> {}}"
+ proof -
+ have closb: "closed_segment ?m c \<subseteq>
+ {z \<in> closed_segment ?m c. closed_segment a z \<inter> S \<noteq> {}} \<union> {z \<in> closed_segment ?m c. closed_segment z b \<inter> S \<noteq> {}}"
+ using that by blast
+ have *: "countable {z \<in> closed_segment ?m c. closed_segment z u \<inter> S \<noteq> {}}"
+ if "u \<in> U" "u \<notin> S" and ncoll: "\<not> collinear {?m, c, u}" for u
+ proof -
+ have **: False if x1: "x1 \<in> closed_segment ?m c" and x2: "x2 \<in> closed_segment ?m c"
+ and "x1 \<noteq> x2" "x1 \<noteq> u"
+ and w: "w \<in> closed_segment x1 u" "w \<in> closed_segment x2 u"
+ and "w \<in> S" for x1 x2 w
+ proof -
+ have "x1 \<in> affine hull {?m,c}" "x2 \<in> affine hull {?m,c}"
+ using segment_as_ball x1 x2 by auto
+ then have coll_x1: "collinear {x1, ?m, c}" and coll_x2: "collinear {?m, c, x2}"
+ by (simp_all add: affine_hull_3_imp_collinear) (metis affine_hull_3_imp_collinear insert_commute)
+ have "\<not> collinear {x1, u, x2}"
+ proof
+ assume "collinear {x1, u, x2}"
+ then have "collinear {?m, c, u}"
+ by (metis (full_types) \<open>c \<noteq> ?m\<close> coll_x1 coll_x2 collinear_3_trans insert_commute ncoll \<open>x1 \<noteq> x2\<close>)
+ with ncoll show False ..
+ qed
+ then have "closed_segment x1 u \<inter> closed_segment u x2 = {u}"
+ by (blast intro!: Int_closed_segment)
+ then have "w = u"
+ using closed_segment_commute w by auto
+ show ?thesis
+ using \<open>u \<notin> S\<close> \<open>w = u\<close> that(7) by auto
+ qed
+ then have disj: "disjoint ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}))"
+ by (fastforce simp: pairwise_def disjnt_def)
+ have cou: "countable ((\<Union>z \<in> closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
+ apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
+ apply (rule countable_subset [OF _ \<open>countable S\<close>], auto)
+ done
+ define f where "f \<equiv> \<lambda>X. (THE z. z \<in> closed_segment ?m c \<and> X = closed_segment z u \<inter> S)"
+ show ?thesis
+ proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
+ fix x
+ assume x: "x \<in> closed_segment ?m c" "closed_segment x u \<inter> S \<noteq> {}"
+ show "x \<in> f ` ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
+ proof (rule_tac x="closed_segment x u \<inter> S" in image_eqI)
+ show "x = f (closed_segment x u \<inter> S)"
+ unfolding f_def
+ apply (rule the_equality [symmetric])
+ using x apply (auto simp: dest: **)
+ done
+ qed (use x in auto)
+ qed
+ qed
+ have "uncountable (closed_segment ?m c)"
+ by (metis \<open>c \<noteq> ?m\<close> uncountable_closed_segment)
+ then show False
+ using closb * [OF \<open>a \<in> U\<close> \<open>a \<notin> S\<close> ncoll_mca] * [OF \<open>b \<in> U\<close> \<open>b \<notin> S\<close> ncoll_mcb]
+ apply (simp add: closed_segment_commute)
+ by (simp add: countable_subset)
+ qed
+ then show ?thesis
+ by (force intro: that)
+ qed
+ show ?thesis
+ proof (intro exI conjI)
+ have "path_image (linepath a z +++ linepath z b) \<subseteq> U"
+ by (metis \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>closed_segment ?m c \<subseteq> U\<close> z \<open>convex U\<close> closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
+ with disjS show "path_image (linepath a z +++ linepath z b) \<subseteq> U - S"
+ by (force simp: path_image_join)
+ qed auto
+ qed
+qed
+
+
+corollary connected_convex_diff_countable:
+ fixes U :: "'a::euclidean_space set"
+ assumes "convex U" "\<not> collinear U" "countable S"
+ shows "connected(U - S)"
+ by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)
+
+lemma path_connected_punctured_convex:
+ assumes "convex S" and aff: "aff_dim S \<noteq> 1"
+ shows "path_connected(S - {a})"
+proof -
+ consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S \<ge> 2"
+ using assms aff_dim_geq [of S] by linarith
+ then show ?thesis
+ proof cases
+ assume "aff_dim S = -1"
+ then show ?thesis
+ by (metis aff_dim_empty empty_Diff path_connected_empty)
+ next
+ assume "aff_dim S = 0"
+ then show ?thesis
+ by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
+ next
+ assume ge2: "aff_dim S \<ge> 2"
+ then have "\<not> collinear S"
+ proof (clarsimp simp add: collinear_affine_hull)
+ fix u v
+ assume "S \<subseteq> affine hull {u, v}"
+ then have "aff_dim S \<le> aff_dim {u, v}"
+ by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
+ with ge2 show False
+ by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
+ qed
+ then show ?thesis
+ apply (rule path_connected_convex_diff_countable [OF \<open>convex S\<close>])
+ by simp
+ qed
+qed
+
+lemma connected_punctured_convex:
+ shows "\<lbrakk>convex S; aff_dim S \<noteq> 1\<rbrakk> \<Longrightarrow> connected(S - {a})"
+ using path_connected_imp_connected path_connected_punctured_convex by blast
+
+lemma path_connected_complement_countable:
+ fixes S :: "'a::euclidean_space set"
+ assumes "2 \<le> DIM('a)" "countable S"
+ shows "path_connected(- S)"
+proof -
+ have "path_connected(UNIV - S)"
+ apply (rule path_connected_convex_diff_countable)
+ using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
+ then show ?thesis
+ by (simp add: Compl_eq_Diff_UNIV)
+qed
+
+proposition path_connected_openin_diff_countable:
+ fixes S :: "'a::euclidean_space set"
+ assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
+ and "\<not> collinear S" "countable T"
+ shows "path_connected(S - T)"
+proof (clarsimp simp add: path_connected_component)
+ fix x y
+ assume xy: "x \<in> S" "x \<notin> T" "y \<in> S" "y \<notin> T"
+ show "path_component (S - T) x y"
+ proof (rule connected_equivalence_relation_gen [OF \<open>connected S\<close>, where P = "\<lambda>x. x \<notin> T"])
+ show "\<exists>z. z \<in> U \<and> z \<notin> T" if opeU: "openin (subtopology euclidean S) U" and "x \<in> U" for U x
+ proof -
+ have "openin (subtopology euclidean (affine hull S)) U"
+ using opeU ope openin_trans by blast
+ with \<open>x \<in> U\<close> obtain r where Usub: "U \<subseteq> affine hull S" and "r > 0"
+ and subU: "ball x r \<inter> affine hull S \<subseteq> U"
+ by (auto simp: openin_contains_ball)
+ with \<open>x \<in> U\<close> have x: "x \<in> ball x r \<inter> affine hull S"
+ by auto
+ have "\<not> S \<subseteq> {x}"
+ using \<open>\<not> collinear S\<close> collinear_subset by blast
+ then obtain x' where "x' \<noteq> x" "x' \<in> S"
+ by blast
+ obtain y where y: "y \<noteq> x" "y \<in> ball x r \<inter> affine hull S"
+ proof
+ show "x + (r / 2 / norm(x' - x)) *\<^sub>R (x' - x) \<noteq> x"
+ using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> by auto
+ show "x + (r / 2 / norm (x' - x)) *\<^sub>R (x' - x) \<in> ball x r \<inter> affine hull S"
+ using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> \<open>x' \<in> S\<close> x
+ by (simp add: dist_norm mem_affine_3_minus hull_inc)
+ qed
+ have "convex (ball x r \<inter> affine hull S)"
+ by (simp add: affine_imp_convex convex_Int)
+ with x y subU have "uncountable U"
+ by (meson countable_subset uncountable_convex)
+ then have "\<not> U \<subseteq> T"
+ using \<open>countable T\<close> countable_subset by blast
+ then show ?thesis by blast
+ qed
+ show "\<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and>
+ (\<forall>x\<in>U. \<forall>y\<in>U. x \<notin> T \<and> y \<notin> T \<longrightarrow> path_component (S - T) x y)"
+ if "x \<in> S" for x
+ proof -
+ obtain r where Ssub: "S \<subseteq> affine hull S" and "r > 0"
+ and subS: "ball x r \<inter> affine hull S \<subseteq> S"
+ using ope \<open>x \<in> S\<close> by (auto simp: openin_contains_ball)
+ then have conv: "convex (ball x r \<inter> affine hull S)"
+ by (simp add: affine_imp_convex convex_Int)
+ have "\<not> aff_dim (affine hull S) \<le> 1"
+ using \<open>\<not> collinear S\<close> collinear_aff_dim by auto
+ then have "\<not> collinear (ball x r \<inter> affine hull S)"
+ apply (simp add: collinear_aff_dim)
+ by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball \<open>0 < r\<close> aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
+ then have *: "path_connected ((ball x r \<inter> affine hull S) - T)"
+ by (rule path_connected_convex_diff_countable [OF conv _ \<open>countable T\<close>])
+ have ST: "ball x r \<inter> affine hull S - T \<subseteq> S - T"
+ using subS by auto
+ show ?thesis
+ proof (intro exI conjI)
+ show "x \<in> ball x r \<inter> affine hull S"
+ using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
+ have "openin (subtopology euclidean (affine hull S)) (ball x r \<inter> affine hull S)"
+ by (subst inf.commute) (simp add: openin_Int_open)
+ then show "openin (subtopology euclidean S) (ball x r \<inter> affine hull S)"
+ by (rule openin_subset_trans [OF _ subS Ssub])
+ qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
+ qed
+ qed (use xy path_component_trans in auto)
+qed
+
+corollary connected_openin_diff_countable:
+ fixes S :: "'a::euclidean_space set"
+ assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
+ and "\<not> collinear S" "countable T"
+ shows "connected(S - T)"
+ by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])
+
+corollary path_connected_open_diff_countable:
+ fixes S :: "'a::euclidean_space set"
+ assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
+ shows "path_connected(S - T)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (simp add: path_connected_empty)
+next
+ case False
+ show ?thesis
+ proof (rule path_connected_openin_diff_countable)
+ show "openin (subtopology euclidean (affine hull S)) S"
+ by (simp add: assms hull_subset open_subset)
+ show "\<not> collinear S"
+ using assms False by (simp add: collinear_aff_dim aff_dim_open)
+ qed (simp_all add: assms)
+qed
+
+corollary connected_open_diff_countable:
+ fixes S :: "'a::euclidean_space set"
+ assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
+ shows "connected(S - T)"
+by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)
+
+
+
+subsection%unimportant \<open>Self-homeomorphisms shuffling points about\<close>
+
+subsubsection%unimportant\<open>The theorem \<open>homeomorphism_moving_points_exists\<close>\<close>
+
+lemma homeomorphism_moving_point_1:
+ fixes a :: "'a::euclidean_space"
+ assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T"
+ obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+ "f a = u" "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = x"
+proof -
+ have nou: "norm (u - a) < r" and "u \<in> T"
+ using u by (auto simp: dist_norm norm_minus_commute)
+ then have "0 < r"
+ by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
+ define f where "f \<equiv> \<lambda>x. (1 - norm(x - a) / r) *\<^sub>R (u - a) + x"
+ have *: "False" if eq: "x + (norm y / r) *\<^sub>R u = y + (norm x / r) *\<^sub>R u"
+ and nou: "norm u < r" and yx: "norm y < norm x" for x y and u::'a
+ proof -
+ have "x = y + (norm x / r - (norm y / r)) *\<^sub>R u"
+ using eq by (simp add: algebra_simps)
+ then have "norm x = norm (y + ((norm x - norm y) / r) *\<^sub>R u)"
+ by (metis diff_divide_distrib)
+ also have "\<dots> \<le> norm y + norm(((norm x - norm y) / r) *\<^sub>R u)"
+ using norm_triangle_ineq by blast
+ also have "\<dots> = norm y + (norm x - norm y) * (norm u / r)"
+ using yx \<open>r > 0\<close>
+ by (simp add: divide_simps)
+ also have "\<dots> < norm y + (norm x - norm y) * 1"
+ apply (subst add_less_cancel_left)
+ apply (rule mult_strict_left_mono)
+ using nou \<open>0 < r\<close> yx
+ apply (simp_all add: field_simps)
+ done
+ also have "\<dots> = norm x"
+ by simp
+ finally show False by simp
+ qed
+ have "inj f"
+ unfolding f_def
+ proof (clarsimp simp: inj_on_def)
+ fix x y
+ assume "(1 - norm (x - a) / r) *\<^sub>R (u - a) + x =
+ (1 - norm (y - a) / r) *\<^sub>R (u - a) + y"
+ then have eq: "(x - a) + (norm (y - a) / r) *\<^sub>R (u - a) = (y - a) + (norm (x - a) / r) *\<^sub>R (u - a)"
+ by (auto simp: algebra_simps)
+ show "x=y"
+ proof (cases "norm (x - a) = norm (y - a)")
+ case True
+ then show ?thesis
+ using eq by auto
+ next
+ case False
+ then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
+ by linarith
+ then have "False"
+ proof cases
+ case 1 show False
+ using * [OF _ nou 1] eq by simp
+ next
+ case 2 with * [OF eq nou] show False
+ by auto
+ qed
+ then show "x=y" ..
+ qed
+ qed
+ then have inj_onf: "inj_on f (cball a r \<inter> T)"
+ using inj_on_Int by fastforce
+ have contf: "continuous_on (cball a r \<inter> T) f"
+ unfolding f_def using \<open>0 < r\<close> by (intro continuous_intros) blast
+ have fim: "f ` (cball a r \<inter> T) = cball a r \<inter> T"
+ proof
+ have *: "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> r" if "norm y \<le> r" "norm u < r" for y u::'a
+ proof -
+ have "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> norm y + norm((1 - norm y / r) *\<^sub>R u)"
+ using norm_triangle_ineq by blast
+ also have "\<dots> = norm y + abs(1 - norm y / r) * norm u"
+ by simp
+ also have "\<dots> \<le> r"
+ proof -
+ have "(r - norm u) * (r - norm y) \<ge> 0"
+ using that by auto
+ then have "r * norm u + r * norm y \<le> r * r + norm u * norm y"
+ by (simp add: algebra_simps)
+ then show ?thesis
+ using that \<open>0 < r\<close> by (simp add: abs_if field_simps)
+ qed
+ finally show ?thesis .
+ qed
+ have "f ` (cball a r) \<subseteq> cball a r"
+ apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
+ using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
+ moreover have "f ` T \<subseteq> T"
+ unfolding f_def using \<open>affine T\<close> \<open>a \<in> T\<close> \<open>u \<in> T\<close>
+ by (force simp: add.commute mem_affine_3_minus)
+ ultimately show "f ` (cball a r \<inter> T) \<subseteq> cball a r \<inter> T"
+ by blast
+ next
+ show "cball a r \<inter> T \<subseteq> f ` (cball a r \<inter> T)"
+ proof (clarsimp simp add: dist_norm norm_minus_commute)
+ fix x
+ assume x: "norm (x - a) \<le> r" and "x \<in> T"
+ have "\<exists>v \<in> {0..1}. ((1 - v) * r - norm ((x - a) - v *\<^sub>R (u - a))) \<bullet> 1 = 0"
+ by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
+ then obtain v where "0\<le>v" "v\<le>1" and v: "(1 - v) * r = norm ((x - a) - v *\<^sub>R (u - a))"
+ by auto
+ show "x \<in> f ` (cball a r \<inter> T)"
+ proof (rule image_eqI)
+ show "x = f (x - v *\<^sub>R (u - a))"
+ using \<open>r > 0\<close> v by (simp add: f_def field_simps)
+ have "x - v *\<^sub>R (u - a) \<in> cball a r"
+ using \<open>r > 0\<close> v \<open>0 \<le> v\<close>
+ apply (simp add: field_simps dist_norm norm_minus_commute)
+ by (metis le_add_same_cancel2 order.order_iff_strict zero_le_mult_iff)
+ moreover have "x - v *\<^sub>R (u - a) \<in> T"
+ by (simp add: f_def \<open>affine T\<close> \<open>u \<in> T\<close> \<open>x \<in> T\<close> assms mem_affine_3_minus2)
+ ultimately show "x - v *\<^sub>R (u - a) \<in> cball a r \<inter> T"
+ by blast
+ qed
+ qed
+ qed
+ have "\<exists>g. homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+ apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
+ apply (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
+ done
+ then show ?thesis
+ apply (rule exE)
+ apply (erule_tac f=f in that)
+ using \<open>r > 0\<close>
+ apply (simp_all add: f_def dist_norm norm_minus_commute)
+ done
+qed
+
+corollary%unimportant homeomorphism_moving_point_2:
+ fixes a :: "'a::euclidean_space"
+ assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
+ obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+ "f u = v" "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
+proof -
+ have "0 < r"
+ by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
+ obtain f1 g1 where hom1: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f1 g1"
+ and "f1 a = u" and f1: "\<And>x. x \<in> sphere a r \<Longrightarrow> f1 x = x"
+ using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u] by blast
+ obtain f2 g2 where hom2: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f2 g2"
+ and "f2 a = v" and f2: "\<And>x. x \<in> sphere a r \<Longrightarrow> f2 x = x"
+ using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> v] by blast
+ show ?thesis
+ proof
+ show "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) (f2 \<circ> g1) (f1 \<circ> g2)"
+ by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
+ have "g1 u = a"
+ using \<open>0 < r\<close> \<open>f1 a = u\<close> assms hom1 homeomorphism_apply1 by fastforce
+ then show "(f2 \<circ> g1) u = v"
+ by (simp add: \<open>f2 a = v\<close>)
+ show "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> (f2 \<circ> g1) x = x"
+ using f1 f2 hom1 homeomorphism_apply1 by fastforce
+ qed
+qed
+
+
+corollary%unimportant homeomorphism_moving_point_3:
+ fixes a :: "'a::euclidean_space"
+ assumes "affine T" "a \<in> T" and ST: "ball a r \<inter> T \<subseteq> S" "S \<subseteq> T"
+ and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
+ obtains f g where "homeomorphism S S f g"
+ "f u = v" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> ball a r \<inter> T"
+proof -
+ obtain f g where hom: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+ and "f u = v" and fid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
+ using homeomorphism_moving_point_2 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u v] by blast
+ have gid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> g x = x"
+ using fid hom homeomorphism_apply1 by fastforce
+ define ff where "ff \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then f x else x"
+ define gg where "gg \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then g x else x"
+ show ?thesis
+ proof
+ show "homeomorphism S S ff gg"
+ proof (rule homeomorphismI)
+ have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) ff"
+ apply (simp add: ff_def)
+ apply (rule continuous_on_cases)
+ using homeomorphism_cont1 [OF hom]
+ apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id fid)
+ done
+ then show "continuous_on S ff"
+ apply (rule continuous_on_subset)
+ using ST by auto
+ have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
+ apply (simp add: gg_def)
+ apply (rule continuous_on_cases)
+ using homeomorphism_cont2 [OF hom]
+ apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id gid)
+ done
+ then show "continuous_on S gg"
+ apply (rule continuous_on_subset)
+ using ST by auto
+ show "ff ` S \<subseteq> S"
+ proof (clarsimp simp add: ff_def)
+ fix x
+ assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
+ then have "f x \<in> cball a r \<inter> T"
+ using homeomorphism_image1 [OF hom] by force
+ then show "f x \<in> S"
+ using ST(1) \<open>x \<in> T\<close> gid hom homeomorphism_def x by fastforce
+ qed
+ show "gg ` S \<subseteq> S"
+ proof (clarsimp simp add: gg_def)
+ fix x
+ assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
+ then have "g x \<in> cball a r \<inter> T"
+ using homeomorphism_image2 [OF hom] by force
+ then have "g x \<in> ball a r"
+ using homeomorphism_apply2 [OF hom]
+ by (metis Diff_Diff_Int Diff_iff \<open>x \<in> T\<close> cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
+ then show "g x \<in> S"
+ using ST(1) \<open>g x \<in> cball a r \<inter> T\<close> by force
+ qed
+ show "\<And>x. x \<in> S \<Longrightarrow> gg (ff x) = x"
+ apply (simp add: ff_def gg_def)
+ using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
+ apply auto
+ apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
+ done
+ show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
+ apply (simp add: ff_def gg_def)
+ using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
+ apply auto
+ apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
+ done
+ qed
+ show "ff u = v"
+ using u by (auto simp: ff_def \<open>f u = v\<close>)
+ show "{x. \<not> (ff x = x \<and> gg x = x)} \<subseteq> ball a r \<inter> T"
+ by (auto simp: ff_def gg_def)
+ qed
+qed
+
+
+proposition%unimportant homeomorphism_moving_point:
+ fixes a :: "'a::euclidean_space"
+ assumes ope: "openin (subtopology euclidean (affine hull S)) S"
+ and "S \<subseteq> T"
+ and TS: "T \<subseteq> affine hull S"
+ and S: "connected S" "a \<in> S" "b \<in> S"
+ obtains f g where "homeomorphism T T f g" "f a = b"
+ "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+ "bounded {x. \<not> (f x = x \<and> g x = x)}"
+proof -
+ have 1: "\<exists>h k. homeomorphism T T h k \<and> h (f d) = d \<and>
+ {x. \<not> (h x = x \<and> k x = x)} \<subseteq> S \<and> bounded {x. \<not> (h x = x \<and> k x = x)}"
+ if "d \<in> S" "f d \<in> S" and homfg: "homeomorphism T T f g"
+ and S: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+ and bo: "bounded {x. \<not> (f x = x \<and> g x = x)}" for d f g
+ proof (intro exI conjI)
+ show homgf: "homeomorphism T T g f"
+ by (metis homeomorphism_symD homfg)
+ then show "g (f d) = d"
+ by (meson \<open>S \<subseteq> T\<close> homeomorphism_def subsetD \<open>d \<in> S\<close>)
+ show "{x. \<not> (g x = x \<and> f x = x)} \<subseteq> S"
+ using S by blast
+ show "bounded {x. \<not> (g x = x \<and> f x = x)}"
+ using bo by (simp add: conj_commute)
+ qed
+ have 2: "\<exists>f g. homeomorphism T T f g \<and> f x = f2 (f1 x) \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+ if "x \<in> S" "f1 x \<in> S" "f2 (f1 x) \<in> S"
+ and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
+ and sub: "{x. \<not> (f1 x = x \<and> g1 x = x)} \<subseteq> S" "{x. \<not> (f2 x = x \<and> g2 x = x)} \<subseteq> S"
+ and bo: "bounded {x. \<not> (f1 x = x \<and> g1 x = x)}" "bounded {x. \<not> (f2 x = x \<and> g2 x = x)}"
+ for x f1 f2 g1 g2
+ proof (intro exI conjI)
+ show homgf: "homeomorphism T T (f2 \<circ> f1) (g1 \<circ> g2)"
+ by (metis homeomorphism_compose hom)
+ then show "(f2 \<circ> f1) x = f2 (f1 x)"
+ by force
+ show "{x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)} \<subseteq> S"
+ using sub by force
+ have "bounded ({x. \<not>(f1 x = x \<and> g1 x = x)} \<union> {x. \<not>(f2 x = x \<and> g2 x = x)})"
+ using bo by simp
+ then show "bounded {x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)}"
+ by (rule bounded_subset) auto
+ qed
+ have 3: "\<exists>U. openin (subtopology euclidean S) U \<and>
+ d \<in> U \<and>
+ (\<forall>x\<in>U.
+ \<exists>f g. homeomorphism T T f g \<and> f d = x \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
+ bounded {x. \<not> (f x = x \<and> g x = x)})"
+ if "d \<in> S" for d
+ proof -
+ obtain r where "r > 0" and r: "ball d r \<inter> affine hull S \<subseteq> S"
+ by (metis \<open>d \<in> S\<close> ope openin_contains_ball)
+ have *: "\<exists>f g. homeomorphism T T f g \<and> f d = e \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
+ bounded {x. \<not> (f x = x \<and> g x = x)}" if "e \<in> S" "e \<in> ball d r" for e
+ apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
+ using r \<open>S \<subseteq> T\<close> TS that
+ apply (auto simp: \<open>d \<in> S\<close> \<open>0 < r\<close> hull_inc)
+ using bounded_subset by blast
+ show ?thesis
+ apply (rule_tac x="S \<inter> ball d r" in exI)
+ apply (intro conjI)
+ apply (simp add: openin_open_Int)
+ apply (simp add: \<open>0 < r\<close> that)
+ apply (blast intro: *)
+ done
+ qed
+ have "\<exists>f g. homeomorphism T T f g \<and> f a = b \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+ apply (rule connected_equivalence_relation [OF S], safe)
+ apply (blast intro: 1 2 3)+
+ done
+ then show ?thesis
+ using that by auto
+qed
+
+
+lemma homeomorphism_moving_points_exists_gen:
+ assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+ "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
+ and "2 \<le> aff_dim S"
+ and ope: "openin (subtopology euclidean (affine hull S)) S"
+ and "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+ shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+ using assms
+proof (induction K)
+ case empty
+ then show ?case
+ by (force simp: homeomorphism_ident)
+next
+ case (insert i K)
+ then have xney: "\<And>j. \<lbrakk>j \<in> K; j \<noteq> i\<rbrakk> \<Longrightarrow> x i \<noteq> x j \<and> y i \<noteq> y j"
+ and pw: "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"
+ and "x i \<in> S" "y i \<in> S"
+ and xyS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+ by (simp_all add: pairwise_insert)
+ obtain f g where homfg: "homeomorphism T T f g" and feq: "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
+ and fg_sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+ and bo_fg: "bounded {x. \<not> (f x = x \<and> g x = x)}"
+ using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
+ then have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+ using insert by blast
+ have aff_eq: "affine hull (S - y ` K) = affine hull S"
+ apply (rule affine_hull_Diff)
+ apply (auto simp: insert)
+ using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
+ have f_in_S: "f x \<in> S" if "x \<in> S" for x
+ using homfg fg_sub homeomorphism_apply1 \<open>S \<subseteq> T\<close>
+ proof -
+ have "(f (f x) \<noteq> f x \<or> g (f x) \<noteq> f x) \<or> f x \<in> S"
+ by (metis \<open>S \<subseteq> T\<close> homfg subsetD homeomorphism_apply1 that)
+ then show ?thesis
+ using fg_sub by force
+ qed
+ obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
+ and hk_sub: "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
+ and bo_hk: "bounded {x. \<not> (h x = x \<and> k x = x)}"
+ proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
+ show "openin (subtopology euclidean (affine hull (S - y ` K))) (S - y ` K)"
+ by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
+ show "S - y ` K \<subseteq> T"
+ using \<open>S \<subseteq> T\<close> by auto
+ show "T \<subseteq> affine hull (S - y ` K)"
+ using insert by (simp add: aff_eq)
+ show "connected (S - y ` K)"
+ proof (rule connected_openin_diff_countable [OF \<open>connected S\<close> ope])
+ show "\<not> collinear S"
+ using collinear_aff_dim \<open>2 \<le> aff_dim S\<close> by force
+ show "countable (y ` K)"
+ using countable_finite insert.hyps(1) by blast
+ qed
+ show "f (x i) \<in> S - y ` K"
+ apply (auto simp: f_in_S \<open>x i \<in> S\<close>)
+ by (metis feq homfg \<open>x i \<in> S\<close> homeomorphism_def \<open>S \<subseteq> T\<close> \<open>i \<notin> K\<close> subsetCE xney xyS)
+ show "y i \<in> S - y ` K"
+ using insert.hyps xney by (auto simp: \<open>y i \<in> S\<close>)
+ qed blast
+ show ?case
+ proof (intro exI conjI)
+ show "homeomorphism T T (h \<circ> f) (g \<circ> k)"
+ using homfg homhk homeomorphism_compose by blast
+ show "\<forall>i \<in> insert i K. (h \<circ> f) (x i) = y i"
+ using feq hk_sub by (auto simp: heq)
+ show "{x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)} \<subseteq> S"
+ using fg_sub hk_sub by force
+ have "bounded ({x. \<not>(f x = x \<and> g x = x)} \<union> {x. \<not>(h x = x \<and> k x = x)})"
+ using bo_fg bo_hk bounded_Un by blast
+ then show "bounded {x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)}"
+ by (rule bounded_subset) auto
+ qed
+qed
+
+proposition%unimportant homeomorphism_moving_points_exists:
+ fixes S :: "'a::euclidean_space set"
+ assumes 2: "2 \<le> DIM('a)" "open S" "connected S" "S \<subseteq> T" "finite K"
+ and KS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+ and pw: "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
+ and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+ obtains f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
+ "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. (\<not> (f x = x \<and> g x = x))}"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ using KS homeomorphism_ident that by fastforce
+next
+ case False
+ then have affS: "affine hull S = UNIV"
+ by (simp add: affine_hull_open \<open>open S\<close>)
+ then have ope: "openin (subtopology euclidean (affine hull S)) S"
+ using \<open>open S\<close> open_openin by auto
+ have "2 \<le> DIM('a)" by (rule 2)
+ also have "\<dots> = aff_dim (UNIV :: 'a set)"
+ by simp
+ also have "\<dots> \<le> aff_dim S"
+ by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
+ finally have "2 \<le> aff_dim S"
+ by linarith
+ then show ?thesis
+ using homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> KS pw _ ope S] that by fastforce
+qed
+
+subsubsection%unimportant\<open>The theorem \<open>homeomorphism_grouping_points_exists\<close>\<close>
+
+lemma homeomorphism_grouping_point_1:
+ fixes a::real and c::real
+ assumes "a < b" "c < d"
+ obtains f g where "homeomorphism (cbox a b) (cbox c d) f g" "f a = c" "f b = d"
+proof -
+ define f where "f \<equiv> \<lambda>x. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
+ have "\<exists>g. homeomorphism (cbox a b) (cbox c d) f g"
+ proof (rule homeomorphism_compact)
+ show "continuous_on (cbox a b) f"
+ apply (simp add: f_def)
+ apply (intro continuous_intros)
+ using assms by auto
+ have "f ` {a..b} = {c..d}"
+ unfolding f_def image_affinity_atLeastAtMost
+ using assms sum_sqs_eq by (auto simp: divide_simps algebra_simps)
+ then show "f ` cbox a b = cbox c d"
+ by auto
+ show "inj_on f (cbox a b)"
+ unfolding f_def inj_on_def using assms by auto
+ qed auto
+ then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
+ then show ?thesis
+ proof
+ show "f a = c"
+ by (simp add: f_def)
+ show "f b = d"
+ using assms sum_sqs_eq [of a b] by (auto simp: f_def divide_simps algebra_simps)
+ qed
+qed
+
+lemma homeomorphism_grouping_point_2:
+ fixes a::real and w::real
+ assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
+ and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
+ and "b \<in> cbox a c" "v \<in> cbox u w"
+ and eq: "f1 a = u" "f1 b = v" "f2 b = v" "f2 c = w"
+ obtains f g where "homeomorphism (cbox a c) (cbox u w) f g" "f a = u" "f c = w"
+ "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = f1 x" "\<And>x. x \<in> cbox b c \<Longrightarrow> f x = f2 x"
+proof -
+ have le: "a \<le> b" "b \<le> c" "u \<le> v" "v \<le> w"
+ using assms by simp_all
+ then have ac: "cbox a c = cbox a b \<union> cbox b c" and uw: "cbox u w = cbox u v \<union> cbox v w"
+ by auto
+ define f where "f \<equiv> \<lambda>x. if x \<le> b then f1 x else f2 x"
+ have "\<exists>g. homeomorphism (cbox a c) (cbox u w) f g"
+ proof (rule homeomorphism_compact)
+ have cf1: "continuous_on (cbox a b) f1"
+ using hom_ab homeomorphism_cont1 by blast
+ have cf2: "continuous_on (cbox b c) f2"
+ using hom_bc homeomorphism_cont1 by blast
+ show "continuous_on (cbox a c) f"
+ apply (simp add: f_def)
+ apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
+ using le eq apply (force simp: continuous_on_id)+
+ done
+ have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
+ unfolding f_def using eq by force+
+ then show "f ` cbox a c = cbox u w"
+ apply (simp only: ac uw image_Un)
+ by (metis hom_ab hom_bc homeomorphism_def)
+ have neq12: "f1 x \<noteq> f2 y" if x: "a \<le> x" "x \<le> b" and y: "b < y" "y \<le> c" for x y
+ proof -
+ have "f1 x \<in> cbox u v"
+ by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
+ moreover have "f2 y \<in> cbox v w"
+ by (metis (full_types) hom_bc homeomorphism_def image_subset_iff mem_box_real(2) not_le not_less_iff_gr_or_eq order_refl y)
+ moreover have "f2 y \<noteq> f2 b"
+ by (metis cancel_comm_monoid_add_class.diff_cancel diff_gt_0_iff_gt hom_bc homeomorphism_def le(2) less_imp_le less_numeral_extra(3) mem_box_real(2) order_refl y)
+ ultimately show ?thesis
+ using le eq by simp
+ qed
+ have "inj_on f1 (cbox a b)"
+ by (metis (full_types) hom_ab homeomorphism_def inj_onI)
+ moreover have "inj_on f2 (cbox b c)"
+ by (metis (full_types) hom_bc homeomorphism_def inj_onI)
+ ultimately show "inj_on f (cbox a c)"
+ apply (simp (no_asm) add: inj_on_def)
+ apply (simp add: f_def inj_on_eq_iff)
+ using neq12 apply force
+ done
+ qed auto
+ then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
+ then show ?thesis
+ apply (rule that)
+ using eq le by (auto simp: f_def)
+qed
+
+lemma homeomorphism_grouping_point_3:
+ fixes a::real
+ assumes cbox_sub: "cbox c d \<subseteq> box a b" "cbox u v \<subseteq> box a b"
+ and box_ne: "box c d \<noteq> {}" "box u v \<noteq> {}"
+ obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
+ "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v"
+proof -
+ have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d \<noteq> {}"
+ using assms
+ by (simp_all add: cbox_sub subset_eq)
+ obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
+ and f1_eq: "f1 a = a" "f1 c = u"
+ using homeomorphism_grouping_point_1 [OF \<open>a < c\<close> \<open>a < u\<close>] .
+ obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
+ and f2_eq: "f2 c = u" "f2 d = v"
+ using homeomorphism_grouping_point_1 [OF \<open>c < d\<close> \<open>u < v\<close>] .
+ obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
+ and f3_eq: "f3 d = v" "f3 b = b"
+ using homeomorphism_grouping_point_1 [OF \<open>d < b\<close> \<open>v < b\<close>] .
+ obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
+ and f4_eq: "\<And>x. x \<in> cbox a c \<Longrightarrow> f4 x = f1 x" "\<And>x. x \<in> cbox c d \<Longrightarrow> f4 x = f2 x"
+ using homeomorphism_grouping_point_2 [OF 1 2] less by (auto simp: f1_eq f2_eq)
+ obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
+ and f_eq: "\<And>x. x \<in> cbox a d \<Longrightarrow> f x = f4 x" "\<And>x. x \<in> cbox d b \<Longrightarrow> f x = f3 x"
+ using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
+ show ?thesis
+ apply (rule that [OF fg])
+ using f4_eq f_eq homeomorphism_image1 [OF 2]
+ apply simp
+ by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
+qed
+
+
+lemma homeomorphism_grouping_point_4:
+ fixes T :: "real set"
+ assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
+ obtains f g where "homeomorphism T T f g"
+ "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+ "bounded {x. (\<not> (f x = x \<and> g x = x))}"
+proof -
+ obtain c d where "box c d \<noteq> {}" "cbox c d \<subseteq> U"
+ proof -
+ obtain u where "u \<in> U"
+ using \<open>U \<noteq> {}\<close> by blast
+ then obtain e where "e > 0" "cball u e \<subseteq> U"
+ using \<open>open U\<close> open_contains_cball by blast
+ then show ?thesis
+ by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
+ qed
+ have "compact K"
+ by (simp add: \<open>finite K\<close> finite_imp_compact)
+ obtain a b where "box a b \<noteq> {}" "K \<subseteq> cbox a b" "cbox a b \<subseteq> S"
+ proof (cases "K = {}")
+ case True then show ?thesis
+ using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> that by blast
+ next
+ case False
+ then obtain a b where "a \<in> K" "b \<in> K"
+ and a: "\<And>x. x \<in> K \<Longrightarrow> a \<le> x" and b: "\<And>x. x \<in> K \<Longrightarrow> x \<le> b"
+ using compact_attains_inf compact_attains_sup by (metis \<open>compact K\<close>)+
+ obtain e where "e > 0" "cball b e \<subseteq> S"
+ using \<open>open S\<close> open_contains_cball
+ by (metis \<open>b \<in> K\<close> \<open>K \<subseteq> S\<close> subsetD)
+ show ?thesis
+ proof
+ show "box a (b + e) \<noteq> {}"
+ using \<open>0 < e\<close> \<open>b \<in> K\<close> a by force
+ show "K \<subseteq> cbox a (b + e)"
+ using \<open>0 < e\<close> a b by fastforce
+ have "a \<in> S"
+ using \<open>a \<in> K\<close> assms(6) by blast
+ have "b + e \<in> S"
+ using \<open>0 < e\<close> \<open>cball b e \<subseteq> S\<close> by (force simp: dist_norm)
+ show "cbox a (b + e) \<subseteq> S"
+ using \<open>a \<in> S\<close> \<open>b + e \<in> S\<close> \<open>connected S\<close> connected_contains_Icc by auto
+ qed
+ qed
+ obtain w z where "cbox w z \<subseteq> S" and sub_wz: "cbox a b \<union> cbox c d \<subseteq> box w z"
+ proof -
+ have "a \<in> S" "b \<in> S"
+ using \<open>box a b \<noteq> {}\<close> \<open>cbox a b \<subseteq> S\<close> by auto
+ moreover have "c \<in> S" "d \<in> S"
+ using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> by force+
+ ultimately have "min a c \<in> S" "max b d \<in> S"
+ by linarith+
+ then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 \<subseteq> S" "e2 > 0" "cball (max b d) e2 \<subseteq> S"
+ using \<open>open S\<close> open_contains_cball by metis
+ then have *: "min a c - e1 \<in> S" "max b d + e2 \<in> S"
+ by (auto simp: dist_norm)
+ show ?thesis
+ proof
+ show "cbox (min a c - e1) (max b d+ e2) \<subseteq> S"
+ using * \<open>connected S\<close> connected_contains_Icc by auto
+ show "cbox a b \<union> cbox c d \<subseteq> box (min a c - e1) (max b d + e2)"
+ using \<open>0 < e1\<close> \<open>0 < e2\<close> by auto
+ qed
+ qed
+ then
+ obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
+ and "f w = w" "f z = z"
+ and fin: "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<in> cbox c d"
+ using homeomorphism_grouping_point_3 [of a b w z c d]
+ using \<open>box a b \<noteq> {}\<close> \<open>box c d \<noteq> {}\<close> by blast
+ have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
+ using hom homeomorphism_def by blast+
+ define f' where "f' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
+ define g' where "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
+ show ?thesis
+ proof
+ have T: "cbox w z \<union> (T - box w z) = T"
+ using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
+ show "homeomorphism T T f' g'"
+ proof
+ have clo: "closedin (subtopology euclidean (cbox w z \<union> (T - box w z))) (T - box w z)"
+ by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
+ have "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
+ unfolding f'_def g'_def
+ apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
+ apply (simp_all add: closed_subset)
+ using \<open>f w = w\<close> \<open>f z = z\<close> apply force
+ by (metis \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_def less_eq_real_def mem_box_real(2))
+ then show "continuous_on T f'" "continuous_on T g'"
+ by (simp_all only: T)
+ show "f' ` T \<subseteq> T"
+ unfolding f'_def
+ by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
+ show "g' ` T \<subseteq> T"
+ unfolding g'_def
+ by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
+ show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
+ unfolding f'_def g'_def
+ using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom] by fastforce
+ show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
+ unfolding f'_def g'_def
+ using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom] by fastforce
+ qed
+ show "\<And>x. x \<in> K \<Longrightarrow> f' x \<in> U"
+ using fin sub_wz \<open>K \<subseteq> cbox a b\<close> \<open>cbox c d \<subseteq> U\<close> by (force simp: f'_def)
+ show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
+ using \<open>cbox w z \<subseteq> S\<close> by (auto simp: f'_def g'_def)
+ show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
+ apply (rule bounded_subset [of "cbox w z"])
+ using bounded_cbox apply blast
+ apply (auto simp: f'_def g'_def)
+ done
+ qed
+qed
+
+proposition%unimportant homeomorphism_grouping_points_exists:
+ fixes S :: "'a::euclidean_space set"
+ assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
+ obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+ "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
+proof (cases "2 \<le> DIM('a)")
+ case True
+ have TS: "T \<subseteq> affine hull S"
+ using affine_hull_open assms by blast
+ have "infinite U"
+ using \<open>open U\<close> \<open>U \<noteq> {}\<close> finite_imp_not_open by blast
+ then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
+ using infinite_arbitrarily_large by metis
+ then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
+ using \<open>finite K\<close> finite_same_card_bij by blast
+ obtain f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f (id i) = \<gamma> i" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. \<not> (f x = x \<and> g x = x)}"
+ proof (rule homeomorphism_moving_points_exists [OF True \<open>open S\<close> \<open>connected S\<close> \<open>S \<subseteq> T\<close> \<open>finite K\<close>])
+ show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
+ using \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> \<open>U \<subseteq> S\<close> bij_betwE by blast
+ show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
+ using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
+ qed (use affine_hull_open assms that in auto)
+ then show ?thesis
+ using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
+next
+ case False
+ with DIM_positive have "DIM('a) = 1"
+ by (simp add: dual_order.antisym)
+ then obtain h::"'a \<Rightarrow>real" and j
+ where "linear h" "linear j"
+ and noh: "\<And>x. norm(h x) = norm x" and noj: "\<And>y. norm(j y) = norm y"
+ and hj: "\<And>x. j(h x) = x" "\<And>y. h(j y) = y"
+ and ranh: "surj h"
+ using isomorphisms_UNIV_UNIV
+ by (metis (mono_tags, hide_lams) DIM_real UNIV_eq_I range_eqI)
+ obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
+ and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
+ and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
+ and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
+ apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
+ by (simp_all add: assms image_mono \<open>linear h\<close> open_surjective_linear_image connected_linear_image ranh)
+ have jf: "j (f (h x)) = x \<longleftrightarrow> f (h x) = h x" for x
+ by (metis hj)
+ have jg: "j (g (h x)) = x \<longleftrightarrow> g (h x) = h x" for x
+ by (metis hj)
+ have cont_hj: "continuous_on X h" "continuous_on Y j" for X Y
+ by (simp_all add: \<open>linear h\<close> \<open>linear j\<close> linear_linear linear_continuous_on)
+ show ?thesis
+ proof
+ show "homeomorphism T T (j \<circ> f \<circ> h) (j \<circ> g \<circ> h)"
+ proof
+ show "continuous_on T (j \<circ> f \<circ> h)" "continuous_on T (j \<circ> g \<circ> h)"
+ using hom homeomorphism_def
+ by (blast intro: continuous_on_compose cont_hj)+
+ show "(j \<circ> f \<circ> h) ` T \<subseteq> T" "(j \<circ> g \<circ> h) ` T \<subseteq> T"
+ by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
+ show "\<And>x. x \<in> T \<Longrightarrow> (j \<circ> g \<circ> h) ((j \<circ> f \<circ> h) x) = x"
+ using hj hom homeomorphism_apply1 by fastforce
+ show "\<And>y. y \<in> T \<Longrightarrow> (j \<circ> f \<circ> h) ((j \<circ> g \<circ> h) y) = y"
+ using hj hom homeomorphism_apply2 by fastforce
+ qed
+ show "{x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} \<subseteq> S"
+ apply (clarsimp simp: jf jg hj)
+ using sub hj
+ apply (drule_tac c="h x" in subsetD, force)
+ by (metis imageE)
+ have "bounded (j ` {x. (\<not> (f x = x \<and> g x = x))})"
+ by (rule bounded_linear_image [OF bou]) (use \<open>linear j\<close> linear_conv_bounded_linear in auto)
+ moreover
+ have *: "{x. \<not>((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
+ using hj by (auto simp: jf jg image_iff, metis+)
+ ultimately show "bounded {x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)}"
+ by metis
+ show "\<And>x. x \<in> K \<Longrightarrow> (j \<circ> f \<circ> h) x \<in> U"
+ using f hj by fastforce
+ qed
+qed
+
+
+proposition%unimportant homeomorphism_grouping_points_exists_gen:
+ fixes S :: "'a::euclidean_space set"
+ assumes opeU: "openin (subtopology euclidean S) U"
+ and opeS: "openin (subtopology euclidean (affine hull S)) S"
+ and "U \<noteq> {}" "finite K" "K \<subseteq> S" and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+ obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+ "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
+proof (cases "2 \<le> aff_dim S")
+ case True
+ have opeU': "openin (subtopology euclidean (affine hull S)) U"
+ using opeS opeU openin_trans by blast
+ obtain u where "u \<in> U" "u \<in> S"
+ using \<open>U \<noteq> {}\<close> opeU openin_imp_subset by fastforce+
+ have "infinite U"
+ apply (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
+ apply (rule connected_imp_perfect_aff_dim [OF \<open>connected S\<close> _ \<open>u \<in> S\<close>])
+ using True apply simp
+ done
+ then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
+ using infinite_arbitrarily_large by metis
+ then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
+ using \<open>finite K\<close> finite_same_card_bij by blast
+ have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(id i) = \<gamma> i) \<and>
+ {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+ proof (rule homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> _ _ True opeS S])
+ show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
+ by (metis id_apply opeU openin_contains_cball subsetCE \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> bij_betwE)
+ show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
+ using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
+ qed
+ then show ?thesis
+ using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
+next
+ case False
+ with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
+ then show ?thesis
+ proof cases
+ assume "aff_dim S = -1"
+ then have "S = {}"
+ using aff_dim_empty by blast
+ then have "False"
+ using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
+ then show ?thesis ..
+ next
+ assume "aff_dim S = 0"
+ then obtain a where "S = {a}"
+ using aff_dim_eq_0 by blast
+ then have "K \<subseteq> U"
+ using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
+ show ?thesis
+ apply (rule that [of id id])
+ using \<open>K \<subseteq> U\<close> by (auto simp: continuous_on_id intro: homeomorphismI)
+ next
+ assume "aff_dim S = 1"
+ then have "affine hull S homeomorphic (UNIV :: real set)"
+ by (auto simp: homeomorphic_affine_sets)
+ then obtain h::"'a\<Rightarrow>real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
+ using homeomorphic_def by blast
+ then have h: "\<And>x. x \<in> affine hull S \<Longrightarrow> j(h(x)) = x" and j: "\<And>y. j y \<in> affine hull S \<and> h(j y) = y"
+ by (auto simp: homeomorphism_def)
+ have connh: "connected (h ` S)"
+ by (meson Topological_Spaces.connected_continuous_image \<open>connected S\<close> homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
+ have hUS: "h ` U \<subseteq> h ` S"
+ by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
+ have opn: "openin (subtopology euclidean (affine hull S)) U \<Longrightarrow> open (h ` U)" for U
+ using homeomorphism_imp_open_map [OF homhj] by simp
+ have "open (h ` U)" "open (h ` S)"
+ by (auto intro: opeS opeU openin_trans opn)
+ then obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
+ and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
+ and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
+ and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
+ apply (rule homeomorphism_grouping_points_exists [of "h ` U" "h ` S" "h ` K" "h ` T"])
+ using assms by (auto simp: connh hUS)
+ have jf: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (f (h x)) = x \<longleftrightarrow> f (h x) = h x"
+ by (metis h j)
+ have jg: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (g (h x)) = x \<longleftrightarrow> g (h x) = h x"
+ by (metis h j)
+ have cont_hj: "continuous_on T h" "continuous_on Y j" for Y
+ apply (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
+ using homeomorphism_def homhj apply blast
+ by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
+ define f' where "f' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> f \<circ> h) x else x"
+ define g' where "g' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> g \<circ> h) x else x"
+ show ?thesis
+ proof
+ show "homeomorphism T T f' g'"
+ proof
+ have "continuous_on T (j \<circ> f \<circ> h)"
+ apply (intro continuous_on_compose cont_hj)
+ using hom homeomorphism_def by blast
+ then show "continuous_on T f'"
+ apply (rule continuous_on_eq)
+ using \<open>T \<subseteq> affine hull S\<close> f'_def by auto
+ have "continuous_on T (j \<circ> g \<circ> h)"
+ apply (intro continuous_on_compose cont_hj)
+ using hom homeomorphism_def by blast
+ then show "continuous_on T g'"
+ apply (rule continuous_on_eq)
+ using \<open>T \<subseteq> affine hull S\<close> g'_def by auto
+ show "f' ` T \<subseteq> T"
+ proof (clarsimp simp: f'_def)
+ fix x assume "x \<in> T"
+ then have "f (h x) \<in> h ` T"
+ by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
+ then show "j (f (h x)) \<in> T"
+ using \<open>T \<subseteq> affine hull S\<close> h by auto
+ qed
+ show "g' ` T \<subseteq> T"
+ proof (clarsimp simp: g'_def)
+ fix x assume "x \<in> T"
+ then have "g (h x) \<in> h ` T"
+ by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
+ then show "j (g (h x)) \<in> T"
+ using \<open>T \<subseteq> affine hull S\<close> h by auto
+ qed
+ show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
+ using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
+ show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
+ using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
+ qed
+ next
+ show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
+ apply (clarsimp simp: f'_def g'_def jf jg)
+ apply (rule imageE [OF subsetD [OF sub]], force)
+ by (metis h hull_inc)
+ next
+ have "compact (j ` closure {x. \<not> (f x = x \<and> g x = x)})"
+ using bou by (auto simp: compact_continuous_image cont_hj)
+ then have "bounded (j ` {x. \<not> (f x = x \<and> g x = x)})"
+ by (rule bounded_closure_image [OF compact_imp_bounded])
+ moreover
+ have *: "{x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
+ using h j by (auto simp: image_iff; metis)
+ ultimately have "bounded {x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x}"
+ by metis
+ then show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
+ by (simp add: f'_def g'_def Collect_mono bounded_subset)
+ next
+ show "f' x \<in> U" if "x \<in> K" for x
+ proof -
+ have "U \<subseteq> S"
+ using opeU openin_imp_subset by blast
+ then have "j (f (h x)) \<in> U"
+ using f h hull_subset that by fastforce
+ then show "f' x \<in> U"
+ using \<open>K \<subseteq> S\<close> S f'_def that by auto
+ qed
+ qed
+ qed
+qed
+
+
+subsection\<open>Nullhomotopic mappings\<close>
+
+text\<open> A mapping out of a sphere is nullhomotopic iff it extends to the ball.
+This even works out in the degenerate cases when the radius is \<open>\<le>\<close> 0, and
+we also don't need to explicitly assume continuity since it's already implicit
+in both sides of the equivalence.\<close>
+
+lemma nullhomotopic_from_lemma:
+ assumes contg: "continuous_on (cball a r - {a}) g"
+ and fa: "\<And>e. 0 < e
+ \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>x. x \<noteq> a \<and> norm(x - a) < d \<longrightarrow> norm(g x - f a) < e)"
+ and r: "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> f x = g x"
+ shows "continuous_on (cball a r) f"
+proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
+ fix x
+ assume x: "dist a x \<le> r"
+ show "continuous (at x within cball a r) f"
+ proof (cases "x=a")
+ case True
+ then show ?thesis
+ by (metis continuous_within_eps_delta fa dist_norm dist_self r)
+ next
+ case False
+ show ?thesis
+ proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
+ have "\<exists>d>0. \<forall>x'\<in>cball a r.
+ dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
+ proof -
+ obtain d where "d > 0"
+ and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
+ dist (g x') (g x) < e"
+ using contg False x \<open>e>0\<close>
+ unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
+ show ?thesis
+ using \<open>d > 0\<close> \<open>x \<noteq> a\<close>
+ by (rule_tac x="min d (norm(x - a))" in exI)
+ (auto simp: dist_commute dist_norm [symmetric] intro!: d)
+ qed
+ then show "continuous (at x within cball a r) g"
+ using contg False by (auto simp: continuous_within_eps_delta)
+ show "0 < norm (x - a)"
+ using False by force
+ show "x \<in> cball a r"
+ by (simp add: x)
+ show "\<And>x'. \<lbrakk>x' \<in> cball a r; dist x' x < norm (x - a)\<rbrakk>
+ \<Longrightarrow> g x' = f x'"
+ by (metis dist_commute dist_norm less_le r)
+ qed
+ qed
+qed
+
+proposition nullhomotopic_from_sphere_extension:
+ fixes f :: "'M::euclidean_space \<Rightarrow> 'a::real_normed_vector"
+ shows "(\<exists>c. homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) \<longleftrightarrow>
+ (\<exists>g. continuous_on (cball a r) g \<and> g ` (cball a r) \<subseteq> S \<and>
+ (\<forall>x \<in> sphere a r. g x = f x))"
+ (is "?lhs = ?rhs")
+proof (cases r "0::real" rule: linorder_cases)
+ case equal
+ then show ?thesis
+ apply (auto simp: homotopic_with)
+ apply (rule_tac x="\<lambda>x. h (0, a)" in exI)
+ apply (fastforce simp add:)
+ using continuous_on_const by blast
+next
+ case greater
+ let ?P = "continuous_on {x. norm(x - a) = r} f \<and> f ` {x. norm(x - a) = r} \<subseteq> S"
+ have ?P if ?lhs using that
+ proof
+ fix c
+ assume c: "homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
+ then have contf: "continuous_on (sphere a r) f" and fim: "f ` sphere a r \<subseteq> S"
+ by (auto simp: homotopic_with_imp_subset1 homotopic_with_imp_continuous)
+ show ?P
+ using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
+ qed
+ moreover have ?P if ?rhs using that
+ proof
+ fix g
+ assume g: "continuous_on (cball a r) g \<and> g ` cball a r \<subseteq> S \<and> (\<forall>xa\<in>sphere a r. g xa = f xa)"
+ then
+ show ?P
+ apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
+ apply (auto simp: dist_norm norm_minus_commute)
+ by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
+ qed
+ moreover have ?thesis if ?P
+ proof
+ assume ?lhs
+ then obtain c where "homotopic_with (\<lambda>x. True) (sphere a r) S (\<lambda>x. c) f"
+ using homotopic_with_sym by blast
+ then obtain h where conth: "continuous_on ({0..1::real} \<times> sphere a r) h"
+ and him: "h ` ({0..1} \<times> sphere a r) \<subseteq> S"
+ and h: "\<And>x. h(0, x) = c" "\<And>x. h(1, x) = f x"
+ by (auto simp: homotopic_with_def)
+ obtain b1::'M where "b1 \<in> Basis"
+ using SOME_Basis by auto
+ have "c \<in> S"
+ apply (rule him [THEN subsetD])
+ apply (rule_tac x = "(0, a + r *\<^sub>R b1)" in image_eqI)
+ using h greater \<open>b1 \<in> Basis\<close>
+ apply (auto simp: dist_norm)
+ done
+ have uconth: "uniformly_continuous_on ({0..1::real} \<times> (sphere a r)) h"
+ by (force intro: compact_Times conth compact_uniformly_continuous)
+ let ?g = "\<lambda>x. h (norm (x - a)/r,
+ a + (if x = a then r *\<^sub>R b1 else (r / norm(x - a)) *\<^sub>R (x - a)))"
+ let ?g' = "\<lambda>x. h (norm (x - a)/r, a + (r / norm(x - a)) *\<^sub>R (x - a))"
+ show ?rhs
+ proof (intro exI conjI)
+ have "continuous_on (cball a r - {a}) ?g'"
+ apply (rule continuous_on_compose2 [OF conth])
+ apply (intro continuous_intros)
+ using greater apply (auto simp: dist_norm norm_minus_commute)
+ done
+ then show "continuous_on (cball a r) ?g"
+ proof (rule nullhomotopic_from_lemma)
+ show "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (?g' x - ?g a) < e" if "0 < e" for e
+ proof -
+ obtain d where "0 < d"
+ and d: "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> sphere a r; x' \<in> {0..1} \<times> sphere a r; dist x' x < d\<rbrakk>
+ \<Longrightarrow> dist (h x') (h x) < e"
+ using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by auto
+ have *: "norm (h (norm (x - a) / r,
+ a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e"
+ if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
+ proof -
+ have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) =
+ norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
+ by (simp add: h)
+ also have "\<dots> < e"
+ apply (rule d [unfolded dist_norm])
+ using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
+ by (auto simp: dist_norm divide_simps)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule_tac x = "min r (d * r)" in exI)
+ using greater \<open>0 < d\<close> by (auto simp: *)
+ qed
+ show "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> ?g x = ?g' x"
+ by auto
+ qed
+ next
+ show "?g ` cball a r \<subseteq> S"
+ using greater him \<open>c \<in> S\<close>
+ by (force simp: h dist_norm norm_minus_commute)
+ next
+ show "\<forall>x\<in>sphere a r. ?g x = f x"
+ using greater by (auto simp: h dist_norm norm_minus_commute)
+ qed
+ next
+ assume ?rhs
+ then obtain g where contg: "continuous_on (cball a r) g"
+ and gim: "g ` cball a r \<subseteq> S"
+ and gf: "\<forall>x \<in> sphere a r. g x = f x"
+ by auto
+ let ?h = "\<lambda>y. g (a + (fst y) *\<^sub>R (snd y - a))"
+ have "continuous_on ({0..1} \<times> sphere a r) ?h"
+ apply (rule continuous_on_compose2 [OF contg])
+ apply (intro continuous_intros)
+ apply (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
+ done
+ moreover
+ have "?h ` ({0..1} \<times> sphere a r) \<subseteq> S"
+ by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
+ moreover
+ have "\<forall>x\<in>sphere a r. ?h (0, x) = g a" "\<forall>x\<in>sphere a r. ?h (1, x) = f x"
+ by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
+ ultimately
+ show ?lhs
+ apply (subst homotopic_with_sym)
+ apply (rule_tac x="g a" in exI)
+ apply (auto simp: homotopic_with)
+ done
+ qed
+ ultimately
+ show ?thesis by meson
+qed simp
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Path_Connected.thy Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Path_Connected.thy Mon Jan 07 14:57:45 2019 +0100
@@ -2,10 +2,10 @@
Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
*)
-section \<open>Continuous Paths\<close>
+section \<open>Path-Connectedness\<close>
theory Path_Connected
- imports Continuous_Extension Continuum_Not_Denumerable
+ imports Starlike
begin
subsection \<open>Paths and Arcs\<close>
@@ -295,7 +295,7 @@
qed
-section%unimportant \<open>Path Images\<close>
+subsection%unimportant \<open>Path Images\<close>
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
by (simp add: compact_imp_bounded compact_path_image)
@@ -1421,8 +1421,6 @@
by (rule_tac x="e/2" in exI) auto
qed
-section "Path-Connectedness" (* TODO: separate theory? *)
-
subsection \<open>Path component\<close>
text \<open>Original formalization by Tom Hales\<close>
@@ -2531,7 +2529,7 @@
by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
-section\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
+subsection\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
text%important\<open>The inside comprises the points in a bounded connected component of the set's complement.
The outside comprises the points in unbounded connected component of the complement.\<close>
@@ -3386,5156 +3384,4 @@
by (metis dw_le norm_minus_commute not_less order_trans rle wy)
qed
-
-section \<open>Homotopy of Maps\<close> (* TODO separate theory? *)
-
-
-definition%important homotopic_with ::
- "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
-where
- "homotopic_with P X Y p q \<equiv>
- (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
- continuous_on ({0..1} \<times> X) h \<and>
- h ` ({0..1} \<times> X) \<subseteq> Y \<and>
- (\<forall>x. h(0, x) = p x) \<and>
- (\<forall>x. h(1, x) = q x) \<and>
- (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
-
-text\<open>\<open>p\<close>, \<open>q\<close> are functions \<open>X \<rightarrow> Y\<close>, and the property \<open>P\<close> restricts all intermediate maps.
-We often just want to require that \<open>P\<close> fixes some subset, but to include the case of a loop homotopy,
-it is convenient to have a general property \<open>P\<close>.\<close>
-
-text \<open>We often want to just localize the ending function equality or whatever.\<close>
-text%important \<open>%whitespace\<close>
-proposition homotopic_with:
- fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
- assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
- shows "homotopic_with P X Y p q \<longleftrightarrow>
- (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
- continuous_on ({0..1} \<times> X) h \<and>
- h ` ({0..1} \<times> X) \<subseteq> Y \<and>
- (\<forall>x \<in> X. h(0,x) = p x) \<and>
- (\<forall>x \<in> X. h(1,x) = q x) \<and>
- (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
- unfolding homotopic_with_def
- apply (rule iffI, blast, clarify)
- apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
- apply auto
- apply (force elim: continuous_on_eq)
- apply (drule_tac x=t in bspec, force)
- apply (subst assms; simp)
- done
-
-proposition homotopic_with_eq:
- assumes h: "homotopic_with P X Y f g"
- and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
- and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
- and P: "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
- shows "homotopic_with P X Y f' g'"
- using h unfolding homotopic_with_def
- apply safe
- apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
- apply (simp add: f' g', safe)
- apply (fastforce intro: continuous_on_eq, fastforce)
- apply (subst P; fastforce)
- done
-
-proposition homotopic_with_equal:
- assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
- and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
- and P: "P f" "P g"
- shows "homotopic_with P X Y f g"
- unfolding homotopic_with_def
- apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
- using assms
- apply (intro conjI)
- apply (rule continuous_on_eq [where f = "f \<circ> snd"])
- apply (rule continuous_intros | force)+
- apply clarify
- apply (case_tac "t=1"; force)
- done
-
-
-lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
- by auto
-
-lemma homotopic_constant_maps:
- "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
-proof (cases "s = {} \<or> t = {}")
- case True with continuous_on_const show ?thesis
- by (auto simp: homotopic_with path_component_def)
-next
- case False
- then obtain c where "c \<in> s" by blast
- show ?thesis
- proof
- assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
- then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
- where conth: "continuous_on ({0..1} \<times> s) h"
- and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
- by (auto simp: homotopic_with)
- have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
- apply (rule continuous_intros conth | simp add: image_Pair_const)+
- apply (blast intro: \<open>c \<in> s\<close> continuous_on_subset [OF conth])
- done
- with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
- apply (simp_all add: homotopic_with path_component_def, auto)
- apply (drule_tac x="h \<circ> (\<lambda>t. (t, c))" in spec)
- apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
- done
- next
- assume "s = {} \<or> path_component t a b"
- with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
- apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
- apply (rule_tac x="g \<circ> fst" in exI)
- apply (rule conjI continuous_intros | force)+
- done
- qed
-qed
-
-
-subsection%unimportant\<open>Trivial properties\<close>
-
-lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
- unfolding homotopic_with_def Ball_def
- apply clarify
- apply (frule_tac x=0 in spec)
- apply (drule_tac x=1 in spec, auto)
- done
-
-lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h \<circ> Pair t)"
- by (fast intro: continuous_intros elim!: continuous_on_subset)
-
-lemma homotopic_with_imp_continuous:
- assumes "homotopic_with P X Y f g"
- shows "continuous_on X f \<and> continuous_on X g"
-proof -
- obtain h :: "real \<times> 'a \<Rightarrow> 'b"
- where conth: "continuous_on ({0..1} \<times> X) h"
- and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
- using assms by (auto simp: homotopic_with_def)
- have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h \<circ> (\<lambda>x. (t,x)))" for t
- by (rule continuous_intros continuous_on_subset [OF conth] | force)+
- show ?thesis
- using h *[of 0] *[of 1] by auto
-qed
-
-proposition homotopic_with_imp_subset1:
- "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
- by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
-
-proposition homotopic_with_imp_subset2:
- "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
- by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
-
-proposition homotopic_with_mono:
- assumes hom: "homotopic_with P X Y f g"
- and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
- shows "homotopic_with Q X Y f g"
- using hom
- apply (simp add: homotopic_with_def)
- apply (erule ex_forward)
- apply (force simp: intro!: Q dest: continuous_on_o_Pair)
- done
-
-proposition homotopic_with_subset_left:
- "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
- apply (simp add: homotopic_with_def)
- apply (fast elim!: continuous_on_subset ex_forward)
- done
-
-proposition homotopic_with_subset_right:
- "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
- apply (simp add: homotopic_with_def)
- apply (fast elim!: continuous_on_subset ex_forward)
- done
-
-proposition homotopic_with_compose_continuous_right:
- "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
- \<Longrightarrow> homotopic_with p W Y (f \<circ> h) (g \<circ> h)"
- apply (clarsimp simp add: homotopic_with_def)
- apply (rename_tac k)
- apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
- apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
- apply (erule continuous_on_subset)
- apply (fastforce simp: o_def)+
- done
-
-proposition homotopic_compose_continuous_right:
- "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
- \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
- using homotopic_with_compose_continuous_right by fastforce
-
-proposition homotopic_with_compose_continuous_left:
- "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
- \<Longrightarrow> homotopic_with p X Z (h \<circ> f) (h \<circ> g)"
- apply (clarsimp simp add: homotopic_with_def)
- apply (rename_tac k)
- apply (rule_tac x="h \<circ> k" in exI)
- apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
- apply (erule continuous_on_subset)
- apply (fastforce simp: o_def)+
- done
-
-proposition homotopic_compose_continuous_left:
- "\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g;
- continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
- \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
- using homotopic_with_compose_continuous_left by fastforce
-
-proposition homotopic_with_Pair:
- assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
- and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
- shows "homotopic_with q (s \<times> s') (t \<times> t')
- (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
- using hom
- apply (clarsimp simp add: homotopic_with_def)
- apply (rename_tac k k')
- apply (rule_tac x="\<lambda>z. ((k \<circ> (\<lambda>x. (fst x, fst (snd x)))) z, (k' \<circ> (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
- apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
- apply (auto intro!: q [unfolded case_prod_unfold])
- done
-
-lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g"
- by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
-
-
-text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
- though this only affects reflexivity.\<close>
-
-
-proposition homotopic_with_refl:
- "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
- apply (rule iffI)
- using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
- apply (simp add: homotopic_with_def)
- apply (rule_tac x="f \<circ> snd" in exI)
- apply (rule conjI continuous_intros | force)+
- done
-
-lemma homotopic_with_symD:
- fixes X :: "'a::real_normed_vector set"
- assumes "homotopic_with P X Y f g"
- shows "homotopic_with P X Y g f"
- using assms
- apply (clarsimp simp add: homotopic_with_def)
- apply (rename_tac h)
- apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
- apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
- done
-
-proposition homotopic_with_sym:
- fixes X :: "'a::real_normed_vector set"
- shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
- using homotopic_with_symD by blast
-
-lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
- by force
-
-lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
- by force
-
-proposition homotopic_with_trans:
- fixes X :: "'a::real_normed_vector set"
- assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
- shows "homotopic_with P X Y f h"
-proof -
- have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
- apply (simp add: closedin_closed split_01_prod [symmetric])
- apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
- apply (force simp: closed_Times)
- done
- have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
- apply (simp add: closedin_closed split_01_prod [symmetric])
- apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
- apply (force simp: closed_Times)
- done
- { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
- assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
- and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
- and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
- and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
- and P: "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
- define k where "k y =
- (if fst y \<le> 1 / 2
- then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
- else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y
- have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2" for u v
- by (simp add: geq that)
- have "continuous_on ({0..1} \<times> X) k"
- using cont
- apply (simp add: split_01_prod k_def)
- apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
- apply (force simp: keq)
- done
- moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
- using Y by (force simp: k_def)
- moreover have "\<forall>x. k (0, x) = f x"
- by (simp add: k_def k12)
- moreover have "(\<forall>x. k (1, x) = h x)"
- by (simp add: k_def k12)
- moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
- using P
- apply (clarsimp simp add: k_def)
- apply (case_tac "t \<le> 1/2", auto)
- done
- ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
- continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
- (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
- by blast
- } note * = this
- show ?thesis
- using assms by (auto intro: * simp add: homotopic_with_def)
-qed
-
-proposition homotopic_compose:
- fixes s :: "'a::real_normed_vector set"
- shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
- \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g \<circ> f) (g' \<circ> f')"
- apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
- apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
- by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
-
-
-text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
-lemma homotopic_triviality:
- fixes S :: "'a::real_normed_vector set"
- shows "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
- continuous_on S g \<and> g ` S \<subseteq> T
- \<longrightarrow> homotopic_with (\<lambda>x. True) S T f g) \<longleftrightarrow>
- (S = {} \<or> path_connected T) \<and>
- (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)))"
- (is "?lhs = ?rhs")
-proof (cases "S = {} \<or> T = {}")
- case True then show ?thesis by auto
-next
- case False show ?thesis
- proof
- assume LHS [rule_format]: ?lhs
- have pab: "path_component T a b" if "a \<in> T" "b \<in> T" for a b
- proof -
- have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
- by (simp add: LHS continuous_on_const image_subset_iff that)
- then show ?thesis
- using False homotopic_constant_maps by blast
- qed
- moreover
- have "\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
- by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
- ultimately show ?rhs
- by (simp add: path_connected_component)
- next
- assume RHS: ?rhs
- with False have T: "path_connected T"
- by blast
- show ?lhs
- proof clarify
- fix f g
- assume "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
- obtain c d where c: "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with (\<lambda>x. True) S T g (\<lambda>x. d)"
- using False \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> T\<close> RHS \<open>continuous_on S g\<close> \<open>g ` S \<subseteq> T\<close> by blast
- then have "c \<in> T" "d \<in> T"
- using False homotopic_with_imp_subset2 by fastforce+
- with T have "path_component T c d"
- using path_connected_component by blast
- then have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
- by (simp add: homotopic_constant_maps)
- with c d show "homotopic_with (\<lambda>x. True) S T f g"
- by (meson homotopic_with_symD homotopic_with_trans)
- qed
- qed
-qed
-
-
-subsection\<open>Homotopy of paths, maintaining the same endpoints\<close>
-
-
-definition%important homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
- where
- "homotopic_paths s p q \<equiv>
- homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
-
-lemma homotopic_paths:
- "homotopic_paths s p q \<longleftrightarrow>
- (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
- h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
- (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
- (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
- (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
- pathfinish(h \<circ> Pair t) = pathfinish p))"
- by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
-
-proposition homotopic_paths_imp_pathstart:
- "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
- by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
-
-proposition homotopic_paths_imp_pathfinish:
- "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
- by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
-
-lemma homotopic_paths_imp_path:
- "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
- using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
-
-lemma homotopic_paths_imp_subset:
- "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
- by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
-
-proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
-by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
-
-proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p"
- by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
-
-proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
- by (metis homotopic_paths_sym)
-
-proposition homotopic_paths_trans [trans]:
- "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
- apply (simp add: homotopic_paths_def)
- apply (rule homotopic_with_trans, assumption)
- by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
-
-proposition homotopic_paths_eq:
- "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
- apply (simp add: homotopic_paths_def)
- apply (rule homotopic_with_eq)
- apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
- done
-
-proposition homotopic_paths_reparametrize:
- assumes "path p"
- and pips: "path_image p \<subseteq> s"
- and contf: "continuous_on {0..1} f"
- and f01:"f ` {0..1} \<subseteq> {0..1}"
- and [simp]: "f(0) = 0" "f(1) = 1"
- and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
- shows "homotopic_paths s p q"
-proof -
- have contp: "continuous_on {0..1} p"
- by (metis \<open>path p\<close> path_def)
- then have "continuous_on {0..1} (p \<circ> f)"
- using contf continuous_on_compose continuous_on_subset f01 by blast
- then have "path q"
- by (simp add: path_def) (metis q continuous_on_cong)
- have piqs: "path_image q \<subseteq> s"
- by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
- have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
- using f01 by force
- have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
- using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
- have "homotopic_paths s q p"
- proof (rule homotopic_paths_trans)
- show "homotopic_paths s q (p \<circ> f)"
- using q by (force intro: homotopic_paths_eq [OF \<open>path q\<close> piqs])
- next
- show "homotopic_paths s (p \<circ> f) p"
- apply (simp add: homotopic_paths_def homotopic_with_def)
- apply (rule_tac x="p \<circ> (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f \<circ> snd) y) + (fst y) *\<^sub>R snd y)" in exI)
- apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
- using pips [unfolded path_image_def]
- apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
- done
- qed
- then show ?thesis
- by (simp add: homotopic_paths_sym)
-qed
-
-lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
- using homotopic_paths_def homotopic_with_subset_right by blast
-
-
-text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
-lemma homotopic_join_lemma:
- fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
- assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
- and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
- and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
- shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
-proof -
- have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
- by (rule ext) (simp)
- have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
- by (rule ext) (simp)
- show ?thesis
- apply (simp add: joinpaths_def)
- apply (rule continuous_on_cases_le)
- apply (simp_all only: 1 2)
- apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
- using pf
- apply (auto simp: mult.commute pathstart_def pathfinish_def)
- done
-qed
-
-text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
-
-lemma homotopic_paths_reversepath_D:
- assumes "homotopic_paths s p q"
- shows "homotopic_paths s (reversepath p) (reversepath q)"
- using assms
- apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
- apply (rule_tac x="h \<circ> (\<lambda>x. (fst x, 1 - snd x))" in exI)
- apply (rule conjI continuous_intros)+
- apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
- done
-
-proposition homotopic_paths_reversepath:
- "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
- using homotopic_paths_reversepath_D by force
-
-
-proposition homotopic_paths_join:
- "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
- apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
- apply (rename_tac k1 k2)
- apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
- apply (rule conjI continuous_intros homotopic_join_lemma)+
- apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
- done
-
-proposition homotopic_paths_continuous_image:
- "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
- unfolding homotopic_paths_def
- apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
- apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
- done
-
-
-subsection\<open>Group properties for homotopy of paths\<close>
-
-text%important\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
-
-proposition homotopic_paths_rid:
- "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
- apply (subst homotopic_paths_sym)
- apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
- apply (simp_all del: le_divide_eq_numeral1)
- apply (subst split_01)
- apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
- done
-
-proposition homotopic_paths_lid:
- "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
- using homotopic_paths_rid [of "reversepath p" s]
- by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
- pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
-
-proposition homotopic_paths_assoc:
- "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
- pathfinish q = pathstart r\<rbrakk>
- \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
- apply (subst homotopic_paths_sym)
- apply (rule homotopic_paths_reparametrize
- [where f = "\<lambda>t. if t \<le> 1 / 2 then inverse 2 *\<^sub>R t
- else if t \<le> 3 / 4 then t - (1 / 4)
- else 2 *\<^sub>R t - 1"])
- apply (simp_all del: le_divide_eq_numeral1)
- apply (simp add: subset_path_image_join)
- apply (rule continuous_on_cases_1 continuous_intros)+
- apply (auto simp: joinpaths_def)
- done
-
-proposition homotopic_paths_rinv:
- assumes "path p" "path_image p \<subseteq> s"
- shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
-proof -
- have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
- using assms
- apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
- apply (rule continuous_on_cases_le)
- apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
- apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
- apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
- apply (force elim!: continuous_on_subset simp add: mult_le_one)+
- done
- then show ?thesis
- using assms
- apply (subst homotopic_paths_sym_eq)
- unfolding homotopic_paths_def homotopic_with_def
- apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
- apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
- apply (force simp: mult_le_one)
- done
-qed
-
-proposition homotopic_paths_linv:
- assumes "path p" "path_image p \<subseteq> s"
- shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
- using homotopic_paths_rinv [of "reversepath p" s] assms by simp
-
-
-subsection\<open>Homotopy of loops without requiring preservation of endpoints\<close>
-
-definition%important homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool" where
- "homotopic_loops s p q \<equiv>
- homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
-
-lemma homotopic_loops:
- "homotopic_loops s p q \<longleftrightarrow>
- (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
- image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
- (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
- (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
- (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
- by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
-
-proposition homotopic_loops_imp_loop:
- "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
-using homotopic_with_imp_property homotopic_loops_def by blast
-
-proposition homotopic_loops_imp_path:
- "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
- unfolding homotopic_loops_def path_def
- using homotopic_with_imp_continuous by blast
-
-proposition homotopic_loops_imp_subset:
- "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
- unfolding homotopic_loops_def path_image_def
- by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
-
-proposition homotopic_loops_refl:
- "homotopic_loops s p p \<longleftrightarrow>
- path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
- by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
-
-proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
- by (simp add: homotopic_loops_def homotopic_with_sym)
-
-proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
- by (metis homotopic_loops_sym)
-
-proposition homotopic_loops_trans:
- "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
- unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
-
-proposition homotopic_loops_subset:
- "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
- by (simp add: homotopic_loops_def homotopic_with_subset_right)
-
-proposition homotopic_loops_eq:
- "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
- \<Longrightarrow> homotopic_loops s p q"
- unfolding homotopic_loops_def
- apply (rule homotopic_with_eq)
- apply (rule homotopic_with_refl [where f = p, THEN iffD2])
- apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
- done
-
-proposition homotopic_loops_continuous_image:
- "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
- unfolding homotopic_loops_def
- apply (rule homotopic_with_compose_continuous_left)
- apply (erule homotopic_with_mono)
- by (simp add: pathfinish_def pathstart_def)
-
-
-subsection\<open>Relations between the two variants of homotopy\<close>
-
-proposition homotopic_paths_imp_homotopic_loops:
- "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
- by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
-
-proposition homotopic_loops_imp_homotopic_paths_null:
- assumes "homotopic_loops s p (linepath a a)"
- shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
-proof -
- have "path p" by (metis assms homotopic_loops_imp_path)
- have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
- have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
- obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
- and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
- and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
- and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
- and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
- using assms by (auto simp: homotopic_loops homotopic_with)
- have conth0: "path (\<lambda>u. h (u, 0))"
- unfolding path_def
- apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
- apply (force intro: continuous_intros continuous_on_subset [OF conth])+
- done
- have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
- using hs by (force simp: path_image_def)
- have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
- apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
- apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
- done
- have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
- apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
- apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
- apply (rule continuous_on_subset [OF conth])
- apply (auto simp: algebra_simps add_increasing2 mult_left_le)
- done
- have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
- using ends by (simp add: pathfinish_def pathstart_def)
- have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
- proof -
- have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
- with \<open>c \<le> 1\<close> show ?thesis by fastforce
- qed
- have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
- (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
- (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
- pathstart(reversepath p) = a) \<and> pathstart p = x
- \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
- by (metis homotopic_paths_lid homotopic_paths_join
- homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
- have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
- using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast
- moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
- (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
- apply (rule homotopic_paths_sym)
- using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
- by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
- moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
- ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
- apply (simp add: homotopic_paths_def homotopic_with_def)
- apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
- apply (simp add: subpath_reversepath)
- apply (intro conjI homotopic_join_lemma)
- using ploop
- apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
- apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
- done
- moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
- (linepath (pathstart p) (pathstart p))"
- apply (rule *)
- apply (simp add: pih0 pathstart_def pathfinish_def conth0)
- apply (simp add: reversepath_def joinpaths_def)
- done
- ultimately show ?thesis
- by (blast intro: homotopic_paths_trans)
-qed
-
-proposition homotopic_loops_conjugate:
- fixes s :: "'a::real_normed_vector set"
- assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
- and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
- shows "homotopic_loops s (p +++ q +++ reversepath p) q"
-proof -
- have contp: "continuous_on {0..1} p" using \<open>path p\<close> [unfolded path_def] by blast
- have contq: "continuous_on {0..1} q" using \<open>path q\<close> [unfolded path_def] by blast
- have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
- apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
- apply (force simp: mult_le_one intro!: continuous_intros)
- apply (rule continuous_on_subset [OF contp])
- apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
- done
- have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
- apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
- apply (force simp: mult_le_one intro!: continuous_intros)
- apply (rule continuous_on_subset [OF contp])
- apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
- done
- have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s"
- using sum_le_prod1
- by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
- have ps2: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
- apply (rule pip [unfolded path_image_def, THEN subsetD])
- apply (rule image_eqI, blast)
- apply (simp add: algebra_simps)
- by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
- add.commute zero_le_numeral)
- have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s"
- using path_image_def piq by fastforce
- have "homotopic_loops s (p +++ q +++ reversepath p)
- (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
- apply (simp add: homotopic_loops_def homotopic_with_def)
- apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
- apply (simp add: subpath_refl subpath_reversepath)
- apply (intro conjI homotopic_join_lemma)
- using papp qloop
- apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
- apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
- apply (auto simp: ps1 ps2 qs)
- done
- moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
- proof -
- have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
- using \<open>path q\<close> homotopic_paths_lid qloop piq by auto
- hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
- using homotopic_paths_trans by blast
- hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
- proof -
- have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
- by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
- thus ?thesis
- by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
- homotopic_paths_trans qloop pathfinish_linepath piq)
- qed
- thus ?thesis
- by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
- qed
- ultimately show ?thesis
- by (blast intro: homotopic_loops_trans)
-qed
-
-lemma homotopic_paths_loop_parts:
- assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
- shows "homotopic_paths S p q"
-proof -
- have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
- using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
- then have "path p"
- using \<open>path q\<close> homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
- show ?thesis
- proof (cases "pathfinish p = pathfinish q")
- case True
- have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
- by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> homotopic_loops_imp_subset homotopic_paths_imp_path loops
- path_image_join path_image_reversepath path_imp_reversepath path_join_eq)+
- have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
- using \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> homotopic_paths_rid homotopic_paths_sym by blast
- moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
- by (simp add: True \<open>path p\<close> \<open>path q\<close> pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
- moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
- by (simp add: True \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
- moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
- by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
- moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
- by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
- ultimately show ?thesis
- using homotopic_paths_trans by metis
- next
- case False
- then show ?thesis
- using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
- qed
-qed
-
-
-subsection%unimportant\<open>Homotopy of "nearby" function, paths and loops\<close>
-
-lemma homotopic_with_linear:
- fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
- assumes contf: "continuous_on s f"
- and contg:"continuous_on s g"
- and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
- shows "homotopic_with (\<lambda>z. True) s t f g"
- apply (simp add: homotopic_with_def)
- apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
- apply (intro conjI)
- apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
- continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
- using sub closed_segment_def apply fastforce+
- done
-
-lemma homotopic_paths_linear:
- fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
- assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
- "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
- shows "homotopic_paths s g h"
- using assms
- unfolding path_def
- apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
- apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g \<circ> snd) y + (fst y) *\<^sub>R (h \<circ> snd) y)" in exI)
- apply (intro conjI subsetI continuous_intros; force)
- done
-
-lemma homotopic_loops_linear:
- fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
- assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
- "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
- shows "homotopic_loops s g h"
- using assms
- unfolding path_def
- apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
- apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
- apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
- apply (force simp: closed_segment_def)
- done
-
-lemma homotopic_paths_nearby_explicit:
- assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
- and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
- shows "homotopic_paths s g h"
- apply (rule homotopic_paths_linear [OF assms(1-4)])
- by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-
-lemma homotopic_loops_nearby_explicit:
- assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
- and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
- shows "homotopic_loops s g h"
- apply (rule homotopic_loops_linear [OF assms(1-4)])
- by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-
-lemma homotopic_nearby_paths:
- fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
- assumes "path g" "open s" "path_image g \<subseteq> s"
- shows "\<exists>e. 0 < e \<and>
- (\<forall>h. path h \<and>
- pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
- (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
-proof -
- obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
- using separate_compact_closed [of "path_image g" "-s"] assms by force
- show ?thesis
- apply (intro exI conjI)
- using e [unfolded dist_norm]
- apply (auto simp: intro!: homotopic_paths_nearby_explicit assms \<open>e > 0\<close>)
- by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
-qed
-
-lemma homotopic_nearby_loops:
- fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
- assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
- shows "\<exists>e. 0 < e \<and>
- (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
- (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
-proof -
- obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
- using separate_compact_closed [of "path_image g" "-s"] assms by force
- show ?thesis
- apply (intro exI conjI)
- using e [unfolded dist_norm]
- apply (auto simp: intro!: homotopic_loops_nearby_explicit assms \<open>e > 0\<close>)
- by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
-qed
-
-
-subsection\<open> Homotopy and subpaths\<close>
-
-lemma homotopic_join_subpaths1:
- assumes "path g" and pag: "path_image g \<subseteq> s"
- and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
- shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
-proof -
- have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
- using affine_ineq \<open>u \<le> v\<close> by fastforce
- have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
- by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
- have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
- show ?thesis
- apply (rule homotopic_paths_subset [OF _ pag])
- using assms
- apply (cases "w = u")
- using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
- apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
- apply (rule homotopic_paths_sym)
- apply (rule homotopic_paths_reparametrize
- [where f = "\<lambda>t. if t \<le> 1 / 2
- then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
- else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
- using \<open>path g\<close> path_subpath u w apply blast
- using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
- apply simp_all
- apply (subst split_01)
- apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
- apply (simp_all add: field_simps not_le)
- apply (force dest!: t2)
- apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
- apply (simp add: joinpaths_def subpath_def)
- apply (force simp: algebra_simps)
- done
-qed
-
-lemma homotopic_join_subpaths2:
- assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
- shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
-by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
-
-lemma homotopic_join_subpaths3:
- assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
- and "path g" and pag: "path_image g \<subseteq> s"
- and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
- shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
-proof -
- have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
- apply (rule homotopic_paths_join)
- using hom homotopic_paths_sym_eq apply blast
- apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
- done
- also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
- apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
- using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
- also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
- (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
- apply (rule homotopic_paths_join)
- apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
- apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
- apply simp
- done
- also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
- apply (rule homotopic_paths_rid)
- using \<open>path g\<close> path_subpath u v apply blast
- apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
- done
- finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
- then show ?thesis
- using homotopic_join_subpaths2 by blast
-qed
-
-proposition homotopic_join_subpaths:
- "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
- \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
- apply (rule le_cases3 [of u v w])
-using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
-
-text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
-
-lemma path_component_imp_homotopic_points:
- "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
-apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
- pathstart_def pathfinish_def path_image_def path_def, clarify)
-apply (rule_tac x="g \<circ> fst" in exI)
-apply (intro conjI continuous_intros continuous_on_compose)+
-apply (auto elim!: continuous_on_subset)
-done
-
-lemma homotopic_loops_imp_path_component_value:
- "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
- \<Longrightarrow> path_component S (p t) (q t)"
-apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
- pathstart_def pathfinish_def path_image_def path_def, clarify)
-apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
-apply (intro conjI continuous_intros continuous_on_compose)+
-apply (auto elim!: continuous_on_subset)
-done
-
-lemma homotopic_points_eq_path_component:
- "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
- path_component S a b"
-by (auto simp: path_component_imp_homotopic_points
- dest: homotopic_loops_imp_path_component_value [where t=1])
-
-lemma path_connected_eq_homotopic_points:
- "path_connected S \<longleftrightarrow>
- (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
-by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
-
-
-subsection\<open>Simply connected sets\<close>
-
-text%important\<open>defined as "all loops are homotopic (as loops)\<close>
-
-definition%important simply_connected where
- "simply_connected S \<equiv>
- \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and>
- path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S
- \<longrightarrow> homotopic_loops S p q"
-
-lemma simply_connected_empty [iff]: "simply_connected {}"
- by (simp add: simply_connected_def)
-
-lemma simply_connected_imp_path_connected:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<Longrightarrow> path_connected S"
-by (simp add: simply_connected_def path_connected_eq_homotopic_points)
-
-lemma simply_connected_imp_connected:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<Longrightarrow> connected S"
-by (simp add: path_connected_imp_connected simply_connected_imp_path_connected)
-
-lemma simply_connected_eq_contractible_loop_any:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<longleftrightarrow>
- (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and>
- pathfinish p = pathstart p \<and> a \<in> S
- \<longrightarrow> homotopic_loops S p (linepath a a))"
-apply (simp add: simply_connected_def)
-apply (rule iffI, force, clarify)
-apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
-apply (fastforce simp add:)
-using homotopic_loops_sym apply blast
-done
-
-lemma simply_connected_eq_contractible_loop_some:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<longleftrightarrow>
- path_connected S \<and>
- (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
- \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
-apply (rule iffI)
- apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
-apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
-apply (drule_tac x=p in spec)
-using homotopic_loops_trans path_connected_eq_homotopic_points
- apply blast
-done
-
-lemma simply_connected_eq_contractible_loop_all:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<longleftrightarrow>
- S = {} \<or>
- (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
- \<longrightarrow> homotopic_loops S p (linepath a a))"
- (is "?lhs = ?rhs")
-proof (cases "S = {}")
- case True then show ?thesis by force
-next
- case False
- then obtain a where "a \<in> S" by blast
- show ?thesis
- proof
- assume "simply_connected S"
- then show ?rhs
- using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
- by blast
- next
- assume ?rhs
- then show "simply_connected S"
- apply (simp add: simply_connected_eq_contractible_loop_any False)
- by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
- path_component_imp_homotopic_points path_component_refl)
- qed
-qed
-
-lemma simply_connected_eq_contractible_path:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<longleftrightarrow>
- path_connected S \<and>
- (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
- \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
-apply (rule iffI)
- apply (simp add: simply_connected_imp_path_connected)
- apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
-by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
- simply_connected_eq_contractible_loop_some subset_iff)
-
-lemma simply_connected_eq_homotopic_paths:
- fixes S :: "_::real_normed_vector set"
- shows "simply_connected S \<longleftrightarrow>
- path_connected S \<and>
- (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
- path q \<and> path_image q \<subseteq> S \<and>
- pathstart q = pathstart p \<and> pathfinish q = pathfinish p
- \<longrightarrow> homotopic_paths S p q)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have pc: "path_connected S"
- and *: "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
- pathfinish p = pathstart p\<rbrakk>
- \<Longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))"
- by (auto simp: simply_connected_eq_contractible_path)
- have "homotopic_paths S p q"
- if "path p" "path_image p \<subseteq> S" "path q"
- "path_image q \<subseteq> S" "pathstart q = pathstart p"
- "pathfinish q = pathfinish p" for p q
- proof -
- have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
- by (simp add: homotopic_paths_rid homotopic_paths_sym that)
- also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
- (p +++ reversepath q +++ q)"
- using that
- by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
- also have "homotopic_paths S (p +++ reversepath q +++ q)
- ((p +++ reversepath q) +++ q)"
- by (simp add: that homotopic_paths_assoc)
- also have "homotopic_paths S ((p +++ reversepath q) +++ q)
- (linepath (pathstart q) (pathstart q) +++ q)"
- using * [of "p +++ reversepath q"] that
- by (simp add: homotopic_paths_join path_image_join)
- also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q"
- using that homotopic_paths_lid by blast
- finally show ?thesis .
- qed
- then show ?rhs
- by (blast intro: pc *)
-next
- assume ?rhs
- then show ?lhs
- by (force simp: simply_connected_eq_contractible_path)
-qed
-
-proposition simply_connected_Times:
- fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
- assumes S: "simply_connected S" and T: "simply_connected T"
- shows "simply_connected(S \<times> T)"
-proof -
- have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
- if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
- for p a b
- proof -
- have "path (fst \<circ> p)"
- apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
- apply (rule continuous_intros)+
- done
- moreover have "path_image (fst \<circ> p) \<subseteq> S"
- using that apply (simp add: path_image_def) by force
- ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
- using S that
- apply (simp add: simply_connected_eq_contractible_loop_any)
- apply (drule_tac x="fst \<circ> p" in spec)
- apply (drule_tac x=a in spec)
- apply (auto simp: pathstart_def pathfinish_def)
- done
- have "path (snd \<circ> p)"
- apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
- apply (rule continuous_intros)+
- done
- moreover have "path_image (snd \<circ> p) \<subseteq> T"
- using that apply (simp add: path_image_def) by force
- ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
- using T that
- apply (simp add: simply_connected_eq_contractible_loop_any)
- apply (drule_tac x="snd \<circ> p" in spec)
- apply (drule_tac x=b in spec)
- apply (auto simp: pathstart_def pathfinish_def)
- done
- show ?thesis
- using p1 p2
- apply (simp add: homotopic_loops, clarify)
- apply (rename_tac h k)
- apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
- apply (intro conjI continuous_intros | assumption)+
- apply (auto simp: pathstart_def pathfinish_def)
- done
- qed
- with assms show ?thesis
- by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
-qed
-
-
-subsection\<open>Contractible sets\<close>
-
-definition%important contractible where
- "contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
-
-proposition contractible_imp_simply_connected:
- fixes S :: "_::real_normed_vector set"
- assumes "contractible S" shows "simply_connected S"
-proof (cases "S = {}")
- case True then show ?thesis by force
-next
- case False
- obtain a where a: "homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
- using assms by (force simp: contractible_def)
- then have "a \<in> S"
- by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
- show ?thesis
- apply (simp add: simply_connected_eq_contractible_loop_all False)
- apply (rule bexI [OF _ \<open>a \<in> S\<close>])
- using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
- apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
- apply (intro conjI continuous_on_compose continuous_intros)
- apply (erule continuous_on_subset | force)+
- done
-qed
-
-corollary contractible_imp_connected:
- fixes S :: "_::real_normed_vector set"
- shows "contractible S \<Longrightarrow> connected S"
-by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)
-
-lemma contractible_imp_path_connected:
- fixes S :: "_::real_normed_vector set"
- shows "contractible S \<Longrightarrow> path_connected S"
-by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)
-
-lemma nullhomotopic_through_contractible:
- fixes S :: "_::topological_space set"
- assumes f: "continuous_on S f" "f ` S \<subseteq> T"
- and g: "continuous_on T g" "g ` T \<subseteq> U"
- and T: "contractible T"
- obtains c where "homotopic_with (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
-proof -
- obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)"
- using assms by (force simp: contractible_def)
- have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
- by (rule homotopic_compose_continuous_left [OF b g])
- then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
- by (rule homotopic_compose_continuous_right [OF _ f])
- then show ?thesis
- by (simp add: comp_def that)
-qed
-
-lemma nullhomotopic_into_contractible:
- assumes f: "continuous_on S f" "f ` S \<subseteq> T"
- and T: "contractible T"
- obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
-apply (rule nullhomotopic_through_contractible [OF f, of id T])
-using assms
-apply (auto simp: continuous_on_id)
-done
-
-lemma nullhomotopic_from_contractible:
- assumes f: "continuous_on S f" "f ` S \<subseteq> T"
- and S: "contractible S"
- obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
-apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
-using assms
-apply (auto simp: comp_def)
-done
-
-lemma homotopic_through_contractible:
- fixes S :: "_::real_normed_vector set"
- assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
- "continuous_on T g1" "g1 ` T \<subseteq> U"
- "continuous_on S f2" "f2 ` S \<subseteq> T"
- "continuous_on T g2" "g2 ` T \<subseteq> U"
- "contractible T" "path_connected U"
- shows "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
-proof -
- obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
- apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
- using assms apply auto
- done
- obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
- apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
- using assms apply auto
- done
- have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
- proof (cases "S = {}")
- case True then show ?thesis by force
- next
- case False
- with c1 c2 have "c1 \<in> U" "c2 \<in> U"
- using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
- with \<open>path_connected U\<close> show ?thesis by blast
- qed
- show ?thesis
- apply (rule homotopic_with_trans [OF c1])
- apply (rule homotopic_with_symD)
- apply (rule homotopic_with_trans [OF c2])
- apply (simp add: path_component homotopic_constant_maps *)
- done
-qed
-
-lemma homotopic_into_contractible:
- fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
- assumes f: "continuous_on S f" "f ` S \<subseteq> T"
- and g: "continuous_on S g" "g ` S \<subseteq> T"
- and T: "contractible T"
- shows "homotopic_with (\<lambda>h. True) S T f g"
-using homotopic_through_contractible [of S f T id T g id]
-by (simp add: assms contractible_imp_path_connected continuous_on_id)
-
-lemma homotopic_from_contractible:
- fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
- assumes f: "continuous_on S f" "f ` S \<subseteq> T"
- and g: "continuous_on S g" "g ` S \<subseteq> T"
- and "contractible S" "path_connected T"
- shows "homotopic_with (\<lambda>h. True) S T f g"
-using homotopic_through_contractible [of S id S f T id g]
-by (simp add: assms contractible_imp_path_connected continuous_on_id)
-
-lemma starlike_imp_contractible_gen:
- fixes S :: "'a::real_normed_vector set"
- assumes S: "starlike S"
- and P: "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P(\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)"
- obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)"
-proof -
- obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
- using S by (auto simp: starlike_def)
- have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
- apply clarify
- apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
- apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
- done
- then show ?thesis
- apply (rule_tac a=a in that)
- using \<open>a \<in> S\<close>
- apply (simp add: homotopic_with_def)
- apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
- apply (intro conjI ballI continuous_on_compose continuous_intros)
- apply (simp_all add: P)
- done
-qed
-
-lemma starlike_imp_contractible:
- fixes S :: "'a::real_normed_vector set"
- shows "starlike S \<Longrightarrow> contractible S"
-using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)
-
-lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
- by (simp add: starlike_imp_contractible)
-
-lemma starlike_imp_simply_connected:
- fixes S :: "'a::real_normed_vector set"
- shows "starlike S \<Longrightarrow> simply_connected S"
-by (simp add: contractible_imp_simply_connected starlike_imp_contractible)
-
-lemma convex_imp_simply_connected:
- fixes S :: "'a::real_normed_vector set"
- shows "convex S \<Longrightarrow> simply_connected S"
-using convex_imp_starlike starlike_imp_simply_connected by blast
-
-lemma starlike_imp_path_connected:
- fixes S :: "'a::real_normed_vector set"
- shows "starlike S \<Longrightarrow> path_connected S"
-by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)
-
-lemma starlike_imp_connected:
- fixes S :: "'a::real_normed_vector set"
- shows "starlike S \<Longrightarrow> connected S"
-by (simp add: path_connected_imp_connected starlike_imp_path_connected)
-
-lemma is_interval_simply_connected_1:
- fixes S :: "real set"
- shows "is_interval S \<longleftrightarrow> simply_connected S"
-using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
-
-lemma contractible_empty [simp]: "contractible {}"
- by (simp add: contractible_def homotopic_with)
-
-lemma contractible_convex_tweak_boundary_points:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S"
- shows "contractible T"
-proof (cases "S = {}")
- case True
- with assms show ?thesis
- by (simp add: subsetCE)
-next
- case False
- show ?thesis
- apply (rule starlike_imp_contractible)
- apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
- done
-qed
-
-lemma convex_imp_contractible:
- fixes S :: "'a::real_normed_vector set"
- shows "convex S \<Longrightarrow> contractible S"
- using contractible_empty convex_imp_starlike starlike_imp_contractible by blast
-
-lemma contractible_sing [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "contractible {a}"
-by (rule convex_imp_contractible [OF convex_singleton])
-
-lemma is_interval_contractible_1:
- fixes S :: "real set"
- shows "is_interval S \<longleftrightarrow> contractible S"
-using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
- is_interval_simply_connected_1 by auto
-
-lemma contractible_Times:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes S: "contractible S" and T: "contractible T"
- shows "contractible (S \<times> T)"
-proof -
- obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
- and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
- and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
- and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (1::real, x) = a"
- using S by (auto simp: contractible_def homotopic_with)
- obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
- and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
- and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
- and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (1::real, x) = b"
- using T by (auto simp: contractible_def homotopic_with)
- show ?thesis
- apply (simp add: contractible_def homotopic_with)
- apply (rule exI [where x=a])
- apply (rule exI [where x=b])
- apply (rule exI [where x = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
- apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
- using hsub ksub
- apply auto
- done
-qed
-
-lemma homotopy_dominated_contractibility:
- fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
- assumes S: "contractible S"
- and f: "continuous_on S f" "image f S \<subseteq> T"
- and g: "continuous_on T g" "image g T \<subseteq> S"
- and hom: "homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
- shows "contractible T"
-proof -
- obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)"
- using nullhomotopic_from_contractible [OF f S] .
- then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)"
- by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
- show ?thesis
- apply (simp add: contractible_def)
- apply (rule exI [where x = b])
- apply (rule homotopic_with_symD)
- apply (rule homotopic_with_trans [OF _ hom])
- using homg apply (simp add: o_def)
- done
-qed
-
-
-subsection\<open>Local versions of topological properties in general\<close>
-
-definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
-where
- "locally P S \<equiv>
- \<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w
- \<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
- x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
-
-lemma locallyI:
- assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk>
- \<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
- x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
- shows "locally P S"
-using assms by (force simp: locally_def)
-
-lemma locallyE:
- assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w"
- obtains u v where "openin (subtopology euclidean S) u"
- "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
- using assms unfolding locally_def by meson
-
-lemma locally_mono:
- assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t"
- shows "locally Q S"
-by (metis assms locally_def)
-
-lemma locally_open_subset:
- assumes "locally P S" "openin (subtopology euclidean S) t"
- shows "locally P t"
-using assms
-apply (simp add: locally_def)
-apply (erule all_forward)+
-apply (rule impI)
-apply (erule impCE)
- using openin_trans apply blast
-apply (erule ex_forward)
-by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
-
-lemma locally_diff_closed:
- "\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> locally P (S - t)"
- using locally_open_subset closedin_def by fastforce
-
-lemma locally_empty [iff]: "locally P {}"
- by (simp add: locally_def openin_subtopology)
-
-lemma locally_singleton [iff]:
- fixes a :: "'a::metric_space"
- shows "locally P {a} \<longleftrightarrow> P {a}"
-apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
-using zero_less_one by blast
-
-lemma locally_iff:
- "locally P S \<longleftrightarrow>
- (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))"
-apply (simp add: le_inf_iff locally_def openin_open, safe)
-apply (metis IntE IntI le_inf_iff)
-apply (metis IntI Int_subset_iff)
-done
-
-lemma locally_Int:
- assumes S: "locally P S" and t: "locally P t"
- and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
- shows "locally P (S \<inter> t)"
-using S t unfolding locally_iff
-apply clarify
-apply (drule_tac x=T in spec)+
-apply (drule_tac x=x in spec)+
-apply clarsimp
-apply (rename_tac U1 U2 V1 V2)
-apply (rule_tac x="U1 \<inter> U2" in exI)
-apply (simp add: open_Int)
-apply (rule_tac x="V1 \<inter> V2" in exI)
-apply (auto intro: P)
-done
-
-lemma locally_Times:
- fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
- assumes PS: "locally P S" and QT: "locally Q T" and R: "\<And>S T. P S \<and> Q T \<Longrightarrow> R(S \<times> T)"
- shows "locally R (S \<times> T)"
- unfolding locally_def
-proof (clarify)
- fix W x y
- assume W: "openin (subtopology euclidean (S \<times> T)) W" and xy: "(x, y) \<in> W"
- then obtain U V where "openin (subtopology euclidean S) U" "x \<in> U"
- "openin (subtopology euclidean T) V" "y \<in> V" "U \<times> V \<subseteq> W"
- using Times_in_interior_subtopology by metis
- then obtain U1 U2 V1 V2
- where opeS: "openin (subtopology euclidean S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
- and opeT: "openin (subtopology euclidean T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
- by (meson PS QT locallyE)
- with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (subtopology euclidean (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
- apply (rule_tac x="U1 \<times> V1" in exI)
- apply (rule_tac x="U2 \<times> V2" in exI)
- apply (auto simp: openin_Times R)
- done
-qed
-
-
-proposition homeomorphism_locally_imp:
- fixes S :: "'a::metric_space set" and t :: "'b::t2_space set"
- assumes S: "locally P S" and hom: "homeomorphism S t f g"
- and Q: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
- shows "locally Q t"
-proof (clarsimp simp: locally_def)
- fix W y
- assume "y \<in> W" and "openin (subtopology euclidean t) W"
- then obtain T where T: "open T" "W = t \<inter> T"
- by (force simp: openin_open)
- then have "W \<subseteq> t" by auto
- have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
- and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
- using hom by (auto simp: homeomorphism_def)
- have gw: "g ` W = S \<inter> f -` W"
- using \<open>W \<subseteq> t\<close>
- apply auto
- using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
- using g \<open>W \<subseteq> t\<close> apply auto[1]
- by (simp add: f rev_image_eqI)
- have \<circ>: "openin (subtopology euclidean S) (g ` W)"
- proof -
- have "continuous_on S f"
- using f(3) by blast
- then show "openin (subtopology euclidean S) (g ` W)"
- by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) W\<close> continuous_on_open f(2))
- qed
- then obtain u v
- where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
- using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
- have "v \<subseteq> S" using uv by (simp add: gw)
- have fv: "f ` v = t \<inter> {x. g x \<in> v}"
- using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto
- have "f ` v \<subseteq> W"
- using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
- have contvf: "continuous_on v f"
- using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast
- have contvg: "continuous_on (f ` v) g"
- using \<open>f ` v \<subseteq> W\<close> \<open>W \<subseteq> t\<close> continuous_on_subset [OF g(3)] by blast
- have homv: "homeomorphism v (f ` v) f g"
- using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
- apply (simp add: homeomorphism_def contvf contvg, auto)
- by (metis f(1) rev_image_eqI rev_subsetD)
- have 1: "openin (subtopology euclidean t) (t \<inter> g -` u)"
- apply (rule continuous_on_open [THEN iffD1, rule_format])
- apply (rule \<open>continuous_on t g\<close>)
- using \<open>g ` t = S\<close> apply (simp add: osu)
- done
- have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
- apply (rule_tac x="f ` v" in exI)
- apply (intro conjI Q [OF \<open>P v\<close> homv])
- using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close> \<open>f ` v \<subseteq> W\<close> uv apply (auto simp: fv)
- done
- show "\<exists>U. openin (subtopology euclidean t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
- by (meson 1 2)
-qed
-
-lemma homeomorphism_locally:
- fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes hom: "homeomorphism S t f g"
- and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)"
- shows "locally P S \<longleftrightarrow> locally Q t"
-apply (rule iffI)
-apply (erule homeomorphism_locally_imp [OF _ hom])
-apply (simp add: eq)
-apply (erule homeomorphism_locally_imp)
-using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
-done
-
-lemma homeomorphic_locally:
- fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
- assumes hom: "S homeomorphic T"
- and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
- shows "locally P S \<longleftrightarrow> locally Q T"
-proof -
- obtain f g where hom: "homeomorphism S T f g"
- using assms by (force simp: homeomorphic_def)
- then show ?thesis
- using homeomorphic_def local.iff
- by (blast intro!: homeomorphism_locally)
-qed
-
-lemma homeomorphic_local_compactness:
- fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
- shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
-by (simp add: homeomorphic_compactness homeomorphic_locally)
-
-lemma locally_translation:
- fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
- shows
- "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
- \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
-apply (rule homeomorphism_locally [OF homeomorphism_translation])
-apply (simp add: homeomorphism_def)
-by metis
-
-lemma locally_injective_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
- shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
-apply (rule linear_homeomorphism_image [OF f])
-apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
-by (metis iff homeomorphism_def)
-
-lemma locally_open_map_image:
- fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
- assumes P: "locally P S"
- and f: "continuous_on S f"
- and oo: "\<And>t. openin (subtopology euclidean S) t
- \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)"
- and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
- shows "locally Q (f ` S)"
-proof (clarsimp simp add: locally_def)
- fix W y
- assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y \<in> W"
- then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
- have oivf: "openin (subtopology euclidean S) (S \<inter> f -` W)"
- by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
- then obtain x where "x \<in> S" "f x = y"
- using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
- then obtain U V
- where "openin (subtopology euclidean S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
- using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
- by auto
- then show "\<exists>X. openin (subtopology euclidean (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
- apply (rule_tac x="f ` U" in exI)
- apply (rule conjI, blast intro!: oo)
- apply (rule_tac x="f ` V" in exI)
- apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
- done
-qed
-
-
-subsection\<open>An induction principle for connected sets\<close>
-
-proposition connected_induction:
- assumes "connected S"
- and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
- and opI: "\<And>a. a \<in> S
- \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
- (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
- and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
- shows "Q b"
-proof -
- have 1: "openin (subtopology euclidean S)
- {b. \<exists>T. openin (subtopology euclidean S) T \<and>
- b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
- apply (subst openin_subopen, clarify)
- apply (rule_tac x=T in exI, auto)
- done
- have 2: "openin (subtopology euclidean S)
- {b. \<exists>T. openin (subtopology euclidean S) T \<and>
- b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
- apply (subst openin_subopen, clarify)
- apply (rule_tac x=T in exI, auto)
- done
- show ?thesis
- using \<open>connected S\<close>
- apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
- apply (elim disjE allE)
- apply (blast intro: 1)
- apply (blast intro: 2, simp_all)
- apply clarify apply (metis opI)
- using opD apply (blast intro: etc elim: dest:)
- using opI etc apply meson+
- done
-qed
-
-lemma connected_equivalence_relation_gen:
- assumes "connected S"
- and etc: "a \<in> S" "b \<in> S" "P a" "P b"
- and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
- and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
- and opI: "\<And>a. a \<in> S
- \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
- (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
- shows "R a b"
-proof -
- have "\<And>a b c. \<lbrakk>a \<in> S; P a; b \<in> S; c \<in> S; P b; P c; R a b\<rbrakk> \<Longrightarrow> R a c"
- apply (rule connected_induction [OF \<open>connected S\<close> opD], simp_all)
- by (meson trans opI)
- then show ?thesis by (metis etc opI)
-qed
-
-lemma connected_induction_simple:
- assumes "connected S"
- and etc: "a \<in> S" "b \<in> S" "P a"
- and opI: "\<And>a. a \<in> S
- \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
- (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
- shows "P b"
-apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
-apply (frule opI)
-using etc apply simp_all
-done
-
-lemma connected_equivalence_relation:
- assumes "connected S"
- and etc: "a \<in> S" "b \<in> S"
- and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
- and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
- and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
- shows "R a b"
-proof -
- have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
- apply (rule connected_induction_simple [OF \<open>connected S\<close>], simp_all)
- by (meson local.sym local.trans opI openin_imp_subset subsetCE)
- then show ?thesis by (metis etc opI)
-qed
-
-lemma locally_constant_imp_constant:
- assumes "connected S"
- and opI: "\<And>a. a \<in> S
- \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
- shows "f constant_on S"
-proof -
- have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
- apply (rule connected_equivalence_relation [OF \<open>connected S\<close>], simp_all)
- by (metis opI)
- then show ?thesis
- by (metis constant_on_def)
-qed
-
-lemma locally_constant:
- "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
-apply (simp add: locally_def)
-apply (rule iffI)
- apply (rule locally_constant_imp_constant, assumption)
- apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
-by (meson constant_on_subset openin_imp_subset order_refl)
-
-
-subsection\<open>Basic properties of local compactness\<close>
-
-proposition locally_compact:
- fixes s :: "'a :: metric_space set"
- shows
- "locally compact s \<longleftrightarrow>
- (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
- openin (subtopology euclidean s) u \<and> compact v)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply clarify
- apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
- by auto
-next
- assume r [rule_format]: ?rhs
- have *: "\<exists>u v.
- openin (subtopology euclidean s) u \<and>
- compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
- if "open T" "x \<in> s" "x \<in> T" for x T
- proof -
- obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u"
- using r [OF \<open>x \<in> s\<close>] by auto
- obtain e where "e>0" and e: "cball x e \<subseteq> T"
- using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
- show ?thesis
- apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
- apply (rule_tac x="cball x e \<inter> v" in exI)
- using that \<open>e > 0\<close> e uv
- apply auto
- done
- qed
- show ?lhs
- apply (rule locallyI)
- apply (subst (asm) openin_open)
- apply (blast intro: *)
- done
-qed
-
-lemma locally_compactE:
- fixes s :: "'a :: metric_space set"
- assumes "locally compact s"
- obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
- openin (subtopology euclidean s) (u x) \<and> compact (v x)"
-using assms
-unfolding locally_compact by metis
-
-lemma locally_compact_alt:
- fixes s :: "'a :: heine_borel set"
- shows "locally compact s \<longleftrightarrow>
- (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
- openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
-apply (simp add: locally_compact)
-apply (intro ball_cong ex_cong refl iffI)
-apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
-by (meson closure_subset compact_closure)
-
-lemma locally_compact_Int_cball:
- fixes s :: "'a :: heine_borel set"
- shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (simp add: locally_compact openin_contains_cball)
- apply (clarify | assumption | drule bspec)+
- by (metis (no_types, lifting) compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
-next
- assume ?rhs
- then show ?lhs
- apply (simp add: locally_compact openin_contains_cball)
- apply (clarify | assumption | drule bspec)+
- apply (rule_tac x="ball x e \<inter> s" in exI, simp)
- apply (rule_tac x="cball x e \<inter> s" in exI)
- using compact_eq_bounded_closed
- apply auto
- apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
- done
-qed
-
-lemma locally_compact_compact:
- fixes s :: "'a :: heine_borel set"
- shows "locally compact s \<longleftrightarrow>
- (\<forall>k. k \<subseteq> s \<and> compact k
- \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
- openin (subtopology euclidean s) u \<and> compact v))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then obtain u v where
- uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
- openin (subtopology euclidean s) (u x) \<and> compact (v x)"
- by (metis locally_compactE)
- have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
- if "k \<subseteq> s" "compact k" for k
- proof -
- have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
- \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
- using that by (simp add: compact_eq_openin_cover)
- moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c"
- using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
- moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
- using that by clarsimp (meson subsetCE uv)
- ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D"
- by metis
- then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)"
- by (metis finite_subset_image)
- have Tuv: "\<Union>(u ` T) \<subseteq> \<Union>(v ` T)"
- using T that by (force simp: dest!: uv)
- show ?thesis
- apply (rule_tac x="\<Union>(u ` T)" in exI)
- apply (rule_tac x="\<Union>(v ` T)" in exI)
- apply (simp add: Tuv)
- using T that
- apply (auto simp: dest!: uv)
- done
- qed
- show ?rhs
- by (blast intro: *)
-next
- assume ?rhs
- then show ?lhs
- apply (clarsimp simp add: locally_compact)
- apply (drule_tac x="{x}" in spec, simp)
- done
-qed
-
-lemma open_imp_locally_compact:
- fixes s :: "'a :: heine_borel set"
- assumes "open s"
- shows "locally compact s"
-proof -
- have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
- if "x \<in> s" for x
- proof -
- obtain e where "e>0" and e: "cball x e \<subseteq> s"
- using open_contains_cball assms \<open>x \<in> s\<close> by blast
- have ope: "openin (subtopology euclidean s) (ball x e)"
- by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
- show ?thesis
- apply (rule_tac x="ball x e" in exI)
- apply (rule_tac x="cball x e" in exI)
- using \<open>e > 0\<close> e apply (auto simp: ope)
- done
- qed
- show ?thesis
- unfolding locally_compact
- by (blast intro: *)
-qed
-
-lemma closed_imp_locally_compact:
- fixes s :: "'a :: heine_borel set"
- assumes "closed s"
- shows "locally compact s"
-proof -
- have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
- openin (subtopology euclidean s) u \<and> compact v"
- if "x \<in> s" for x
- proof -
- show ?thesis
- apply (rule_tac x = "s \<inter> ball x 1" in exI)
- apply (rule_tac x = "s \<inter> cball x 1" in exI)
- using \<open>x \<in> s\<close> assms apply auto
- done
- qed
- show ?thesis
- unfolding locally_compact
- by (blast intro: *)
-qed
-
-lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
- by (simp add: closed_imp_locally_compact)
-
-lemma locally_compact_Int:
- fixes s :: "'a :: t2_space set"
- shows "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
-by (simp add: compact_Int locally_Int)
-
-lemma locally_compact_closedin:
- fixes s :: "'a :: heine_borel set"
- shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk>
- \<Longrightarrow> locally compact t"
-unfolding closedin_closed
-using closed_imp_locally_compact locally_compact_Int by blast
-
-lemma locally_compact_delete:
- fixes s :: "'a :: t1_space set"
- shows "locally compact s \<Longrightarrow> locally compact (s - {a})"
- by (auto simp: openin_delete locally_open_subset)
-
-lemma locally_closed:
- fixes s :: "'a :: heine_borel set"
- shows "locally closed s \<longleftrightarrow> locally compact s"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (simp only: locally_def)
- apply (erule all_forward imp_forward asm_rl exE)+
- apply (rule_tac x = "u \<inter> ball x 1" in exI)
- apply (rule_tac x = "v \<inter> cball x 1" in exI)
- apply (force intro: openin_trans)
- done
-next
- assume ?rhs then show ?lhs
- using compact_eq_bounded_closed locally_mono by blast
-qed
-
-lemma locally_compact_openin_Un:
- fixes S :: "'a::euclidean_space set"
- assumes LCS: "locally compact S" and LCT:"locally compact T"
- and opS: "openin (subtopology euclidean (S \<union> T)) S"
- and opT: "openin (subtopology euclidean (S \<union> T)) T"
- shows "locally compact (S \<union> T)"
-proof -
- have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
- proof -
- obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
- using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
- moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> S"
- by (meson \<open>x \<in> S\<close> opS openin_contains_cball)
- then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> S"
- by force
- ultimately show ?thesis
- apply (rule_tac x="min e1 e2" in exI)
- apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
- by (metis closed_Int closed_cball inf_left_commute)
- qed
- moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
- proof -
- obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
- using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
- moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
- by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
- then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
- by force
- ultimately show ?thesis
- apply (rule_tac x="min e1 e2" in exI)
- apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
- by (metis closed_Int closed_cball inf_left_commute)
- qed
- ultimately show ?thesis
- by (force simp: locally_compact_Int_cball)
-qed
-
-lemma locally_compact_closedin_Un:
- fixes S :: "'a::euclidean_space set"
- assumes LCS: "locally compact S" and LCT:"locally compact T"
- and clS: "closedin (subtopology euclidean (S \<union> T)) S"
- and clT: "closedin (subtopology euclidean (S \<union> T)) T"
- shows "locally compact (S \<union> T)"
-proof -
- have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
- proof -
- obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
- using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
- moreover
- obtain e2 where "e2 > 0" and e2: "closed (cball x e2 \<inter> T)"
- using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
- ultimately show ?thesis
- apply (rule_tac x="min e1 e2" in exI)
- apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
- by (metis closed_Int closed_Un closed_cball inf_left_commute)
- qed
- moreover
- have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
- proof -
- obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
- using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
- moreover
- obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
- using clT x by (fastforce simp: openin_contains_cball closedin_def)
- then have "closed (cball x e2 \<inter> T)"
- proof -
- have "{} = T - (T - cball x e2)"
- using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
- then show ?thesis
- by (simp add: Diff_Diff_Int inf_commute)
- qed
- ultimately show ?thesis
- apply (rule_tac x="min e1 e2" in exI)
- apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
- by (metis closed_Int closed_Un closed_cball inf_left_commute)
- qed
- moreover
- have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
- proof -
- obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
- using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
- moreover
- obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
- using clS x by (fastforce simp: openin_contains_cball closedin_def)
- then have "closed (cball x e2 \<inter> S)"
- by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
- ultimately show ?thesis
- apply (rule_tac x="min e1 e2" in exI)
- apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
- by (metis closed_Int closed_Un closed_cball inf_left_commute)
- qed
- ultimately show ?thesis
- by (auto simp: locally_compact_Int_cball)
-qed
-
-lemma locally_compact_Times:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- shows "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
- by (auto simp: compact_Times locally_Times)
-
-lemma locally_compact_compact_subopen:
- fixes S :: "'a :: heine_borel set"
- shows
- "locally compact S \<longleftrightarrow>
- (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
- \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
- openin (subtopology euclidean S) U \<and> compact V))"
- (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- show ?rhs
- proof clarify
- fix K :: "'a set" and T :: "'a set"
- assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
- obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
- and ope: "openin (subtopology euclidean S) U"
- using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
- show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
- openin (subtopology euclidean S) U \<and> compact V"
- proof (intro exI conjI)
- show "K \<subseteq> U \<inter> T"
- by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
- show "U \<inter> T \<subseteq> closure(U \<inter> T)"
- by (rule closure_subset)
- show "closure (U \<inter> T) \<subseteq> S"
- by (metis \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S\<close> \<open>compact V\<close> closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
- show "openin (subtopology euclidean S) (U \<inter> T)"
- by (simp add: \<open>open T\<close> ope openin_Int_open)
- show "compact (closure (U \<inter> T))"
- by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> bounded_subset compact_closure compact_eq_bounded_closed)
- qed auto
- qed
-next
- assume ?rhs then show ?lhs
- unfolding locally_compact_compact
- by (metis open_openin openin_topspace subtopology_superset top.extremum topspace_euclidean_subtopology)
-qed
-
-
-subsection\<open>Sura-Bura's results about compact components of sets\<close>
-
-proposition Sura_Bura_compact:
- fixes S :: "'a::euclidean_space set"
- assumes "compact S" and C: "C \<in> components S"
- shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean S) T \<and>
- closedin (subtopology euclidean S) T}"
- (is "C = \<Inter>?\<T>")
-proof
- obtain x where x: "C = connected_component_set S x" and "x \<in> S"
- using C by (auto simp: components_def)
- have "C \<subseteq> S"
- by (simp add: C in_components_subset)
- have "\<Inter>?\<T> \<subseteq> connected_component_set S x"
- proof (rule connected_component_maximal)
- have "x \<in> C"
- by (simp add: \<open>x \<in> S\<close> x)
- then show "x \<in> \<Inter>?\<T>"
- by blast
- have clo: "closed (\<Inter>?\<T>)"
- by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
- have False
- if K1: "closedin (subtopology euclidean (\<Inter>?\<T>)) K1" and
- K2: "closedin (subtopology euclidean (\<Inter>?\<T>)) K2" and
- K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
- for K1 K2
- proof -
- have "closed K1" "closed K2"
- using closedin_closed_trans clo K1 K2 by blast+
- then obtain V1 V2 where "open V1" "open V2" "K1 \<subseteq> V1" "K2 \<subseteq> V2" and V12: "V1 \<inter> V2 = {}"
- using separation_normal \<open>K1 \<inter> K2 = {}\<close> by metis
- have SV12_ne: "(S - (V1 \<union> V2)) \<inter> (\<Inter>?\<T>) \<noteq> {}"
- proof (rule compact_imp_fip)
- show "compact (S - (V1 \<union> V2))"
- by (simp add: \<open>open V1\<close> \<open>open V2\<close> \<open>compact S\<close> compact_diff open_Un)
- show clo\<T>: "closed T" if "T \<in> ?\<T>" for T
- using that \<open>compact S\<close>
- by (force intro: closedin_closed_trans simp add: compact_imp_closed)
- show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
- proof
- assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
- obtain D where opeD: "openin (subtopology euclidean S) D"
- and cloD: "closedin (subtopology euclidean S) D"
- and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
- proof (cases "\<F> = {}")
- case True
- with \<open>C \<subseteq> S\<close> djo that show ?thesis
- by force
- next
- case False show ?thesis
- proof
- show ope: "openin (subtopology euclidean S) (\<Inter>\<F>)"
- using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
- then show "closedin (subtopology euclidean S) (\<Inter>\<F>)"
- by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
- show "C \<subseteq> \<Inter>\<F>"
- using \<F> by auto
- show "\<Inter>\<F> \<subseteq> V1 \<union> V2"
- using ope djo openin_imp_subset by fastforce
- qed
- qed
- have "connected C"
- by (simp add: x)
- have "closed D"
- using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
- have cloV1: "closedin (subtopology euclidean D) (D \<inter> closure V1)"
- and cloV2: "closedin (subtopology euclidean D) (D \<inter> closure V2)"
- by (simp_all add: closedin_closed_Int)
- moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
- apply safe
- using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
- apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
- done
- ultimately have cloDV1: "closedin (subtopology euclidean D) (D \<inter> V1)"
- and cloDV2: "closedin (subtopology euclidean D) (D \<inter> V2)"
- by metis+
- then obtain U1 U2 where "closed U1" "closed U2"
- and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
- by (auto simp: closedin_closed)
- have "D \<inter> U1 \<inter> C \<noteq> {}"
- proof
- assume "D \<inter> U1 \<inter> C = {}"
- then have *: "C \<subseteq> D \<inter> V2"
- using D1 DV12 \<open>C \<subseteq> D\<close> by auto
- have "\<Inter>?\<T> \<subseteq> D \<inter> V2"
- apply (rule Inter_lower)
- using * apply simp
- by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
- then show False
- using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
- qed
- moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
- proof
- assume "D \<inter> U2 \<inter> C = {}"
- then have *: "C \<subseteq> D \<inter> V1"
- using D2 DV12 \<open>C \<subseteq> D\<close> by auto
- have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
- apply (rule Inter_lower)
- using * apply simp
- by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
- then show False
- using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
- qed
- ultimately show False
- using \<open>connected C\<close> unfolding connected_closed
- apply (simp only: not_ex)
- apply (drule_tac x="D \<inter> U1" in spec)
- apply (drule_tac x="D \<inter> U2" in spec)
- using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
- by blast
- qed
- qed
- show False
- by (metis (full_types) DiffE UnE Un_upper2 SV12_ne \<open>K1 \<subseteq> V1\<close> \<open>K2 \<subseteq> V2\<close> disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
- qed
- then show "connected (\<Inter>?\<T>)"
- by (auto simp: connected_closedin_eq)
- show "\<Inter>?\<T> \<subseteq> S"
- by (fastforce simp: C in_components_subset)
- qed
- with x show "\<Inter>?\<T> \<subseteq> C" by simp
-qed auto
-
-
-corollary Sura_Bura_clopen_subset:
- fixes S :: "'a::euclidean_space set"
- assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
- and U: "open U" "C \<subseteq> U"
- obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
-proof (rule ccontr)
- assume "\<not> thesis"
- with that have neg: "\<nexists>K. openin (subtopology euclidean S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
- by metis
- obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
- and opeSV: "openin (subtopology euclidean S) V"
- using S U \<open>compact C\<close>
- apply (simp add: locally_compact_compact_subopen)
- by (meson C in_components_subset)
- let ?\<T> = "{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> compact T \<and> T \<subseteq> K}"
- have CK: "C \<in> components K"
- by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
- with \<open>compact K\<close>
- have "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> closedin (subtopology euclidean K) T}"
- by (simp add: Sura_Bura_compact)
- then have Ceq: "C = \<Inter>?\<T>"
- by (simp add: closedin_compact_eq \<open>compact K\<close>)
- obtain W where "open W" and W: "V = S \<inter> W"
- using opeSV by (auto simp: openin_open)
- have "-(U \<inter> W) \<inter> \<Inter>?\<T> \<noteq> {}"
- proof (rule closed_imp_fip_compact)
- show "- (U \<inter> W) \<inter> \<Inter>\<F> \<noteq> {}"
- if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
- proof (cases "\<F> = {}")
- case True
- have False if "U = UNIV" "W = UNIV"
- proof -
- have "V = S"
- by (simp add: W \<open>W = UNIV\<close>)
- with neg show False
- using \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> \<open>V \<subseteq> U\<close> \<open>compact K\<close> by auto
- qed
- with True show ?thesis
- by auto
- next
- case False
- show ?thesis
- proof
- assume "- (U \<inter> W) \<inter> \<Inter>\<F> = {}"
- then have FUW: "\<Inter>\<F> \<subseteq> U \<inter> W"
- by blast
- have "C \<subseteq> \<Inter>\<F>"
- using \<F> by auto
- moreover have "compact (\<Inter>\<F>)"
- by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
- moreover have "\<Inter>\<F> \<subseteq> K"
- using False that(2) by fastforce
- moreover have opeKF: "openin (subtopology euclidean K) (\<Inter>\<F>)"
- using False \<F> \<open>finite \<F>\<close> by blast
- then have opeVF: "openin (subtopology euclidean V) (\<Inter>\<F>)"
- using W \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> opeKF \<open>\<Inter>\<F> \<subseteq> K\<close> FUW openin_subset_trans by fastforce
- then have "openin (subtopology euclidean S) (\<Inter>\<F>)"
- by (metis opeSV openin_trans)
- moreover have "\<Inter>\<F> \<subseteq> U"
- by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
- ultimately show False
- using neg by blast
- qed
- qed
- qed (use \<open>open W\<close> \<open>open U\<close> in auto)
- with W Ceq \<open>C \<subseteq> V\<close> \<open>C \<subseteq> U\<close> show False
- by auto
-qed
-
-
-corollary Sura_Bura_clopen_subset_alt:
- fixes S :: "'a::euclidean_space set"
- assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
- and opeSU: "openin (subtopology euclidean S) U" and "C \<subseteq> U"
- obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
-proof -
- obtain V where "open V" "U = S \<inter> V"
- using opeSU by (auto simp: openin_open)
- with \<open>C \<subseteq> U\<close> have "C \<subseteq> V"
- by auto
- then show ?thesis
- using Sura_Bura_clopen_subset [OF S C \<open>compact C\<close> \<open>open V\<close>]
- by (metis \<open>U = S \<inter> V\<close> inf.bounded_iff openin_imp_subset that)
-qed
-
-corollary Sura_Bura:
- fixes S :: "'a::euclidean_space set"
- assumes "locally compact S" "C \<in> components S" "compact C"
- shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (subtopology euclidean S) K}"
- (is "C = ?rhs")
-proof
- show "?rhs \<subseteq> C"
- proof (clarsimp, rule ccontr)
- fix x
- assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (subtopology euclidean S) X \<longrightarrow> x \<in> X"
- and "x \<notin> C"
- obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
- using separation_normal [of "{x}" C]
- by (metis Int_empty_left \<open>x \<notin> C\<close> \<open>compact C\<close> closed_empty closed_insert compact_imp_closed insert_disjoint(1))
- have "x \<notin> V"
- using \<open>U \<inter> V = {}\<close> \<open>{x} \<subseteq> U\<close> by blast
- then show False
- by (meson "*" Sura_Bura_clopen_subset \<open>C \<subseteq> V\<close> \<open>open V\<close> assms(1) assms(2) assms(3) subsetCE)
- qed
-qed blast
-
-
-subsection\<open>Special cases of local connectedness and path connectedness\<close>
-
-lemma locally_connected_1:
- assumes
- "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
- \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and>
- connected u \<and> x \<in> u \<and> u \<subseteq> v"
- shows "locally connected S"
-apply (clarsimp simp add: locally_def)
-apply (drule assms; blast)
-done
-
-lemma locally_connected_2:
- assumes "locally connected S"
- "openin (subtopology euclidean S) t"
- "x \<in> t"
- shows "openin (subtopology euclidean S) (connected_component_set t x)"
-proof -
- { fix y :: 'a
- let ?SS = "subtopology euclidean S"
- assume 1: "openin ?SS t"
- "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
- and "connected_component t x y"
- then have "y \<in> t" and y: "y \<in> connected_component_set t x"
- using connected_component_subset by blast+
- obtain F where
- "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
- by moura
- then obtain G where
- "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
- by moura
- then have *: "openin ?SS (F y t) \<and> connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
- using 1 \<open>y \<in> t\<close> by presburger
- have "G y t \<subseteq> connected_component_set t y"
- by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
- then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> connected_component_set t x"
- by (metis (no_types) * connected_component_eq dual_order.trans y)
- }
- then show ?thesis
- using assms openin_subopen by (force simp: locally_def)
-qed
-
-lemma locally_connected_3:
- assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean S)
- (connected_component_set t x)"
- "openin (subtopology euclidean S) v" "x \<in> v"
- shows "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
-using assms connected_component_subset by fastforce
-
-lemma locally_connected:
- "locally connected S \<longleftrightarrow>
- (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
- \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
-by (metis locally_connected_1 locally_connected_2 locally_connected_3)
-
-lemma locally_connected_open_connected_component:
- "locally connected S \<longleftrightarrow>
- (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
- \<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))"
-by (metis locally_connected_1 locally_connected_2 locally_connected_3)
-
-lemma locally_path_connected_1:
- assumes
- "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
- \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
- shows "locally path_connected S"
-apply (clarsimp simp add: locally_def)
-apply (drule assms; blast)
-done
-
-lemma locally_path_connected_2:
- assumes "locally path_connected S"
- "openin (subtopology euclidean S) t"
- "x \<in> t"
- shows "openin (subtopology euclidean S) (path_component_set t x)"
-proof -
- { fix y :: 'a
- let ?SS = "subtopology euclidean S"
- assume 1: "openin ?SS t"
- "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
- and "path_component t x y"
- then have "y \<in> t" and y: "y \<in> path_component_set t x"
- using path_component_mem(2) by blast+
- obtain F where
- "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. path_connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. path_connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
- by moura
- then obtain G where
- "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. path_connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> path_connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
- by moura
- then have *: "openin ?SS (F y t) \<and> path_connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
- using 1 \<open>y \<in> t\<close> by presburger
- have "G y t \<subseteq> path_component_set t y"
- using * path_component_maximal set_rev_mp by blast
- then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x"
- by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> dual_order.trans path_component_eq y)
- }
- then show ?thesis
- using assms openin_subopen by (force simp: locally_def)
-qed
-
-lemma locally_path_connected_3:
- assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)"
- "openin (subtopology euclidean S) v" "x \<in> v"
- shows "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
-proof -
- have "path_component v x x"
- by (meson assms(3) path_component_refl)
- then show ?thesis
- by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
-qed
-
-proposition locally_path_connected:
- "locally path_connected S \<longleftrightarrow>
- (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
- \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))"
- by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
-
-proposition locally_path_connected_open_path_component:
- "locally path_connected S \<longleftrightarrow>
- (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
- \<longrightarrow> openin (subtopology euclidean S) (path_component_set t x))"
- by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
-
-lemma locally_connected_open_component:
- "locally connected S \<longleftrightarrow>
- (\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t
- \<longrightarrow> openin (subtopology euclidean S) c)"
-by (metis components_iff locally_connected_open_connected_component)
-
-proposition locally_connected_im_kleinen:
- "locally connected S \<longleftrightarrow>
- (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
- \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
- x \<in> u \<and> u \<subseteq> v \<and>
- (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (fastforce simp add: locally_connected)
-next
- assume ?rhs
- have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c"
- if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
- proof -
- from that \<open>?rhs\<close> [rule_format, of t x]
- obtain u where u:
- "openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
- (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
- using in_components_subset by auto
- obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
- "\<forall>x y. (\<exists>z. z \<in> x \<and> y = connected_component_set x z) = (F x y \<in> x \<and> y = connected_component_set x (F x y))"
- by moura
- then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)"
- by (meson components_iff c)
- obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
- G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)"
- by moura
- have "G c u \<notin> u \<or> G c u \<in> c"
- using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3))
- then show ?thesis
- using G u by auto
- qed
- show ?lhs
- apply (clarsimp simp add: locally_connected_open_component)
- apply (subst openin_subopen)
- apply (blast intro: *)
- done
-qed
-
-proposition locally_path_connected_im_kleinen:
- "locally path_connected S \<longleftrightarrow>
- (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
- \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
- x \<in> u \<and> u \<subseteq> v \<and>
- (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
- pathstart p = x \<and> pathfinish p = y))))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (simp add: locally_path_connected path_connected_def)
- apply (erule all_forward ex_forward imp_forward conjE | simp)+
- by (meson dual_order.trans)
-next
- assume ?rhs
- have *: "\<exists>T. openin (subtopology euclidean S) T \<and>
- x \<in> T \<and> T \<subseteq> path_component_set u z"
- if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x
- proof -
- have "x \<in> u"
- by (meson c path_component_mem(2))
- with that \<open>?rhs\<close> [rule_format, of u x]
- obtain U where U:
- "openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
- (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
- by blast
- show ?thesis
- apply (rule_tac x=U in exI)
- apply (auto simp: U)
- apply (metis U c path_component_trans path_component_def)
- done
- qed
- show ?lhs
- apply (clarsimp simp add: locally_path_connected_open_path_component)
- apply (subst openin_subopen)
- apply (blast intro: *)
- done
-qed
-
-lemma locally_path_connected_imp_locally_connected:
- "locally path_connected S \<Longrightarrow> locally connected S"
-using locally_mono path_connected_imp_connected by blast
-
-lemma locally_connected_components:
- "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
-by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
-
-lemma locally_path_connected_components:
- "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally path_connected c"
-by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)
-
-lemma locally_path_connected_connected_component:
- "locally path_connected S \<Longrightarrow> locally path_connected (connected_component_set S x)"
-by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)
-
-lemma open_imp_locally_path_connected:
- fixes S :: "'a :: real_normed_vector set"
- shows "open S \<Longrightarrow> locally path_connected S"
-apply (rule locally_mono [of convex])
-apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
-apply (meson open_ball centre_in_ball convex_ball openE order_trans)
-done
-
-lemma open_imp_locally_connected:
- fixes S :: "'a :: real_normed_vector set"
- shows "open S \<Longrightarrow> locally connected S"
-by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)
-
-lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
- by (simp add: open_imp_locally_path_connected)
-
-lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
- by (simp add: open_imp_locally_connected)
-
-lemma openin_connected_component_locally_connected:
- "locally connected S
- \<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)"
-apply (simp add: locally_connected_open_connected_component)
-by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
-
-lemma openin_components_locally_connected:
- "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c"
- using locally_connected_open_component openin_subtopology_self by blast
-
-lemma openin_path_component_locally_path_connected:
- "locally path_connected S
- \<Longrightarrow> openin (subtopology euclidean S) (path_component_set S x)"
-by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
-
-lemma closedin_path_component_locally_path_connected:
- "locally path_connected S
- \<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)"
-apply (simp add: closedin_def path_component_subset complement_path_component_Union)
-apply (rule openin_Union)
-using openin_path_component_locally_path_connected by auto
-
-lemma convex_imp_locally_path_connected:
- fixes S :: "'a:: real_normed_vector set"
- shows "convex S \<Longrightarrow> locally path_connected S"
-apply (clarsimp simp add: locally_path_connected)
-apply (subst (asm) openin_open)
-apply clarify
-apply (erule (1) openE)
-apply (rule_tac x = "S \<inter> ball x e" in exI)
-apply (force simp: convex_Int convex_imp_path_connected)
-done
-
-lemma convex_imp_locally_connected:
- fixes S :: "'a:: real_normed_vector set"
- shows "convex S \<Longrightarrow> locally connected S"
- by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
-
-
-subsection\<open>Relations between components and path components\<close>
-
-lemma path_component_eq_connected_component:
- assumes "locally path_connected S"
- shows "(path_component S x = connected_component S x)"
-proof (cases "x \<in> S")
- case True
- have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
- apply (rule openin_subset_trans [of S])
- apply (intro conjI openin_path_component_locally_path_connected [OF assms])
- using path_component_subset_connected_component apply (auto simp: connected_component_subset)
- done
- moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
- apply (rule closedin_subset_trans [of S])
- apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
- using path_component_subset_connected_component apply (auto simp: connected_component_subset)
- done
- ultimately have *: "path_component_set S x = connected_component_set S x"
- by (metis connected_connected_component connected_clopen True path_component_eq_empty)
- then show ?thesis
- by blast
-next
- case False then show ?thesis
- by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
-qed
-
-lemma path_component_eq_connected_component_set:
- "locally path_connected S \<Longrightarrow> (path_component_set S x = connected_component_set S x)"
-by (simp add: path_component_eq_connected_component)
-
-lemma locally_path_connected_path_component:
- "locally path_connected S \<Longrightarrow> locally path_connected (path_component_set S x)"
-using locally_path_connected_connected_component path_component_eq_connected_component by fastforce
-
-lemma open_path_connected_component:
- fixes S :: "'a :: real_normed_vector set"
- shows "open S \<Longrightarrow> path_component S x = connected_component S x"
-by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)
-
-lemma open_path_connected_component_set:
- fixes S :: "'a :: real_normed_vector set"
- shows "open S \<Longrightarrow> path_component_set S x = connected_component_set S x"
-by (simp add: open_path_connected_component)
-
-proposition locally_connected_quotient_image:
- assumes lcS: "locally connected S"
- and oo: "\<And>T. T \<subseteq> f ` S
- \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
- openin (subtopology euclidean (f ` S)) T"
- shows "locally connected (f ` S)"
-proof (clarsimp simp: locally_connected_open_component)
- fix U C
- assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C \<in> components U"
- then have "C \<subseteq> U" "U \<subseteq> f ` S"
- by (meson in_components_subset openin_imp_subset)+
- then have "openin (subtopology euclidean (f ` S)) C \<longleftrightarrow>
- openin (subtopology euclidean S) (S \<inter> f -` C)"
- by (auto simp: oo)
- moreover have "openin (subtopology euclidean S) (S \<inter> f -` C)"
- proof (subst openin_subopen, clarify)
- fix x
- assume "x \<in> S" "f x \<in> C"
- show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
- proof (intro conjI exI)
- show "openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
- proof (rule ccontr)
- assume **: "\<not> openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
- then have "x \<notin> (S \<inter> f -` U)"
- using \<open>U \<subseteq> f ` S\<close> opefSU lcS locally_connected_2 oo by blast
- with ** show False
- by (metis (no_types) connected_component_eq_empty empty_iff openin_subopen)
- qed
- next
- show "x \<in> connected_component_set (S \<inter> f -` U) x"
- using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by auto
- next
- have contf: "continuous_on S f"
- by (simp add: continuous_on_open oo openin_imp_subset)
- then have "continuous_on (connected_component_set (S \<inter> f -` U) x) f"
- apply (rule continuous_on_subset)
- using connected_component_subset apply blast
- done
- then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
- by (rule connected_continuous_image [OF _ connected_connected_component])
- moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
- using connected_component_in by blast
- moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
- using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
- ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
- by (rule components_maximal [OF \<open>C \<in> components U\<close>])
- have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
- using connected_component_subset fC by blast
- have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
- proof -
- { assume "x \<in> connected_component_set (S \<inter> f -` U) x"
- then have ?thesis
- using cUC connected_component_idemp connected_component_mono by blast }
- then show ?thesis
- using connected_component_eq_empty by auto
- qed
- also have "\<dots> \<subseteq> (S \<inter> f -` C)"
- by (rule connected_component_subset)
- finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
- qed
- qed
- ultimately show "openin (subtopology euclidean (f ` S)) C"
- by metis
-qed
-
-text\<open>The proof resembles that above but is not identical!\<close>
-proposition locally_path_connected_quotient_image:
- assumes lcS: "locally path_connected S"
- and oo: "\<And>T. T \<subseteq> f ` S
- \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow> openin (subtopology euclidean (f ` S)) T"
- shows "locally path_connected (f ` S)"
-proof (clarsimp simp: locally_path_connected_open_path_component)
- fix U y
- assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "y \<in> U"
- then have "path_component_set U y \<subseteq> U" "U \<subseteq> f ` S"
- by (meson path_component_subset openin_imp_subset)+
- then have "openin (subtopology euclidean (f ` S)) (path_component_set U y) \<longleftrightarrow>
- openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
- proof -
- have "path_component_set U y \<subseteq> f ` S"
- using \<open>U \<subseteq> f ` S\<close> \<open>path_component_set U y \<subseteq> U\<close> by blast
- then show ?thesis
- using oo by blast
- qed
- moreover have "openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
- proof (subst openin_subopen, clarify)
- fix x
- assume "x \<in> S" and Uyfx: "path_component U y (f x)"
- then have "f x \<in> U"
- using path_component_mem by blast
- show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
- proof (intro conjI exI)
- show "openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
- proof (rule ccontr)
- assume **: "\<not> openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
- then have "x \<notin> (S \<inter> f -` U)"
- by (metis (no_types, lifting) \<open>U \<subseteq> f ` S\<close> opefSU lcS oo locally_path_connected_open_path_component)
- then show False
- using ** \<open>path_component_set U y \<subseteq> U\<close> \<open>x \<in> S\<close> \<open>path_component U y (f x)\<close> by blast
- qed
- next
- show "x \<in> path_component_set (S \<inter> f -` U) x"
- by (simp add: \<open>f x \<in> U\<close> \<open>x \<in> S\<close> path_component_refl)
- next
- have contf: "continuous_on S f"
- by (simp add: continuous_on_open oo openin_imp_subset)
- then have "continuous_on (path_component_set (S \<inter> f -` U) x) f"
- apply (rule continuous_on_subset)
- using path_component_subset apply blast
- done
- then have "path_connected (f ` path_component_set (S \<inter> f -` U) x)"
- by (simp add: path_connected_continuous_image)
- moreover have "f ` path_component_set (S \<inter> f -` U) x \<subseteq> U"
- using path_component_mem by fastforce
- moreover have "f x \<in> f ` path_component_set (S \<inter> f -` U) x"
- by (force simp: \<open>x \<in> S\<close> \<open>f x \<in> U\<close> path_component_refl_eq)
- ultimately have "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U (f x)"
- by (meson path_component_maximal)
- also have "\<dots> \<subseteq> path_component_set U y"
- by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
- finally have fC: "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U y" .
- have cUC: "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)"
- using path_component_subset fC by blast
- have "path_component_set (S \<inter> f -` U) x \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) x"
- proof -
- have "\<And>a. path_component_set (path_component_set (S \<inter> f -` U) x) a \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) a"
- using cUC path_component_mono by blast
- then show ?thesis
- using path_component_path_component by blast
- qed
- also have "\<dots> \<subseteq> (S \<inter> f -` path_component_set U y)"
- by (rule path_component_subset)
- finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
- qed
- qed
- ultimately show "openin (subtopology euclidean (f ` S)) (path_component_set U y)"
- by metis
-qed
-
-subsection%unimportant\<open>Components, continuity, openin, closedin\<close>
-
-lemma continuous_on_components_gen:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes "\<And>c. c \<in> components S \<Longrightarrow>
- openin (subtopology euclidean S) c \<and> continuous_on c f"
- shows "continuous_on S f"
-proof (clarsimp simp: continuous_openin_preimage_eq)
- fix t :: "'b set"
- assume "open t"
- have *: "S \<inter> f -` t = (\<Union>c \<in> components S. c \<inter> f -` t)"
- by auto
- show "openin (subtopology euclidean S) (S \<inter> f -` t)"
- unfolding * using \<open>open t\<close> assms continuous_openin_preimage_gen openin_trans openin_Union by blast
-qed
-
-lemma continuous_on_components:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes "locally connected S "
- "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
- shows "continuous_on S f"
-apply (rule continuous_on_components_gen)
-apply (auto simp: assms intro: openin_components_locally_connected)
-done
-
-lemma continuous_on_components_eq:
- "locally connected S
- \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
-by (meson continuous_on_components continuous_on_subset in_components_subset)
-
-lemma continuous_on_components_open:
- fixes S :: "'a::real_normed_vector set"
- assumes "open S "
- "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
- shows "continuous_on S f"
-using continuous_on_components open_imp_locally_connected assms by blast
-
-lemma continuous_on_components_open_eq:
- fixes S :: "'a::real_normed_vector set"
- shows "open S \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
-using continuous_on_subset in_components_subset
-by (blast intro: continuous_on_components_open)
-
-lemma closedin_union_complement_components:
- assumes u: "locally connected u"
- and S: "closedin (subtopology euclidean u) S"
- and cuS: "c \<subseteq> components(u - S)"
- shows "closedin (subtopology euclidean u) (S \<union> \<Union>c)"
-proof -
- have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c'
- by (simp add: disjnt_def) blast
- have "S \<subseteq> u"
- using S closedin_imp_subset by blast
- moreover have "u - S = \<Union>c \<union> \<Union>(components (u - S) - c)"
- by (metis Diff_partition Union_components Union_Un_distrib assms(3))
- moreover have "disjnt (\<Union>c) (\<Union>(components (u - S) - c))"
- apply (rule di)
- by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
- ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))"
- by (auto simp: disjnt_def)
- have *: "openin (subtopology euclidean u) (\<Union>(components (u - S) - c))"
- apply (rule openin_Union)
- apply (rule openin_trans [of "u - S"])
- apply (simp add: u S locally_diff_closed openin_components_locally_connected)
- apply (simp add: openin_diff S)
- done
- have "openin (subtopology euclidean u) (u - (u - \<Union>(components (u - S) - c)))"
- apply (rule openin_diff, simp)
- apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
- done
- then show ?thesis
- by (force simp: eq closedin_def)
-qed
-
-lemma closed_union_complement_components:
- fixes S :: "'a::real_normed_vector set"
- assumes S: "closed S" and c: "c \<subseteq> components(- S)"
- shows "closed(S \<union> \<Union> c)"
-proof -
- have "closedin (subtopology euclidean UNIV) (S \<union> \<Union>c)"
- apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
- using S c apply (simp_all add: Compl_eq_Diff_UNIV)
- done
- then show ?thesis by simp
-qed
-
-lemma closedin_Un_complement_component:
- fixes S :: "'a::real_normed_vector set"
- assumes u: "locally connected u"
- and S: "closedin (subtopology euclidean u) S"
- and c: " c \<in> components(u - S)"
- shows "closedin (subtopology euclidean u) (S \<union> c)"
-proof -
- have "closedin (subtopology euclidean u) (S \<union> \<Union>{c})"
- using c by (blast intro: closedin_union_complement_components [OF u S])
- then show ?thesis
- by simp
-qed
-
-lemma closed_Un_complement_component:
- fixes S :: "'a::real_normed_vector set"
- assumes S: "closed S" and c: " c \<in> components(-S)"
- shows "closed (S \<union> c)"
- by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
- locally_connected_UNIV subtopology_UNIV)
-
-
-subsection\<open>Existence of isometry between subspaces of same dimension\<close>
-
-lemma isometry_subset_subspace:
- fixes S :: "'a::euclidean_space set"
- and T :: "'b::euclidean_space set"
- assumes S: "subspace S"
- and T: "subspace T"
- and d: "dim S \<le> dim T"
- obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
-proof -
- obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
- and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
- and "independent B" "finite B" "card B = dim S" "span B = S"
- by (metis orthonormal_basis_subspace [OF S] independent_finite)
- obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
- and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
- and "independent C" "finite C" "card C = dim T" "span C = T"
- by (metis orthonormal_basis_subspace [OF T] independent_finite)
- obtain fb where "fb ` B \<subseteq> C" "inj_on fb B"
- by (metis \<open>card B = dim S\<close> \<open>card C = dim T\<close> \<open>finite B\<close> \<open>finite C\<close> card_le_inj d)
- then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
- using Corth
- apply (auto simp: pairwise_def orthogonal_clauses)
- by (meson subsetD image_eqI inj_on_def)
- obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
- using linear_independent_extend \<open>independent B\<close> by fastforce
- have "span (f ` B) \<subseteq> span C"
- by (metis \<open>fb ` B \<subseteq> C\<close> ffb image_cong span_mono)
- then have "f ` S \<subseteq> T"
- unfolding \<open>span B = S\<close> \<open>span C = T\<close> span_linear_image[OF \<open>linear f\<close>] .
- have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
- using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto
- have "norm (f x) = norm x" if "x \<in> S" for x
- proof -
- interpret linear f by fact
- obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
- using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
- have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by (simp add: sum scale ffb x)
- also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
- apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
- apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
- done
- also have "\<dots> = norm x ^2"
- by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
- finally show ?thesis
- by (simp add: norm_eq_sqrt_inner)
- qed
- then show ?thesis
- by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
-qed
-
-proposition isometries_subspaces:
- fixes S :: "'a::euclidean_space set"
- and T :: "'b::euclidean_space set"
- assumes S: "subspace S"
- and T: "subspace T"
- and d: "dim S = dim T"
- obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
- "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
- "\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x"
- "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
- "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
-proof -
- obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
- and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
- and "independent B" "finite B" "card B = dim S" "span B = S"
- by (metis orthonormal_basis_subspace [OF S] independent_finite)
- obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
- and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
- and "independent C" "finite C" "card C = dim T" "span C = T"
- by (metis orthonormal_basis_subspace [OF T] independent_finite)
- obtain fb where "bij_betw fb B C"
- by (metis \<open>finite B\<close> \<open>finite C\<close> bij_betw_iff_card \<open>card B = dim S\<close> \<open>card C = dim T\<close> d)
- then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
- using Corth
- apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
- by (meson subsetD image_eqI inj_on_def)
- obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
- using linear_independent_extend \<open>independent B\<close> by fastforce
- interpret f: linear f by fact
- define gb where "gb \<equiv> inv_into B fb"
- then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C"
- using Borth
- apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
- by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
- obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x"
- using linear_independent_extend \<open>independent C\<close> by fastforce
- interpret g: linear g by fact
- have "span (f ` B) \<subseteq> span C"
- by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb image_cong)
- then have "f ` S \<subseteq> T"
- unfolding \<open>span B = S\<close> \<open>span C = T\<close>
- span_linear_image[OF \<open>linear f\<close>] .
- have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
- using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce
- have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x
- proof -
- obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
- using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
- have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
- using linear_sum [OF \<open>linear f\<close>] x by auto
- also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
- by (simp add: f.sum f.scale)
- also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
- by (simp add: ffb cong: sum.cong)
- finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
- then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
- also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
- apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
- apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
- done
- also have "\<dots> = (norm x)\<^sup>2"
- by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
- finally show "norm (f x) = norm x"
- by (simp add: norm_eq_sqrt_inner)
- have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
- also have "\<dots> = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
- by (simp add: g.sum g.scale)
- also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
- by (simp add: g.scale)
- also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R v)"
- apply (rule sum.cong [OF refl])
- using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
- also have "\<dots> = x"
- using x by blast
- finally show "g (f x) = x" .
- qed
- have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x"
- by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into)
- have g [simp]: "f (g x) = x" if "x \<in> T" for x
- proof -
- obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
- using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
- have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
- by (simp add: x g.sum)
- also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
- by (simp add: g.scale)
- also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
- by (simp add: ggb cong: sum.cong)
- finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
- also have "\<dots> = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
- by (simp add: f.scale f.sum)
- also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
- by (simp add: f.scale f.sum)
- also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R v)"
- using \<open>bij_betw fb B C\<close>
- by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
- also have "\<dots> = x"
- using x by blast
- finally show "f (g x) = x" .
- qed
- have gim: "g ` T = S"
- by (metis (full_types) S T \<open>f ` S \<subseteq> T\<close> d dim_eq_span dim_image_le f(2) g.linear_axioms
- image_iff linear_subspace_image span_eq_iff subset_iff)
- have fim: "f ` S = T"
- using \<open>g ` T = S\<close> image_iff by fastforce
- have [simp]: "norm (g x) = norm x" if "x \<in> T" for x
- using fim that by auto
- show ?thesis
- apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>])
- apply (simp_all add: fim gim)
- done
-qed
-
-corollary isometry_subspaces:
- fixes S :: "'a::euclidean_space set"
- and T :: "'b::euclidean_space set"
- assumes S: "subspace S"
- and T: "subspace T"
- and d: "dim S = dim T"
- obtains f where "linear f" "f ` S = T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
-using isometries_subspaces [OF assms]
-by metis
-
-corollary isomorphisms_UNIV_UNIV:
- assumes "DIM('M) = DIM('N)"
- obtains f::"'M::euclidean_space \<Rightarrow>'N::euclidean_space" and g
- where "linear f" "linear g"
- "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y"
- "\<And>x. g (f x) = x" "\<And>y. f(g y) = y"
- using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])
-
-lemma homeomorphic_subspaces:
- fixes S :: "'a::euclidean_space set"
- and T :: "'b::euclidean_space set"
- assumes S: "subspace S"
- and T: "subspace T"
- and d: "dim S = dim T"
- shows "S homeomorphic T"
-proof -
- obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
- "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
- by (blast intro: isometries_subspaces [OF assms])
- then show ?thesis
- apply (simp add: homeomorphic_def homeomorphism_def)
- apply (rule_tac x=f in exI)
- apply (rule_tac x=g in exI)
- apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
- done
-qed
-
-lemma homeomorphic_affine_sets:
- assumes "affine S" "affine T" "aff_dim S = aff_dim T"
- shows "S homeomorphic T"
-proof (cases "S = {} \<or> T = {}")
- case True with assms aff_dim_empty homeomorphic_empty show ?thesis
- by metis
-next
- case False
- then obtain a b where ab: "a \<in> S" "b \<in> T" by auto
- then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
- using affine_diffs_subspace assms by blast+
- have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
- using assms ab by (simp add: aff_dim_eq_dim [OF hull_inc] image_def)
- have "S homeomorphic ((+) (- a) ` S)"
- by (simp add: homeomorphic_translation)
- also have "\<dots> homeomorphic ((+) (- b) ` T)"
- by (rule homeomorphic_subspaces [OF ss dd])
- also have "\<dots> homeomorphic T"
- using homeomorphic_sym homeomorphic_translation by auto
- finally show ?thesis .
-qed
-
-
-subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
-
-locale%important Retracts =
- fixes s h t k
- assumes conth: "continuous_on s h"
- and imh: "h ` s = t"
- and contk: "continuous_on t k"
- and imk: "k ` t \<subseteq> s"
- and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
-
-begin
-
-lemma homotopically_trivial_retraction_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
- and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
- and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f;
- continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk>
- \<Longrightarrow> homotopic_with P u s f g"
- and contf: "continuous_on u f" and imf: "f ` u \<subseteq> t" and Qf: "Q f"
- and contg: "continuous_on u g" and img: "g ` u \<subseteq> t" and Qg: "Q g"
- shows "homotopic_with Q u t f g"
-proof -
- have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
- have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto
- have "continuous_on u (k \<circ> f)"
- using contf continuous_on_compose continuous_on_subset contk imf by blast
- moreover have "(k \<circ> f) ` u \<subseteq> s"
- using imf imk by fastforce
- moreover have "P (k \<circ> f)"
- by (simp add: P Qf contf imf)
- moreover have "continuous_on u (k \<circ> g)"
- using contg continuous_on_compose continuous_on_subset contk img by blast
- moreover have "(k \<circ> g) ` u \<subseteq> s"
- using img imk by fastforce
- moreover have "P (k \<circ> g)"
- by (simp add: P Qg contg img)
- ultimately have "homotopic_with P u s (k \<circ> f) (k \<circ> g)"
- by (rule hom)
- then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
- apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
- using Q by (auto simp: conth imh)
- then show ?thesis
- apply (rule homotopic_with_eq)
- apply (metis feq)
- apply (metis geq)
- apply (metis Qeq)
- done
-qed
-
-lemma homotopically_trivial_retraction_null_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
- and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
- and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk>
- \<Longrightarrow> \<exists>c. homotopic_with P u s f (\<lambda>x. c)"
- and contf: "continuous_on u f" and imf:"f ` u \<subseteq> t" and Qf: "Q f"
- obtains c where "homotopic_with Q u t f (\<lambda>x. c)"
-proof -
- have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
- have "continuous_on u (k \<circ> f)"
- using contf continuous_on_compose continuous_on_subset contk imf by blast
- moreover have "(k \<circ> f) ` u \<subseteq> s"
- using imf imk by fastforce
- moreover have "P (k \<circ> f)"
- by (simp add: P Qf contf imf)
- ultimately obtain c where "homotopic_with P u s (k \<circ> f) (\<lambda>x. c)"
- by (metis hom)
- then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
- apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
- using Q by (auto simp: conth imh)
- then show ?thesis
- apply (rule_tac c = "h c" in that)
- apply (erule homotopic_with_eq)
- apply (metis feq, simp)
- apply (metis Qeq)
- done
-qed
-
-lemma cohomotopically_trivial_retraction_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
- and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
- and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f;
- continuous_on s g; g ` s \<subseteq> u; P g\<rbrakk>
- \<Longrightarrow> homotopic_with P s u f g"
- and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
- and contg: "continuous_on t g" and img: "g ` t \<subseteq> u" and Qg: "Q g"
- shows "homotopic_with Q t u f g"
-proof -
- have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
- have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto
- have "continuous_on s (f \<circ> h)"
- using contf conth continuous_on_compose imh by blast
- moreover have "(f \<circ> h) ` s \<subseteq> u"
- using imf imh by fastforce
- moreover have "P (f \<circ> h)"
- by (simp add: P Qf contf imf)
- moreover have "continuous_on s (g \<circ> h)"
- using contg continuous_on_compose continuous_on_subset conth imh by blast
- moreover have "(g \<circ> h) ` s \<subseteq> u"
- using img imh by fastforce
- moreover have "P (g \<circ> h)"
- by (simp add: P Qg contg img)
- ultimately have "homotopic_with P s u (f \<circ> h) (g \<circ> h)"
- by (rule hom)
- then have "homotopic_with Q t u (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
- apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
- using Q by (auto simp: contk imk)
- then show ?thesis
- apply (rule homotopic_with_eq)
- apply (metis feq)
- apply (metis geq)
- apply (metis Qeq)
- done
-qed
-
-lemma cohomotopically_trivial_retraction_null_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
- and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
- and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk>
- \<Longrightarrow> \<exists>c. homotopic_with P s u f (\<lambda>x. c)"
- and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
- obtains c where "homotopic_with Q t u f (\<lambda>x. c)"
-proof -
- have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
- have "continuous_on s (f \<circ> h)"
- using contf conth continuous_on_compose imh by blast
- moreover have "(f \<circ> h) ` s \<subseteq> u"
- using imf imh by fastforce
- moreover have "P (f \<circ> h)"
- by (simp add: P Qf contf imf)
- ultimately obtain c where "homotopic_with P s u (f \<circ> h) (\<lambda>x. c)"
- by (metis hom)
- then have "homotopic_with Q t u (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
- apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
- using Q by (auto simp: contk imk)
- then show ?thesis
- apply (rule_tac c = c in that)
- apply (erule homotopic_with_eq)
- apply (metis feq, simp)
- apply (metis Qeq)
- done
-qed
-
end
-
-lemma simply_connected_retraction_gen:
- shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T;
- continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
- \<Longrightarrow> simply_connected T"
-apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
-apply (rule Retracts.homotopically_trivial_retraction_gen
- [of S h _ k _ "\<lambda>p. pathfinish p = pathstart p" "\<lambda>p. pathfinish p = pathstart p"])
-apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
-done
-
-lemma homeomorphic_simply_connected:
- "\<lbrakk>S homeomorphic T; simply_connected S\<rbrakk> \<Longrightarrow> simply_connected T"
- by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)
-
-lemma homeomorphic_simply_connected_eq:
- "S homeomorphic T \<Longrightarrow> (simply_connected S \<longleftrightarrow> simply_connected T)"
- by (metis homeomorphic_simply_connected homeomorphic_sym)
-
-
-subsection\<open>Homotopy equivalence\<close>
-
-definition%important homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
- (infix "homotopy'_eqv" 50)
- where "S homotopy_eqv T \<equiv>
- \<exists>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
- continuous_on T g \<and> g ` T \<subseteq> S \<and>
- homotopic_with (\<lambda>x. True) S S (g \<circ> f) id \<and>
- homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
-
-lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
- unfolding homeomorphic_def homotopy_eqv_def homeomorphism_def
- by (fastforce intro!: homotopic_with_equal continuous_on_compose)
-
-lemma homotopy_eqv_refl: "S homotopy_eqv S"
- by (rule homeomorphic_imp_homotopy_eqv homeomorphic_refl)+
-
-lemma homotopy_eqv_sym: "S homotopy_eqv T \<longleftrightarrow> T homotopy_eqv S"
- by (auto simp: homotopy_eqv_def)
-
-lemma homotopy_eqv_trans [trans]:
- fixes S :: "'a::real_normed_vector set" and U :: "'c::real_normed_vector set"
- assumes ST: "S homotopy_eqv T" and TU: "T homotopy_eqv U"
- shows "S homotopy_eqv U"
-proof -
- obtain f1 g1 where f1: "continuous_on S f1" "f1 ` S \<subseteq> T"
- and g1: "continuous_on T g1" "g1 ` T \<subseteq> S"
- and hom1: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> f1) id"
- "homotopic_with (\<lambda>x. True) T T (f1 \<circ> g1) id"
- using ST by (auto simp: homotopy_eqv_def)
- obtain f2 g2 where f2: "continuous_on T f2" "f2 ` T \<subseteq> U"
- and g2: "continuous_on U g2" "g2 ` U \<subseteq> T"
- and hom2: "homotopic_with (\<lambda>x. True) T T (g2 \<circ> f2) id"
- "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
- using TU by (auto simp: homotopy_eqv_def)
- have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
- by (rule homotopic_with_compose_continuous_right hom2 f1)+
- then have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
- by (simp add: o_assoc)
- then have "homotopic_with (\<lambda>x. True) S S
- (g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
- by (simp add: g1 homotopic_with_compose_continuous_left)
- moreover have "homotopic_with (\<lambda>x. True) S S (g1 \<circ> id \<circ> f1) id"
- using hom1 by simp
- ultimately have SS: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
- apply (simp add: o_assoc)
- apply (blast intro: homotopic_with_trans)
- done
- have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
- by (rule homotopic_with_compose_continuous_right hom1 g2)+
- then have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> (g1 \<circ> g2)) (id \<circ> g2)"
- by (simp add: o_assoc)
- then have "homotopic_with (\<lambda>x. True) U U
- (f2 \<circ> (f1 \<circ> (g1 \<circ> g2))) (f2 \<circ> (id \<circ> g2))"
- by (simp add: f2 homotopic_with_compose_continuous_left)
- moreover have "homotopic_with (\<lambda>x. True) U U (f2 \<circ> id \<circ> g2) id"
- using hom2 by simp
- ultimately have UU: "homotopic_with (\<lambda>x. True) U U (f2 \<circ> f1 \<circ> (g1 \<circ> g2)) id"
- apply (simp add: o_assoc)
- apply (blast intro: homotopic_with_trans)
- done
- show ?thesis
- unfolding homotopy_eqv_def
- apply (rule_tac x = "f2 \<circ> f1" in exI)
- apply (rule_tac x = "g1 \<circ> g2" in exI)
- apply (intro conjI continuous_on_compose SS UU)
- using f1 f2 g1 g2 apply (force simp: elim!: continuous_on_subset)+
- done
-qed
-
-lemma homotopy_eqv_inj_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "linear f" "inj f"
- shows "(f ` S) homotopy_eqv S"
-apply (rule homeomorphic_imp_homotopy_eqv)
-using assms homeomorphic_sym linear_homeomorphic_image by auto
-
-lemma homotopy_eqv_translation:
- fixes S :: "'a::real_normed_vector set"
- shows "(+) a ` S homotopy_eqv S"
- apply (rule homeomorphic_imp_homotopy_eqv)
- using homeomorphic_translation homeomorphic_sym by blast
-
-lemma homotopy_eqv_homotopic_triviality_imp:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- and f: "continuous_on U f" "f ` U \<subseteq> T"
- and g: "continuous_on U g" "g ` U \<subseteq> T"
- and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
- continuous_on U g; g ` U \<subseteq> S\<rbrakk>
- \<Longrightarrow> homotopic_with (\<lambda>x. True) U S f g"
- shows "homotopic_with (\<lambda>x. True) U T f g"
-proof -
- obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
- and k: "continuous_on T k" "k ` T \<subseteq> S"
- and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
- "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
- using assms by (auto simp: homotopy_eqv_def)
- have "homotopic_with (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
- apply (rule homUS)
- using f g k
- apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
- apply (force simp: o_def)+
- done
- then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
- apply (rule homotopic_with_compose_continuous_left)
- apply (simp_all add: h)
- done
- moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
- apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
- apply (auto simp: hom f)
- done
- moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
- apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
- apply (auto simp: hom g)
- done
- ultimately show "homotopic_with (\<lambda>x. True) U T f g"
- apply (simp add: o_assoc)
- using homotopic_with_trans homotopic_with_sym by blast
-qed
-
-lemma homotopy_eqv_homotopic_triviality:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
- continuous_on U g \<and> g ` U \<subseteq> S
- \<longrightarrow> homotopic_with (\<lambda>x. True) U S f g) \<longleftrightarrow>
- (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and>
- continuous_on U g \<and> g ` U \<subseteq> T
- \<longrightarrow> homotopic_with (\<lambda>x. True) U T f g)"
-apply (rule iffI)
-apply (metis assms homotopy_eqv_homotopic_triviality_imp)
-by (metis (no_types) assms homotopy_eqv_homotopic_triviality_imp homotopy_eqv_sym)
-
-lemma homotopy_eqv_cohomotopic_triviality_null_imp:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- and f: "continuous_on T f" "f ` T \<subseteq> U"
- and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
- \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c)"
- obtains c where "homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)"
-proof -
- obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
- and k: "continuous_on T k" "k ` T \<subseteq> S"
- and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
- "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
- using assms by (auto simp: homotopy_eqv_def)
- obtain c where "homotopic_with (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
- apply (rule exE [OF homSU [of "f \<circ> h"]])
- apply (intro continuous_on_compose h)
- using h f apply (force elim!: continuous_on_subset)+
- done
- then have "homotopic_with (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
- apply (rule homotopic_with_compose_continuous_right [where X=S])
- using k by auto
- moreover have "homotopic_with (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
- apply (rule homotopic_with_compose_continuous_left [where Y=T])
- apply (simp add: hom homotopic_with_symD)
- using f apply auto
- done
- ultimately show ?thesis
- apply (rule_tac c=c in that)
- apply (simp add: o_def)
- using homotopic_with_trans by blast
-qed
-
-lemma homotopy_eqv_cohomotopic_triviality_null:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c))) \<longleftrightarrow>
- (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)))"
-apply (rule iffI)
-apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
-by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
-
-lemma homotopy_eqv_homotopic_triviality_null_imp:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- and f: "continuous_on U f" "f ` U \<subseteq> T"
- and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
- \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
- shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
-proof -
- obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
- and k: "continuous_on T k" "k ` T \<subseteq> S"
- and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
- "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
- using assms by (auto simp: homotopy_eqv_def)
- obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
- apply (rule exE [OF homSU [of "k \<circ> f"]])
- apply (intro continuous_on_compose h)
- using k f apply (force elim!: continuous_on_subset)+
- done
- then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
- apply (rule homotopic_with_compose_continuous_left [where Y=S])
- using h by auto
- moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
- apply (rule homotopic_with_compose_continuous_right [where X=T])
- apply (simp add: hom homotopic_with_symD)
- using f apply auto
- done
- ultimately show ?thesis
- using homotopic_with_trans by (fastforce simp add: o_def)
-qed
-
-lemma homotopy_eqv_homotopic_triviality_null:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- and U :: "'c::real_normed_vector set"
- assumes "S homotopy_eqv T"
- shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
- (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
-apply (rule iffI)
-apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
-by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
-
-lemma homotopy_eqv_contractible_sets:
- fixes S :: "'a::real_normed_vector set"
- and T :: "'b::real_normed_vector set"
- assumes "contractible S" "contractible T" "S = {} \<longleftrightarrow> T = {}"
- shows "S homotopy_eqv T"
-proof (cases "S = {}")
- case True with assms show ?thesis
- by (simp add: homeomorphic_imp_homotopy_eqv)
-next
- case False
- with assms obtain a b where "a \<in> S" "b \<in> T"
- by auto
- then show ?thesis
- unfolding homotopy_eqv_def
- apply (rule_tac x="\<lambda>x. b" in exI)
- apply (rule_tac x="\<lambda>x. a" in exI)
- apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
- apply (auto simp: o_def continuous_on_const)
- done
-qed
-
-lemma homotopy_eqv_empty1 [simp]:
- fixes S :: "'a::real_normed_vector set"
- shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}"
-apply (rule iffI)
-using homotopy_eqv_def apply fastforce
-by (simp add: homotopy_eqv_contractible_sets)
-
-lemma homotopy_eqv_empty2 [simp]:
- fixes S :: "'a::real_normed_vector set"
- shows "({}::'b::real_normed_vector set) homotopy_eqv S \<longleftrightarrow> S = {}"
-by (metis homotopy_eqv_empty1 homotopy_eqv_sym)
-
-lemma homotopy_eqv_contractibility:
- fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
- shows "S homotopy_eqv T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
-unfolding homotopy_eqv_def
-by (blast intro: homotopy_dominated_contractibility)
-
-lemma homotopy_eqv_sing:
- fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
- shows "S homotopy_eqv {a} \<longleftrightarrow> S \<noteq> {} \<and> contractible S"
-proof (cases "S = {}")
- case True then show ?thesis
- by simp
-next
- case False then show ?thesis
- by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
-qed
-
-lemma homeomorphic_contractible_eq:
- fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
- shows "S homeomorphic T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
-by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)
-
-lemma homeomorphic_contractible:
- fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
- shows "\<lbrakk>contractible S; S homeomorphic T\<rbrakk> \<Longrightarrow> contractible T"
- by (metis homeomorphic_contractible_eq)
-
-
-subsection%unimportant\<open>Misc other results\<close>
-
-lemma bounded_connected_Compl_real:
- fixes S :: "real set"
- assumes "bounded S" and conn: "connected(- S)"
- shows "S = {}"
-proof -
- obtain a b where "S \<subseteq> box a b"
- by (meson assms bounded_subset_box_symmetric)
- then have "a \<notin> S" "b \<notin> S"
- by auto
- then have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> - S"
- by (meson Compl_iff conn connected_iff_interval)
- then show ?thesis
- using \<open>S \<subseteq> box a b\<close> by auto
-qed
-
-lemma bounded_connected_Compl_1:
- fixes S :: "'a::{euclidean_space} set"
- assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
- shows "S = {}"
-proof -
- have "DIM('a) = DIM(real)"
- by (simp add: "1")
- then obtain f::"'a \<Rightarrow> real" and g
- where "linear f" "\<And>x. norm(f x) = norm x" "\<And>x. g(f x) = x" "\<And>y. f(g y) = y"
- by (rule isomorphisms_UNIV_UNIV) blast
- with \<open>bounded S\<close> have "bounded (f ` S)"
- using bounded_linear_image linear_linear by blast
- have "connected (f ` (-S))"
- using connected_linear_image assms \<open>linear f\<close> by blast
- moreover have "f ` (-S) = - (f ` S)"
- apply (rule bij_image_Compl_eq)
- apply (auto simp: bij_def)
- apply (metis \<open>\<And>x. g (f x) = x\<close> injI)
- by (metis UNIV_I \<open>\<And>y. f (g y) = y\<close> image_iff)
- finally have "connected (- (f ` S))"
- by simp
- then have "f ` S = {}"
- using \<open>bounded (f ` S)\<close> bounded_connected_Compl_real by blast
- then show ?thesis
- by blast
-qed
-
-
-subsection%unimportant\<open>Some Uncountable Sets\<close>
-
-lemma uncountable_closed_segment:
- fixes a :: "'a::real_normed_vector"
- assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
-unfolding path_image_linepath [symmetric] path_image_def
- using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
- countable_image_inj_on by auto
-
-lemma uncountable_open_segment:
- fixes a :: "'a::real_normed_vector"
- assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
- by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
-
-lemma uncountable_convex:
- fixes a :: "'a::real_normed_vector"
- assumes "convex S" "a \<in> S" "b \<in> S" "a \<noteq> b"
- shows "uncountable S"
-proof -
- have "uncountable (closed_segment a b)"
- by (simp add: uncountable_closed_segment assms)
- then show ?thesis
- by (meson assms convex_contains_segment countable_subset)
-qed
-
-lemma uncountable_ball:
- fixes a :: "'a::euclidean_space"
- assumes "r > 0"
- shows "uncountable (ball a r)"
-proof -
- have "uncountable (open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)))"
- by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
- moreover have "open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)) \<subseteq> ball a r"
- using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
- ultimately show ?thesis
- by (metis countable_subset)
-qed
-
-lemma ball_minus_countable_nonempty:
- assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
- shows "ball z r - A \<noteq> {}"
-proof
- assume *: "ball z r - A = {}"
- have "uncountable (ball z r - A)"
- by (intro uncountable_minus_countable assms uncountable_ball)
- thus False by (subst (asm) *) auto
-qed
-
-lemma uncountable_cball:
- fixes a :: "'a::euclidean_space"
- assumes "r > 0"
- shows "uncountable (cball a r)"
- using assms countable_subset uncountable_ball by auto
-
-lemma pairwise_disjnt_countable:
- fixes \<N> :: "nat set set"
- assumes "pairwise disjnt \<N>"
- shows "countable \<N>"
-proof -
- have "inj_on (\<lambda>X. SOME n. n \<in> X) (\<N> - {{}})"
- apply (clarsimp simp add: inj_on_def)
- by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
- then show ?thesis
- by (metis countable_Diff_eq countable_def)
-qed
-
-lemma pairwise_disjnt_countable_Union:
- assumes "countable (\<Union>\<N>)" and pwd: "pairwise disjnt \<N>"
- shows "countable \<N>"
-proof -
- obtain f :: "_ \<Rightarrow> nat" where f: "inj_on f (\<Union>\<N>)"
- using assms by blast
- then have "pairwise disjnt (\<Union> X \<in> \<N>. {f ` X})"
- using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
- then have "countable (\<Union> X \<in> \<N>. {f ` X})"
- using pairwise_disjnt_countable by blast
- then show ?thesis
- by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
-qed
-
-lemma connected_uncountable:
- fixes S :: "'a::metric_space set"
- assumes "connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
-proof -
- have "continuous_on S (dist a)"
- by (intro continuous_intros)
- then have "connected (dist a ` S)"
- by (metis connected_continuous_image \<open>connected S\<close>)
- then have "closed_segment 0 (dist a b) \<subseteq> (dist a ` S)"
- by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
- then have "uncountable (dist a ` S)"
- by (metis \<open>a \<noteq> b\<close> countable_subset dist_eq_0_iff uncountable_closed_segment)
- then show ?thesis
- by blast
-qed
-
-lemma path_connected_uncountable:
- fixes S :: "'a::metric_space set"
- assumes "path_connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
- using path_connected_imp_connected assms connected_uncountable by metis
-
-lemma connected_finite_iff_sing:
- fixes S :: "'a::metric_space set"
- assumes "connected S"
- shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})" (is "_ = ?rhs")
-proof -
- have "uncountable S" if "\<not> ?rhs"
- using connected_uncountable assms that by blast
- then show ?thesis
- using uncountable_infinite by auto
-qed
-
-lemma connected_card_eq_iff_nontrivial:
- fixes S :: "'a::metric_space set"
- shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
- apply (auto simp: countable_finite finite_subset)
- by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
-
-lemma simple_path_image_uncountable:
- fixes g :: "real \<Rightarrow> 'a::metric_space"
- assumes "simple_path g"
- shows "uncountable (path_image g)"
-proof -
- have "g 0 \<in> path_image g" "g (1/2) \<in> path_image g"
- by (simp_all add: path_defs)
- moreover have "g 0 \<noteq> g (1/2)"
- using assms by (fastforce simp add: simple_path_def)
- ultimately show ?thesis
- apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
- by blast
-qed
-
-lemma arc_image_uncountable:
- fixes g :: "real \<Rightarrow> 'a::metric_space"
- assumes "arc g"
- shows "uncountable (path_image g)"
- by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
-
-
-subsection%unimportant\<open> Some simple positive connection theorems\<close>
-
-proposition path_connected_convex_diff_countable:
- fixes U :: "'a::euclidean_space set"
- assumes "convex U" "\<not> collinear U" "countable S"
- shows "path_connected(U - S)"
-proof (clarsimp simp add: path_connected_def)
- fix a b
- assume "a \<in> U" "a \<notin> S" "b \<in> U" "b \<notin> S"
- let ?m = "midpoint a b"
- show "\<exists>g. path g \<and> path_image g \<subseteq> U - S \<and> pathstart g = a \<and> pathfinish g = b"
- proof (cases "a = b")
- case True
- then show ?thesis
- by (metis DiffI \<open>a \<in> U\<close> \<open>a \<notin> S\<close> path_component_def path_component_refl)
- next
- case False
- then have "a \<noteq> ?m" "b \<noteq> ?m"
- using midpoint_eq_endpoint by fastforce+
- have "?m \<in> U"
- using \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>convex U\<close> convex_contains_segment by force
- obtain c where "c \<in> U" and nc_abc: "\<not> collinear {a,b,c}"
- by (metis False \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>\<not> collinear U\<close> collinear_triples insert_absorb)
- have ncoll_mca: "\<not> collinear {?m,c,a}"
- by (metis (full_types) \<open>a \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
- have ncoll_mcb: "\<not> collinear {?m,c,b}"
- by (metis (full_types) \<open>b \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
- have "c \<noteq> ?m"
- by (metis collinear_midpoint insert_commute nc_abc)
- then have "closed_segment ?m c \<subseteq> U"
- by (simp add: \<open>c \<in> U\<close> \<open>?m \<in> U\<close> \<open>convex U\<close> closed_segment_subset)
- then obtain z where z: "z \<in> closed_segment ?m c"
- and disjS: "(closed_segment a z \<union> closed_segment z b) \<inter> S = {}"
- proof -
- have False if "closed_segment ?m c \<subseteq> {z. (closed_segment a z \<union> closed_segment z b) \<inter> S \<noteq> {}}"
- proof -
- have closb: "closed_segment ?m c \<subseteq>
- {z \<in> closed_segment ?m c. closed_segment a z \<inter> S \<noteq> {}} \<union> {z \<in> closed_segment ?m c. closed_segment z b \<inter> S \<noteq> {}}"
- using that by blast
- have *: "countable {z \<in> closed_segment ?m c. closed_segment z u \<inter> S \<noteq> {}}"
- if "u \<in> U" "u \<notin> S" and ncoll: "\<not> collinear {?m, c, u}" for u
- proof -
- have **: False if x1: "x1 \<in> closed_segment ?m c" and x2: "x2 \<in> closed_segment ?m c"
- and "x1 \<noteq> x2" "x1 \<noteq> u"
- and w: "w \<in> closed_segment x1 u" "w \<in> closed_segment x2 u"
- and "w \<in> S" for x1 x2 w
- proof -
- have "x1 \<in> affine hull {?m,c}" "x2 \<in> affine hull {?m,c}"
- using segment_as_ball x1 x2 by auto
- then have coll_x1: "collinear {x1, ?m, c}" and coll_x2: "collinear {?m, c, x2}"
- by (simp_all add: affine_hull_3_imp_collinear) (metis affine_hull_3_imp_collinear insert_commute)
- have "\<not> collinear {x1, u, x2}"
- proof
- assume "collinear {x1, u, x2}"
- then have "collinear {?m, c, u}"
- by (metis (full_types) \<open>c \<noteq> ?m\<close> coll_x1 coll_x2 collinear_3_trans insert_commute ncoll \<open>x1 \<noteq> x2\<close>)
- with ncoll show False ..
- qed
- then have "closed_segment x1 u \<inter> closed_segment u x2 = {u}"
- by (blast intro!: Int_closed_segment)
- then have "w = u"
- using closed_segment_commute w by auto
- show ?thesis
- using \<open>u \<notin> S\<close> \<open>w = u\<close> that(7) by auto
- qed
- then have disj: "disjoint ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}))"
- by (fastforce simp: pairwise_def disjnt_def)
- have cou: "countable ((\<Union>z \<in> closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
- apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
- apply (rule countable_subset [OF _ \<open>countable S\<close>], auto)
- done
- define f where "f \<equiv> \<lambda>X. (THE z. z \<in> closed_segment ?m c \<and> X = closed_segment z u \<inter> S)"
- show ?thesis
- proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
- fix x
- assume x: "x \<in> closed_segment ?m c" "closed_segment x u \<inter> S \<noteq> {}"
- show "x \<in> f ` ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
- proof (rule_tac x="closed_segment x u \<inter> S" in image_eqI)
- show "x = f (closed_segment x u \<inter> S)"
- unfolding f_def
- apply (rule the_equality [symmetric])
- using x apply (auto simp: dest: **)
- done
- qed (use x in auto)
- qed
- qed
- have "uncountable (closed_segment ?m c)"
- by (metis \<open>c \<noteq> ?m\<close> uncountable_closed_segment)
- then show False
- using closb * [OF \<open>a \<in> U\<close> \<open>a \<notin> S\<close> ncoll_mca] * [OF \<open>b \<in> U\<close> \<open>b \<notin> S\<close> ncoll_mcb]
- apply (simp add: closed_segment_commute)
- by (simp add: countable_subset)
- qed
- then show ?thesis
- by (force intro: that)
- qed
- show ?thesis
- proof (intro exI conjI)
- have "path_image (linepath a z +++ linepath z b) \<subseteq> U"
- by (metis \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>closed_segment ?m c \<subseteq> U\<close> z \<open>convex U\<close> closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
- with disjS show "path_image (linepath a z +++ linepath z b) \<subseteq> U - S"
- by (force simp: path_image_join)
- qed auto
- qed
-qed
-
-
-corollary connected_convex_diff_countable:
- fixes U :: "'a::euclidean_space set"
- assumes "convex U" "\<not> collinear U" "countable S"
- shows "connected(U - S)"
- by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)
-
-lemma path_connected_punctured_convex:
- assumes "convex S" and aff: "aff_dim S \<noteq> 1"
- shows "path_connected(S - {a})"
-proof -
- consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S \<ge> 2"
- using assms aff_dim_geq [of S] by linarith
- then show ?thesis
- proof cases
- assume "aff_dim S = -1"
- then show ?thesis
- by (metis aff_dim_empty empty_Diff path_connected_empty)
- next
- assume "aff_dim S = 0"
- then show ?thesis
- by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
- next
- assume ge2: "aff_dim S \<ge> 2"
- then have "\<not> collinear S"
- proof (clarsimp simp add: collinear_affine_hull)
- fix u v
- assume "S \<subseteq> affine hull {u, v}"
- then have "aff_dim S \<le> aff_dim {u, v}"
- by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
- with ge2 show False
- by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
- qed
- then show ?thesis
- apply (rule path_connected_convex_diff_countable [OF \<open>convex S\<close>])
- by simp
- qed
-qed
-
-lemma connected_punctured_convex:
- shows "\<lbrakk>convex S; aff_dim S \<noteq> 1\<rbrakk> \<Longrightarrow> connected(S - {a})"
- using path_connected_imp_connected path_connected_punctured_convex by blast
-
-lemma path_connected_complement_countable:
- fixes S :: "'a::euclidean_space set"
- assumes "2 \<le> DIM('a)" "countable S"
- shows "path_connected(- S)"
-proof -
- have "path_connected(UNIV - S)"
- apply (rule path_connected_convex_diff_countable)
- using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
- then show ?thesis
- by (simp add: Compl_eq_Diff_UNIV)
-qed
-
-proposition path_connected_openin_diff_countable:
- fixes S :: "'a::euclidean_space set"
- assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
- and "\<not> collinear S" "countable T"
- shows "path_connected(S - T)"
-proof (clarsimp simp add: path_connected_component)
- fix x y
- assume xy: "x \<in> S" "x \<notin> T" "y \<in> S" "y \<notin> T"
- show "path_component (S - T) x y"
- proof (rule connected_equivalence_relation_gen [OF \<open>connected S\<close>, where P = "\<lambda>x. x \<notin> T"])
- show "\<exists>z. z \<in> U \<and> z \<notin> T" if opeU: "openin (subtopology euclidean S) U" and "x \<in> U" for U x
- proof -
- have "openin (subtopology euclidean (affine hull S)) U"
- using opeU ope openin_trans by blast
- with \<open>x \<in> U\<close> obtain r where Usub: "U \<subseteq> affine hull S" and "r > 0"
- and subU: "ball x r \<inter> affine hull S \<subseteq> U"
- by (auto simp: openin_contains_ball)
- with \<open>x \<in> U\<close> have x: "x \<in> ball x r \<inter> affine hull S"
- by auto
- have "\<not> S \<subseteq> {x}"
- using \<open>\<not> collinear S\<close> collinear_subset by blast
- then obtain x' where "x' \<noteq> x" "x' \<in> S"
- by blast
- obtain y where y: "y \<noteq> x" "y \<in> ball x r \<inter> affine hull S"
- proof
- show "x + (r / 2 / norm(x' - x)) *\<^sub>R (x' - x) \<noteq> x"
- using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> by auto
- show "x + (r / 2 / norm (x' - x)) *\<^sub>R (x' - x) \<in> ball x r \<inter> affine hull S"
- using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> \<open>x' \<in> S\<close> x
- by (simp add: dist_norm mem_affine_3_minus hull_inc)
- qed
- have "convex (ball x r \<inter> affine hull S)"
- by (simp add: affine_imp_convex convex_Int)
- with x y subU have "uncountable U"
- by (meson countable_subset uncountable_convex)
- then have "\<not> U \<subseteq> T"
- using \<open>countable T\<close> countable_subset by blast
- then show ?thesis by blast
- qed
- show "\<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and>
- (\<forall>x\<in>U. \<forall>y\<in>U. x \<notin> T \<and> y \<notin> T \<longrightarrow> path_component (S - T) x y)"
- if "x \<in> S" for x
- proof -
- obtain r where Ssub: "S \<subseteq> affine hull S" and "r > 0"
- and subS: "ball x r \<inter> affine hull S \<subseteq> S"
- using ope \<open>x \<in> S\<close> by (auto simp: openin_contains_ball)
- then have conv: "convex (ball x r \<inter> affine hull S)"
- by (simp add: affine_imp_convex convex_Int)
- have "\<not> aff_dim (affine hull S) \<le> 1"
- using \<open>\<not> collinear S\<close> collinear_aff_dim by auto
- then have "\<not> collinear (ball x r \<inter> affine hull S)"
- apply (simp add: collinear_aff_dim)
- by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball \<open>0 < r\<close> aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
- then have *: "path_connected ((ball x r \<inter> affine hull S) - T)"
- by (rule path_connected_convex_diff_countable [OF conv _ \<open>countable T\<close>])
- have ST: "ball x r \<inter> affine hull S - T \<subseteq> S - T"
- using subS by auto
- show ?thesis
- proof (intro exI conjI)
- show "x \<in> ball x r \<inter> affine hull S"
- using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
- have "openin (subtopology euclidean (affine hull S)) (ball x r \<inter> affine hull S)"
- by (subst inf.commute) (simp add: openin_Int_open)
- then show "openin (subtopology euclidean S) (ball x r \<inter> affine hull S)"
- by (rule openin_subset_trans [OF _ subS Ssub])
- qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
- qed
- qed (use xy path_component_trans in auto)
-qed
-
-corollary connected_openin_diff_countable:
- fixes S :: "'a::euclidean_space set"
- assumes "connected S" and ope: "openin (subtopology euclidean (affine hull S)) S"
- and "\<not> collinear S" "countable T"
- shows "connected(S - T)"
- by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])
-
-corollary path_connected_open_diff_countable:
- fixes S :: "'a::euclidean_space set"
- assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
- shows "path_connected(S - T)"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (simp add: path_connected_empty)
-next
- case False
- show ?thesis
- proof (rule path_connected_openin_diff_countable)
- show "openin (subtopology euclidean (affine hull S)) S"
- by (simp add: assms hull_subset open_subset)
- show "\<not> collinear S"
- using assms False by (simp add: collinear_aff_dim aff_dim_open)
- qed (simp_all add: assms)
-qed
-
-corollary connected_open_diff_countable:
- fixes S :: "'a::euclidean_space set"
- assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
- shows "connected(S - T)"
-by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)
-
-
-
-subsection%unimportant \<open>Self-homeomorphisms shuffling points about\<close>
-
-subsubsection%unimportant\<open>The theorem \<open>homeomorphism_moving_points_exists\<close>\<close>
-
-lemma homeomorphism_moving_point_1:
- fixes a :: "'a::euclidean_space"
- assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T"
- obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
- "f a = u" "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = x"
-proof -
- have nou: "norm (u - a) < r" and "u \<in> T"
- using u by (auto simp: dist_norm norm_minus_commute)
- then have "0 < r"
- by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
- define f where "f \<equiv> \<lambda>x. (1 - norm(x - a) / r) *\<^sub>R (u - a) + x"
- have *: "False" if eq: "x + (norm y / r) *\<^sub>R u = y + (norm x / r) *\<^sub>R u"
- and nou: "norm u < r" and yx: "norm y < norm x" for x y and u::'a
- proof -
- have "x = y + (norm x / r - (norm y / r)) *\<^sub>R u"
- using eq by (simp add: algebra_simps)
- then have "norm x = norm (y + ((norm x - norm y) / r) *\<^sub>R u)"
- by (metis diff_divide_distrib)
- also have "\<dots> \<le> norm y + norm(((norm x - norm y) / r) *\<^sub>R u)"
- using norm_triangle_ineq by blast
- also have "\<dots> = norm y + (norm x - norm y) * (norm u / r)"
- using yx \<open>r > 0\<close>
- by (simp add: divide_simps)
- also have "\<dots> < norm y + (norm x - norm y) * 1"
- apply (subst add_less_cancel_left)
- apply (rule mult_strict_left_mono)
- using nou \<open>0 < r\<close> yx
- apply (simp_all add: field_simps)
- done
- also have "\<dots> = norm x"
- by simp
- finally show False by simp
- qed
- have "inj f"
- unfolding f_def
- proof (clarsimp simp: inj_on_def)
- fix x y
- assume "(1 - norm (x - a) / r) *\<^sub>R (u - a) + x =
- (1 - norm (y - a) / r) *\<^sub>R (u - a) + y"
- then have eq: "(x - a) + (norm (y - a) / r) *\<^sub>R (u - a) = (y - a) + (norm (x - a) / r) *\<^sub>R (u - a)"
- by (auto simp: algebra_simps)
- show "x=y"
- proof (cases "norm (x - a) = norm (y - a)")
- case True
- then show ?thesis
- using eq by auto
- next
- case False
- then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
- by linarith
- then have "False"
- proof cases
- case 1 show False
- using * [OF _ nou 1] eq by simp
- next
- case 2 with * [OF eq nou] show False
- by auto
- qed
- then show "x=y" ..
- qed
- qed
- then have inj_onf: "inj_on f (cball a r \<inter> T)"
- using inj_on_Int by fastforce
- have contf: "continuous_on (cball a r \<inter> T) f"
- unfolding f_def using \<open>0 < r\<close> by (intro continuous_intros) blast
- have fim: "f ` (cball a r \<inter> T) = cball a r \<inter> T"
- proof
- have *: "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> r" if "norm y \<le> r" "norm u < r" for y u::'a
- proof -
- have "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> norm y + norm((1 - norm y / r) *\<^sub>R u)"
- using norm_triangle_ineq by blast
- also have "\<dots> = norm y + abs(1 - norm y / r) * norm u"
- by simp
- also have "\<dots> \<le> r"
- proof -
- have "(r - norm u) * (r - norm y) \<ge> 0"
- using that by auto
- then have "r * norm u + r * norm y \<le> r * r + norm u * norm y"
- by (simp add: algebra_simps)
- then show ?thesis
- using that \<open>0 < r\<close> by (simp add: abs_if field_simps)
- qed
- finally show ?thesis .
- qed
- have "f ` (cball a r) \<subseteq> cball a r"
- apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
- using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
- moreover have "f ` T \<subseteq> T"
- unfolding f_def using \<open>affine T\<close> \<open>a \<in> T\<close> \<open>u \<in> T\<close>
- by (force simp: add.commute mem_affine_3_minus)
- ultimately show "f ` (cball a r \<inter> T) \<subseteq> cball a r \<inter> T"
- by blast
- next
- show "cball a r \<inter> T \<subseteq> f ` (cball a r \<inter> T)"
- proof (clarsimp simp add: dist_norm norm_minus_commute)
- fix x
- assume x: "norm (x - a) \<le> r" and "x \<in> T"
- have "\<exists>v \<in> {0..1}. ((1 - v) * r - norm ((x - a) - v *\<^sub>R (u - a))) \<bullet> 1 = 0"
- by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
- then obtain v where "0\<le>v" "v\<le>1" and v: "(1 - v) * r = norm ((x - a) - v *\<^sub>R (u - a))"
- by auto
- show "x \<in> f ` (cball a r \<inter> T)"
- proof (rule image_eqI)
- show "x = f (x - v *\<^sub>R (u - a))"
- using \<open>r > 0\<close> v by (simp add: f_def field_simps)
- have "x - v *\<^sub>R (u - a) \<in> cball a r"
- using \<open>r > 0\<close> v \<open>0 \<le> v\<close>
- apply (simp add: field_simps dist_norm norm_minus_commute)
- by (metis le_add_same_cancel2 order.order_iff_strict zero_le_mult_iff)
- moreover have "x - v *\<^sub>R (u - a) \<in> T"
- by (simp add: f_def \<open>affine T\<close> \<open>u \<in> T\<close> \<open>x \<in> T\<close> assms mem_affine_3_minus2)
- ultimately show "x - v *\<^sub>R (u - a) \<in> cball a r \<inter> T"
- by blast
- qed
- qed
- qed
- have "\<exists>g. homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
- apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
- apply (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
- done
- then show ?thesis
- apply (rule exE)
- apply (erule_tac f=f in that)
- using \<open>r > 0\<close>
- apply (simp_all add: f_def dist_norm norm_minus_commute)
- done
-qed
-
-corollary%unimportant homeomorphism_moving_point_2:
- fixes a :: "'a::euclidean_space"
- assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
- obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
- "f u = v" "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
-proof -
- have "0 < r"
- by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
- obtain f1 g1 where hom1: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f1 g1"
- and "f1 a = u" and f1: "\<And>x. x \<in> sphere a r \<Longrightarrow> f1 x = x"
- using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u] by blast
- obtain f2 g2 where hom2: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f2 g2"
- and "f2 a = v" and f2: "\<And>x. x \<in> sphere a r \<Longrightarrow> f2 x = x"
- using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> v] by blast
- show ?thesis
- proof
- show "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) (f2 \<circ> g1) (f1 \<circ> g2)"
- by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
- have "g1 u = a"
- using \<open>0 < r\<close> \<open>f1 a = u\<close> assms hom1 homeomorphism_apply1 by fastforce
- then show "(f2 \<circ> g1) u = v"
- by (simp add: \<open>f2 a = v\<close>)
- show "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> (f2 \<circ> g1) x = x"
- using f1 f2 hom1 homeomorphism_apply1 by fastforce
- qed
-qed
-
-
-corollary%unimportant homeomorphism_moving_point_3:
- fixes a :: "'a::euclidean_space"
- assumes "affine T" "a \<in> T" and ST: "ball a r \<inter> T \<subseteq> S" "S \<subseteq> T"
- and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
- obtains f g where "homeomorphism S S f g"
- "f u = v" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> ball a r \<inter> T"
-proof -
- obtain f g where hom: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
- and "f u = v" and fid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
- using homeomorphism_moving_point_2 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u v] by blast
- have gid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> g x = x"
- using fid hom homeomorphism_apply1 by fastforce
- define ff where "ff \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then f x else x"
- define gg where "gg \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then g x else x"
- show ?thesis
- proof
- show "homeomorphism S S ff gg"
- proof (rule homeomorphismI)
- have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) ff"
- apply (simp add: ff_def)
- apply (rule continuous_on_cases)
- using homeomorphism_cont1 [OF hom]
- apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id fid)
- done
- then show "continuous_on S ff"
- apply (rule continuous_on_subset)
- using ST by auto
- have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
- apply (simp add: gg_def)
- apply (rule continuous_on_cases)
- using homeomorphism_cont2 [OF hom]
- apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id gid)
- done
- then show "continuous_on S gg"
- apply (rule continuous_on_subset)
- using ST by auto
- show "ff ` S \<subseteq> S"
- proof (clarsimp simp add: ff_def)
- fix x
- assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
- then have "f x \<in> cball a r \<inter> T"
- using homeomorphism_image1 [OF hom] by force
- then show "f x \<in> S"
- using ST(1) \<open>x \<in> T\<close> gid hom homeomorphism_def x by fastforce
- qed
- show "gg ` S \<subseteq> S"
- proof (clarsimp simp add: gg_def)
- fix x
- assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
- then have "g x \<in> cball a r \<inter> T"
- using homeomorphism_image2 [OF hom] by force
- then have "g x \<in> ball a r"
- using homeomorphism_apply2 [OF hom]
- by (metis Diff_Diff_Int Diff_iff \<open>x \<in> T\<close> cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
- then show "g x \<in> S"
- using ST(1) \<open>g x \<in> cball a r \<inter> T\<close> by force
- qed
- show "\<And>x. x \<in> S \<Longrightarrow> gg (ff x) = x"
- apply (simp add: ff_def gg_def)
- using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
- apply auto
- apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
- done
- show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
- apply (simp add: ff_def gg_def)
- using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
- apply auto
- apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
- done
- qed
- show "ff u = v"
- using u by (auto simp: ff_def \<open>f u = v\<close>)
- show "{x. \<not> (ff x = x \<and> gg x = x)} \<subseteq> ball a r \<inter> T"
- by (auto simp: ff_def gg_def)
- qed
-qed
-
-
-proposition%unimportant homeomorphism_moving_point:
- fixes a :: "'a::euclidean_space"
- assumes ope: "openin (subtopology euclidean (affine hull S)) S"
- and "S \<subseteq> T"
- and TS: "T \<subseteq> affine hull S"
- and S: "connected S" "a \<in> S" "b \<in> S"
- obtains f g where "homeomorphism T T f g" "f a = b"
- "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
- "bounded {x. \<not> (f x = x \<and> g x = x)}"
-proof -
- have 1: "\<exists>h k. homeomorphism T T h k \<and> h (f d) = d \<and>
- {x. \<not> (h x = x \<and> k x = x)} \<subseteq> S \<and> bounded {x. \<not> (h x = x \<and> k x = x)}"
- if "d \<in> S" "f d \<in> S" and homfg: "homeomorphism T T f g"
- and S: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
- and bo: "bounded {x. \<not> (f x = x \<and> g x = x)}" for d f g
- proof (intro exI conjI)
- show homgf: "homeomorphism T T g f"
- by (metis homeomorphism_symD homfg)
- then show "g (f d) = d"
- by (meson \<open>S \<subseteq> T\<close> homeomorphism_def subsetD \<open>d \<in> S\<close>)
- show "{x. \<not> (g x = x \<and> f x = x)} \<subseteq> S"
- using S by blast
- show "bounded {x. \<not> (g x = x \<and> f x = x)}"
- using bo by (simp add: conj_commute)
- qed
- have 2: "\<exists>f g. homeomorphism T T f g \<and> f x = f2 (f1 x) \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
- if "x \<in> S" "f1 x \<in> S" "f2 (f1 x) \<in> S"
- and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
- and sub: "{x. \<not> (f1 x = x \<and> g1 x = x)} \<subseteq> S" "{x. \<not> (f2 x = x \<and> g2 x = x)} \<subseteq> S"
- and bo: "bounded {x. \<not> (f1 x = x \<and> g1 x = x)}" "bounded {x. \<not> (f2 x = x \<and> g2 x = x)}"
- for x f1 f2 g1 g2
- proof (intro exI conjI)
- show homgf: "homeomorphism T T (f2 \<circ> f1) (g1 \<circ> g2)"
- by (metis homeomorphism_compose hom)
- then show "(f2 \<circ> f1) x = f2 (f1 x)"
- by force
- show "{x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)} \<subseteq> S"
- using sub by force
- have "bounded ({x. \<not>(f1 x = x \<and> g1 x = x)} \<union> {x. \<not>(f2 x = x \<and> g2 x = x)})"
- using bo by simp
- then show "bounded {x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)}"
- by (rule bounded_subset) auto
- qed
- have 3: "\<exists>U. openin (subtopology euclidean S) U \<and>
- d \<in> U \<and>
- (\<forall>x\<in>U.
- \<exists>f g. homeomorphism T T f g \<and> f d = x \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
- bounded {x. \<not> (f x = x \<and> g x = x)})"
- if "d \<in> S" for d
- proof -
- obtain r where "r > 0" and r: "ball d r \<inter> affine hull S \<subseteq> S"
- by (metis \<open>d \<in> S\<close> ope openin_contains_ball)
- have *: "\<exists>f g. homeomorphism T T f g \<and> f d = e \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
- bounded {x. \<not> (f x = x \<and> g x = x)}" if "e \<in> S" "e \<in> ball d r" for e
- apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
- using r \<open>S \<subseteq> T\<close> TS that
- apply (auto simp: \<open>d \<in> S\<close> \<open>0 < r\<close> hull_inc)
- using bounded_subset by blast
- show ?thesis
- apply (rule_tac x="S \<inter> ball d r" in exI)
- apply (intro conjI)
- apply (simp add: openin_open_Int)
- apply (simp add: \<open>0 < r\<close> that)
- apply (blast intro: *)
- done
- qed
- have "\<exists>f g. homeomorphism T T f g \<and> f a = b \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
- apply (rule connected_equivalence_relation [OF S], safe)
- apply (blast intro: 1 2 3)+
- done
- then show ?thesis
- using that by auto
-qed
-
-
-lemma homeomorphism_moving_points_exists_gen:
- assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
- "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
- and "2 \<le> aff_dim S"
- and ope: "openin (subtopology euclidean (affine hull S)) S"
- and "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
- shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
- using assms
-proof (induction K)
- case empty
- then show ?case
- by (force simp: homeomorphism_ident)
-next
- case (insert i K)
- then have xney: "\<And>j. \<lbrakk>j \<in> K; j \<noteq> i\<rbrakk> \<Longrightarrow> x i \<noteq> x j \<and> y i \<noteq> y j"
- and pw: "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"
- and "x i \<in> S" "y i \<in> S"
- and xyS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
- by (simp_all add: pairwise_insert)
- obtain f g where homfg: "homeomorphism T T f g" and feq: "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
- and fg_sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
- and bo_fg: "bounded {x. \<not> (f x = x \<and> g x = x)}"
- using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
- then have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
- using insert by blast
- have aff_eq: "affine hull (S - y ` K) = affine hull S"
- apply (rule affine_hull_Diff)
- apply (auto simp: insert)
- using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
- have f_in_S: "f x \<in> S" if "x \<in> S" for x
- using homfg fg_sub homeomorphism_apply1 \<open>S \<subseteq> T\<close>
- proof -
- have "(f (f x) \<noteq> f x \<or> g (f x) \<noteq> f x) \<or> f x \<in> S"
- by (metis \<open>S \<subseteq> T\<close> homfg subsetD homeomorphism_apply1 that)
- then show ?thesis
- using fg_sub by force
- qed
- obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
- and hk_sub: "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
- and bo_hk: "bounded {x. \<not> (h x = x \<and> k x = x)}"
- proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
- show "openin (subtopology euclidean (affine hull (S - y ` K))) (S - y ` K)"
- by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
- show "S - y ` K \<subseteq> T"
- using \<open>S \<subseteq> T\<close> by auto
- show "T \<subseteq> affine hull (S - y ` K)"
- using insert by (simp add: aff_eq)
- show "connected (S - y ` K)"
- proof (rule connected_openin_diff_countable [OF \<open>connected S\<close> ope])
- show "\<not> collinear S"
- using collinear_aff_dim \<open>2 \<le> aff_dim S\<close> by force
- show "countable (y ` K)"
- using countable_finite insert.hyps(1) by blast
- qed
- show "f (x i) \<in> S - y ` K"
- apply (auto simp: f_in_S \<open>x i \<in> S\<close>)
- by (metis feq homfg \<open>x i \<in> S\<close> homeomorphism_def \<open>S \<subseteq> T\<close> \<open>i \<notin> K\<close> subsetCE xney xyS)
- show "y i \<in> S - y ` K"
- using insert.hyps xney by (auto simp: \<open>y i \<in> S\<close>)
- qed blast
- show ?case
- proof (intro exI conjI)
- show "homeomorphism T T (h \<circ> f) (g \<circ> k)"
- using homfg homhk homeomorphism_compose by blast
- show "\<forall>i \<in> insert i K. (h \<circ> f) (x i) = y i"
- using feq hk_sub by (auto simp: heq)
- show "{x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)} \<subseteq> S"
- using fg_sub hk_sub by force
- have "bounded ({x. \<not>(f x = x \<and> g x = x)} \<union> {x. \<not>(h x = x \<and> k x = x)})"
- using bo_fg bo_hk bounded_Un by blast
- then show "bounded {x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)}"
- by (rule bounded_subset) auto
- qed
-qed
-
-proposition%unimportant homeomorphism_moving_points_exists:
- fixes S :: "'a::euclidean_space set"
- assumes 2: "2 \<le> DIM('a)" "open S" "connected S" "S \<subseteq> T" "finite K"
- and KS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
- and pw: "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
- and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
- obtains f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
- "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. (\<not> (f x = x \<and> g x = x))}"
-proof (cases "S = {}")
- case True
- then show ?thesis
- using KS homeomorphism_ident that by fastforce
-next
- case False
- then have affS: "affine hull S = UNIV"
- by (simp add: affine_hull_open \<open>open S\<close>)
- then have ope: "openin (subtopology euclidean (affine hull S)) S"
- using \<open>open S\<close> open_openin by auto
- have "2 \<le> DIM('a)" by (rule 2)
- also have "\<dots> = aff_dim (UNIV :: 'a set)"
- by simp
- also have "\<dots> \<le> aff_dim S"
- by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
- finally have "2 \<le> aff_dim S"
- by linarith
- then show ?thesis
- using homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> KS pw _ ope S] that by fastforce
-qed
-
-subsubsection%unimportant\<open>The theorem \<open>homeomorphism_grouping_points_exists\<close>\<close>
-
-lemma homeomorphism_grouping_point_1:
- fixes a::real and c::real
- assumes "a < b" "c < d"
- obtains f g where "homeomorphism (cbox a b) (cbox c d) f g" "f a = c" "f b = d"
-proof -
- define f where "f \<equiv> \<lambda>x. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
- have "\<exists>g. homeomorphism (cbox a b) (cbox c d) f g"
- proof (rule homeomorphism_compact)
- show "continuous_on (cbox a b) f"
- apply (simp add: f_def)
- apply (intro continuous_intros)
- using assms by auto
- have "f ` {a..b} = {c..d}"
- unfolding f_def image_affinity_atLeastAtMost
- using assms sum_sqs_eq by (auto simp: divide_simps algebra_simps)
- then show "f ` cbox a b = cbox c d"
- by auto
- show "inj_on f (cbox a b)"
- unfolding f_def inj_on_def using assms by auto
- qed auto
- then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
- then show ?thesis
- proof
- show "f a = c"
- by (simp add: f_def)
- show "f b = d"
- using assms sum_sqs_eq [of a b] by (auto simp: f_def divide_simps algebra_simps)
- qed
-qed
-
-lemma homeomorphism_grouping_point_2:
- fixes a::real and w::real
- assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
- and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
- and "b \<in> cbox a c" "v \<in> cbox u w"
- and eq: "f1 a = u" "f1 b = v" "f2 b = v" "f2 c = w"
- obtains f g where "homeomorphism (cbox a c) (cbox u w) f g" "f a = u" "f c = w"
- "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = f1 x" "\<And>x. x \<in> cbox b c \<Longrightarrow> f x = f2 x"
-proof -
- have le: "a \<le> b" "b \<le> c" "u \<le> v" "v \<le> w"
- using assms by simp_all
- then have ac: "cbox a c = cbox a b \<union> cbox b c" and uw: "cbox u w = cbox u v \<union> cbox v w"
- by auto
- define f where "f \<equiv> \<lambda>x. if x \<le> b then f1 x else f2 x"
- have "\<exists>g. homeomorphism (cbox a c) (cbox u w) f g"
- proof (rule homeomorphism_compact)
- have cf1: "continuous_on (cbox a b) f1"
- using hom_ab homeomorphism_cont1 by blast
- have cf2: "continuous_on (cbox b c) f2"
- using hom_bc homeomorphism_cont1 by blast
- show "continuous_on (cbox a c) f"
- apply (simp add: f_def)
- apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
- using le eq apply (force simp: continuous_on_id)+
- done
- have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
- unfolding f_def using eq by force+
- then show "f ` cbox a c = cbox u w"
- apply (simp only: ac uw image_Un)
- by (metis hom_ab hom_bc homeomorphism_def)
- have neq12: "f1 x \<noteq> f2 y" if x: "a \<le> x" "x \<le> b" and y: "b < y" "y \<le> c" for x y
- proof -
- have "f1 x \<in> cbox u v"
- by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
- moreover have "f2 y \<in> cbox v w"
- by (metis (full_types) hom_bc homeomorphism_def image_subset_iff mem_box_real(2) not_le not_less_iff_gr_or_eq order_refl y)
- moreover have "f2 y \<noteq> f2 b"
- by (metis cancel_comm_monoid_add_class.diff_cancel diff_gt_0_iff_gt hom_bc homeomorphism_def le(2) less_imp_le less_numeral_extra(3) mem_box_real(2) order_refl y)
- ultimately show ?thesis
- using le eq by simp
- qed
- have "inj_on f1 (cbox a b)"
- by (metis (full_types) hom_ab homeomorphism_def inj_onI)
- moreover have "inj_on f2 (cbox b c)"
- by (metis (full_types) hom_bc homeomorphism_def inj_onI)
- ultimately show "inj_on f (cbox a c)"
- apply (simp (no_asm) add: inj_on_def)
- apply (simp add: f_def inj_on_eq_iff)
- using neq12 apply force
- done
- qed auto
- then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
- then show ?thesis
- apply (rule that)
- using eq le by (auto simp: f_def)
-qed
-
-lemma homeomorphism_grouping_point_3:
- fixes a::real
- assumes cbox_sub: "cbox c d \<subseteq> box a b" "cbox u v \<subseteq> box a b"
- and box_ne: "box c d \<noteq> {}" "box u v \<noteq> {}"
- obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
- "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v"
-proof -
- have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d \<noteq> {}"
- using assms
- by (simp_all add: cbox_sub subset_eq)
- obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
- and f1_eq: "f1 a = a" "f1 c = u"
- using homeomorphism_grouping_point_1 [OF \<open>a < c\<close> \<open>a < u\<close>] .
- obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
- and f2_eq: "f2 c = u" "f2 d = v"
- using homeomorphism_grouping_point_1 [OF \<open>c < d\<close> \<open>u < v\<close>] .
- obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
- and f3_eq: "f3 d = v" "f3 b = b"
- using homeomorphism_grouping_point_1 [OF \<open>d < b\<close> \<open>v < b\<close>] .
- obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
- and f4_eq: "\<And>x. x \<in> cbox a c \<Longrightarrow> f4 x = f1 x" "\<And>x. x \<in> cbox c d \<Longrightarrow> f4 x = f2 x"
- using homeomorphism_grouping_point_2 [OF 1 2] less by (auto simp: f1_eq f2_eq)
- obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
- and f_eq: "\<And>x. x \<in> cbox a d \<Longrightarrow> f x = f4 x" "\<And>x. x \<in> cbox d b \<Longrightarrow> f x = f3 x"
- using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
- show ?thesis
- apply (rule that [OF fg])
- using f4_eq f_eq homeomorphism_image1 [OF 2]
- apply simp
- by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
-qed
-
-
-lemma homeomorphism_grouping_point_4:
- fixes T :: "real set"
- assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
- obtains f g where "homeomorphism T T f g"
- "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
- "bounded {x. (\<not> (f x = x \<and> g x = x))}"
-proof -
- obtain c d where "box c d \<noteq> {}" "cbox c d \<subseteq> U"
- proof -
- obtain u where "u \<in> U"
- using \<open>U \<noteq> {}\<close> by blast
- then obtain e where "e > 0" "cball u e \<subseteq> U"
- using \<open>open U\<close> open_contains_cball by blast
- then show ?thesis
- by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
- qed
- have "compact K"
- by (simp add: \<open>finite K\<close> finite_imp_compact)
- obtain a b where "box a b \<noteq> {}" "K \<subseteq> cbox a b" "cbox a b \<subseteq> S"
- proof (cases "K = {}")
- case True then show ?thesis
- using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> that by blast
- next
- case False
- then obtain a b where "a \<in> K" "b \<in> K"
- and a: "\<And>x. x \<in> K \<Longrightarrow> a \<le> x" and b: "\<And>x. x \<in> K \<Longrightarrow> x \<le> b"
- using compact_attains_inf compact_attains_sup by (metis \<open>compact K\<close>)+
- obtain e where "e > 0" "cball b e \<subseteq> S"
- using \<open>open S\<close> open_contains_cball
- by (metis \<open>b \<in> K\<close> \<open>K \<subseteq> S\<close> subsetD)
- show ?thesis
- proof
- show "box a (b + e) \<noteq> {}"
- using \<open>0 < e\<close> \<open>b \<in> K\<close> a by force
- show "K \<subseteq> cbox a (b + e)"
- using \<open>0 < e\<close> a b by fastforce
- have "a \<in> S"
- using \<open>a \<in> K\<close> assms(6) by blast
- have "b + e \<in> S"
- using \<open>0 < e\<close> \<open>cball b e \<subseteq> S\<close> by (force simp: dist_norm)
- show "cbox a (b + e) \<subseteq> S"
- using \<open>a \<in> S\<close> \<open>b + e \<in> S\<close> \<open>connected S\<close> connected_contains_Icc by auto
- qed
- qed
- obtain w z where "cbox w z \<subseteq> S" and sub_wz: "cbox a b \<union> cbox c d \<subseteq> box w z"
- proof -
- have "a \<in> S" "b \<in> S"
- using \<open>box a b \<noteq> {}\<close> \<open>cbox a b \<subseteq> S\<close> by auto
- moreover have "c \<in> S" "d \<in> S"
- using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> by force+
- ultimately have "min a c \<in> S" "max b d \<in> S"
- by linarith+
- then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 \<subseteq> S" "e2 > 0" "cball (max b d) e2 \<subseteq> S"
- using \<open>open S\<close> open_contains_cball by metis
- then have *: "min a c - e1 \<in> S" "max b d + e2 \<in> S"
- by (auto simp: dist_norm)
- show ?thesis
- proof
- show "cbox (min a c - e1) (max b d+ e2) \<subseteq> S"
- using * \<open>connected S\<close> connected_contains_Icc by auto
- show "cbox a b \<union> cbox c d \<subseteq> box (min a c - e1) (max b d + e2)"
- using \<open>0 < e1\<close> \<open>0 < e2\<close> by auto
- qed
- qed
- then
- obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
- and "f w = w" "f z = z"
- and fin: "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<in> cbox c d"
- using homeomorphism_grouping_point_3 [of a b w z c d]
- using \<open>box a b \<noteq> {}\<close> \<open>box c d \<noteq> {}\<close> by blast
- have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
- using hom homeomorphism_def by blast+
- define f' where "f' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
- define g' where "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
- show ?thesis
- proof
- have T: "cbox w z \<union> (T - box w z) = T"
- using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
- show "homeomorphism T T f' g'"
- proof
- have clo: "closedin (subtopology euclidean (cbox w z \<union> (T - box w z))) (T - box w z)"
- by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
- have "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
- unfolding f'_def g'_def
- apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
- apply (simp_all add: closed_subset)
- using \<open>f w = w\<close> \<open>f z = z\<close> apply force
- by (metis \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_def less_eq_real_def mem_box_real(2))
- then show "continuous_on T f'" "continuous_on T g'"
- by (simp_all only: T)
- show "f' ` T \<subseteq> T"
- unfolding f'_def
- by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
- show "g' ` T \<subseteq> T"
- unfolding g'_def
- by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
- show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
- unfolding f'_def g'_def
- using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom] by fastforce
- show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
- unfolding f'_def g'_def
- using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom] by fastforce
- qed
- show "\<And>x. x \<in> K \<Longrightarrow> f' x \<in> U"
- using fin sub_wz \<open>K \<subseteq> cbox a b\<close> \<open>cbox c d \<subseteq> U\<close> by (force simp: f'_def)
- show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
- using \<open>cbox w z \<subseteq> S\<close> by (auto simp: f'_def g'_def)
- show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
- apply (rule bounded_subset [of "cbox w z"])
- using bounded_cbox apply blast
- apply (auto simp: f'_def g'_def)
- done
- qed
-qed
-
-proposition%unimportant homeomorphism_grouping_points_exists:
- fixes S :: "'a::euclidean_space set"
- assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
- obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
- "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
-proof (cases "2 \<le> DIM('a)")
- case True
- have TS: "T \<subseteq> affine hull S"
- using affine_hull_open assms by blast
- have "infinite U"
- using \<open>open U\<close> \<open>U \<noteq> {}\<close> finite_imp_not_open by blast
- then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
- using infinite_arbitrarily_large by metis
- then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
- using \<open>finite K\<close> finite_same_card_bij by blast
- obtain f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f (id i) = \<gamma> i" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. \<not> (f x = x \<and> g x = x)}"
- proof (rule homeomorphism_moving_points_exists [OF True \<open>open S\<close> \<open>connected S\<close> \<open>S \<subseteq> T\<close> \<open>finite K\<close>])
- show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
- using \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> \<open>U \<subseteq> S\<close> bij_betwE by blast
- show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
- using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
- qed (use affine_hull_open assms that in auto)
- then show ?thesis
- using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
-next
- case False
- with DIM_positive have "DIM('a) = 1"
- by (simp add: dual_order.antisym)
- then obtain h::"'a \<Rightarrow>real" and j
- where "linear h" "linear j"
- and noh: "\<And>x. norm(h x) = norm x" and noj: "\<And>y. norm(j y) = norm y"
- and hj: "\<And>x. j(h x) = x" "\<And>y. h(j y) = y"
- and ranh: "surj h"
- using isomorphisms_UNIV_UNIV
- by (metis (mono_tags, hide_lams) DIM_real UNIV_eq_I range_eqI)
- obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
- and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
- and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
- and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
- apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
- by (simp_all add: assms image_mono \<open>linear h\<close> open_surjective_linear_image connected_linear_image ranh)
- have jf: "j (f (h x)) = x \<longleftrightarrow> f (h x) = h x" for x
- by (metis hj)
- have jg: "j (g (h x)) = x \<longleftrightarrow> g (h x) = h x" for x
- by (metis hj)
- have cont_hj: "continuous_on X h" "continuous_on Y j" for X Y
- by (simp_all add: \<open>linear h\<close> \<open>linear j\<close> linear_linear linear_continuous_on)
- show ?thesis
- proof
- show "homeomorphism T T (j \<circ> f \<circ> h) (j \<circ> g \<circ> h)"
- proof
- show "continuous_on T (j \<circ> f \<circ> h)" "continuous_on T (j \<circ> g \<circ> h)"
- using hom homeomorphism_def
- by (blast intro: continuous_on_compose cont_hj)+
- show "(j \<circ> f \<circ> h) ` T \<subseteq> T" "(j \<circ> g \<circ> h) ` T \<subseteq> T"
- by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
- show "\<And>x. x \<in> T \<Longrightarrow> (j \<circ> g \<circ> h) ((j \<circ> f \<circ> h) x) = x"
- using hj hom homeomorphism_apply1 by fastforce
- show "\<And>y. y \<in> T \<Longrightarrow> (j \<circ> f \<circ> h) ((j \<circ> g \<circ> h) y) = y"
- using hj hom homeomorphism_apply2 by fastforce
- qed
- show "{x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} \<subseteq> S"
- apply (clarsimp simp: jf jg hj)
- using sub hj
- apply (drule_tac c="h x" in subsetD, force)
- by (metis imageE)
- have "bounded (j ` {x. (\<not> (f x = x \<and> g x = x))})"
- by (rule bounded_linear_image [OF bou]) (use \<open>linear j\<close> linear_conv_bounded_linear in auto)
- moreover
- have *: "{x. \<not>((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
- using hj by (auto simp: jf jg image_iff, metis+)
- ultimately show "bounded {x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)}"
- by metis
- show "\<And>x. x \<in> K \<Longrightarrow> (j \<circ> f \<circ> h) x \<in> U"
- using f hj by fastforce
- qed
-qed
-
-
-proposition%unimportant homeomorphism_grouping_points_exists_gen:
- fixes S :: "'a::euclidean_space set"
- assumes opeU: "openin (subtopology euclidean S) U"
- and opeS: "openin (subtopology euclidean (affine hull S)) S"
- and "U \<noteq> {}" "finite K" "K \<subseteq> S" and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
- obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
- "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
-proof (cases "2 \<le> aff_dim S")
- case True
- have opeU': "openin (subtopology euclidean (affine hull S)) U"
- using opeS opeU openin_trans by blast
- obtain u where "u \<in> U" "u \<in> S"
- using \<open>U \<noteq> {}\<close> opeU openin_imp_subset by fastforce+
- have "infinite U"
- apply (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
- apply (rule connected_imp_perfect_aff_dim [OF \<open>connected S\<close> _ \<open>u \<in> S\<close>])
- using True apply simp
- done
- then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
- using infinite_arbitrarily_large by metis
- then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
- using \<open>finite K\<close> finite_same_card_bij by blast
- have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(id i) = \<gamma> i) \<and>
- {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
- proof (rule homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> _ _ True opeS S])
- show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
- by (metis id_apply opeU openin_contains_cball subsetCE \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> bij_betwE)
- show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
- using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
- qed
- then show ?thesis
- using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
-next
- case False
- with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
- then show ?thesis
- proof cases
- assume "aff_dim S = -1"
- then have "S = {}"
- using aff_dim_empty by blast
- then have "False"
- using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
- then show ?thesis ..
- next
- assume "aff_dim S = 0"
- then obtain a where "S = {a}"
- using aff_dim_eq_0 by blast
- then have "K \<subseteq> U"
- using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
- show ?thesis
- apply (rule that [of id id])
- using \<open>K \<subseteq> U\<close> by (auto simp: continuous_on_id intro: homeomorphismI)
- next
- assume "aff_dim S = 1"
- then have "affine hull S homeomorphic (UNIV :: real set)"
- by (auto simp: homeomorphic_affine_sets)
- then obtain h::"'a\<Rightarrow>real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
- using homeomorphic_def by blast
- then have h: "\<And>x. x \<in> affine hull S \<Longrightarrow> j(h(x)) = x" and j: "\<And>y. j y \<in> affine hull S \<and> h(j y) = y"
- by (auto simp: homeomorphism_def)
- have connh: "connected (h ` S)"
- by (meson Topological_Spaces.connected_continuous_image \<open>connected S\<close> homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
- have hUS: "h ` U \<subseteq> h ` S"
- by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
- have opn: "openin (subtopology euclidean (affine hull S)) U \<Longrightarrow> open (h ` U)" for U
- using homeomorphism_imp_open_map [OF homhj] by simp
- have "open (h ` U)" "open (h ` S)"
- by (auto intro: opeS opeU openin_trans opn)
- then obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
- and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
- and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
- and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
- apply (rule homeomorphism_grouping_points_exists [of "h ` U" "h ` S" "h ` K" "h ` T"])
- using assms by (auto simp: connh hUS)
- have jf: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (f (h x)) = x \<longleftrightarrow> f (h x) = h x"
- by (metis h j)
- have jg: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (g (h x)) = x \<longleftrightarrow> g (h x) = h x"
- by (metis h j)
- have cont_hj: "continuous_on T h" "continuous_on Y j" for Y
- apply (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
- using homeomorphism_def homhj apply blast
- by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
- define f' where "f' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> f \<circ> h) x else x"
- define g' where "g' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> g \<circ> h) x else x"
- show ?thesis
- proof
- show "homeomorphism T T f' g'"
- proof
- have "continuous_on T (j \<circ> f \<circ> h)"
- apply (intro continuous_on_compose cont_hj)
- using hom homeomorphism_def by blast
- then show "continuous_on T f'"
- apply (rule continuous_on_eq)
- using \<open>T \<subseteq> affine hull S\<close> f'_def by auto
- have "continuous_on T (j \<circ> g \<circ> h)"
- apply (intro continuous_on_compose cont_hj)
- using hom homeomorphism_def by blast
- then show "continuous_on T g'"
- apply (rule continuous_on_eq)
- using \<open>T \<subseteq> affine hull S\<close> g'_def by auto
- show "f' ` T \<subseteq> T"
- proof (clarsimp simp: f'_def)
- fix x assume "x \<in> T"
- then have "f (h x) \<in> h ` T"
- by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
- then show "j (f (h x)) \<in> T"
- using \<open>T \<subseteq> affine hull S\<close> h by auto
- qed
- show "g' ` T \<subseteq> T"
- proof (clarsimp simp: g'_def)
- fix x assume "x \<in> T"
- then have "g (h x) \<in> h ` T"
- by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
- then show "j (g (h x)) \<in> T"
- using \<open>T \<subseteq> affine hull S\<close> h by auto
- qed
- show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
- using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
- show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
- using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
- qed
- next
- show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
- apply (clarsimp simp: f'_def g'_def jf jg)
- apply (rule imageE [OF subsetD [OF sub]], force)
- by (metis h hull_inc)
- next
- have "compact (j ` closure {x. \<not> (f x = x \<and> g x = x)})"
- using bou by (auto simp: compact_continuous_image cont_hj)
- then have "bounded (j ` {x. \<not> (f x = x \<and> g x = x)})"
- by (rule bounded_closure_image [OF compact_imp_bounded])
- moreover
- have *: "{x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
- using h j by (auto simp: image_iff; metis)
- ultimately have "bounded {x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x}"
- by metis
- then show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
- by (simp add: f'_def g'_def Collect_mono bounded_subset)
- next
- show "f' x \<in> U" if "x \<in> K" for x
- proof -
- have "U \<subseteq> S"
- using opeU openin_imp_subset by blast
- then have "j (f (h x)) \<in> U"
- using f h hull_subset that by fastforce
- then show "f' x \<in> U"
- using \<open>K \<subseteq> S\<close> S f'_def that by auto
- qed
- qed
- qed
-qed
-
-
-subsection\<open>Nullhomotopic mappings\<close>
-
-text\<open> A mapping out of a sphere is nullhomotopic iff it extends to the ball.
-This even works out in the degenerate cases when the radius is \<open>\<le>\<close> 0, and
-we also don't need to explicitly assume continuity since it's already implicit
-in both sides of the equivalence.\<close>
-
-lemma nullhomotopic_from_lemma:
- assumes contg: "continuous_on (cball a r - {a}) g"
- and fa: "\<And>e. 0 < e
- \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>x. x \<noteq> a \<and> norm(x - a) < d \<longrightarrow> norm(g x - f a) < e)"
- and r: "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> f x = g x"
- shows "continuous_on (cball a r) f"
-proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
- fix x
- assume x: "dist a x \<le> r"
- show "continuous (at x within cball a r) f"
- proof (cases "x=a")
- case True
- then show ?thesis
- by (metis continuous_within_eps_delta fa dist_norm dist_self r)
- next
- case False
- show ?thesis
- proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
- have "\<exists>d>0. \<forall>x'\<in>cball a r.
- dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
- proof -
- obtain d where "d > 0"
- and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
- dist (g x') (g x) < e"
- using contg False x \<open>e>0\<close>
- unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
- show ?thesis
- using \<open>d > 0\<close> \<open>x \<noteq> a\<close>
- by (rule_tac x="min d (norm(x - a))" in exI)
- (auto simp: dist_commute dist_norm [symmetric] intro!: d)
- qed
- then show "continuous (at x within cball a r) g"
- using contg False by (auto simp: continuous_within_eps_delta)
- show "0 < norm (x - a)"
- using False by force
- show "x \<in> cball a r"
- by (simp add: x)
- show "\<And>x'. \<lbrakk>x' \<in> cball a r; dist x' x < norm (x - a)\<rbrakk>
- \<Longrightarrow> g x' = f x'"
- by (metis dist_commute dist_norm less_le r)
- qed
- qed
-qed
-
-proposition nullhomotopic_from_sphere_extension:
- fixes f :: "'M::euclidean_space \<Rightarrow> 'a::real_normed_vector"
- shows "(\<exists>c. homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) \<longleftrightarrow>
- (\<exists>g. continuous_on (cball a r) g \<and> g ` (cball a r) \<subseteq> S \<and>
- (\<forall>x \<in> sphere a r. g x = f x))"
- (is "?lhs = ?rhs")
-proof (cases r "0::real" rule: linorder_cases)
- case equal
- then show ?thesis
- apply (auto simp: homotopic_with)
- apply (rule_tac x="\<lambda>x. h (0, a)" in exI)
- apply (fastforce simp add:)
- using continuous_on_const by blast
-next
- case greater
- let ?P = "continuous_on {x. norm(x - a) = r} f \<and> f ` {x. norm(x - a) = r} \<subseteq> S"
- have ?P if ?lhs using that
- proof
- fix c
- assume c: "homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
- then have contf: "continuous_on (sphere a r) f" and fim: "f ` sphere a r \<subseteq> S"
- by (auto simp: homotopic_with_imp_subset1 homotopic_with_imp_continuous)
- show ?P
- using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
- qed
- moreover have ?P if ?rhs using that
- proof
- fix g
- assume g: "continuous_on (cball a r) g \<and> g ` cball a r \<subseteq> S \<and> (\<forall>xa\<in>sphere a r. g xa = f xa)"
- then
- show ?P
- apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
- apply (auto simp: dist_norm norm_minus_commute)
- by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
- qed
- moreover have ?thesis if ?P
- proof
- assume ?lhs
- then obtain c where "homotopic_with (\<lambda>x. True) (sphere a r) S (\<lambda>x. c) f"
- using homotopic_with_sym by blast
- then obtain h where conth: "continuous_on ({0..1::real} \<times> sphere a r) h"
- and him: "h ` ({0..1} \<times> sphere a r) \<subseteq> S"
- and h: "\<And>x. h(0, x) = c" "\<And>x. h(1, x) = f x"
- by (auto simp: homotopic_with_def)
- obtain b1::'M where "b1 \<in> Basis"
- using SOME_Basis by auto
- have "c \<in> S"
- apply (rule him [THEN subsetD])
- apply (rule_tac x = "(0, a + r *\<^sub>R b1)" in image_eqI)
- using h greater \<open>b1 \<in> Basis\<close>
- apply (auto simp: dist_norm)
- done
- have uconth: "uniformly_continuous_on ({0..1::real} \<times> (sphere a r)) h"
- by (force intro: compact_Times conth compact_uniformly_continuous)
- let ?g = "\<lambda>x. h (norm (x - a)/r,
- a + (if x = a then r *\<^sub>R b1 else (r / norm(x - a)) *\<^sub>R (x - a)))"
- let ?g' = "\<lambda>x. h (norm (x - a)/r, a + (r / norm(x - a)) *\<^sub>R (x - a))"
- show ?rhs
- proof (intro exI conjI)
- have "continuous_on (cball a r - {a}) ?g'"
- apply (rule continuous_on_compose2 [OF conth])
- apply (intro continuous_intros)
- using greater apply (auto simp: dist_norm norm_minus_commute)
- done
- then show "continuous_on (cball a r) ?g"
- proof (rule nullhomotopic_from_lemma)
- show "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (?g' x - ?g a) < e" if "0 < e" for e
- proof -
- obtain d where "0 < d"
- and d: "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> sphere a r; x' \<in> {0..1} \<times> sphere a r; dist x' x < d\<rbrakk>
- \<Longrightarrow> dist (h x') (h x) < e"
- using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by auto
- have *: "norm (h (norm (x - a) / r,
- a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e"
- if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
- proof -
- have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) =
- norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
- by (simp add: h)
- also have "\<dots> < e"
- apply (rule d [unfolded dist_norm])
- using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
- by (auto simp: dist_norm divide_simps)
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule_tac x = "min r (d * r)" in exI)
- using greater \<open>0 < d\<close> by (auto simp: *)
- qed
- show "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> ?g x = ?g' x"
- by auto
- qed
- next
- show "?g ` cball a r \<subseteq> S"
- using greater him \<open>c \<in> S\<close>
- by (force simp: h dist_norm norm_minus_commute)
- next
- show "\<forall>x\<in>sphere a r. ?g x = f x"
- using greater by (auto simp: h dist_norm norm_minus_commute)
- qed
- next
- assume ?rhs
- then obtain g where contg: "continuous_on (cball a r) g"
- and gim: "g ` cball a r \<subseteq> S"
- and gf: "\<forall>x \<in> sphere a r. g x = f x"
- by auto
- let ?h = "\<lambda>y. g (a + (fst y) *\<^sub>R (snd y - a))"
- have "continuous_on ({0..1} \<times> sphere a r) ?h"
- apply (rule continuous_on_compose2 [OF contg])
- apply (intro continuous_intros)
- apply (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
- done
- moreover
- have "?h ` ({0..1} \<times> sphere a r) \<subseteq> S"
- by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
- moreover
- have "\<forall>x\<in>sphere a r. ?h (0, x) = g a" "\<forall>x\<in>sphere a r. ?h (1, x) = f x"
- by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
- ultimately
- show ?lhs
- apply (subst homotopic_with_sym)
- apply (rule_tac x="g a" in exI)
- apply (auto simp: homotopic_with)
- done
- qed
- ultimately
- show ?thesis by meson
-qed simp
-
-end