(* 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 rev_subsetD 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