(* 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 Product_Topology Uncountable_Sets
begin
definition\<^marker>\<open>tag important\<close> homotopic_with
where
"homotopic_with P X Y f g \<equiv>
(\<exists>h. continuous_map (prod_topology (top_of_set {0..1::real}) X) Y h \<and>
(\<forall>x. h(0, x) = f x) \<and>
(\<forall>x. h(1, x) = g 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>
abbreviation homotopic_with_canon ::
"[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
where
"homotopic_with_canon P S T p q \<equiv> homotopic_with P (top_of_set S) (top_of_set T) p q"
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
lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
by auto
lemma fst_o_paired [simp]: "fst \<circ> (\<lambda>(x,y). (f x y, g x y)) = (\<lambda>(x,y). f x y)"
by auto
lemma snd_o_paired [simp]: "snd \<circ> (\<lambda>(x,y). (f x y, g x y)) = (\<lambda>(x,y). g x y)"
by auto
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 continuous_map_o_Pair:
assumes h: "continuous_map (prod_topology X Y) Z h" and t: "t \<in> topspace X"
shows "continuous_map Y Z (h \<circ> Pair t)"
by (intro continuous_map_compose [OF _ h] continuous_intros; simp add: t)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial properties\<close>
text \<open>We often want to just localize the ending function equality or whatever.\<close>
text\<^marker>\<open>tag important\<close> \<open>%whitespace\<close>
proposition homotopic_with:
assumes "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
shows "homotopic_with P X Y p q \<longleftrightarrow>
(\<exists>h. continuous_map (prod_topology (subtopology euclideanreal {0..1}) X) Y h \<and>
(\<forall>x \<in> topspace X. h(0,x) = p x) \<and>
(\<forall>x \<in> topspace 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> topspace X then h(u,v) else if u = 0 then p v else q v" in exI)
apply simp
by (smt (verit, best) SigmaE assms case_prod_conv continuous_map_eq topspace_prod_topology)
lemma homotopic_with_mono:
assumes hom: "homotopic_with P X Y f g"
and Q: "\<And>h. \<lbrakk>continuous_map X Y h; P h\<rbrakk> \<Longrightarrow> Q h"
shows "homotopic_with Q X Y f g"
using hom unfolding homotopic_with_def
by (force simp: o_def dest: continuous_map_o_Pair intro: Q)
lemma homotopic_with_imp_continuous_maps:
assumes "homotopic_with P X Y f g"
shows "continuous_map X Y f \<and> continuous_map X Y g"
proof -
obtain h :: "real \<times> 'a \<Rightarrow> 'b"
where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) Y 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_map X Y (h \<circ> (\<lambda>x. (t,x)))" for t
by (rule continuous_map_compose [OF _ conth]) (simp add: o_def continuous_map_pairwise)
show ?thesis
using h *[of 0] *[of 1] by (simp add: continuous_map_eq)
qed
lemma homotopic_with_imp_continuous:
assumes "homotopic_with_canon P X Y f g"
shows "continuous_on X f \<and> continuous_on X g"
by (meson assms continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
lemma homotopic_with_imp_property:
assumes "homotopic_with P X Y f g"
shows "P f \<and> P g"
proof
obtain h where h: "\<And>x. h(0, x) = f x" "\<And>x. h(1, x) = g x" and P: "\<And>t. t \<in> {0..1::real} \<Longrightarrow> P(\<lambda>x. h(t,x))"
using assms by (force simp: homotopic_with_def)
show "P f" "P g"
using P [of 0] P [of 1] by (force simp: h)+
qed
lemma homotopic_with_equal:
assumes "P f" "P g" and contf: "continuous_map X Y f" and fg: "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x"
shows "homotopic_with P X Y f g"
unfolding homotopic_with_def
proof (intro exI conjI allI ballI)
let ?h = "\<lambda>(t::real,x). if t = 1 then g x else f x"
show "continuous_map (prod_topology (top_of_set {0..1}) X) Y ?h"
proof (rule continuous_map_eq)
show "continuous_map (prod_topology (top_of_set {0..1}) X) Y (f \<circ> snd)"
by (simp add: contf continuous_map_of_snd)
qed (auto simp: fg)
show "P (\<lambda>x. ?h (t, x))" if "t \<in> {0..1}" for t
by (cases "t = 1") (simp_all add: assms)
qed auto
lemma homotopic_with_imp_subset1:
"homotopic_with_canon P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
lemma homotopic_with_imp_subset2:
"homotopic_with_canon P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
lemma homotopic_with_subset_left:
"\<lbrakk>homotopic_with_canon P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with_canon P Z Y f g"
unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)
lemma homotopic_with_subset_right:
"\<lbrakk>homotopic_with_canon P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with_canon P X Z f g"
unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)
subsection\<open>Homotopy with P is an equivalence relation\<close>
text \<open>(on continuous functions mapping X into Y that satisfy P, though this only affects reflexivity)\<close>
lemma homotopic_with_refl [simp]: "homotopic_with P X Y f f \<longleftrightarrow> continuous_map X Y f \<and> P f"
by (metis homotopic_with_equal homotopic_with_imp_continuous_maps homotopic_with_imp_property)
lemma homotopic_with_symD:
assumes "homotopic_with P X Y f g"
shows "homotopic_with P X Y g f"
proof -
let ?I01 = "subtopology euclideanreal {0..1}"
let ?j = "\<lambda>y. (1 - fst y, snd y)"
have 1: "continuous_map (prod_topology ?I01 X) (prod_topology euclideanreal X) ?j"
by (intro continuous_intros; simp add: continuous_map_subtopology_fst prod_topology_subtopology)
have *: "continuous_map (prod_topology ?I01 X) (prod_topology ?I01 X) ?j"
proof -
have "continuous_map (prod_topology ?I01 X) (subtopology (prod_topology euclideanreal X) ({0..1} \<times> topspace X)) ?j"
by (simp add: continuous_map_into_subtopology [OF 1] image_subset_iff)
then show ?thesis
by (simp add: prod_topology_subtopology(1))
qed
show ?thesis
using assms
apply (clarsimp simp: homotopic_with_def)
subgoal for h
by (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI) (simp add: continuous_map_compose [OF *])
done
qed
lemma homotopic_with_sym:
"homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
by (metis homotopic_with_symD)
proposition homotopic_with_trans:
assumes "homotopic_with P X Y f g" "homotopic_with P X Y g h"
shows "homotopic_with P X Y f h"
proof -
let ?X01 = "prod_topology (subtopology euclideanreal {0..1}) X"
obtain k1 k2
where contk1: "continuous_map ?X01 Y k1" and contk2: "continuous_map ?X01 Y k2"
and k12: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
"\<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))"
using assms by (auto simp: homotopic_with_def)
define k where "k \<equiv> \<lambda>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"
have keq: "k1 (2 * u, v) = k2 (2 * u -1, v)" if "u = 1/2" for u v
by (simp add: k12 that)
show ?thesis
unfolding homotopic_with_def
proof (intro exI conjI)
show "continuous_map ?X01 Y k"
unfolding k_def
proof (rule continuous_map_cases_le)
show fst: "continuous_map ?X01 euclideanreal fst"
using continuous_map_fst continuous_map_in_subtopology by blast
show "continuous_map ?X01 euclideanreal (\<lambda>x. 1/2)"
by simp
show "continuous_map (subtopology ?X01 {y \<in> topspace ?X01. fst y \<le> 1/2}) Y
(k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x)))"
apply (intro fst continuous_map_compose [OF _ contk1] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
by (force simp: prod_topology_subtopology)
show "continuous_map (subtopology ?X01 {y \<in> topspace ?X01. 1/2 \<le> fst y}) Y
(k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x)))"
apply (intro fst continuous_map_compose [OF _ contk2] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
by (force simp: prod_topology_subtopology)
show "(k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y = (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y"
if "y \<in> topspace ?X01" and "fst y = 1/2" for y
using that by (simp add: keq)
qed
show "\<forall>x. k (0, x) = f x"
by (simp add: k12 k_def)
show "\<forall>x. k (1, x) = h x"
by (simp add: k12 k_def)
show "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
proof
fix t show "t\<in>{0..1} \<Longrightarrow> P (\<lambda>x. k (t, x))"
by (cases "t \<le> 1/2") (auto simp: k_def P)
qed
qed
qed
lemma homotopic_with_id2:
"(\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x) \<Longrightarrow> homotopic_with (\<lambda>x. True) X X (g \<circ> f) id"
by (metis comp_apply continuous_map_id eq_id_iff homotopic_with_equal homotopic_with_symD)
subsection\<open>Continuity lemmas\<close>
lemma homotopic_with_compose_continuous_map_left:
"\<lbrakk>homotopic_with p X1 X2 f g; continuous_map X2 X3 h; \<And>j. p j \<Longrightarrow> q(h \<circ> j)\<rbrakk>
\<Longrightarrow> homotopic_with q X1 X3 (h \<circ> f) (h \<circ> g)"
unfolding homotopic_with_def
apply clarify
subgoal for k
by (rule_tac x="h \<circ> k" in exI) (rule conjI continuous_map_compose | simp add: o_def)+
done
lemma homotopic_with_compose_continuous_map_right:
assumes hom: "homotopic_with p X2 X3 f g" and conth: "continuous_map X1 X2 h"
and q: "\<And>j. p j \<Longrightarrow> q(j \<circ> h)"
shows "homotopic_with q X1 X3 (f \<circ> h) (g \<circ> h)"
proof -
obtain k
where contk: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) X3 k"
and k: "\<forall>x. k (0, x) = f x" "\<forall>x. k (1, x) = g x" and p: "\<And>t. t\<in>{0..1} \<Longrightarrow> p (\<lambda>x. k (t, x))"
using hom unfolding homotopic_with_def by blast
have hsnd: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X2 (h \<circ> snd)"
by (rule continuous_map_compose [OF continuous_map_snd conth])
let ?h = "k \<circ> (\<lambda>(t,x). (t,h x))"
show ?thesis
unfolding homotopic_with_def
proof (intro exI conjI allI ballI)
have "continuous_map (prod_topology (top_of_set {0..1}) X1)
(prod_topology (top_of_set {0..1::real}) X2) (\<lambda>(t, x). (t, h x))"
by (metis (mono_tags, lifting) case_prod_beta' comp_def continuous_map_eq continuous_map_fst continuous_map_pairedI hsnd)
then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X3 ?h"
by (intro conjI continuous_map_compose [OF _ contk])
show "q (\<lambda>x. ?h (t, x))" if "t \<in> {0..1}" for t
using q [OF p [OF that]] by (simp add: o_def)
qed (auto simp: k)
qed
corollary homotopic_compose:
assumes "homotopic_with (\<lambda>x. True) X Y f f'" "homotopic_with (\<lambda>x. True) Y Z g g'"
shows "homotopic_with (\<lambda>x. True) X Z (g \<circ> f) (g' \<circ> f')"
by (metis assms homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right homotopic_with_imp_continuous_maps homotopic_with_trans)
proposition homotopic_with_compose_continuous_right:
"\<lbrakk>homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
\<Longrightarrow> homotopic_with_canon p W Y (f \<circ> h) (g \<circ> h)"
by (simp add: homotopic_with_compose_continuous_map_right)
proposition homotopic_with_compose_continuous_left:
"\<lbrakk>homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
\<Longrightarrow> homotopic_with_canon p X Z (h \<circ> f) (h \<circ> g)"
by (simp add: homotopic_with_compose_continuous_map_left)
lemma homotopic_from_subtopology:
"homotopic_with P X X' f g \<Longrightarrow> homotopic_with P (subtopology X S) X' f g"
by (metis continuous_map_id_subt homotopic_with_compose_continuous_map_right o_id)
lemma homotopic_on_emptyI:
assumes "topspace X = {}" "P f" "P g"
shows "homotopic_with P X X' f g"
by (metis assms continuous_map_on_empty empty_iff homotopic_with_equal)
lemma homotopic_on_empty:
"topspace X = {} \<Longrightarrow> (homotopic_with P X X' f g \<longleftrightarrow> P f \<and> P g)"
using homotopic_on_emptyI homotopic_with_imp_property by metis
lemma homotopic_with_canon_on_empty [simp]: "homotopic_with_canon (\<lambda>x. True) {} t f g"
by (auto intro: homotopic_with_equal)
lemma homotopic_constant_maps:
"homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow>
topspace X = {} \<or> path_component_of X' a b" (is "?lhs = ?rhs")
proof (cases "topspace X = {}")
case False
then obtain c where c: "c \<in> topspace X"
by blast
have "\<exists>g. continuous_map (top_of_set {0..1::real}) X' g \<and> g 0 = a \<and> g 1 = b"
if "x \<in> topspace X" and hom: "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b)" for x
proof -
obtain h :: "real \<times> 'a \<Rightarrow> 'b"
where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X' h"
and h: "\<And>x. h (0, x) = a" "\<And>x. h (1, x) = b"
using hom by (auto simp: homotopic_with_def)
have cont: "continuous_map (top_of_set {0..1}) X' (h \<circ> (\<lambda>t. (t, c)))"
by (rule continuous_map_compose [OF _ conth] continuous_intros c | simp)+
then show ?thesis
by (force simp: h)
qed
moreover have "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. g 0) (\<lambda>x. g 1)"
if "x \<in> topspace X" "a = g 0" "b = g 1" "continuous_map (top_of_set {0..1}) X' g"
for x and g :: "real \<Rightarrow> 'b"
unfolding homotopic_with_def
by (force intro!: continuous_map_compose continuous_intros c that)
ultimately show ?thesis
using False by (auto simp: path_component_of_def pathin_def)
qed (simp add: homotopic_on_empty)
proposition homotopic_with_eq:
assumes h: "homotopic_with P X Y f g"
and f': "\<And>x. x \<in> topspace X \<Longrightarrow> f' x = f x"
and g': "\<And>x. x \<in> topspace X \<Longrightarrow> g' x = g x"
and P: "(\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> P h \<longleftrightarrow> P k)"
shows "homotopic_with P X Y f' g'"
by (smt (verit, ccfv_SIG) assms homotopic_with)
lemma homotopic_with_prod_topology:
assumes "homotopic_with p X1 Y1 f f'" and "homotopic_with q X2 Y2 g g'"
and r: "\<And>i j. \<lbrakk>p i; q j\<rbrakk> \<Longrightarrow> r(\<lambda>(x,y). (i x, j y))"
shows "homotopic_with r (prod_topology X1 X2) (prod_topology Y1 Y2)
(\<lambda>z. (f(fst z),g(snd z))) (\<lambda>z. (f'(fst z), g'(snd z)))"
proof -
obtain h
where h: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) Y1 h"
and h0: "\<And>x. h (0, x) = f x"
and h1: "\<And>x. h (1, x) = f' x"
and p: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p (\<lambda>x. h (t,x))"
using assms unfolding homotopic_with_def by auto
obtain k
where k: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) Y2 k"
and k0: "\<And>x. k (0, x) = g x"
and k1: "\<And>x. k (1, x) = g' x"
and q: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> q (\<lambda>x. k (t,x))"
using assms unfolding homotopic_with_def by auto
let ?hk = "\<lambda>(t,x,y). (h(t,x), k(t,y))"
show ?thesis
unfolding homotopic_with_def
proof (intro conjI allI exI)
show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (prod_topology X1 X2))
(prod_topology Y1 Y2) ?hk"
unfolding continuous_map_pairwise case_prod_unfold
by (rule conjI continuous_map_pairedI continuous_intros continuous_map_id [unfolded id_def]
continuous_map_fst_of [unfolded o_def] continuous_map_snd_of [unfolded o_def]
continuous_map_compose [OF _ h, unfolded o_def]
continuous_map_compose [OF _ k, unfolded o_def])+
next
fix x
show "?hk (0, x) = (f (fst x), g (snd x))" "?hk (1, x) = (f' (fst x), g' (snd x))"
by (auto simp: case_prod_beta h0 k0 h1 k1)
qed (auto simp: p q r)
qed
lemma homotopic_with_product_topology:
assumes ht: "\<And>i. i \<in> I \<Longrightarrow> homotopic_with (p i) (X i) (Y i) (f i) (g i)"
and pq: "\<And>h. (\<And>i. i \<in> I \<Longrightarrow> p i (h i)) \<Longrightarrow> q(\<lambda>x. (\<lambda>i\<in>I. h i (x i)))"
shows "homotopic_with q (product_topology X I) (product_topology Y I)
(\<lambda>z. (\<lambda>i\<in>I. (f i) (z i))) (\<lambda>z. (\<lambda>i\<in>I. (g i) (z i)))"
proof -
obtain h
where h: "\<And>i. i \<in> I \<Longrightarrow> continuous_map (prod_topology (subtopology euclideanreal {0..1}) (X i)) (Y i) (h i)"
and h0: "\<And>i x. i \<in> I \<Longrightarrow> h i (0, x) = f i x"
and h1: "\<And>i x. i \<in> I \<Longrightarrow> h i (1, x) = g i x"
and p: "\<And>i t. \<lbrakk>i \<in> I; t \<in> {0..1}\<rbrakk> \<Longrightarrow> p i (\<lambda>x. h i (t,x))"
using ht unfolding homotopic_with_def by metis
show ?thesis
unfolding homotopic_with_def
proof (intro conjI allI exI)
let ?h = "\<lambda>(t,z). \<lambda>i\<in>I. h i (t,z i)"
have "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
(Y i) (\<lambda>x. h i (fst x, snd x i))" if "i \<in> I" for i
proof -
have \<section>: "continuous_map (prod_topology (top_of_set {0..1}) (product_topology X I)) (X i) (\<lambda>x. snd x i)"
using continuous_map_componentwise continuous_map_snd that by fastforce
show ?thesis
unfolding continuous_map_pairwise case_prod_unfold
by (intro conjI that \<section> continuous_intros continuous_map_compose [OF _ h, unfolded o_def])
qed
then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
(product_topology Y I) ?h"
by (auto simp: continuous_map_componentwise case_prod_beta)
show "?h (0, x) = (\<lambda>i\<in>I. f i (x i))" "?h (1, x) = (\<lambda>i\<in>I. g i (x i))" for x
by (auto simp: case_prod_beta h0 h1)
show "\<forall>t\<in>{0..1}. q (\<lambda>x. ?h (t, x))"
by (force intro: p pq)
qed
qed
text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
lemma homotopic_triviality:
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_canon (\<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_canon (\<lambda>x. True) S T f (\<lambda>x. c)))"
(is "?lhs = ?rhs")
proof (cases "S = {} \<or> T = {}")
case True then show ?thesis
by (auto simp: homotopic_on_emptyI)
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_canon (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
by (simp add: LHS image_subset_iff that)
then show ?thesis
using False homotopic_constant_maps [of "top_of_set S" "top_of_set T" a b] by auto
qed
moreover
have "\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
using False LHS continuous_on_const that by blast
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_canon (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with_canon (\<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
with T have "path_component T c d"
by (metis False ex_in_conv homotopic_with_imp_subset2 image_subset_iff path_connected_component)
then have "homotopic_with_canon (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
by (simp add: homotopic_constant_maps)
with c d show "homotopic_with_canon (\<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\<^marker>\<open>tag important\<close> homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
where
"homotopic_paths S p q \<equiv>
homotopic_with_canon (\<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_maps path_def continuous_map_subtopology_eu 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 (metis (mono_tags) continuous_map_subtopology_eu homotopic_paths_def homotopic_with_imp_continuous_maps 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 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]:
assumes "homotopic_paths S p q" "homotopic_paths S q r"
shows "homotopic_paths S p r"
using assms homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart unfolding homotopic_paths_def
by (smt (verit, ccfv_SIG) homotopic_with_mono homotopic_with_trans)
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"
by (smt (verit, best) homotopic_paths homotopic_paths_refl)
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, opaque_lifting) 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"
using pips [unfolded path_image_def]
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)+
by (auto simp: fb0 fb1 pathstart_def pathfinish_def)
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"
unfolding homotopic_paths by fast
text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
lemma continuous_on_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))" (is "continuous_on ?A ?p")
and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))" (is "continuous_on ?A ?q")
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 \<section>: "(\<lambda>t. p (fst t) (2 * snd t)) = ?p \<circ> (\<lambda>y. (fst y, 2 * snd y))"
"(\<lambda>t. q (fst t) (2 * snd t - 1)) = ?q \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
by force+
show ?thesis
unfolding joinpaths_def
proof (rule continuous_on_cases_le)
show "continuous_on {y \<in> ?A. snd y \<le> 1/2} (\<lambda>t. p (fst t) (2 * snd t))"
"continuous_on {y \<in> ?A. 1/2 \<le> snd y} (\<lambda>t. q (fst t) (2 * snd t - 1))"
"continuous_on ?A snd"
unfolding \<section>
by (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
qed (use pf in \<open>auto simp: mult.commute pathstart_def pathfinish_def\<close>)
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 (clarsimp simp: homotopic_paths_def homotopic_with_def)
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 (intro conjI continuous_intros continuous_on_homotopic_join_lemma; force 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
by (simp add: homotopic_with_compose_continuous_map_left pathfinish_compose pathstart_compose)
subsection\<open>Group properties for homotopy of paths\<close>
text\<^marker>\<open>tag important\<close>\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
proposition homotopic_paths_rid:
assumes "path p" "path_image p \<subseteq> S"
shows "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p)) p"
proof -
have \<section>: "continuous_on {0..1} (\<lambda>t::real. if t \<le> 1/2 then 2 *\<^sub>R t else 1)"
unfolding split_01
by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
show ?thesis
apply (rule homotopic_paths_sym)
using assms unfolding pathfinish_def joinpaths_def
by (intro \<section> continuous_on_cases continuous_intros homotopic_paths_reparametrize [where f = "\<lambda>t. if t \<le> 1/2 then 2 *\<^sub>R t else 1"]; force)
qed
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 add: subset_path_image_join)
apply (rule continuous_on_cases_1 continuous_intros | 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 p: "continuous_on {0..1} p"
using assms by (auto simp: path_def)
let ?A = "{0..1} \<times> {0..1}"
have "continuous_on ?A (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
unfolding joinpaths_def subpath_def reversepath_def path_def add_0_right diff_0_right
proof (rule continuous_on_cases_le)
show "continuous_on {x \<in> ?A. snd x \<le> 1/2} (\<lambda>t. p (fst t * (2 * snd t)))"
"continuous_on {x \<in> ?A. 1/2 \<le> snd x} (\<lambda>t. p (fst t * (1 - (2 * snd t - 1))))"
"continuous_on ?A snd"
by (intro continuous_on_compose2 [OF p] continuous_intros; auto simp: mult_le_one)+
qed (auto simp: algebra_simps)
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 (force simp: mult_le_one path_defs joinpaths_def subpath_def reversepath_def)
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\<^marker>\<open>tag important\<close> 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_canon (\<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_maps continuous_map_subtopology_eu 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 (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
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 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 (fastforce simp add: homotopic_loops)
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 path_image_def path_def pathstart_def pathfinish_def
by (auto intro: homotopic_with_eq [OF homotopic_with_refl [where f = p, THEN iffD2]])
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
by (simp add: homotopic_with_compose_continuous_map_left 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_with_def homotopic_paths_def homotopic_loops_def)
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)
let ?A = "{0..1::real} \<times> {0..1::real}"
obtain h where conth: "continuous_on ?A h"
and hs: "h ` ?A \<subseteq> S"
and h0[simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
and h1[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
proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
show "continuous_on ((\<lambda>x. (x, 0)) ` {0..1}) h"
by (force intro: continuous_on_subset [OF conth])
qed (force intro: continuous_intros)
have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> S"
using hs by (force simp: path_image_def)
have c1: "continuous_on ?A (\<lambda>x. h (fst x * snd x, 0))"
proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
show "continuous_on ((\<lambda>x. (fst x * snd x, 0)) ` ?A) h"
by (force simp: mult_le_one intro: continuous_on_subset [OF conth])
qed (force intro: continuous_intros)+
have c2: "continuous_on ?A (\<lambda>x. h (fst x - fst x * snd x, 0))"
proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
show "continuous_on ((\<lambda>x. (fst x - fst x * snd x, 0)) ` ?A) h"
by (auto simp: algebra_simps add_increasing2 mult_left_le intro: continuous_on_subset [OF conth])
qed (force intro: continuous_intros)
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. \<lbrakk>path p \<and> path(reversepath p);
path_image p \<subseteq> S \<and> path_image(reversepath p) \<subseteq> S;
pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
pathstart(reversepath p) = a \<and> pathstart p = x\<rbrakk>
\<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))"
using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" S]
by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_sym pathstart_join)
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)))"
unfolding homotopic_paths_def homotopic_with_def
proof (intro exI strip conjI)
let ?h = "\<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)"
have "continuous_on ?A ?h"
by (intro continuous_on_homotopic_join_lemma; simp add: path_defs joinpaths_def subpath_def conth c1 c2)
moreover have "?h ` ?A \<subseteq> S"
unfolding joinpaths_def subpath_def
by (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
ultimately show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1}))
(top_of_set S) ?h"
by (simp add: subpath_reversepath)
qed (use ploop in \<open>simp_all add: reversepath_def path_defs joinpaths_def o_def subpath_def conth c1 c2\<close>)
moreover have "homotopic_paths S ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
(linepath (pathstart p) (pathstart p))"
by (rule *; simp add: pih0 pathstart_def pathfinish_def conth0; simp add: reversepath_def joinpaths_def)
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 pq: "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
let ?A = "{0..1::real} \<times> {0..1::real}"
have c1: "continuous_on ?A (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
show "continuous_on ((\<lambda>x. (1 - fst x) * snd x + fst x) ` ?A) p"
by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
qed (force intro: continuous_intros)
have c2: "continuous_on ?A (\<lambda>x. p ((fst x - 1) * snd x + 1))"
proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
show "continuous_on ((\<lambda>x. (fst x - 1) * snd x + 1) ` ?A) p"
by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_left_le_one_le)
qed (force intro: continuous_intros)
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 affine_ineq linear mult.commute mult.left_neutral mult_right_mono
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))"
unfolding homotopic_loops_def homotopic_with_def
proof (intro exI strip conjI)
let ?h = "(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))"
have "continuous_on ?A (\<lambda>y. q (snd y))"
by (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
then have "continuous_on ?A ?h"
using pq qloop
by (intro continuous_on_homotopic_join_lemma) (auto simp: path_defs joinpaths_def subpath_def c1 c2)
then show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set S) ?h"
by (auto simp: joinpaths_def subpath_def ps1 ps2 qs)
show "?h (1,x) = (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) x" for x
using pq by (simp add: pathfinish_def subpath_refl)
qed (auto simp: subpath_reversepath)
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"
by (smt (verit, best) \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_lid
homotopic_paths_rid homotopic_paths_trans pathstart_join piq qloop)
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
obtain 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)
ultimately show ?thesis
by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid homotopic_paths_trans path_join_path_ends pathfinish_linepath pipq(2))
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\<^marker>\<open>tag unimportant\<close>\<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_canon (\<lambda>z. True) S t f g"
unfolding 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)
using sub closed_segment_def
by (fastforce intro: continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
continuous_on_subset [OF contg] continuous_on_compose2 [where g=g])
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_defs 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)
by (force simp: closed_segment_def intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
lemma homotopic_paths_nearby_explicit:
assumes \<section>: "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"
using homotopic_paths_linear [OF \<section>] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)
lemma homotopic_loops_nearby_explicit:
assumes \<section>: "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"
using homotopic_loops_linear [OF \<section>] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)
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
using e [unfolded dist_norm] \<open>e > 0\<close>
by (fastforce simp: path_image_def intro!: homotopic_paths_nearby_explicit assms exI)
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
using e [unfolded dist_norm] \<open>e > 0\<close>
by (fastforce simp: path_image_def intro!: homotopic_loops_nearby_explicit assms exI)
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
have "homotopic_paths (path_image g) (subpath u v g +++ subpath v w g) (subpath u w g)"
proof (cases "w = u")
case True
then show ?thesis
by (metis \<open>path g\<close> homotopic_paths_rinv path_image_subpath_subset path_subpath pathstart_subpath reversepath_subpath subpath_refl u v)
next
case False
let ?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))"
show ?thesis
proof (rule homotopic_paths_sym [OF homotopic_paths_reparametrize [where f = ?f]])
show "path (subpath u w g)"
using assms(1) path_subpath u w(1) by blast
show "path_image (subpath u w g) \<subseteq> path_image g"
by (meson path_image_subpath_subset u w(1))
show "continuous_on {0..1} ?f"
unfolding split_01
by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def dest!: t2)+
show "?f ` {0..1} \<subseteq> {0..1}"
using False assms
by (force simp: field_simps not_le mult_left_mono affine_ineq dest!: 1 2)
show "(subpath u v g +++ subpath v w g) t = subpath u w g (?f t)" if "t \<in> {0..1}" for t
using assms
unfolding joinpaths_def subpath_def by (auto simp: divide_simps add.commute mult.commute mult.left_commute)
qed (use False in auto)
qed
then show ?thesis
by (rule homotopic_paths_subset [OF _ pag])
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 -
obtain wvg: "path (subpath w v g)" "path_image (subpath w v g) \<subseteq> S"
and wug: "path (subpath w u g)" "path_image (subpath w u g) \<subseteq> S"
and vug: "path (subpath v u g)" "path_image (subpath v u g) \<subseteq> S"
by (meson \<open>path g\<close> pag path_image_subpath_subset path_subpath subset_trans u v w)
have "homotopic_paths S (subpath u w g +++ subpath w v g)
((subpath u v g +++ subpath v w g) +++ subpath w v g)"
by (simp add: hom homotopic_paths_join homotopic_paths_sym wvg)
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)"
using wvg vug \<open>path g\<close>
by (metis homotopic_paths_assoc homotopic_paths_sym path_image_subpath_commute path_subpath
pathfinish_subpath pathstart_subpath u v w)
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)))"
using wvg vug \<open>path g\<close>
by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl path_image_subpath_commute
path_subpath pathfinish_subpath pathstart_join pathstart_subpath reversepath_subpath u v)
also have "homotopic_paths S (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
using vug \<open>path g\<close> by (metis homotopic_paths_rid path_image_subpath_commute path_subpath u v)
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)"
by (smt (verit, del_insts) homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3)
text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
lemma path_component_imp_homotopic_points:
assumes "path_component S a b"
shows "homotopic_loops S (linepath a a) (linepath b b)"
proof -
obtain g :: "real \<Rightarrow> 'a" where g: "continuous_on {0..1} g" "g ` {0..1} \<subseteq> S" "g 0 = a" "g 1 = b"
using assms by (auto simp: path_defs)
then have "continuous_on ({0..1} \<times> {0..1}) (g \<circ> fst)"
by (fastforce intro!: continuous_intros)+
with g show ?thesis
by (auto simp: homotopic_loops_def homotopic_with_def path_defs image_subset_iff)
qed
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 (clarsimp simp: homotopic_loops_def homotopic_with_def path_defs)
apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
apply (fastforce elim!: continuous_on_subset intro!: continuous_intros)
done
lemma homotopic_points_eq_path_component:
"homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow> path_component S a b"
using homotopic_loops_imp_path_component_value path_component_imp_homotopic_points by fastforce
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\<^marker>\<open>tag important\<close>\<open>defined as "all loops are homotopic (as loops)\<close>
definition\<^marker>\<open>tag important\<close> 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))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
unfolding simply_connected_def by force
next
assume ?rhs then show ?lhs
unfolding simply_connected_def
by (metis pathfinish_in_path_image subsetD homotopic_loops_trans homotopic_loops_sym)
qed
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)))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
using simply_connected_eq_contractible_loop_any by (blast intro: simply_connected_imp_path_connected)
next
assume ?rhs
then show ?lhs
by (meson homotopic_loops_trans path_connected_eq_homotopic_points simply_connected_eq_contractible_loop_any)
qed
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
then show ?thesis
by (meson False homotopic_loops_sym homotopic_loops_trans simply_connected_def simply_connected_eq_contractible_loop_any)
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)))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding simply_connected_imp_path_connected
by (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
next
assume ?rhs
then show ?lhs
using homotopic_paths_imp_homotopic_loops simply_connected_eq_contractible_loop_some by fastforce
qed
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 +++ reversepath q +++ q)"
using that
by (smt (verit, best) homotopic_paths_join homotopic_paths_linv homotopic_paths_rid homotopic_paths_sym
homotopic_paths_trans 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_assoc 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)"
by (simp add: continuous_on_fst Path_Connected.path_continuous_image [OF \<open>path p\<close>])
moreover have "path_image (fst \<circ> p) \<subseteq> S"
using that by (force simp add: path_image_def)
ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
using S that
by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
have "path (snd \<circ> p)"
by (simp add: continuous_on_snd Path_Connected.path_continuous_image [OF \<open>path p\<close>])
moreover have "path_image (snd \<circ> p) \<subseteq> T"
using that by (force simp: path_image_def)
ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
using T that
by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
show ?thesis
using p1 p2 unfolding homotopic_loops
apply clarify
subgoal for h k
by (rule_tac x="\<lambda>z. (h z, k z)" in exI) (force intro: continuous_intros simp: path_defs)
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\<^marker>\<open>tag important\<close> contractible where
"contractible S \<equiv> \<exists>a. homotopic_with_canon (\<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_canon (\<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_in_topspace topspace_euclidean_subtopology)
have "\<forall>p. path p \<and>
path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow>
homotopic_loops S p (linepath a a)"
using a apply (clarsimp simp: homotopic_loops_def homotopic_with_def path_defs)
apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
apply (intro conjI continuous_on_compose continuous_intros; force elim: continuous_on_subset)
done
with \<open>a \<in> S\<close> show ?thesis
by (auto simp: simply_connected_eq_contractible_loop_all False)
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_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
proof -
obtain b where b: "homotopic_with_canon (\<lambda>x. True) T T id (\<lambda>x. b)"
using assms by (force simp: contractible_def)
have "homotopic_with_canon (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
by (metis Abstract_Topology.continuous_map_subtopology_eu b g homotopic_with_compose_continuous_map_left)
then have "homotopic_with_canon (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
by (simp add: f homotopic_with_compose_continuous_map_right)
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_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
by (rule nullhomotopic_through_contractible [OF f, of id T]) (use assms in auto)
lemma nullhomotopic_from_contractible:
assumes f: "continuous_on S f" "f ` S \<subseteq> T"
and S: "contractible S"
obtains c where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
by (auto simp: comp_def intro: nullhomotopic_through_contractible [OF continuous_on_id _ f S])
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_canon (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
proof -
obtain c1 where c1: "homotopic_with_canon (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
by (rule nullhomotopic_through_contractible [of S f1 T g1 U]) (use assms in auto)
obtain c2 where c2: "homotopic_with_canon (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
by (rule nullhomotopic_through_contractible [of S f2 T g2 U]) (use assms in auto)
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_continuous_maps by fastforce+
with \<open>path_connected U\<close> show ?thesis by blast
qed
then have "homotopic_with_canon (\<lambda>h. True) S U (\<lambda>x. c2) (\<lambda>x. c1)"
by (simp add: path_component homotopic_constant_maps)
then show ?thesis
using c1 c2 homotopic_with_symD homotopic_with_trans by blast
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_canon (\<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)
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_canon (\<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)
subsection\<open>Starlike sets\<close>
definition\<^marker>\<open>tag important\<close> "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
lemma starlike_UNIV [simp]: "starlike UNIV"
by (simp add: starlike_def)
lemma convex_imp_starlike:
"convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
unfolding convex_contains_segment starlike_def by auto
lemma starlike_convex_tweak_boundary_points:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
shows "starlike T"
proof -
have "rel_interior S \<noteq> {}"
by (simp add: assms rel_interior_eq_empty)
with ST obtain a where a: "a \<in> rel_interior S" and "a \<in> T" by blast
have "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
by (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a]) (use assms in auto)
then have "\<forall>x\<in>T. a \<in> T \<and> open_segment a x \<subseteq> T"
using ST by (blast intro: a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
then show ?thesis
unfolding starlike_def using bexI [OF _ \<open>a \<in> T\<close>]
by (simp add: closed_segment_eq_open)
qed
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_canon 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 "\<And>t b. 0 \<le> t \<and> t \<le> 1 \<Longrightarrow>
\<exists>u. (1 - t) *\<^sub>R b + t *\<^sub>R a = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
by (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
then have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
using a [unfolded closed_segment_def] by force
then have "homotopic_with_canon P S S (\<lambda>x. x) (\<lambda>x. a)"
using \<open>a \<in> S\<close>
unfolding homotopic_with_def
apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
apply (force simp add: P intro: continuous_intros)
done
then show ?thesis
using that by blast
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"
by (meson convex_imp_simply_connected is_interval_connected_1 is_interval_convex_1 simply_connected_imp_connected)
lemma contractible_empty [simp]: "contractible {}"
by (simp add: contractible_def homotopic_on_emptyI)
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"
by (metis assms closure_eq_empty contractible_empty empty_subsetI
starlike_convex_tweak_boundary_points starlike_imp_contractible subset_antisym)
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)))"])
using hsub ksub
apply (fastforce intro!: continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
done
qed
subsection\<open>Local versions of topological properties in general\<close>
definition\<^marker>\<open>tag important\<close> locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
where
"locally P S \<equiv>
\<forall>w x. openin (top_of_set S) w \<and> x \<in> w
\<longrightarrow> (\<exists>u v. openin (top_of_set 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 (top_of_set S) w; x \<in> w\<rbrakk>
\<Longrightarrow> \<exists>u v. openin (top_of_set 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 (top_of_set S) w" "x \<in> w"
obtains u v where "openin (top_of_set 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 (top_of_set S) t"
shows "locally P t"
by (smt (verit, ccfv_SIG) assms order.trans locally_def openin_imp_subset openin_subset_trans openin_trans)
lemma locally_diff_closed:
"\<lbrakk>locally P S; closedin (top_of_set 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}"
proof -
have "\<forall>x::real. \<not> 0 < x \<Longrightarrow> P {a}"
using zero_less_one by blast
then show ?thesis
unfolding locally_def
by (auto simp: openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR)
qed
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)))"
by (smt (verit) locally_def openin_open)
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)"
unfolding locally_iff
proof clarify
fix A x
assume "open A" "x \<in> A" "x \<in> S" "x \<in> T"
then obtain U1 V1 U2 V2
where "open U1" "P V1" "x \<in> S \<inter> U1" "S \<inter> U1 \<subseteq> V1 \<and> V1 \<subseteq> S \<inter> A"
"open U2" "P V2" "x \<in> T \<inter> U2" "T \<inter> U2 \<subseteq> V2 \<and> V2 \<subseteq> T \<inter> A"
using S T unfolding locally_iff by (meson IntI)
then have "S \<inter> T \<inter> (U1 \<inter> U2) \<subseteq> V1 \<inter> V2" "V1 \<inter> V2 \<subseteq> S \<inter> T \<inter> A" "x \<in> S \<inter> T \<inter> (U1 \<inter> U2)"
by blast+
moreover have "P (V1 \<inter> V2)"
by (simp add: P \<open>P V1\<close> \<open>P V2\<close>)
ultimately show "\<exists>U. open U \<and> (\<exists>V. P V \<and> x \<in> S \<inter> T \<inter> U \<and> S \<inter> T \<inter> U \<subseteq> V \<and> V \<subseteq> S \<inter> T \<inter> A)"
using \<open>open U1\<close> \<open>open U2\<close> by blast
qed
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 (top_of_set (S \<times> T)) W" and xy: "(x, y) \<in> W"
then obtain U V where "openin (top_of_set S) U" "x \<in> U"
"openin (top_of_set 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 (top_of_set S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
and opeT: "openin (top_of_set T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
by (meson PS QT locallyE)
then have "openin (top_of_set (S \<times> T)) (U1 \<times> V1)"
by (simp add: openin_Times)
moreover have "R (U2 \<times> V2)"
by (simp add: R opeS opeT)
moreover have "U1 \<times> V1 \<subseteq> U2 \<times> V2 \<and> U2 \<times> V2 \<subseteq> W"
using opeS opeT \<open>U \<times> V \<subseteq> W\<close> by auto
ultimately show "\<exists>U V. openin (top_of_set (S \<times> T)) U \<and> R V \<and> (x,y) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
using opeS opeT by auto
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 (top_of_set T) W"
then obtain A where T: "open A" "W = T \<inter> A"
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> g by force
have "openin (top_of_set S) (g ` W)"
using \<open>openin (top_of_set T) W\<close> continuous_on_open f gw by auto
then obtain U V
where osu: "openin (top_of_set S) U" and uv: "P V" "g y \<in> U" "U \<subseteq> V" "V \<subseteq> g ` W"
by (metis S \<open>y \<in> W\<close> image_eqI locallyE)
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 contvf: "continuous_on V f"
using \<open>V \<subseteq> S\<close> continuous_on_subset f(3) by blast
have "f ` V \<subseteq> W"
using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
then have contvg: "continuous_on (f ` V) g"
using \<open>W \<subseteq> T\<close> continuous_on_subset [OF g(3)] by blast
have "V \<subseteq> g ` f ` V"
by (metis \<open>V \<subseteq> S\<close> hom homeomorphism_def homeomorphism_of_subsets order_refl)
then have homv: "homeomorphism V (f ` V) f g"
using \<open>V \<subseteq> S\<close> f by (auto simp: homeomorphism_def contvf contvg)
have "openin (top_of_set (g ` T)) U"
using \<open>g ` T = S\<close> by (simp add: osu)
then have "openin (top_of_set T) (T \<inter> g -` U)"
using \<open>continuous_on T g\<close> continuous_on_open [THEN iffD1] by blast
moreover have "\<exists>V. Q V \<and> y \<in> (T \<inter> g -` U) \<and> (T \<inter> g -` U) \<subseteq> V \<and> V \<subseteq> W"
proof (intro exI conjI)
show "Q (f ` V)"
using Q homv \<open>P V\<close> by blast
show "y \<in> T \<inter> g -` U"
using T(2) \<open>y \<in> W\<close> \<open>g y \<in> U\<close> by blast
show "T \<inter> g -` U \<subseteq> f ` V"
using g(1) image_iff uv(3) by fastforce
show "f ` V \<subseteq> W"
using \<open>f ` V \<subseteq> W\<close> by blast
qed
ultimately show "\<exists>U. openin (top_of_set T) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
by meson
qed
lemma homeomorphism_locally:
fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
assumes "homeomorphism S T f g"
and "\<And>S T. homeomorphism S T f g \<Longrightarrow> (P S \<longleftrightarrow> Q T)"
shows "locally P S \<longleftrightarrow> locally Q T"
by (smt (verit) assms homeomorphism_locally_imp homeomorphism_symD)
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"
by (smt (verit, ccfv_SIG) hom homeomorphic_def homeomorphism_locally homeomorphism_locally_imp iff)
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 ((+) a ` S) = P S) \<Longrightarrow> locally P ((+) a ` S) = locally P S"
using homeomorphism_locally [OF homeomorphism_translation]
by (metis (full_types) homeomorphism_image2)
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"
using homeomorphism_locally [of "f`S" _ _ f] linear_homeomorphism_image [OF f]
by (metis (no_types, lifting) homeomorphism_image2 iff)
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 (top_of_set S) T \<Longrightarrow> openin (top_of_set (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: locally_def)
fix W y
assume oiw: "openin (top_of_set (f ` S)) W" and "y \<in> W"
then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
have oivf: "openin (top_of_set 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 (top_of_set S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
by (metis IntI P \<open>y \<in> W\<close> locallyE oivf vimageI)
then have "openin (top_of_set (f ` S)) (f ` U)"
by (simp add: oo)
then show "\<exists>X. openin (top_of_set (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
using Q \<open>P V\<close> \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S \<inter> f -` W\<close> \<open>f x = y\<close> \<open>x \<in> U\<close> by blast
qed
subsection\<open>An induction principle for connected sets\<close>
proposition connected_induction:
assumes "connected S"
and opD: "\<And>T a. \<lbrakk>openin (top_of_set 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 (top_of_set 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 -
let ?A = "{b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
let ?B = "{b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
have "?A \<inter> ?B = {}"
by (clarsimp simp: set_eq_iff) (metis (no_types, opaque_lifting) Int_iff opD openin_Int)
moreover have "S \<subseteq> ?A \<union> ?B"
by clarsimp (meson opI)
moreover have "openin (top_of_set S) ?A"
by (subst openin_subopen, blast)
moreover have "openin (top_of_set S) ?B"
by (subst openin_subopen, blast)
ultimately have "?A = {} \<or> ?B = {}"
by (metis (no_types, lifting) \<open>connected S\<close> connected_openin)
then show ?thesis
by clarsimp (meson opI etc)
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 (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and> a \<in> T \<and>
(\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
shows "P b"
by (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"])
(use opI etc in auto)
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 (top_of_set 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 (top_of_set 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:
assumes "connected S"
shows "locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (smt (verit, del_insts) assms constant_on_def locally_constant_imp_constant locally_def openin_subtopology_self subset_iff)
next
assume ?rhs then show ?lhs
by (metis constant_on_subset locallyI openin_imp_subset order_refl)
qed
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 (top_of_set S) u \<and> compact v)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (meson locallyE openin_subtopology_self)
next
assume r [rule_format]: ?rhs
have *: "\<exists>u v.
openin (top_of_set 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 (top_of_set 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
by (rule locallyI) (metis "*" Int_iff openin_open)
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 (top_of_set 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 (top_of_set S) U \<and> compact(closure U) \<and> closure U \<subseteq> S)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (meson bounded_subset closure_minimal compact_closure compact_imp_bounded
compact_imp_closed dual_order.trans locally_compactE)
next
assume ?rhs then show ?lhs
by (meson closure_subset locally_compact)
qed
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 L: ?lhs
then have "\<And>x U V e. \<lbrakk>U \<subseteq> V; V \<subseteq> S; compact V; 0 < e; cball x e \<inter> S \<subseteq> U\<rbrakk>
\<Longrightarrow> closed (cball x e \<inter> S)"
by (metis compact_Int compact_cball compact_imp_closed inf.absorb_iff2 inf.assoc inf.orderE)
with L show ?rhs
by (meson locally_compactE openin_contains_cball)
next
assume R: ?rhs
show ?lhs unfolding locally_compact
proof
fix x
assume "x \<in> S"
then obtain e where "e>0" and "compact (cball x e \<inter> S)"
by (metis Int_commute compact_Int_closed compact_cball inf.right_idem R)
moreover have "\<forall>y\<in>ball x e \<inter> S. \<exists>\<epsilon>>0. cball y \<epsilon> \<inter> S \<subseteq> ball x e"
by (meson Elementary_Metric_Spaces.open_ball IntD1 le_infI1 open_contains_cball_eq)
moreover have "openin (top_of_set S) (ball x e \<inter> S)"
by (simp add: inf_commute openin_open_Int)
ultimately show "\<exists>U V. x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> S \<and> openin (top_of_set S) U \<and> compact V"
by (metis Int_iff \<open>0 < e\<close> \<open>x \<in> S\<close> ball_subset_cball centre_in_ball inf_commute inf_le1 inf_mono order_refl)
qed
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 (top_of_set 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 (top_of_set 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 (top_of_set S) U \<and> compact V"
if "K \<subseteq> S" "compact K" for K
proof -
have "\<And>C. (\<forall>c\<in>C. openin (top_of_set 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 (top_of_set 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 dest!: uv)
moreover
have "openin (top_of_set S) (\<Union> (u ` T))"
using T that uv by fastforce
moreover
obtain "compact (\<Union> (v ` T))" "\<Union> (v ` T) \<subseteq> S"
by (metis T UN_subset_iff \<open>K \<subseteq> S\<close> compact_UN subset_iff uv)
ultimately show ?thesis
using T by auto
qed
show ?rhs
by (blast intro: *)
next
assume ?rhs
then show ?lhs
apply (clarsimp simp: 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 (top_of_set 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 (top_of_set S) (ball x e)"
by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
show ?thesis
by (meson \<open>0 < e\<close> ball_subset_cball centre_in_ball compact_cball e ope)
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 (top_of_set S) U \<and> compact V"
if "x \<in> S" for x
apply (rule_tac x = "S \<inter> ball x 1" in exI, rule_tac x = "S \<inter> cball x 1" in exI)
using \<open>x \<in> S\<close> assms by auto
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 (top_of_set 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
unfolding locally_def
apply (elim 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 (top_of_set (S \<union> T)) S"
and opT: "openin (top_of_set (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 have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
by (metis (no_types, lifting) cball_min_Int closed_Int closed_cball inf_assoc inf_commute)
then show ?thesis
by (metis \<open>0 < e1\<close> \<open>0 < e2\<close> min_def)
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
moreover have "closed (cball x e1 \<inter> (cball x e2 \<inter> T))"
by (metis closed_Int closed_cball e1 inf_left_commute)
ultimately show ?thesis
by (rule_tac x="min e1 e2" in exI) (simp add: \<open>0 < e2\<close> cball_min_Int inf_assoc)
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 (top_of_set (S \<union> T)) S"
and clT: "closedin (top_of_set (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
moreover have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
by (smt (verit) Int_Un_distrib2 Int_commute cball_min_Int closed_Int closed_Un closed_cball e1 e2 inf_left_commute)
ultimately show ?thesis
by (rule_tac x="min e1 e2" in exI) linarith
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
with e1 have "closed ((cball x e1 \<inter> cball x e2) \<inter> (S \<union> T))"
apply (simp add: inf_commute inf_sup_distrib2)
by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
then have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
by (simp add: cball_min_Int inf_commute)
ultimately show ?thesis
using \<open>0 < e2\<close> by (rule_tac x="min e1 e2" in exI) linarith
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)
with e1 have "closed ((cball x e1 \<inter> cball x e2) \<inter> (S \<union> T))"
apply (simp add: inf_commute inf_sup_distrib2)
by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
then have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
by (auto simp: cball_min_Int)
ultimately show ?thesis
using \<open>0 < e2\<close> by (rule_tac x="min e1 e2" in exI) linarith
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 (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and>
closedin (top_of_set 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 (top_of_set (\<Inter>?\<T>)) K1" and
K2: "closedin (top_of_set (\<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 (top_of_set S) D"
and cloD: "closedin (top_of_set 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 (top_of_set S) (\<Inter>\<F>)"
using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
then show "closedin (top_of_set 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 (top_of_set D) (D \<inter> closure V1)"
and cloV2: "closedin (top_of_set 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"
using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
by (auto simp: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
ultimately have cloDV1: "closedin (top_of_set D) (D \<inter> V1)"
and cloDV2: "closedin (top_of_set 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 1: "openin (top_of_set S) (D \<inter> V2)"
by (simp add: \<open>open V2\<close> opeD openin_Int_open)
have 2: "closedin (top_of_set S) (D \<inter> V2)"
using cloD cloDV2 closedin_trans by blast
have "\<Inter> ?\<T> \<subseteq> D \<inter> V2"
by (rule Inter_lower) (use * 1 2 in simp)
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 1: "openin (top_of_set S) (D \<inter> V1)"
by (simp add: \<open>open V1\<close> opeD openin_Int_open)
have 2: "closedin (top_of_set S) (D \<inter> V1)"
using cloD cloDV1 closedin_trans by blast
have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
by (rule Inter_lower) (use * 1 2 in simp)
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> [unfolded connected_closed, simplified, rule_format, of concl: "D \<inter> U1" "D \<inter> U2"]
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 (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
proof (rule ccontr)
assume "\<not> thesis"
with that have neg: "\<nexists>K. openin (top_of_set 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 (top_of_set S) V"
using S U \<open>compact C\<close> by (meson C in_components_subset locally_compact_compact_subopen)
let ?\<T> = "{T. C \<subseteq> T \<and> openin (top_of_set 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 (top_of_set K) T \<and> closedin (top_of_set 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 (top_of_set K) (\<Inter>\<F>)"
using False \<F> \<open>finite \<F>\<close> by blast
then have opeVF: "openin (top_of_set 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 (top_of_set 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 (top_of_set S) U" and "C \<subseteq> U"
obtains K where "openin (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set S) V; x \<in> V\<rbrakk> \<Longrightarrow> \<exists>U. openin (top_of_set S) U \<and> connected U \<and> x \<in> U \<and> U \<subseteq> V"
shows "locally connected S"
by (metis assms locally_def)
lemma locally_connected_2:
assumes "locally connected S"
"openin (top_of_set S) t"
"x \<in> t"
shows "openin (top_of_set S) (connected_component_set t x)"
proof -
{ fix y :: 'a
let ?SS = "top_of_set 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 (top_of_set S) t; x \<in> t\<rbrakk>
\<Longrightarrow> openin (top_of_set S)
(connected_component_set t x)"
"openin (top_of_set S) v" "x \<in> v"
shows "\<exists>u. openin (top_of_set 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 (top_of_set S) v \<and> x \<in> v
\<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) t \<and> x \<in> t
\<longrightarrow> openin (top_of_set 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 (top_of_set S) v; x \<in> v\<rbrakk>
\<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
shows "locally path_connected S"
by (force simp add: locally_def dest: assms)
lemma locally_path_connected_2:
assumes "locally path_connected S"
"openin (top_of_set S) t"
"x \<in> t"
shows "openin (top_of_set S) (path_component_set t x)"
proof -
{ fix y :: 'a
let ?SS = "top_of_set 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 (top_of_set S) t; x \<in> t\<rbrakk>
\<Longrightarrow> openin (top_of_set S) (path_component_set t x)"
"openin (top_of_set S) v" "x \<in> v"
shows "\<exists>u. openin (top_of_set 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 mem_Collect_eq path_component_subset path_connected_path_component)
qed
proposition locally_path_connected:
"locally path_connected S \<longleftrightarrow>
(\<forall>V x. openin (top_of_set S) V \<and> x \<in> V
\<longrightarrow> (\<exists>U. openin (top_of_set 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 (top_of_set S) t \<and> x \<in> t
\<longrightarrow> openin (top_of_set 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 (top_of_set S) t \<and> c \<in> components t
\<longrightarrow> openin (top_of_set S) c)"
by (metis components_iff locally_connected_open_connected_component)
proposition locally_connected_im_kleinen:
"locally connected S \<longleftrightarrow>
(\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
\<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> c"
if "openin (top_of_set 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 (top_of_set 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
unfolding locally_connected_open_component by (meson "*" openin_subopen)
qed
proposition locally_path_connected_im_kleinen:
"locally path_connected S \<longleftrightarrow>
(\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
\<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) T \<and>
x \<in> T \<and> T \<subseteq> path_component_set u z"
if "openin (top_of_set 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 (top_of_set 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
by (metis U c mem_Collect_eq path_component_def path_component_eq subsetI)
qed
show ?lhs
unfolding locally_path_connected_open_path_component
using "*" openin_subopen by fastforce
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"
assumes "open S"
shows "locally path_connected S"
proof (rule locally_mono)
show "locally convex S"
using assms unfolding locally_def
by (meson open_ball centre_in_ball convex_ball openE open_subset openin_imp_subset openin_open_trans subset_trans)
show "\<And>T::'a set. convex T \<Longrightarrow> path_connected T"
using convex_imp_path_connected by blast
qed
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 (top_of_set S) (connected_component_set S x)"
by (metis connected_component_eq_empty locally_connected_2 openin_empty openin_subtopology_self)
lemma openin_components_locally_connected:
"\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (top_of_set 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 (top_of_set 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:
assumes "locally path_connected S"
shows "closedin (top_of_set S) (path_component_set S x)"
proof -
have "openin (top_of_set S) (\<Union> ({path_component_set S y |y. y \<in> S} - {path_component_set S x}))"
using locally_path_connected_2 assms by fastforce
then show ?thesis
by (simp add: closedin_def path_component_subset complement_path_component_Union)
qed
lemma convex_imp_locally_path_connected:
fixes S :: "'a:: real_normed_vector set"
assumes "convex S"
shows "locally path_connected S"
proof (clarsimp simp: locally_path_connected)
fix V x
assume "openin (top_of_set S) V" and "x \<in> V"
then obtain T e where "V = S \<inter> T" "x \<in> S" "0 < e" "ball x e \<subseteq> T"
by (metis Int_iff openE openin_open)
then have "openin (top_of_set S) (S \<inter> ball x e)" "path_connected (S \<inter> ball x e)"
by (simp_all add: assms convex_Int convex_imp_path_connected openin_open_Int)
then show "\<exists>U. openin (top_of_set S) U \<and> path_connected U \<and> x \<in> U \<and> U \<subseteq> V"
using \<open>0 < e\<close> \<open>V = S \<inter> T\<close> \<open>ball x e \<subseteq> T\<close> \<open>x \<in> S\<close> by auto
qed
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 (top_of_set (connected_component_set S x)) (path_component_set S x)"
proof (rule openin_subset_trans)
show "openin (top_of_set S) (path_component_set S x)"
by (simp add: True assms locally_path_connected_2)
show "connected_component_set S x \<subseteq> S"
by (simp add: connected_component_subset)
qed (simp add: path_component_subset_connected_component)
moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
proof (rule closedin_subset_trans [of S])
show "closedin (top_of_set S) (path_component_set S x)"
by (simp add: assms closedin_path_component_locally_path_connected)
show "connected_component_set S x \<subseteq> S"
by (simp add: connected_component_subset)
qed (simp add: path_component_subset_connected_component)
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 (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
openin (top_of_set (f ` S)) T"
shows "locally connected (f ` S)"
proof (clarsimp simp: locally_connected_open_component)
fix U C
assume opefSU: "openin (top_of_set (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 (top_of_set (f ` S)) C \<longleftrightarrow>
openin (top_of_set S) (S \<inter> f -` C)"
by (auto simp: oo)
moreover have "openin (top_of_set S) (S \<inter> f -` C)"
proof (subst openin_subopen, clarify)
fix x
assume "x \<in> S" "f x \<in> C"
show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
proof (intro conjI exI)
show "openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
proof (rule ccontr)
assume **: "\<not> openin (top_of_set 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"
by (meson connected_component_subset continuous_on_subset inf.boundedE)
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 (top_of_set (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 (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow> openin (top_of_set (f ` S)) T"
shows "locally path_connected (f ` S)"
proof (clarsimp simp: locally_path_connected_open_path_component)
fix U y
assume opefSU: "openin (top_of_set (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 (top_of_set (f ` S)) (path_component_set U y) \<longleftrightarrow>
openin (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
proof (intro conjI exI)
show "openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
proof (rule ccontr)
assume **: "\<not> openin (top_of_set 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"
by (meson Int_lower1 continuous_on_subset path_component_subset)
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 (top_of_set (f ` S)) (path_component_set U y)"
by metis
qed
subsection\<^marker>\<open>tag unimportant\<close>\<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 (top_of_set 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 (top_of_set 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"
proof (rule continuous_on_components_gen)
fix C
assume "C \<in> components S"
then show "openin (top_of_set S) C \<and> continuous_on C f"
by (simp add: assms openin_components_locally_connected)
qed
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 (top_of_set U) S"
and cuS: "c \<subseteq> components(U - S)"
shows "closedin (top_of_set 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 di 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 (top_of_set U) (\<Union>(components (U - S) - c))"
proof (rule openin_Union [OF openin_trans [of "U - S"]])
show "openin (top_of_set (U - S)) T" if "T \<in> components (U - S) - c" for T
using that by (simp add: U S locally_diff_closed openin_components_locally_connected)
show "openin (top_of_set U) (U - S)" if "T \<in> components (U - S) - c" for T
using that by (simp add: openin_diff S)
qed
have "closedin (top_of_set U) (U - \<Union> (components (U - S) - c))"
by (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
then have "openin (top_of_set U) (U - (U - \<Union>(components (U - S) - c)))"
by (simp add: openin_diff)
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 (top_of_set UNIV) (S \<union> \<Union>c)"
by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_union_complement_components locally_connected_UNIV subtopology_UNIV)
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 (top_of_set u) S"
and c: " c \<in> components(u - S)"
shows "closedin (top_of_set u) (S \<union> c)"
proof -
have "closedin (top_of_set 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 unfolding pairwise_def inj_on_def
by (blast intro: orthogonal_clauses)
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)"
proof (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
show "pairwise (\<lambda>v j. orthogonal (a v *\<^sub>R fb v) (a j *\<^sub>R fb j)) B"
by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
qed
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 unfolding pairwise_def inj_on_def bij_betw_def
by (blast intro: orthogonal_clauses)
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 \<open>bij_betw fb B C\<close> unfolding pairwise_def bij_betw_def by force
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)"
proof (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
show "pairwise (\<lambda>v j. orthogonal (a v *\<^sub>R fb v) (a j *\<^sub>R fb j)) B"
by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
qed
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)"
proof (rule sum.cong [OF refl])
show "a x *\<^sub>R g (fb x) = a x *\<^sub>R x" if "x \<in> B" for x
using that \<open>bij_betw fb B C\<close> bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
qed
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
by (rule that [OF \<open>linear f\<close> \<open>linear g\<close>]) (simp_all add: fim gim)
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
unfolding homeomorphic_def homeomorphism_def
apply (rule_tac x=f in exI, 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 (fact homeomorphic_translation)
also have "\<dots> homeomorphic ((+) (- b) ` T)"
by (rule homeomorphic_subspaces [OF ss dd])
also have "\<dots> homeomorphic T"
using homeomorphic_translation [of T "- b"] by (simp add: homeomorphic_sym [of T])
finally show ?thesis .
qed
subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
locale\<^marker>\<open>tag important\<close> 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_canon 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_canon Q U t f g"
proof -
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_canon P U S (k \<circ> f) (k \<circ> g)"
by (rule hom)
then have "homotopic_with_canon 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
proof (rule homotopic_with_eq; simp)
show "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
using Qeq topspace_euclidean_subtopology by blast
show "\<And>x. x \<in> U \<Longrightarrow> f x = h (k (f x))" "\<And>x. x \<in> U \<Longrightarrow> g x = h (k (g x))"
using idhk imf img by auto
qed
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_canon 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_canon 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_canon P U S (k \<circ> f) (\<lambda>x. c)"
by (metis hom)
then have "homotopic_with_canon 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 have "homotopic_with_canon Q U t f (\<lambda>x. h c)"
proof (rule homotopic_with_eq)
show "\<And>x. x \<in> topspace (top_of_set U) \<Longrightarrow> f x = (h \<circ> (k \<circ> f)) x"
using feq by auto
show "\<And>h k. (\<And>x. x \<in> topspace (top_of_set U) \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
using Qeq topspace_euclidean_subtopology by blast
qed auto
then show ?thesis
using that by blast
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_canon 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_canon 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_canon P S U (f \<circ> h) (g \<circ> h)"
by (rule hom)
then have "homotopic_with_canon 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
proof (rule homotopic_with_eq)
show "f x = (f \<circ> h \<circ> k) x" "g x = (g \<circ> h \<circ> k) x"
if "x \<in> topspace (top_of_set t)" for x
using feq geq that by force+
qed (use Qeq topspace_euclidean_subtopology in blast)
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_canon 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_canon 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_canon P S U (f \<circ> h) (\<lambda>x. c)"
by (metis hom)
then have \<section>: "homotopic_with_canon Q t U (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
proof (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
show "\<And>h. \<lbrakk>continuous_map (top_of_set S) (top_of_set U) h; P h\<rbrakk> \<Longrightarrow> Q (h \<circ> k)"
using Q by auto
qed (auto simp: contk imk)
moreover have "homotopic_with_canon Q t U f (\<lambda>x. c)"
using homotopic_with_eq [OF \<section>] feq Qeq by fastforce
ultimately show ?thesis
using that by blast
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>
subsection\<open>Homotopy equivalence of topological spaces.\<close>
definition\<^marker>\<open>tag important\<close> homotopy_equivalent_space
(infix "homotopy'_equivalent'_space" 50)
where "X homotopy_equivalent_space Y \<equiv>
(\<exists>f g. continuous_map X Y f \<and>
continuous_map Y X g \<and>
homotopic_with (\<lambda>x. True) X X (g \<circ> f) id \<and>
homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id)"
lemma homeomorphic_imp_homotopy_equivalent_space:
"X homeomorphic_space Y \<Longrightarrow> X homotopy_equivalent_space Y"
unfolding homeomorphic_space_def homotopy_equivalent_space_def
apply (erule ex_forward)+
by (simp add: homotopic_with_equal homotopic_with_sym homeomorphic_maps_def)
lemma homotopy_equivalent_space_refl:
"X homotopy_equivalent_space X"
by (simp add: homeomorphic_imp_homotopy_equivalent_space homeomorphic_space_refl)
lemma homotopy_equivalent_space_sym:
"X homotopy_equivalent_space Y \<longleftrightarrow> Y homotopy_equivalent_space X"
by (meson homotopy_equivalent_space_def)
lemma homotopy_eqv_trans [trans]:
assumes 1: "X homotopy_equivalent_space Y" and 2: "Y homotopy_equivalent_space U"
shows "X homotopy_equivalent_space U"
proof -
obtain f1 g1 where f1: "continuous_map X Y f1"
and g1: "continuous_map Y X g1"
and hom1: "homotopic_with (\<lambda>x. True) X X (g1 \<circ> f1) id"
"homotopic_with (\<lambda>x. True) Y Y (f1 \<circ> g1) id"
using 1 by (auto simp: homotopy_equivalent_space_def)
obtain f2 g2 where f2: "continuous_map Y U f2"
and g2: "continuous_map U Y g2"
and hom2: "homotopic_with (\<lambda>x. True) Y Y (g2 \<circ> f2) id"
"homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
using 2 by (auto simp: homotopy_equivalent_space_def)
have "homotopic_with (\<lambda>f. True) X Y (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
using f1 hom2(1) homotopic_with_compose_continuous_map_right by metis
then have "homotopic_with (\<lambda>f. True) X Y (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
by (simp add: o_assoc)
then have "homotopic_with (\<lambda>x. True) X X
(g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
by (simp add: g1 homotopic_with_compose_continuous_map_left)
moreover have "homotopic_with (\<lambda>x. True) X X (g1 \<circ> id \<circ> f1) id"
using hom1 by simp
ultimately have SS: "homotopic_with (\<lambda>x. True) X X (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
by (metis comp_assoc homotopic_with_trans id_comp)
have "homotopic_with (\<lambda>f. True) U Y (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
using g2 hom1(2) homotopic_with_compose_continuous_map_right by fastforce
then have "homotopic_with (\<lambda>f. True) U Y (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_map_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"
by (simp add: fun.map_comp hom2(2) homotopic_with_trans)
show ?thesis
unfolding homotopy_equivalent_space_def
by (blast intro: f1 f2 g1 g2 continuous_map_compose SS UU)
qed
lemma deformation_retraction_imp_homotopy_equivalent_space:
"\<lbrakk>homotopic_with (\<lambda>x. True) X X (S \<circ> r) id; retraction_maps X Y r S\<rbrakk>
\<Longrightarrow> X homotopy_equivalent_space Y"
unfolding homotopy_equivalent_space_def retraction_maps_def
using homotopic_with_id2 by fastforce
lemma deformation_retract_imp_homotopy_equivalent_space:
"\<lbrakk>homotopic_with (\<lambda>x. True) X X r id; retraction_maps X Y r id\<rbrakk>
\<Longrightarrow> X homotopy_equivalent_space Y"
using deformation_retraction_imp_homotopy_equivalent_space by force
lemma deformation_retract_of_space:
"S \<subseteq> topspace X \<and>
(\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id) \<longleftrightarrow>
S retract_of_space X \<and> (\<exists>f. homotopic_with (\<lambda>x. True) X X id f \<and> f ` (topspace X) \<subseteq> S)"
proof (cases "S \<subseteq> topspace X")
case True
moreover have "(\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id)
\<longleftrightarrow> (S retract_of_space X \<and> (\<exists>f. homotopic_with (\<lambda>x. True) X X id f \<and> f ` topspace X \<subseteq> S))"
unfolding retract_of_space_def
proof safe
fix f r
assume f: "homotopic_with (\<lambda>x. True) X X id f"
and fS: "f ` topspace X \<subseteq> S"
and r: "continuous_map X (subtopology X S) r"
and req: "\<forall>x\<in>S. r x = x"
show "\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id"
proof (intro exI conjI)
have "homotopic_with (\<lambda>x. True) X X f r"
proof (rule homotopic_with_eq)
show "homotopic_with (\<lambda>x. True) X X (r \<circ> f) (r \<circ> id)"
by (metis continuous_map_into_fulltopology f homotopic_with_compose_continuous_map_left homotopic_with_symD r)
show "f x = (r \<circ> f) x" if "x \<in> topspace X" for x
using that fS req by auto
qed auto
then show "homotopic_with (\<lambda>x. True) X X id r"
by (rule homotopic_with_trans [OF f])
next
show "retraction_maps X (subtopology X S) r id"
by (simp add: r req retraction_maps_def)
qed
qed (use True in \<open>auto simp: retraction_maps_def topspace_subtopology_subset continuous_map_in_subtopology\<close>)
ultimately show ?thesis by simp
qed (auto simp: retract_of_space_def retraction_maps_def)
subsection\<open>Contractible spaces\<close>
text\<open>The definition (which agrees with "contractible" on subsets of Euclidean space)
is a little cryptic because we don't in fact assume that the constant "a" is in the space.
This forces the convention that the empty space / set is contractible, avoiding some special cases. \<close>
definition contractible_space where
"contractible_space X \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
lemma contractible_space_top_of_set [simp]:"contractible_space (top_of_set S) \<longleftrightarrow> contractible S"
by (auto simp: contractible_space_def contractible_def)
lemma contractible_space_empty:
"topspace X = {} \<Longrightarrow> contractible_space X"
unfolding contractible_space_def homotopic_with_def
apply (rule_tac x=undefined in exI)
apply (rule_tac x="\<lambda>(t,x). if t = 0 then x else undefined" in exI)
apply (auto simp: continuous_map_on_empty)
done
lemma contractible_space_singleton:
"topspace X = {a} \<Longrightarrow> contractible_space X"
unfolding contractible_space_def homotopic_with_def
apply (rule_tac x=a in exI)
apply (rule_tac x="\<lambda>(t,x). if t = 0 then x else a" in exI)
apply (auto intro: continuous_map_eq [where f = "\<lambda>z. a"])
done
lemma contractible_space_subset_singleton:
"topspace X \<subseteq> {a} \<Longrightarrow> contractible_space X"
by (meson contractible_space_empty contractible_space_singleton subset_singletonD)
lemma contractible_space_subtopology_singleton:
"contractible_space(subtopology X {a})"
by (meson contractible_space_subset_singleton insert_subset path_connectedin_singleton path_connectedin_subtopology subsetI)
lemma contractible_space:
"contractible_space X \<longleftrightarrow>
topspace X = {} \<or>
(\<exists>a \<in> topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))"
proof (cases "topspace X = {}")
case False
then show ?thesis
using homotopic_with_imp_continuous_maps by (fastforce simp: contractible_space_def)
qed (simp add: contractible_space_empty)
lemma contractible_imp_path_connected_space:
assumes "contractible_space X" shows "path_connected_space X"
proof (cases "topspace X = {}")
case False
have *: "path_connected_space X"
if "a \<in> topspace X" and conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X h"
and h: "\<forall>x. h (0, x) = x" "\<forall>x. h (1, x) = a"
for a and h :: "real \<times> 'a \<Rightarrow> 'a"
proof -
have "path_component_of X b a" if "b \<in> topspace X" for b
unfolding path_component_of_def
proof (intro exI conjI)
let ?g = "h \<circ> (\<lambda>x. (x,b))"
show "pathin X ?g"
unfolding pathin_def
proof (rule continuous_map_compose [OF _ conth])
show "continuous_map (top_of_set {0..1}) (prod_topology (top_of_set {0..1}) X) (\<lambda>x. (x, b))"
using that by (auto intro!: continuous_intros)
qed
qed (use h in auto)
then show ?thesis
by (metis path_component_of_equiv path_connected_space_iff_path_component)
qed
show ?thesis
using assms False by (auto simp: contractible_space homotopic_with_def *)
qed (simp add: path_connected_space_topspace_empty)
lemma contractible_imp_connected_space:
"contractible_space X \<Longrightarrow> connected_space X"
by (simp add: contractible_imp_path_connected_space path_connected_imp_connected_space)
lemma contractible_space_alt:
"contractible_space X \<longleftrightarrow> (\<forall>a \<in> topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))" (is "?lhs = ?rhs")
proof
assume X: ?lhs
then obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
by (auto simp: contractible_space_def)
show ?rhs
proof
show "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. b)" if "b \<in> topspace X" for b
proof (rule homotopic_with_trans [OF a])
show "homotopic_with (\<lambda>x. True) X X (\<lambda>x. a) (\<lambda>x. b)"
using homotopic_constant_maps path_connected_space_imp_path_component_of
by (metis (full_types) X a continuous_map_const contractible_imp_path_connected_space homotopic_with_imp_continuous_maps that)
qed
qed
next
assume R: ?rhs
then show ?lhs
unfolding contractible_space_def by (metis equals0I homotopic_on_emptyI)
qed
lemma compose_const [simp]: "f \<circ> (\<lambda>x. a) = (\<lambda>x. f a)" "(\<lambda>x. a) \<circ> g = (\<lambda>x. a)"
by (simp_all add: o_def)
lemma nullhomotopic_through_contractible_space:
assumes f: "continuous_map X Y f" and g: "continuous_map Y Z g" and Y: "contractible_space Y"
obtains c where "homotopic_with (\<lambda>h. True) X Z (g \<circ> f) (\<lambda>x. c)"
proof -
obtain b where b: "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. b)"
using Y by (auto simp: contractible_space_def)
show thesis
using homotopic_with_compose_continuous_map_right
[OF homotopic_with_compose_continuous_map_left [OF b g] f]
by (force simp add: that)
qed
lemma nullhomotopic_into_contractible_space:
assumes f: "continuous_map X Y f" and Y: "contractible_space Y"
obtains c where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
using nullhomotopic_through_contractible_space [OF f _ Y]
by (metis continuous_map_id id_comp)
lemma nullhomotopic_from_contractible_space:
assumes f: "continuous_map X Y f" and X: "contractible_space X"
obtains c where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
using nullhomotopic_through_contractible_space [OF _ f X]
by (metis comp_id continuous_map_id)
lemma homotopy_dominated_contractibility:
assumes f: "continuous_map X Y f" and g: "continuous_map Y X g"
and hom: "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id" and X: "contractible_space X"
shows "contractible_space Y"
proof -
obtain c where c: "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
using nullhomotopic_from_contractible_space [OF f X] .
have "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) (\<lambda>x. c)"
using homotopic_with_compose_continuous_map_right [OF c g] by fastforce
then have "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. c)"
using homotopic_with_trans [OF _ hom] homotopic_with_symD by blast
then show ?thesis
unfolding contractible_space_def ..
qed
lemma homotopy_equivalent_space_contractibility:
"X homotopy_equivalent_space Y \<Longrightarrow> (contractible_space X \<longleftrightarrow> contractible_space Y)"
unfolding homotopy_equivalent_space_def
by (blast intro: homotopy_dominated_contractibility)
lemma homeomorphic_space_contractibility:
"X homeomorphic_space Y
\<Longrightarrow> (contractible_space X \<longleftrightarrow> contractible_space Y)"
by (simp add: homeomorphic_imp_homotopy_equivalent_space homotopy_equivalent_space_contractibility)
lemma contractible_eq_homotopy_equivalent_singleton_subtopology:
"contractible_space X \<longleftrightarrow>
topspace X = {} \<or> (\<exists>a \<in> topspace X. X homotopy_equivalent_space (subtopology X {a}))"(is "?lhs = ?rhs")
proof (cases "topspace X = {}")
case False
show ?thesis
proof
assume ?lhs
then obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
by (auto simp: contractible_space_def)
then have "a \<in> topspace X"
by (metis False continuous_map_const homotopic_with_imp_continuous_maps)
then have "homotopic_with (\<lambda>x. True) (subtopology X {a}) (subtopology X {a}) id (\<lambda>x. a)"
using connectedin_absolute connectedin_sing contractible_space_alt contractible_space_subtopology_singleton by fastforce
then have "X homotopy_equivalent_space subtopology X {a}"
unfolding homotopy_equivalent_space_def using \<open>a \<in> topspace X\<close>
by (metis (full_types) a comp_id continuous_map_const continuous_map_id_subt empty_subsetI homotopic_with_symD
id_comp insertI1 insert_subset topspace_subtopology_subset)
with \<open>a \<in> topspace X\<close> show ?rhs
by blast
next
assume ?rhs
then show ?lhs
by (meson False contractible_space_subtopology_singleton homotopy_equivalent_space_contractibility)
qed
qed (simp add: contractible_space_empty)
lemma contractible_space_retraction_map_image:
assumes "retraction_map X Y f" and X: "contractible_space X"
shows "contractible_space Y"
proof -
obtain g where f: "continuous_map X Y f" and g: "continuous_map Y X g" and fg: "\<forall>y \<in> topspace Y. f(g y) = y"
using assms by (auto simp: retraction_map_def retraction_maps_def)
obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
using X by (auto simp: contractible_space_def)
have "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. f a)"
proof (rule homotopic_with_eq)
show "homotopic_with (\<lambda>x. True) Y Y (f \<circ> id \<circ> g) (f \<circ> (\<lambda>x. a) \<circ> g)"
using f g a homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right by metis
qed (use fg in auto)
then show ?thesis
unfolding contractible_space_def by blast
qed
lemma contractible_space_prod_topology:
"contractible_space(prod_topology X Y) \<longleftrightarrow>
topspace X = {} \<or> topspace Y = {} \<or> contractible_space X \<and> contractible_space Y"
proof (cases "topspace X = {} \<or> topspace Y = {}")
case True
then have "topspace (prod_topology X Y) = {}"
by simp
then show ?thesis
by (auto simp: contractible_space_empty)
next
case False
have "contractible_space(prod_topology X Y) \<longleftrightarrow> contractible_space X \<and> contractible_space Y"
proof safe
assume XY: "contractible_space (prod_topology X Y)"
with False have "retraction_map (prod_topology X Y) X fst"
by (auto simp: contractible_space False retraction_map_fst)
then show "contractible_space X"
by (rule contractible_space_retraction_map_image [OF _ XY])
have "retraction_map (prod_topology X Y) Y snd"
using False XY by (auto simp: contractible_space False retraction_map_snd)
then show "contractible_space Y"
by (rule contractible_space_retraction_map_image [OF _ XY])
next
assume "contractible_space X" and "contractible_space Y"
with False obtain a b
where "a \<in> topspace X" and a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
and "b \<in> topspace Y" and b: "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. b)"
by (auto simp: contractible_space)
with False show "contractible_space (prod_topology X Y)"
apply (simp add: contractible_space)
apply (rule_tac x=a in bexI)
apply (rule_tac x=b in bexI)
using homotopic_with_prod_topology [OF a b]
apply (metis (no_types, lifting) case_prod_Pair case_prod_beta' eq_id_iff)
apply auto
done
qed
with False show ?thesis
by auto
qed
lemma contractible_space_product_topology:
"contractible_space(product_topology X I) \<longleftrightarrow>
topspace (product_topology X I) = {} \<or> (\<forall>i \<in> I. contractible_space(X i))"
proof (cases "topspace (product_topology X I) = {}")
case False
have 1: "contractible_space (X i)"
if XI: "contractible_space (product_topology X I)" and "i \<in> I"
for i
proof (rule contractible_space_retraction_map_image [OF _ XI])
show "retraction_map (product_topology X I) (X i) (\<lambda>x. x i)"
using False by (simp add: retraction_map_product_projection \<open>i \<in> I\<close>)
qed
have 2: "contractible_space (product_topology X I)"
if "x \<in> topspace (product_topology X I)" and cs: "\<forall>i\<in>I. contractible_space (X i)"
for x :: "'a \<Rightarrow> 'b"
proof -
obtain f where f: "\<And>i. i\<in>I \<Longrightarrow> homotopic_with (\<lambda>x. True) (X i) (X i) id (\<lambda>x. f i)"
using cs unfolding contractible_space_def by metis
have "homotopic_with (\<lambda>x. True)
(product_topology X I) (product_topology X I) id (\<lambda>x. restrict f I)"
by (rule homotopic_with_eq [OF homotopic_with_product_topology [OF f]]) (auto)
then show ?thesis
by (auto simp: contractible_space_def)
qed
show ?thesis
using False 1 2 by blast
qed (simp add: contractible_space_empty)
lemma contractible_space_subtopology_euclideanreal [simp]:
"contractible_space(subtopology euclideanreal S) \<longleftrightarrow> is_interval S"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "path_connectedin (subtopology euclideanreal S) S"
using contractible_imp_path_connected_space path_connectedin_topspace path_connectedin_absolute
by (simp add: contractible_imp_path_connected)
then show ?rhs
by (simp add: is_interval_path_connected_1)
next
assume ?rhs
then have "convex S"
by (simp add: is_interval_convex_1)
show ?lhs
proof (cases "S = {}")
case False
then obtain z where "z \<in> S"
by blast
show ?thesis
unfolding contractible_space_def homotopic_with_def
proof (intro exI conjI allI)
note \<section> = convexD [OF \<open>convex S\<close>, simplified]
show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set S)
(\<lambda>(t,x). (1 - t) * x + t * z)"
using \<open>z \<in> S\<close>
by (auto simp: case_prod_unfold intro!: continuous_intros \<section>)
qed auto
qed (simp add: contractible_space_empty)
qed
corollary contractible_space_euclideanreal: "contractible_space euclideanreal"
proof -
have "contractible_space (subtopology euclideanreal UNIV)"
using contractible_space_subtopology_euclideanreal by blast
then show ?thesis
by simp
qed
abbreviation\<^marker>\<open>tag important\<close> homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
(infix "homotopy'_eqv" 50)
where "S homotopy_eqv T \<equiv> top_of_set S homotopy_equivalent_space top_of_set T"
lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
unfolding homeomorphic_def homeomorphism_def homotopy_equivalent_space_def
by (metis continuous_map_subtopology_eu homotopic_with_id2 openin_imp_subset openin_subtopology_self topspace_euclidean_subtopology)
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"
by (metis assms homeomorphic_sym homeomorphic_imp_homotopy_eqv linear_homeomorphic_image)
lemma homotopy_eqv_translation:
fixes S :: "'a::real_normed_vector set"
shows "(+) a ` S homotopy_eqv S"
using homeomorphic_imp_homotopy_eqv 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_canon (\<lambda>x. True) U S f g"
shows "homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
"homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
using assms by (auto simp: homotopy_equivalent_space_def)
have "homotopic_with_canon (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
proof (rule homUS)
show "continuous_on U (k \<circ> f)" "continuous_on U (k \<circ> g)"
using continuous_on_compose continuous_on_subset f g k by blast+
qed (use f g k in \<open>(force simp: o_def)+\<close> )
then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
by (rule homotopic_with_compose_continuous_left; simp add: h)
moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom f)
moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom g)
ultimately show "homotopic_with_canon (\<lambda>x. True) U T f g"
unfolding o_assoc
by (metis homotopic_with_trans homotopic_with_sym id_comp)
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_canon (\<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_canon (\<lambda>x. True) U T f g)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (metis assms homotopy_eqv_homotopic_triviality_imp)
next
assume ?rhs
moreover
have "T homotopy_eqv S"
using assms homotopy_equivalent_space_sym by blast
ultimately show ?lhs
by (blast intro: homotopy_eqv_homotopic_triviality_imp)
qed
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_canon (\<lambda>x. True) S U f (\<lambda>x. c)"
obtains c where "homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
"homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
using assms by (auto simp: homotopy_equivalent_space_def)
obtain c where "homotopic_with_canon (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
proof (rule exE [OF homSU])
show "continuous_on S (f \<circ> h)"
using continuous_on_compose continuous_on_subset f h by blast
qed (use f h in force)
then have "homotopic_with_canon (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
by (rule homotopic_with_compose_continuous_right [where X=S]) (use k in auto)
moreover have "homotopic_with_canon (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
by (rule homotopic_with_compose_continuous_left [where Y=T])
(use f in \<open>auto simp: hom homotopic_with_symD\<close>)
ultimately show ?thesis
using that homotopic_with_trans by (fastforce simp add: o_def)
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_canon (\<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_canon (\<lambda>x. True) T U f (\<lambda>x. c)))"
by (rule iffI; metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_equivalent_space_sym)
text \<open>Similar to the proof above\<close>
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_canon (\<lambda>x. True) U S f (\<lambda>x. c)"
shows "\<exists>c. homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
"homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
using assms by (auto simp: homotopy_equivalent_space_def)
obtain c::'a where "homotopic_with_canon (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
proof (rule exE [OF homSU [of "k \<circ> f"]])
show "continuous_on U (k \<circ> f)"
using continuous_on_compose continuous_on_subset f k by blast
qed (use f k in force)
then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
by (rule homotopic_with_compose_continuous_left [where Y=S]) (use h in auto)
moreover have "homotopic_with_canon (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
by (rule homotopic_with_compose_continuous_right [where X=T])
(use f in \<open>auto simp: hom homotopic_with_symD\<close>)
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_canon (\<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_canon (\<lambda>x. True) U T f (\<lambda>x. c)))"
by (rule iffI; metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_equivalent_space_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_equivalent_space_def
apply (rule_tac x="\<lambda>x. b" in exI, rule_tac x="\<lambda>x. a" in exI)
apply (intro assms conjI continuous_on_id' homotopic_into_contractible; force)
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 = {}" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis continuous_map_subtopology_eu empty_iff equalityI homotopy_equivalent_space_def image_subset_iff subsetI)
qed (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 = {}"
using homotopy_equivalent_space_sym homotopy_eqv_empty1 by blast
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)"
by (meson contractible_space_top_of_set homotopy_equivalent_space_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 False then show ?thesis
by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
qed simp
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\<^marker>\<open>tag unimportant\<close>\<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
corollary bounded_path_connected_Compl_real:
fixes S :: "real set"
assumes "bounded S" "path_connected(- S)" shows "S = {}"
by (simp add: assms bounded_connected_Compl_real path_connected_imp_connected)
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 fg: "\<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 "bij f" by (metis fg bijI')
have "connected (f ` (-S))"
using connected_linear_image assms \<open>linear f\<close> by blast
moreover have "f ` (-S) = - (f ` S)"
by (simp add: \<open>bij f\<close> bij_image_Compl_eq)
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
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})"
by (metis connected_uncountable finite.emptyI finite.insertI rev_finite_subset singleton_iff subsetI uncountable_infinite)
lemma connected_finite_iff_sing:
fixes S :: "'a::metric_space set"
assumes "connected S"
shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
using assms connected_uncountable countable_finite by blast
subsection\<^marker>\<open>tag unimportant\<close>\<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: 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
by (rule the_equality [symmetric]) (use x in \<open>auto dest: **\<close>)
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]
by (simp add: closed_segment_commute 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: 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
moreover have "countable {a}"
by simp
ultimately show ?thesis
by (metis path_connected_convex_diff_countable [OF \<open>convex S\<close>])
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 "\<not> collinear (UNIV::'a set)"
using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
then have "path_connected(UNIV - S)"
by (simp add: \<open>countable S\<close> path_connected_convex_diff_countable)
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 (top_of_set (affine hull S)) S"
and "\<not> collinear S" "countable T"
shows "path_connected(S - T)"
proof (clarsimp simp: 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 (top_of_set S) U" and "x \<in> U" for U x
proof -
have "openin (top_of_set (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 (top_of_set 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> aff_dim (ball x r \<inter> affine hull S) \<le> 1"
by (metis (no_types, opaque_lifting) 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 "\<not> collinear (ball x r \<inter> affine hull S)"
by (simp add: collinear_aff_dim)
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 (top_of_set (affine hull S)) (ball x r \<inter> affine hull S)"
by (subst inf.commute) (simp add: openin_Int_open)
then show "openin (top_of_set 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 (top_of_set (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)
next
case False
show ?thesis
proof (rule path_connected_openin_diff_countable)
show "openin (top_of_set (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\<^marker>\<open>tag unimportant\<close> \<open>Self-homeomorphisms shuffling points about\<close>
subsubsection\<^marker>\<open>tag unimportant\<close>\<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: field_split_simps)
also have "\<dots> < norm y + (norm x - norm y) * 1"
proof (subst add_less_cancel_left)
show "(norm x - norm y) * (norm u / r) < (norm x - norm y) * 1"
proof (rule mult_strict_left_mono)
show "norm u / r < 1"
using \<open>0 < r\<close> divide_less_eq_1_pos nou by blast
qed (simp add: yx)
qed
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"
using * nou
apply (clarsimp simp: dist_norm norm_minus_commute f_def)
by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute)
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: 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
then have n: "norm (a - (x - v *\<^sub>R (u - a))) = r - r * v"
by (simp add: field_simps norm_minus_commute)
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) (simp add: field_simps)
have "x - v *\<^sub>R (u - a) \<in> cball a r"
using \<open>r > 0\<close>\<open>0 \<le> v\<close>
by (simp add: dist_norm n)
moreover have "x - v *\<^sub>R (u - a) \<in> T"
by (simp add: f_def \<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 "compact (cball a r \<inter> T)"
by (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
then obtain g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
by (metis homeomorphism_compact [OF _ contf fim inj_onf])
then show thesis
apply (rule_tac f=f in that)
using \<open>r > 0\<close> by (simp_all add: f_def dist_norm norm_minus_commute)
qed
corollary\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> 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"
unfolding ff_def
using homeomorphism_cont1 [OF hom]
by (intro continuous_on_cases) (auto simp: affine_closed \<open>affine T\<close> fid)
then show "continuous_on S ff"
by (rule continuous_on_subset) (use ST in auto)
have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
unfolding gg_def
using homeomorphism_cont2 [OF hom]
by (intro continuous_on_cases) (auto simp: affine_closed \<open>affine T\<close> gid)
then show "continuous_on S gg"
by (rule continuous_on_subset) (use ST in auto)
show "ff ` S \<subseteq> S"
proof (clarsimp simp: 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: 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"
unfolding ff_def gg_def
using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
by simp (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
unfolding ff_def gg_def
using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
by simp (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
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\<^marker>\<open>tag unimportant\<close> homeomorphism_moving_point:
fixes a :: "'a::euclidean_space"
assumes ope: "openin (top_of_set (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 (top_of_set 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
by (rule_tac x="S \<inter> ball d r" in exI) (fastforce simp: openin_open_Int \<open>0 < r\<close> that intro: *)
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)}"
by (rule connected_equivalence_relation [OF S]; blast intro: 1 2 3)
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 (top_of_set (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"
proof (rule affine_hull_Diff [OF ope])
show "finite (y ` K)"
by (simp add: insert.hyps(1))
show "y ` K \<subset> S"
using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
qed
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 (top_of_set (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
have "\<And>k. \<lbrakk>f (x i) = y k; k \<in> K\<rbrakk> \<Longrightarrow> False"
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)
then show "f (x i) \<in> S - y ` K"
by (auto simp: f_in_S \<open>x i \<in> S\<close>)
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\<^marker>\<open>tag unimportant\<close> 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 (top_of_set (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\<^marker>\<open>tag unimportant\<close>\<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"
unfolding f_def by (intro continuous_intros)
have "f ` {a..b} = {c..d}"
unfolding f_def image_affinity_atLeastAtMost
using assms sum_sqs_eq by (auto simp: field_split_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 field_split_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"
unfolding f_def using le eq
by (force intro: continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
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"
unfolding 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 by force
qed auto
then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
then show ?thesis
using eq f_def le that by force
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
proof (rule that [OF fg])
show "f x \<in> cbox u v" if "x \<in> cbox c d" for x
using that f4_eq f_eq homeomorphism_image1 [OF 2]
by (metis atLeastAtMost_iff box_real(2) image_eqI less(1) less_eq_real_def order_trans)
qed
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 (top_of_set (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 "\<And>x. \<lbrakk>w \<le> x \<and> x \<le> z; w < x \<longrightarrow> \<not> x < z\<rbrakk> \<Longrightarrow> f x = x"
using \<open>f w = w\<close> \<open>f z = z\<close> by auto
moreover have "\<And>x. \<lbrakk>w \<le> x \<and> x \<le> z; w < x \<longrightarrow> \<not> x < z\<rbrakk> \<Longrightarrow> g x = x"
using \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_apply1 by fastforce
ultimately
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
by (intro continuous_on_cases_local contfg continuous_on_id clo; auto simp: closed_subset)+
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)}"
proof (rule bounded_subset [of "cbox w z"])
show "bounded (cbox w z)"
using bounded_cbox by blast
show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> cbox w z"
by (auto simp: f'_def g'_def)
qed
qed
qed
proposition\<^marker>\<open>tag unimportant\<close> 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, opaque_lifting) 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, opaque_lifting) 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"
proof (clarsimp simp: jf jg hj)
show "f (h x) = h x \<longrightarrow> g (h x) \<noteq> h x \<Longrightarrow> x \<in> S" for x
using sub [THEN subsetD, of "h x"] hj by simp (metis imageE)
qed
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\<^marker>\<open>tag unimportant\<close> homeomorphism_grouping_points_exists_gen:
fixes S :: "'a::euclidean_space set"
assumes opeU: "openin (top_of_set S) U"
and opeS: "openin (top_of_set (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 (top_of_set (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"
proof (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
show "u islimpt S"
using True \<open>u \<in> S\<close> assms(8) connected_imp_perfect_aff_dim by fastforce
qed
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
using \<open>K \<subseteq> U\<close> by (intro that [of id id]) (auto 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 (top_of_set (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
proof (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
show "continuous_on (affine hull S) h"
using homeomorphism_def homhj by blast
qed (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)"
using hom homeomorphism_def by (intro continuous_on_compose cont_hj) 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)"
using hom homeomorphism_def by (intro continuous_on_compose cont_hj) 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
have \<section>: "\<And>x y. \<lbrakk>x \<in> affine hull S; h x = h y; y \<in> S\<rbrakk> \<Longrightarrow> x \<in> S"
by (metis h hull_inc)
show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
using sub by (simp add: f'_def g'_def jf jg) (force elim: \<section>)
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_canon (\<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 less
then show ?thesis
by (simp add: homotopic_on_emptyI)
next
case equal
show ?thesis
proof
assume L: ?lhs
with equal have [simp]: "f a \<in> S"
using homotopic_with_imp_subset1 by fastforce
obtain h:: "real \<times> 'M \<Rightarrow> 'a"
where h: "continuous_on ({0..1} \<times> {a}) h" "h ` ({0..1} \<times> {a}) \<subseteq> S" "h (0, a) = f a"
using L equal by (auto simp: homotopic_with)
then have "continuous_on (cball a r) (\<lambda>x. h (0, a))" "(\<lambda>x. h (0, a)) ` cball a r \<subseteq> S"
by (auto simp: equal)
then show ?rhs
using h(3) local.equal by force
next
assume ?rhs
then show ?lhs
using equal continuous_on_const by (force simp add: homotopic_with)
qed
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_canon (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
then have contf: "continuous_on (sphere a r) f"
by (metis homotopic_with_imp_continuous)
moreover have fim: "f ` sphere a r \<subseteq> S"
by (meson continuous_map_subtopology_eu c homotopic_with_imp_continuous_maps)
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 have "f ` {x. norm (x - a) = r} \<subseteq> S"
using sphere_cball [of a r] unfolding image_subset_iff sphere_def
by (metis dist_commute dist_norm mem_Collect_eq subset_eq)
with g show ?P
by (auto simp: dist_norm norm_minus_commute elim!: continuous_on_eq [OF continuous_on_subset])
qed
moreover have ?thesis if ?P
proof
assume ?lhs
then obtain c where "homotopic_with_canon (\<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> h ` ({0..1} \<times> sphere a r)"
proof
show "c = h (0, a + r *\<^sub>R b1)"
by (simp add: h)
show "(0, a + r *\<^sub>R b1) \<in> {0..1::real} \<times> sphere a r"
using greater \<open>b1 \<in> Basis\<close> by (auto simp: dist_norm)
qed
then have "c \<in> S"
using him by blast
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'"
using greater
by (force simp: dist_norm norm_minus_commute intro: continuous_on_compose2 [OF conth] continuous_intros)
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; norm ( x' - x) < d\<rbrakk>
\<Longrightarrow> norm (h x' - h x) < e"
using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by (auto simp: dist_norm)
have *: "norm (h (norm (x - a) / r,
a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e" (is "norm (?ha - ?hb) < e")
if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
proof -
have "norm (?ha - ?hb) = norm (?ha - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
by (simp add: h)
also have "\<dots> < e"
using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
by (intro d) (simp_all add: dist_norm, simp add: field_simps)
finally show ?thesis .
qed
show ?thesis
using greater \<open>0 < d\<close>
by (rule_tac x = "min r (d * r)" in exI) (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"
proof (rule continuous_on_compose2 [OF contg])
show "continuous_on ({0..1} \<times> sphere a r) (\<lambda>x. a + fst x *\<^sub>R (snd x - a))"
by (intro continuous_intros)
qed (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
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 have "homotopic_with_canon (\<lambda>x. True) (sphere a r) S (\<lambda>x. g a) f"
by (auto simp: homotopic_with)
then show ?lhs
using homotopic_with_symD by blast
qed
ultimately
show ?thesis by meson
qed
end