src/HOL/Analysis/Homotopy.thy
 author paulson Tue Mar 19 16:14:51 2019 +0000 (2 months ago) changeset 69922 4a9167f377b0 parent 69918 eddcc7c726f3 child 69986 f2d327275065 permissions -rw-r--r--
new material about topology, etc.; also fixes for yesterday's
```     1 (*  Title:      HOL/Analysis/Path_Connected.thy
```
```     2     Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
```
```     3 *)
```
```     4
```
```     5 section \<open>Homotopy of Maps\<close>
```
```     6
```
```     7 theory Homotopy
```
```     8   imports Path_Connected Continuum_Not_Denumerable
```
```     9 begin
```
```    10
```
```    11 definition%important homotopic_with ::
```
```    12   "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
```
```    13 where
```
```    14  "homotopic_with P X Y p q \<equiv>
```
```    15    (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
```
```    16        continuous_on ({0..1} \<times> X) h \<and>
```
```    17        h ` ({0..1} \<times> X) \<subseteq> Y \<and>
```
```    18        (\<forall>x. h(0, x) = p x) \<and>
```
```    19        (\<forall>x. h(1, x) = q x) \<and>
```
```    20        (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
```
```    21
```
```    22 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.
```
```    23 We often just want to require that \<open>P\<close> fixes some subset, but to include the case of a loop homotopy,
```
```    24 it is convenient to have a general property \<open>P\<close>.\<close>
```
```    25
```
```    26 text \<open>We often want to just localize the ending function equality or whatever.\<close>
```
```    27 text%important \<open>%whitespace\<close>
```
```    28 proposition homotopic_with:
```
```    29   fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
```
```    30   assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
```
```    31   shows "homotopic_with P X Y p q \<longleftrightarrow>
```
```    32            (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
```
```    33               continuous_on ({0..1} \<times> X) h \<and>
```
```    34               h ` ({0..1} \<times> X) \<subseteq> Y \<and>
```
```    35               (\<forall>x \<in> X. h(0,x) = p x) \<and>
```
```    36               (\<forall>x \<in> X. h(1,x) = q x) \<and>
```
```    37               (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
```
```    38   unfolding homotopic_with_def
```
```    39   apply (rule iffI, blast, clarify)
```
```    40   apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
```
```    41   apply auto
```
```    42   apply (force elim: continuous_on_eq)
```
```    43   apply (drule_tac x=t in bspec, force)
```
```    44   apply (subst assms; simp)
```
```    45   done
```
```    46
```
```    47 proposition homotopic_with_eq:
```
```    48    assumes h: "homotopic_with P X Y f g"
```
```    49        and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
```
```    50        and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
```
```    51        and P:  "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
```
```    52    shows "homotopic_with P X Y f' g'"
```
```    53   using h unfolding homotopic_with_def
```
```    54   apply safe
```
```    55   apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
```
```    56   apply (simp add: f' g', safe)
```
```    57   apply (fastforce intro: continuous_on_eq, fastforce)
```
```    58   apply (subst P; fastforce)
```
```    59   done
```
```    60
```
```    61 proposition homotopic_with_equal:
```
```    62    assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
```
```    63        and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
```
```    64        and P:  "P f" "P g"
```
```    65    shows "homotopic_with P X Y f g"
```
```    66   unfolding homotopic_with_def
```
```    67   apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
```
```    68   using assms
```
```    69   apply (intro conjI)
```
```    70   apply (rule continuous_on_eq [where f = "f \<circ> snd"])
```
```    71   apply (rule continuous_intros | force)+
```
```    72   apply clarify
```
```    73   apply (case_tac "t=1"; force)
```
```    74   done
```
```    75
```
```    76
```
```    77 lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
```
```    78   by auto
```
```    79
```
```    80 lemma homotopic_constant_maps:
```
```    81    "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
```
```    82 proof (cases "s = {} \<or> t = {}")
```
```    83   case True with continuous_on_const show ?thesis
```
```    84     by (auto simp: homotopic_with path_component_def)
```
```    85 next
```
```    86   case False
```
```    87   then obtain c where "c \<in> s" by blast
```
```    88   show ?thesis
```
```    89   proof
```
```    90     assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
```
```    91     then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
```
```    92         where conth: "continuous_on ({0..1} \<times> s) h"
```
```    93           and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
```
```    94       by (auto simp: homotopic_with)
```
```    95     have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
```
```    96       apply (rule continuous_intros conth | simp add: image_Pair_const)+
```
```    97       apply (blast intro:  \<open>c \<in> s\<close> continuous_on_subset [OF conth])
```
```    98       done
```
```    99     with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
```
```   100       apply (simp_all add: homotopic_with path_component_def, auto)
```
```   101       apply (drule_tac x="h \<circ> (\<lambda>t. (t, c))" in spec)
```
```   102       apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
```
```   103       done
```
```   104   next
```
```   105     assume "s = {} \<or> path_component t a b"
```
```   106     with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
```
```   107       apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
```
```   108       apply (rule_tac x="g \<circ> fst" in exI)
```
```   109       apply (rule conjI continuous_intros | force)+
```
```   110       done
```
```   111   qed
```
```   112 qed
```
```   113
```
```   114
```
```   115 subsection%unimportant\<open>Trivial properties\<close>
```
```   116
```
```   117 lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
```
```   118   unfolding homotopic_with_def Ball_def
```
```   119   apply clarify
```
```   120   apply (frule_tac x=0 in spec)
```
```   121   apply (drule_tac x=1 in spec, auto)
```
```   122   done
```
```   123
```
```   124 lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h \<circ> Pair t)"
```
```   125   by (fast intro: continuous_intros elim!: continuous_on_subset)
```
```   126
```
```   127 lemma homotopic_with_imp_continuous:
```
```   128     assumes "homotopic_with P X Y f g"
```
```   129     shows "continuous_on X f \<and> continuous_on X g"
```
```   130 proof -
```
```   131   obtain h :: "real \<times> 'a \<Rightarrow> 'b"
```
```   132     where conth: "continuous_on ({0..1} \<times> X) h"
```
```   133       and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
```
```   134     using assms by (auto simp: homotopic_with_def)
```
```   135   have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h \<circ> (\<lambda>x. (t,x)))" for t
```
```   136     by (rule continuous_intros continuous_on_subset [OF conth] | force)+
```
```   137   show ?thesis
```
```   138     using h *[of 0] *[of 1] by auto
```
```   139 qed
```
```   140
```
```   141 proposition homotopic_with_imp_subset1:
```
```   142      "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
```
```   143   by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
```
```   144
```
```   145 proposition homotopic_with_imp_subset2:
```
```   146      "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
```
```   147   by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
```
```   148
```
```   149 proposition homotopic_with_mono:
```
```   150     assumes hom: "homotopic_with P X Y f g"
```
```   151         and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
```
```   152       shows "homotopic_with Q X Y f g"
```
```   153   using hom
```
```   154   apply (simp add: homotopic_with_def)
```
```   155   apply (erule ex_forward)
```
```   156   apply (force simp: intro!: Q dest: continuous_on_o_Pair)
```
```   157   done
```
```   158
```
```   159 proposition homotopic_with_subset_left:
```
```   160      "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
```
```   161   apply (simp add: homotopic_with_def)
```
```   162   apply (fast elim!: continuous_on_subset ex_forward)
```
```   163   done
```
```   164
```
```   165 proposition homotopic_with_subset_right:
```
```   166      "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
```
```   167   apply (simp add: homotopic_with_def)
```
```   168   apply (fast elim!: continuous_on_subset ex_forward)
```
```   169   done
```
```   170
```
```   171 proposition homotopic_with_compose_continuous_right:
```
```   172     "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
```
```   173      \<Longrightarrow> homotopic_with p W Y (f \<circ> h) (g \<circ> h)"
```
```   174   apply (clarsimp simp add: homotopic_with_def)
```
```   175   apply (rename_tac k)
```
```   176   apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
```
```   177   apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
```
```   178   apply (erule continuous_on_subset)
```
```   179   apply (fastforce simp: o_def)+
```
```   180   done
```
```   181
```
```   182 proposition homotopic_compose_continuous_right:
```
```   183      "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
```
```   184       \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
```
```   185   using homotopic_with_compose_continuous_right by fastforce
```
```   186
```
```   187 proposition homotopic_with_compose_continuous_left:
```
```   188      "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
```
```   189       \<Longrightarrow> homotopic_with p X Z (h \<circ> f) (h \<circ> g)"
```
```   190   apply (clarsimp simp add: homotopic_with_def)
```
```   191   apply (rename_tac k)
```
```   192   apply (rule_tac x="h \<circ> k" in exI)
```
```   193   apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
```
```   194   apply (erule continuous_on_subset)
```
```   195   apply (fastforce simp: o_def)+
```
```   196   done
```
```   197
```
```   198 proposition homotopic_compose_continuous_left:
```
```   199    "\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g;
```
```   200      continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
```
```   201     \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
```
```   202   using homotopic_with_compose_continuous_left by fastforce
```
```   203
```
```   204 proposition homotopic_with_Pair:
```
```   205    assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
```
```   206        and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
```
```   207      shows "homotopic_with q (s \<times> s') (t \<times> t')
```
```   208                   (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
```
```   209   using hom
```
```   210   apply (clarsimp simp add: homotopic_with_def)
```
```   211   apply (rename_tac k k')
```
```   212   apply (rule_tac x="\<lambda>z. ((k \<circ> (\<lambda>x. (fst x, fst (snd x)))) z, (k' \<circ> (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
```
```   213   apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
```
```   214   apply (auto intro!: q [unfolded case_prod_unfold])
```
```   215   done
```
```   216
```
```   217 lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g"
```
```   218   by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
```
```   219
```
```   220
```
```   221 text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
```
```   222      though this only affects reflexivity.\<close>
```
```   223
```
```   224
```
```   225 proposition homotopic_with_refl:
```
```   226    "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
```
```   227   apply (rule iffI)
```
```   228   using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
```
```   229   apply (simp add: homotopic_with_def)
```
```   230   apply (rule_tac x="f \<circ> snd" in exI)
```
```   231   apply (rule conjI continuous_intros | force)+
```
```   232   done
```
```   233
```
```   234 lemma homotopic_with_symD:
```
```   235   fixes X :: "'a::real_normed_vector set"
```
```   236     assumes "homotopic_with P X Y f g"
```
```   237       shows "homotopic_with P X Y g f"
```
```   238   using assms
```
```   239   apply (clarsimp simp add: homotopic_with_def)
```
```   240   apply (rename_tac h)
```
```   241   apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
```
```   242   apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
```
```   243   done
```
```   244
```
```   245 proposition homotopic_with_sym:
```
```   246     fixes X :: "'a::real_normed_vector set"
```
```   247     shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
```
```   248   using homotopic_with_symD by blast
```
```   249
```
```   250 lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
```
```   251   by force
```
```   252
```
```   253 lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
```
```   254   by force
```
```   255
```
```   256 proposition homotopic_with_trans:
```
```   257     fixes X :: "'a::real_normed_vector set"
```
```   258     assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
```
```   259       shows "homotopic_with P X Y f h"
```
```   260 proof -
```
```   261   have clo1: "closedin (top_of_set ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
```
```   262     apply (simp add: closedin_closed split_01_prod [symmetric])
```
```   263     apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
```
```   264     apply (force simp: closed_Times)
```
```   265     done
```
```   266   have clo2: "closedin (top_of_set ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
```
```   267     apply (simp add: closedin_closed split_01_prod [symmetric])
```
```   268     apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
```
```   269     apply (force simp: closed_Times)
```
```   270     done
```
```   271   { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
```
```   272     assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
```
```   273        and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
```
```   274        and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
```
```   275        and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
```
```   276        and P:   "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
```
```   277     define k where "k y =
```
```   278       (if fst y \<le> 1 / 2
```
```   279        then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
```
```   280        else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y
```
```   281     have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2"  for u v
```
```   282       by (simp add: geq that)
```
```   283     have "continuous_on ({0..1} \<times> X) k"
```
```   284       using cont
```
```   285       apply (simp add: split_01_prod k_def)
```
```   286       apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
```
```   287       apply (force simp: keq)
```
```   288       done
```
```   289     moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
```
```   290       using Y by (force simp: k_def)
```
```   291     moreover have "\<forall>x. k (0, x) = f x"
```
```   292       by (simp add: k_def k12)
```
```   293     moreover have "(\<forall>x. k (1, x) = h x)"
```
```   294       by (simp add: k_def k12)
```
```   295     moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
```
```   296       using P
```
```   297       apply (clarsimp simp add: k_def)
```
```   298       apply (case_tac "t \<le> 1/2", auto)
```
```   299       done
```
```   300     ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
```
```   301                        continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
```
```   302                        (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
```
```   303       by blast
```
```   304   } note * = this
```
```   305   show ?thesis
```
```   306     using assms by (auto intro: * simp add: homotopic_with_def)
```
```   307 qed
```
```   308
```
```   309 proposition homotopic_compose:
```
```   310       fixes s :: "'a::real_normed_vector set"
```
```   311       shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
```
```   312              \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g \<circ> f) (g' \<circ> f')"
```
```   313   apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
```
```   314   apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
```
```   315   by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
```
```   316
```
```   317
```
```   318 text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
```
```   319 lemma homotopic_triviality:
```
```   320   fixes S :: "'a::real_normed_vector set"
```
```   321   shows  "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
```
```   322                  continuous_on S g \<and> g ` S \<subseteq> T
```
```   323                  \<longrightarrow> homotopic_with (\<lambda>x. True) S T f g) \<longleftrightarrow>
```
```   324           (S = {} \<or> path_connected T) \<and>
```
```   325           (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)))"
```
```   326           (is "?lhs = ?rhs")
```
```   327 proof (cases "S = {} \<or> T = {}")
```
```   328   case True then show ?thesis by auto
```
```   329 next
```
```   330   case False show ?thesis
```
```   331   proof
```
```   332     assume LHS [rule_format]: ?lhs
```
```   333     have pab: "path_component T a b" if "a \<in> T" "b \<in> T" for a b
```
```   334     proof -
```
```   335       have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
```
```   336         by (simp add: LHS continuous_on_const image_subset_iff that)
```
```   337       then show ?thesis
```
```   338         using False homotopic_constant_maps by blast
```
```   339     qed
```
```   340       moreover
```
```   341     have "\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
```
```   342       by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
```
```   343     ultimately show ?rhs
```
```   344       by (simp add: path_connected_component)
```
```   345   next
```
```   346     assume RHS: ?rhs
```
```   347     with False have T: "path_connected T"
```
```   348       by blast
```
```   349     show ?lhs
```
```   350     proof clarify
```
```   351       fix f g
```
```   352       assume "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
```
```   353       obtain c d where c: "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with (\<lambda>x. True) S T g (\<lambda>x. d)"
```
```   354         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
```
```   355       then have "c \<in> T" "d \<in> T"
```
```   356         using False homotopic_with_imp_subset2 by fastforce+
```
```   357       with T have "path_component T c d"
```
```   358         using path_connected_component by blast
```
```   359       then have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
```
```   360         by (simp add: homotopic_constant_maps)
```
```   361       with c d show "homotopic_with (\<lambda>x. True) S T f g"
```
```   362         by (meson homotopic_with_symD homotopic_with_trans)
```
```   363     qed
```
```   364   qed
```
```   365 qed
```
```   366
```
```   367
```
```   368 subsection\<open>Homotopy of paths, maintaining the same endpoints\<close>
```
```   369
```
```   370
```
```   371 definition%important homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
```
```   372   where
```
```   373      "homotopic_paths s p q \<equiv>
```
```   374        homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
```
```   375
```
```   376 lemma homotopic_paths:
```
```   377    "homotopic_paths s p q \<longleftrightarrow>
```
```   378       (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
```
```   379           h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
```
```   380           (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
```
```   381           (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
```
```   382           (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
```
```   383                         pathfinish(h \<circ> Pair t) = pathfinish p))"
```
```   384   by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
```
```   385
```
```   386 proposition homotopic_paths_imp_pathstart:
```
```   387      "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
```
```   388   by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
```
```   389
```
```   390 proposition homotopic_paths_imp_pathfinish:
```
```   391      "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
```
```   392   by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
```
```   393
```
```   394 lemma homotopic_paths_imp_path:
```
```   395      "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
```
```   396   using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
```
```   397
```
```   398 lemma homotopic_paths_imp_subset:
```
```   399      "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
```
```   400   by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
```
```   401
```
```   402 proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
```
```   403 by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
```
```   404
```
```   405 proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p"
```
```   406   by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
```
```   407
```
```   408 proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
```
```   409   by (metis homotopic_paths_sym)
```
```   410
```
```   411 proposition homotopic_paths_trans [trans]:
```
```   412      "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
```
```   413   apply (simp add: homotopic_paths_def)
```
```   414   apply (rule homotopic_with_trans, assumption)
```
```   415   by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
```
```   416
```
```   417 proposition homotopic_paths_eq:
```
```   418      "\<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"
```
```   419   apply (simp add: homotopic_paths_def)
```
```   420   apply (rule homotopic_with_eq)
```
```   421   apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
```
```   422   done
```
```   423
```
```   424 proposition homotopic_paths_reparametrize:
```
```   425   assumes "path p"
```
```   426       and pips: "path_image p \<subseteq> s"
```
```   427       and contf: "continuous_on {0..1} f"
```
```   428       and f01:"f ` {0..1} \<subseteq> {0..1}"
```
```   429       and [simp]: "f(0) = 0" "f(1) = 1"
```
```   430       and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
```
```   431     shows "homotopic_paths s p q"
```
```   432 proof -
```
```   433   have contp: "continuous_on {0..1} p"
```
```   434     by (metis \<open>path p\<close> path_def)
```
```   435   then have "continuous_on {0..1} (p \<circ> f)"
```
```   436     using contf continuous_on_compose continuous_on_subset f01 by blast
```
```   437   then have "path q"
```
```   438     by (simp add: path_def) (metis q continuous_on_cong)
```
```   439   have piqs: "path_image q \<subseteq> s"
```
```   440     by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
```
```   441   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"
```
```   442     using f01 by force
```
```   443   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
```
```   444     using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
```
```   445   have "homotopic_paths s q p"
```
```   446   proof (rule homotopic_paths_trans)
```
```   447     show "homotopic_paths s q (p \<circ> f)"
```
```   448       using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
```
```   449   next
```
```   450     show "homotopic_paths s (p \<circ> f) p"
```
```   451       apply (simp add: homotopic_paths_def homotopic_with_def)
```
```   452       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)
```
```   453       apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
```
```   454       using pips [unfolded path_image_def]
```
```   455       apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
```
```   456       done
```
```   457   qed
```
```   458   then show ?thesis
```
```   459     by (simp add: homotopic_paths_sym)
```
```   460 qed
```
```   461
```
```   462 lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
```
```   463   using homotopic_paths_def homotopic_with_subset_right by blast
```
```   464
```
```   465
```
```   466 text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
```
```   467 lemma homotopic_join_lemma:
```
```   468   fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
```
```   469   assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
```
```   470       and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
```
```   471       and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
```
```   472     shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
```
```   473 proof -
```
```   474   have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
```
```   475     by (rule ext) (simp)
```
```   476   have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
```
```   477     by (rule ext) (simp)
```
```   478   show ?thesis
```
```   479     apply (simp add: joinpaths_def)
```
```   480     apply (rule continuous_on_cases_le)
```
```   481     apply (simp_all only: 1 2)
```
```   482     apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
```
```   483     using pf
```
```   484     apply (auto simp: mult.commute pathstart_def pathfinish_def)
```
```   485     done
```
```   486 qed
```
```   487
```
```   488 text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
```
```   489
```
```   490 lemma homotopic_paths_reversepath_D:
```
```   491       assumes "homotopic_paths s p q"
```
```   492       shows   "homotopic_paths s (reversepath p) (reversepath q)"
```
```   493   using assms
```
```   494   apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
```
```   495   apply (rule_tac x="h \<circ> (\<lambda>x. (fst x, 1 - snd x))" in exI)
```
```   496   apply (rule conjI continuous_intros)+
```
```   497   apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
```
```   498   done
```
```   499
```
```   500 proposition homotopic_paths_reversepath:
```
```   501      "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
```
```   502   using homotopic_paths_reversepath_D by force
```
```   503
```
```   504
```
```   505 proposition homotopic_paths_join:
```
```   506     "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
```
```   507   apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
```
```   508   apply (rename_tac k1 k2)
```
```   509   apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
```
```   510   apply (rule conjI continuous_intros homotopic_join_lemma)+
```
```   511   apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
```
```   512   done
```
```   513
```
```   514 proposition homotopic_paths_continuous_image:
```
```   515     "\<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)"
```
```   516   unfolding homotopic_paths_def
```
```   517   apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
```
```   518   apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
```
```   519   done
```
```   520
```
```   521
```
```   522 subsection\<open>Group properties for homotopy of paths\<close>
```
```   523
```
```   524 text%important\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
```
```   525
```
```   526 proposition homotopic_paths_rid:
```
```   527     "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
```
```   528   apply (subst homotopic_paths_sym)
```
```   529   apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if  t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
```
```   530   apply (simp_all del: le_divide_eq_numeral1)
```
```   531   apply (subst split_01)
```
```   532   apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
```
```   533   done
```
```   534
```
```   535 proposition homotopic_paths_lid:
```
```   536    "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
```
```   537   using homotopic_paths_rid [of "reversepath p" s]
```
```   538   by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
```
```   539         pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
```
```   540
```
```   541 proposition homotopic_paths_assoc:
```
```   542    "\<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;
```
```   543      pathfinish q = pathstart r\<rbrakk>
```
```   544     \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
```
```   545   apply (subst homotopic_paths_sym)
```
```   546   apply (rule homotopic_paths_reparametrize
```
```   547            [where f = "\<lambda>t. if  t \<le> 1 / 2 then inverse 2 *\<^sub>R t
```
```   548                            else if  t \<le> 3 / 4 then t - (1 / 4)
```
```   549                            else 2 *\<^sub>R t - 1"])
```
```   550   apply (simp_all del: le_divide_eq_numeral1)
```
```   551   apply (simp add: subset_path_image_join)
```
```   552   apply (rule continuous_on_cases_1 continuous_intros)+
```
```   553   apply (auto simp: joinpaths_def)
```
```   554   done
```
```   555
```
```   556 proposition homotopic_paths_rinv:
```
```   557   assumes "path p" "path_image p \<subseteq> s"
```
```   558     shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
```
```   559 proof -
```
```   560   have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
```
```   561     using assms
```
```   562     apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
```
```   563     apply (rule continuous_on_cases_le)
```
```   564     apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
```
```   565     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
```
```   566     apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
```
```   567     apply (force elim!: continuous_on_subset simp add: mult_le_one)+
```
```   568     done
```
```   569   then show ?thesis
```
```   570     using assms
```
```   571     apply (subst homotopic_paths_sym_eq)
```
```   572     unfolding homotopic_paths_def homotopic_with_def
```
```   573     apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
```
```   574     apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
```
```   575     apply (force simp: mult_le_one)
```
```   576     done
```
```   577 qed
```
```   578
```
```   579 proposition homotopic_paths_linv:
```
```   580   assumes "path p" "path_image p \<subseteq> s"
```
```   581     shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
```
```   582   using homotopic_paths_rinv [of "reversepath p" s] assms by simp
```
```   583
```
```   584
```
```   585 subsection\<open>Homotopy of loops without requiring preservation of endpoints\<close>
```
```   586
```
```   587 definition%important homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool"  where
```
```   588  "homotopic_loops s p q \<equiv>
```
```   589      homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
```
```   590
```
```   591 lemma homotopic_loops:
```
```   592    "homotopic_loops s p q \<longleftrightarrow>
```
```   593       (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
```
```   594           image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
```
```   595           (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
```
```   596           (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
```
```   597           (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
```
```   598   by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
```
```   599
```
```   600 proposition homotopic_loops_imp_loop:
```
```   601      "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
```
```   602 using homotopic_with_imp_property homotopic_loops_def by blast
```
```   603
```
```   604 proposition homotopic_loops_imp_path:
```
```   605      "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
```
```   606   unfolding homotopic_loops_def path_def
```
```   607   using homotopic_with_imp_continuous by blast
```
```   608
```
```   609 proposition homotopic_loops_imp_subset:
```
```   610      "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
```
```   611   unfolding homotopic_loops_def path_image_def
```
```   612   by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
```
```   613
```
```   614 proposition homotopic_loops_refl:
```
```   615      "homotopic_loops s p p \<longleftrightarrow>
```
```   616       path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
```
```   617   by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
```
```   618
```
```   619 proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
```
```   620   by (simp add: homotopic_loops_def homotopic_with_sym)
```
```   621
```
```   622 proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
```
```   623   by (metis homotopic_loops_sym)
```
```   624
```
```   625 proposition homotopic_loops_trans:
```
```   626    "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
```
```   627   unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
```
```   628
```
```   629 proposition homotopic_loops_subset:
```
```   630    "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
```
```   631   by (simp add: homotopic_loops_def homotopic_with_subset_right)
```
```   632
```
```   633 proposition homotopic_loops_eq:
```
```   634    "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
```
```   635           \<Longrightarrow> homotopic_loops s p q"
```
```   636   unfolding homotopic_loops_def
```
```   637   apply (rule homotopic_with_eq)
```
```   638   apply (rule homotopic_with_refl [where f = p, THEN iffD2])
```
```   639   apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
```
```   640   done
```
```   641
```
```   642 proposition homotopic_loops_continuous_image:
```
```   643    "\<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)"
```
```   644   unfolding homotopic_loops_def
```
```   645   apply (rule homotopic_with_compose_continuous_left)
```
```   646   apply (erule homotopic_with_mono)
```
```   647   by (simp add: pathfinish_def pathstart_def)
```
```   648
```
```   649
```
```   650 subsection\<open>Relations between the two variants of homotopy\<close>
```
```   651
```
```   652 proposition homotopic_paths_imp_homotopic_loops:
```
```   653     "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
```
```   654   by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
```
```   655
```
```   656 proposition homotopic_loops_imp_homotopic_paths_null:
```
```   657   assumes "homotopic_loops s p (linepath a a)"
```
```   658     shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
```
```   659 proof -
```
```   660   have "path p" by (metis assms homotopic_loops_imp_path)
```
```   661   have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
```
```   662   have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
```
```   663   obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
```
```   664              and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
```
```   665              and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
```
```   666              and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
```
```   667              and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
```
```   668     using assms by (auto simp: homotopic_loops homotopic_with)
```
```   669   have conth0: "path (\<lambda>u. h (u, 0))"
```
```   670     unfolding path_def
```
```   671     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
```
```   672     apply (force intro: continuous_intros continuous_on_subset [OF conth])+
```
```   673     done
```
```   674   have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
```
```   675     using hs by (force simp: path_image_def)
```
```   676   have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
```
```   677     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
```
```   678     apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
```
```   679     done
```
```   680   have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
```
```   681     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
```
```   682     apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
```
```   683     apply (rule continuous_on_subset [OF conth])
```
```   684     apply (auto simp: algebra_simps add_increasing2 mult_left_le)
```
```   685     done
```
```   686   have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
```
```   687     using ends by (simp add: pathfinish_def pathstart_def)
```
```   688   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
```
```   689   proof -
```
```   690     have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
```
```   691     with \<open>c \<le> 1\<close> show ?thesis by fastforce
```
```   692   qed
```
```   693   have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
```
```   694                   (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
```
```   695                   (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
```
```   696                    pathstart(reversepath p) = a) \<and> pathstart p = x
```
```   697                   \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
```
```   698     by (metis homotopic_paths_lid homotopic_paths_join
```
```   699               homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
```
```   700   have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
```
```   701     using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast
```
```   702   moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
```
```   703                                    (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
```
```   704     apply (rule homotopic_paths_sym)
```
```   705     using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
```
```   706     by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
```
```   707   moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
```
```   708                                    ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
```
```   709     apply (simp add: homotopic_paths_def homotopic_with_def)
```
```   710     apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
```
```   711     apply (simp add: subpath_reversepath)
```
```   712     apply (intro conjI homotopic_join_lemma)
```
```   713     using ploop
```
```   714     apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
```
```   715     apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
```
```   716     done
```
```   717   moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
```
```   718                                    (linepath (pathstart p) (pathstart p))"
```
```   719     apply (rule *)
```
```   720     apply (simp add: pih0 pathstart_def pathfinish_def conth0)
```
```   721     apply (simp add: reversepath_def joinpaths_def)
```
```   722     done
```
```   723   ultimately show ?thesis
```
```   724     by (blast intro: homotopic_paths_trans)
```
```   725 qed
```
```   726
```
```   727 proposition homotopic_loops_conjugate:
```
```   728   fixes s :: "'a::real_normed_vector set"
```
```   729   assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
```
```   730       and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
```
```   731     shows "homotopic_loops s (p +++ q +++ reversepath p) q"
```
```   732 proof -
```
```   733   have contp: "continuous_on {0..1} p"  using \<open>path p\<close> [unfolded path_def] by blast
```
```   734   have contq: "continuous_on {0..1} q"  using \<open>path q\<close> [unfolded path_def] by blast
```
```   735   have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
```
```   736     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
```
```   737     apply (force simp: mult_le_one intro!: continuous_intros)
```
```   738     apply (rule continuous_on_subset [OF contp])
```
```   739     apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
```
```   740     done
```
```   741   have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
```
```   742     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
```
```   743     apply (force simp: mult_le_one intro!: continuous_intros)
```
```   744     apply (rule continuous_on_subset [OF contp])
```
```   745     apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
```
```   746     done
```
```   747   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"
```
```   748     using sum_le_prod1
```
```   749     by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
```
```   750   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"
```
```   751     apply (rule pip [unfolded path_image_def, THEN subsetD])
```
```   752     apply (rule image_eqI, blast)
```
```   753     apply (simp add: algebra_simps)
```
```   754     by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
```
```   755               add.commute zero_le_numeral)
```
```   756   have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s"
```
```   757     using path_image_def piq by fastforce
```
```   758   have "homotopic_loops s (p +++ q +++ reversepath p)
```
```   759                           (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
```
```   760     apply (simp add: homotopic_loops_def homotopic_with_def)
```
```   761     apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
```
```   762     apply (simp add: subpath_refl subpath_reversepath)
```
```   763     apply (intro conjI homotopic_join_lemma)
```
```   764     using papp qloop
```
```   765     apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
```
```   766     apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
```
```   767     apply (auto simp: ps1 ps2 qs)
```
```   768     done
```
```   769   moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
```
```   770   proof -
```
```   771     have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
```
```   772       using \<open>path q\<close> homotopic_paths_lid qloop piq by auto
```
```   773     hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
```
```   774       using homotopic_paths_trans by blast
```
```   775     hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
```
```   776     proof -
```
```   777       have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
```
```   778         by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
```
```   779       thus ?thesis
```
```   780         by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
```
```   781                   homotopic_paths_trans qloop pathfinish_linepath piq)
```
```   782     qed
```
```   783     thus ?thesis
```
```   784       by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
```
```   785   qed
```
```   786   ultimately show ?thesis
```
```   787     by (blast intro: homotopic_loops_trans)
```
```   788 qed
```
```   789
```
```   790 lemma homotopic_paths_loop_parts:
```
```   791   assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
```
```   792   shows "homotopic_paths S p q"
```
```   793 proof -
```
```   794   have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
```
```   795     using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
```
```   796   then have "path p"
```
```   797     using \<open>path q\<close> homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
```
```   798   show ?thesis
```
```   799   proof (cases "pathfinish p = pathfinish q")
```
```   800     case True
```
```   801     have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
```
```   802       by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> homotopic_loops_imp_subset homotopic_paths_imp_path loops
```
```   803            path_image_join path_image_reversepath path_imp_reversepath path_join_eq)+
```
```   804     have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
```
```   805       using \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> homotopic_paths_rid homotopic_paths_sym by blast
```
```   806     moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
```
```   807       by (simp add: True \<open>path p\<close> \<open>path q\<close> pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
```
```   808     moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
```
```   809       by (simp add: True \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
```
```   810     moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
```
```   811       by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
```
```   812     moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
```
```   813       by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
```
```   814     ultimately show ?thesis
```
```   815       using homotopic_paths_trans by metis
```
```   816   next
```
```   817     case False
```
```   818     then show ?thesis
```
```   819       using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
```
```   820   qed
```
```   821 qed
```
```   822
```
```   823
```
```   824 subsection%unimportant\<open>Homotopy of "nearby" function, paths and loops\<close>
```
```   825
```
```   826 lemma homotopic_with_linear:
```
```   827   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   828   assumes contf: "continuous_on s f"
```
```   829       and contg:"continuous_on s g"
```
```   830       and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
```
```   831     shows "homotopic_with (\<lambda>z. True) s t f g"
```
```   832   apply (simp add: homotopic_with_def)
```
```   833   apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
```
```   834   apply (intro conjI)
```
```   835   apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
```
```   836                                             continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
```
```   837   using sub closed_segment_def apply fastforce+
```
```   838   done
```
```   839
```
```   840 lemma homotopic_paths_linear:
```
```   841   fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
```
```   842   assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
```
```   843           "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
```
```   844     shows "homotopic_paths s g h"
```
```   845   using assms
```
```   846   unfolding path_def
```
```   847   apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
```
```   848   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)
```
```   849   apply (intro conjI subsetI continuous_intros; force)
```
```   850   done
```
```   851
```
```   852 lemma homotopic_loops_linear:
```
```   853   fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
```
```   854   assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
```
```   855           "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
```
```   856     shows "homotopic_loops s g h"
```
```   857   using assms
```
```   858   unfolding path_def
```
```   859   apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
```
```   860   apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
```
```   861   apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
```
```   862   apply (force simp: closed_segment_def)
```
```   863   done
```
```   864
```
```   865 lemma homotopic_paths_nearby_explicit:
```
```   866   assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
```
```   867       and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
```
```   868     shows "homotopic_paths s g h"
```
```   869   apply (rule homotopic_paths_linear [OF assms(1-4)])
```
```   870   by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
```
```   871
```
```   872 lemma homotopic_loops_nearby_explicit:
```
```   873   assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
```
```   874       and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
```
```   875     shows "homotopic_loops s g h"
```
```   876   apply (rule homotopic_loops_linear [OF assms(1-4)])
```
```   877   by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
```
```   878
```
```   879 lemma homotopic_nearby_paths:
```
```   880   fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
```
```   881   assumes "path g" "open s" "path_image g \<subseteq> s"
```
```   882     shows "\<exists>e. 0 < e \<and>
```
```   883                (\<forall>h. path h \<and>
```
```   884                     pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
```
```   885                     (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
```
```   886 proof -
```
```   887   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"
```
```   888     using separate_compact_closed [of "path_image g" "-s"] assms by force
```
```   889   show ?thesis
```
```   890     apply (intro exI conjI)
```
```   891     using e [unfolded dist_norm]
```
```   892     apply (auto simp: intro!: homotopic_paths_nearby_explicit assms  \<open>e > 0\<close>)
```
```   893     by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
```
```   894 qed
```
```   895
```
```   896 lemma homotopic_nearby_loops:
```
```   897   fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
```
```   898   assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
```
```   899     shows "\<exists>e. 0 < e \<and>
```
```   900                (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
```
```   901                     (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
```
```   902 proof -
```
```   903   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"
```
```   904     using separate_compact_closed [of "path_image g" "-s"] assms by force
```
```   905   show ?thesis
```
```   906     apply (intro exI conjI)
```
```   907     using e [unfolded dist_norm]
```
```   908     apply (auto simp: intro!: homotopic_loops_nearby_explicit assms  \<open>e > 0\<close>)
```
```   909     by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
```
```   910 qed
```
```   911
```
```   912
```
```   913 subsection\<open> Homotopy and subpaths\<close>
```
```   914
```
```   915 lemma homotopic_join_subpaths1:
```
```   916   assumes "path g" and pag: "path_image g \<subseteq> s"
```
```   917       and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
```
```   918     shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
```
```   919 proof -
```
```   920   have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
```
```   921     using affine_ineq \<open>u \<le> v\<close> by fastforce
```
```   922   have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
```
```   923     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>)
```
```   924   have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
```
```   925   show ?thesis
```
```   926     apply (rule homotopic_paths_subset [OF _ pag])
```
```   927     using assms
```
```   928     apply (cases "w = u")
```
```   929     using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
```
```   930     apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
```
```   931       apply (rule homotopic_paths_sym)
```
```   932       apply (rule homotopic_paths_reparametrize
```
```   933              [where f = "\<lambda>t. if  t \<le> 1 / 2
```
```   934                              then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
```
```   935                              else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
```
```   936       using \<open>path g\<close> path_subpath u w apply blast
```
```   937       using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
```
```   938       apply simp_all
```
```   939       apply (subst split_01)
```
```   940       apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
```
```   941       apply (simp_all add: field_simps not_le)
```
```   942       apply (force dest!: t2)
```
```   943       apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
```
```   944       apply (simp add: joinpaths_def subpath_def)
```
```   945       apply (force simp: algebra_simps)
```
```   946       done
```
```   947 qed
```
```   948
```
```   949 lemma homotopic_join_subpaths2:
```
```   950   assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
```
```   951     shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
```
```   952 by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
```
```   953
```
```   954 lemma homotopic_join_subpaths3:
```
```   955   assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
```
```   956       and "path g" and pag: "path_image g \<subseteq> s"
```
```   957       and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
```
```   958     shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
```
```   959 proof -
```
```   960   have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
```
```   961     apply (rule homotopic_paths_join)
```
```   962     using hom homotopic_paths_sym_eq apply blast
```
```   963     apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
```
```   964     done
```
```   965   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)"
```
```   966     apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
```
```   967     using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
```
```   968   also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
```
```   969                                (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
```
```   970     apply (rule homotopic_paths_join)
```
```   971     apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
```
```   972     apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
```
```   973     apply simp
```
```   974     done
```
```   975   also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
```
```   976     apply (rule homotopic_paths_rid)
```
```   977     using \<open>path g\<close> path_subpath u v apply blast
```
```   978     apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
```
```   979     done
```
```   980   finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
```
```   981   then show ?thesis
```
```   982     using homotopic_join_subpaths2 by blast
```
```   983 qed
```
```   984
```
```   985 proposition homotopic_join_subpaths:
```
```   986    "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
```
```   987     \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
```
```   988   apply (rule le_cases3 [of u v w])
```
```   989 using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
```
```   990
```
```   991 text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
```
```   992
```
```   993 lemma path_component_imp_homotopic_points:
```
```   994     "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
```
```   995 apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
```
```   996                  pathstart_def pathfinish_def path_image_def path_def, clarify)
```
```   997 apply (rule_tac x="g \<circ> fst" in exI)
```
```   998 apply (intro conjI continuous_intros continuous_on_compose)+
```
```   999 apply (auto elim!: continuous_on_subset)
```
```  1000 done
```
```  1001
```
```  1002 lemma homotopic_loops_imp_path_component_value:
```
```  1003    "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
```
```  1004         \<Longrightarrow> path_component S (p t) (q t)"
```
```  1005 apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
```
```  1006                  pathstart_def pathfinish_def path_image_def path_def, clarify)
```
```  1007 apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
```
```  1008 apply (intro conjI continuous_intros continuous_on_compose)+
```
```  1009 apply (auto elim!: continuous_on_subset)
```
```  1010 done
```
```  1011
```
```  1012 lemma homotopic_points_eq_path_component:
```
```  1013    "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
```
```  1014         path_component S a b"
```
```  1015 by (auto simp: path_component_imp_homotopic_points
```
```  1016          dest: homotopic_loops_imp_path_component_value [where t=1])
```
```  1017
```
```  1018 lemma path_connected_eq_homotopic_points:
```
```  1019     "path_connected S \<longleftrightarrow>
```
```  1020       (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
```
```  1021 by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
```
```  1022
```
```  1023
```
```  1024 subsection\<open>Simply connected sets\<close>
```
```  1025
```
```  1026 text%important\<open>defined as "all loops are homotopic (as loops)\<close>
```
```  1027
```
```  1028 definition%important simply_connected where
```
```  1029   "simply_connected S \<equiv>
```
```  1030         \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and>
```
```  1031               path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S
```
```  1032               \<longrightarrow> homotopic_loops S p q"
```
```  1033
```
```  1034 lemma simply_connected_empty [iff]: "simply_connected {}"
```
```  1035   by (simp add: simply_connected_def)
```
```  1036
```
```  1037 lemma simply_connected_imp_path_connected:
```
```  1038   fixes S :: "_::real_normed_vector set"
```
```  1039   shows "simply_connected S \<Longrightarrow> path_connected S"
```
```  1040 by (simp add: simply_connected_def path_connected_eq_homotopic_points)
```
```  1041
```
```  1042 lemma simply_connected_imp_connected:
```
```  1043   fixes S :: "_::real_normed_vector set"
```
```  1044   shows "simply_connected S \<Longrightarrow> connected S"
```
```  1045 by (simp add: path_connected_imp_connected simply_connected_imp_path_connected)
```
```  1046
```
```  1047 lemma simply_connected_eq_contractible_loop_any:
```
```  1048   fixes S :: "_::real_normed_vector set"
```
```  1049   shows "simply_connected S \<longleftrightarrow>
```
```  1050             (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and>
```
```  1051                   pathfinish p = pathstart p \<and> a \<in> S
```
```  1052                   \<longrightarrow> homotopic_loops S p (linepath a a))"
```
```  1053 apply (simp add: simply_connected_def)
```
```  1054 apply (rule iffI, force, clarify)
```
```  1055 apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
```
```  1056 apply (fastforce simp add:)
```
```  1057 using homotopic_loops_sym apply blast
```
```  1058 done
```
```  1059
```
```  1060 lemma simply_connected_eq_contractible_loop_some:
```
```  1061   fixes S :: "_::real_normed_vector set"
```
```  1062   shows "simply_connected S \<longleftrightarrow>
```
```  1063                 path_connected S \<and>
```
```  1064                 (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
```
```  1065                     \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
```
```  1066 apply (rule iffI)
```
```  1067  apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
```
```  1068 apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
```
```  1069 apply (drule_tac x=p in spec)
```
```  1070 using homotopic_loops_trans path_connected_eq_homotopic_points
```
```  1071   apply blast
```
```  1072 done
```
```  1073
```
```  1074 lemma simply_connected_eq_contractible_loop_all:
```
```  1075   fixes S :: "_::real_normed_vector set"
```
```  1076   shows "simply_connected S \<longleftrightarrow>
```
```  1077          S = {} \<or>
```
```  1078          (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
```
```  1079                 \<longrightarrow> homotopic_loops S p (linepath a a))"
```
```  1080         (is "?lhs = ?rhs")
```
```  1081 proof (cases "S = {}")
```
```  1082   case True then show ?thesis by force
```
```  1083 next
```
```  1084   case False
```
```  1085   then obtain a where "a \<in> S" by blast
```
```  1086   show ?thesis
```
```  1087   proof
```
```  1088     assume "simply_connected S"
```
```  1089     then show ?rhs
```
```  1090       using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
```
```  1091       by blast
```
```  1092   next
```
```  1093     assume ?rhs
```
```  1094     then show "simply_connected S"
```
```  1095       apply (simp add: simply_connected_eq_contractible_loop_any False)
```
```  1096       by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
```
```  1097              path_component_imp_homotopic_points path_component_refl)
```
```  1098   qed
```
```  1099 qed
```
```  1100
```
```  1101 lemma simply_connected_eq_contractible_path:
```
```  1102   fixes S :: "_::real_normed_vector set"
```
```  1103   shows "simply_connected S \<longleftrightarrow>
```
```  1104            path_connected S \<and>
```
```  1105            (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
```
```  1106             \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
```
```  1107 apply (rule iffI)
```
```  1108  apply (simp add: simply_connected_imp_path_connected)
```
```  1109  apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
```
```  1110 by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
```
```  1111          simply_connected_eq_contractible_loop_some subset_iff)
```
```  1112
```
```  1113 lemma simply_connected_eq_homotopic_paths:
```
```  1114   fixes S :: "_::real_normed_vector set"
```
```  1115   shows "simply_connected S \<longleftrightarrow>
```
```  1116           path_connected S \<and>
```
```  1117           (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
```
```  1118                 path q \<and> path_image q \<subseteq> S \<and>
```
```  1119                 pathstart q = pathstart p \<and> pathfinish q = pathfinish p
```
```  1120                 \<longrightarrow> homotopic_paths S p q)"
```
```  1121          (is "?lhs = ?rhs")
```
```  1122 proof
```
```  1123   assume ?lhs
```
```  1124   then have pc: "path_connected S"
```
```  1125         and *:  "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
```
```  1126                        pathfinish p = pathstart p\<rbrakk>
```
```  1127                       \<Longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))"
```
```  1128     by (auto simp: simply_connected_eq_contractible_path)
```
```  1129   have "homotopic_paths S p q"
```
```  1130         if "path p" "path_image p \<subseteq> S" "path q"
```
```  1131            "path_image q \<subseteq> S" "pathstart q = pathstart p"
```
```  1132            "pathfinish q = pathfinish p" for p q
```
```  1133   proof -
```
```  1134     have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
```
```  1135       by (simp add: homotopic_paths_rid homotopic_paths_sym that)
```
```  1136     also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
```
```  1137                                  (p +++ reversepath q +++ q)"
```
```  1138       using that
```
```  1139       by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
```
```  1140     also have "homotopic_paths S (p +++ reversepath q +++ q)
```
```  1141                                  ((p +++ reversepath q) +++ q)"
```
```  1142       by (simp add: that homotopic_paths_assoc)
```
```  1143     also have "homotopic_paths S ((p +++ reversepath q) +++ q)
```
```  1144                                  (linepath (pathstart q) (pathstart q) +++ q)"
```
```  1145       using * [of "p +++ reversepath q"] that
```
```  1146       by (simp add: homotopic_paths_join path_image_join)
```
```  1147     also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q"
```
```  1148       using that homotopic_paths_lid by blast
```
```  1149     finally show ?thesis .
```
```  1150   qed
```
```  1151   then show ?rhs
```
```  1152     by (blast intro: pc *)
```
```  1153 next
```
```  1154   assume ?rhs
```
```  1155   then show ?lhs
```
```  1156     by (force simp: simply_connected_eq_contractible_path)
```
```  1157 qed
```
```  1158
```
```  1159 proposition simply_connected_Times:
```
```  1160   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
```
```  1161   assumes S: "simply_connected S" and T: "simply_connected T"
```
```  1162     shows "simply_connected(S \<times> T)"
```
```  1163 proof -
```
```  1164   have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
```
```  1165        if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
```
```  1166        for p a b
```
```  1167   proof -
```
```  1168     have "path (fst \<circ> p)"
```
```  1169       apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
```
```  1170       apply (rule continuous_intros)+
```
```  1171       done
```
```  1172     moreover have "path_image (fst \<circ> p) \<subseteq> S"
```
```  1173       using that apply (simp add: path_image_def) by force
```
```  1174     ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
```
```  1175       using S that
```
```  1176       apply (simp add: simply_connected_eq_contractible_loop_any)
```
```  1177       apply (drule_tac x="fst \<circ> p" in spec)
```
```  1178       apply (drule_tac x=a in spec)
```
```  1179       apply (auto simp: pathstart_def pathfinish_def)
```
```  1180       done
```
```  1181     have "path (snd \<circ> p)"
```
```  1182       apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
```
```  1183       apply (rule continuous_intros)+
```
```  1184       done
```
```  1185     moreover have "path_image (snd \<circ> p) \<subseteq> T"
```
```  1186       using that apply (simp add: path_image_def) by force
```
```  1187     ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
```
```  1188       using T that
```
```  1189       apply (simp add: simply_connected_eq_contractible_loop_any)
```
```  1190       apply (drule_tac x="snd \<circ> p" in spec)
```
```  1191       apply (drule_tac x=b in spec)
```
```  1192       apply (auto simp: pathstart_def pathfinish_def)
```
```  1193       done
```
```  1194     show ?thesis
```
```  1195       using p1 p2
```
```  1196       apply (simp add: homotopic_loops, clarify)
```
```  1197       apply (rename_tac h k)
```
```  1198       apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
```
```  1199       apply (intro conjI continuous_intros | assumption)+
```
```  1200       apply (auto simp: pathstart_def pathfinish_def)
```
```  1201       done
```
```  1202   qed
```
```  1203   with assms show ?thesis
```
```  1204     by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
```
```  1205 qed
```
```  1206
```
```  1207
```
```  1208 subsection\<open>Contractible sets\<close>
```
```  1209
```
```  1210 definition%important contractible where
```
```  1211  "contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
```
```  1212
```
```  1213 proposition contractible_imp_simply_connected:
```
```  1214   fixes S :: "_::real_normed_vector set"
```
```  1215   assumes "contractible S" shows "simply_connected S"
```
```  1216 proof (cases "S = {}")
```
```  1217   case True then show ?thesis by force
```
```  1218 next
```
```  1219   case False
```
```  1220   obtain a where a: "homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)"
```
```  1221     using assms by (force simp: contractible_def)
```
```  1222   then have "a \<in> S"
```
```  1223     by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
```
```  1224   show ?thesis
```
```  1225     apply (simp add: simply_connected_eq_contractible_loop_all False)
```
```  1226     apply (rule bexI [OF _ \<open>a \<in> S\<close>])
```
```  1227     using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
```
```  1228     apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
```
```  1229     apply (intro conjI continuous_on_compose continuous_intros)
```
```  1230     apply (erule continuous_on_subset | force)+
```
```  1231     done
```
```  1232 qed
```
```  1233
```
```  1234 corollary contractible_imp_connected:
```
```  1235   fixes S :: "_::real_normed_vector set"
```
```  1236   shows "contractible S \<Longrightarrow> connected S"
```
```  1237 by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)
```
```  1238
```
```  1239 lemma contractible_imp_path_connected:
```
```  1240   fixes S :: "_::real_normed_vector set"
```
```  1241   shows "contractible S \<Longrightarrow> path_connected S"
```
```  1242 by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)
```
```  1243
```
```  1244 lemma nullhomotopic_through_contractible:
```
```  1245   fixes S :: "_::topological_space set"
```
```  1246   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
```
```  1247       and g: "continuous_on T g" "g ` T \<subseteq> U"
```
```  1248       and T: "contractible T"
```
```  1249     obtains c where "homotopic_with (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
```
```  1250 proof -
```
```  1251   obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)"
```
```  1252     using assms by (force simp: contractible_def)
```
```  1253   have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
```
```  1254     by (rule homotopic_compose_continuous_left [OF b g])
```
```  1255   then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
```
```  1256     by (rule homotopic_compose_continuous_right [OF _ f])
```
```  1257   then show ?thesis
```
```  1258     by (simp add: comp_def that)
```
```  1259 qed
```
```  1260
```
```  1261 lemma nullhomotopic_into_contractible:
```
```  1262   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
```
```  1263       and T: "contractible T"
```
```  1264     obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
```
```  1265 apply (rule nullhomotopic_through_contractible [OF f, of id T])
```
```  1266 using assms
```
```  1267 apply (auto simp: continuous_on_id)
```
```  1268 done
```
```  1269
```
```  1270 lemma nullhomotopic_from_contractible:
```
```  1271   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
```
```  1272       and S: "contractible S"
```
```  1273     obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
```
```  1274 apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
```
```  1275 using assms
```
```  1276 apply (auto simp: comp_def)
```
```  1277 done
```
```  1278
```
```  1279 lemma homotopic_through_contractible:
```
```  1280   fixes S :: "_::real_normed_vector set"
```
```  1281   assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
```
```  1282           "continuous_on T g1" "g1 ` T \<subseteq> U"
```
```  1283           "continuous_on S f2" "f2 ` S \<subseteq> T"
```
```  1284           "continuous_on T g2" "g2 ` T \<subseteq> U"
```
```  1285           "contractible T" "path_connected U"
```
```  1286    shows "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
```
```  1287 proof -
```
```  1288   obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
```
```  1289     apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
```
```  1290     using assms apply auto
```
```  1291     done
```
```  1292   obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
```
```  1293     apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
```
```  1294     using assms apply auto
```
```  1295     done
```
```  1296   have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
```
```  1297   proof (cases "S = {}")
```
```  1298     case True then show ?thesis by force
```
```  1299   next
```
```  1300     case False
```
```  1301     with c1 c2 have "c1 \<in> U" "c2 \<in> U"
```
```  1302       using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
```
```  1303     with \<open>path_connected U\<close> show ?thesis by blast
```
```  1304   qed
```
```  1305   show ?thesis
```
```  1306     apply (rule homotopic_with_trans [OF c1])
```
```  1307     apply (rule homotopic_with_symD)
```
```  1308     apply (rule homotopic_with_trans [OF c2])
```
```  1309     apply (simp add: path_component homotopic_constant_maps *)
```
```  1310     done
```
```  1311 qed
```
```  1312
```
```  1313 lemma homotopic_into_contractible:
```
```  1314   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
```
```  1315   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
```
```  1316       and g: "continuous_on S g" "g ` S \<subseteq> T"
```
```  1317       and T: "contractible T"
```
```  1318     shows "homotopic_with (\<lambda>h. True) S T f g"
```
```  1319 using homotopic_through_contractible [of S f T id T g id]
```
```  1320 by (simp add: assms contractible_imp_path_connected continuous_on_id)
```
```  1321
```
```  1322 lemma homotopic_from_contractible:
```
```  1323   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
```
```  1324   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
```
```  1325       and g: "continuous_on S g" "g ` S \<subseteq> T"
```
```  1326       and "contractible S" "path_connected T"
```
```  1327     shows "homotopic_with (\<lambda>h. True) S T f g"
```
```  1328 using homotopic_through_contractible [of S id S f T id g]
```
```  1329 by (simp add: assms contractible_imp_path_connected continuous_on_id)
```
```  1330
```
```  1331 lemma starlike_imp_contractible_gen:
```
```  1332   fixes S :: "'a::real_normed_vector set"
```
```  1333   assumes S: "starlike S"
```
```  1334       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)"
```
```  1335     obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)"
```
```  1336 proof -
```
```  1337   obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
```
```  1338     using S by (auto simp: starlike_def)
```
```  1339   have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
```
```  1340     apply clarify
```
```  1341     apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
```
```  1342     apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
```
```  1343     done
```
```  1344   then show ?thesis
```
```  1345     apply (rule_tac a=a in that)
```
```  1346     using \<open>a \<in> S\<close>
```
```  1347     apply (simp add: homotopic_with_def)
```
```  1348     apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
```
```  1349     apply (intro conjI ballI continuous_on_compose continuous_intros)
```
```  1350     apply (simp_all add: P)
```
```  1351     done
```
```  1352 qed
```
```  1353
```
```  1354 lemma starlike_imp_contractible:
```
```  1355   fixes S :: "'a::real_normed_vector set"
```
```  1356   shows "starlike S \<Longrightarrow> contractible S"
```
```  1357 using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)
```
```  1358
```
```  1359 lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
```
```  1360   by (simp add: starlike_imp_contractible)
```
```  1361
```
```  1362 lemma starlike_imp_simply_connected:
```
```  1363   fixes S :: "'a::real_normed_vector set"
```
```  1364   shows "starlike S \<Longrightarrow> simply_connected S"
```
```  1365 by (simp add: contractible_imp_simply_connected starlike_imp_contractible)
```
```  1366
```
```  1367 lemma convex_imp_simply_connected:
```
```  1368   fixes S :: "'a::real_normed_vector set"
```
```  1369   shows "convex S \<Longrightarrow> simply_connected S"
```
```  1370 using convex_imp_starlike starlike_imp_simply_connected by blast
```
```  1371
```
```  1372 lemma starlike_imp_path_connected:
```
```  1373   fixes S :: "'a::real_normed_vector set"
```
```  1374   shows "starlike S \<Longrightarrow> path_connected S"
```
```  1375 by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)
```
```  1376
```
```  1377 lemma starlike_imp_connected:
```
```  1378   fixes S :: "'a::real_normed_vector set"
```
```  1379   shows "starlike S \<Longrightarrow> connected S"
```
```  1380 by (simp add: path_connected_imp_connected starlike_imp_path_connected)
```
```  1381
```
```  1382 lemma is_interval_simply_connected_1:
```
```  1383   fixes S :: "real set"
```
```  1384   shows "is_interval S \<longleftrightarrow> simply_connected S"
```
```  1385 using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
```
```  1386
```
```  1387 lemma contractible_empty [simp]: "contractible {}"
```
```  1388   by (simp add: contractible_def homotopic_with)
```
```  1389
```
```  1390 lemma contractible_convex_tweak_boundary_points:
```
```  1391   fixes S :: "'a::euclidean_space set"
```
```  1392   assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S"
```
```  1393   shows "contractible T"
```
```  1394 proof (cases "S = {}")
```
```  1395   case True
```
```  1396   with assms show ?thesis
```
```  1397     by (simp add: subsetCE)
```
```  1398 next
```
```  1399   case False
```
```  1400   show ?thesis
```
```  1401     apply (rule starlike_imp_contractible)
```
```  1402     apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
```
```  1403     done
```
```  1404 qed
```
```  1405
```
```  1406 lemma convex_imp_contractible:
```
```  1407   fixes S :: "'a::real_normed_vector set"
```
```  1408   shows "convex S \<Longrightarrow> contractible S"
```
```  1409   using contractible_empty convex_imp_starlike starlike_imp_contractible by blast
```
```  1410
```
```  1411 lemma contractible_sing [simp]:
```
```  1412   fixes a :: "'a::real_normed_vector"
```
```  1413   shows "contractible {a}"
```
```  1414 by (rule convex_imp_contractible [OF convex_singleton])
```
```  1415
```
```  1416 lemma is_interval_contractible_1:
```
```  1417   fixes S :: "real set"
```
```  1418   shows  "is_interval S \<longleftrightarrow> contractible S"
```
```  1419 using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
```
```  1420       is_interval_simply_connected_1 by auto
```
```  1421
```
```  1422 lemma contractible_Times:
```
```  1423   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
```
```  1424   assumes S: "contractible S" and T: "contractible T"
```
```  1425   shows "contractible (S \<times> T)"
```
```  1426 proof -
```
```  1427   obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
```
```  1428              and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
```
```  1429              and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
```
```  1430              and [simp]: "\<And>x. x \<in> S \<Longrightarrow>  h (1::real, x) = a"
```
```  1431     using S by (auto simp: contractible_def homotopic_with)
```
```  1432   obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
```
```  1433              and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
```
```  1434              and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
```
```  1435              and [simp]: "\<And>x. x \<in> T \<Longrightarrow>  k (1::real, x) = b"
```
```  1436     using T by (auto simp: contractible_def homotopic_with)
```
```  1437   show ?thesis
```
```  1438     apply (simp add: contractible_def homotopic_with)
```
```  1439     apply (rule exI [where x=a])
```
```  1440     apply (rule exI [where x=b])
```
```  1441     apply (rule exI [where x = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
```
```  1442     apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
```
```  1443     using hsub ksub
```
```  1444     apply auto
```
```  1445     done
```
```  1446 qed
```
```  1447
```
```  1448 lemma homotopy_dominated_contractibility:
```
```  1449   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
```
```  1450   assumes S: "contractible S"
```
```  1451       and f: "continuous_on S f" "image f S \<subseteq> T"
```
```  1452       and g: "continuous_on T g" "image g T \<subseteq> S"
```
```  1453       and hom: "homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
```
```  1454     shows "contractible T"
```
```  1455 proof -
```
```  1456   obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)"
```
```  1457     using nullhomotopic_from_contractible [OF f S] .
```
```  1458   then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)"
```
```  1459     by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
```
```  1460   show ?thesis
```
```  1461     apply (simp add: contractible_def)
```
```  1462     apply (rule exI [where x = b])
```
```  1463     apply (rule homotopic_with_symD)
```
```  1464     apply (rule homotopic_with_trans [OF _ hom])
```
```  1465     using homg apply (simp add: o_def)
```
```  1466     done
```
```  1467 qed
```
```  1468
```
```  1469
```
```  1470 subsection\<open>Local versions of topological properties in general\<close>
```
```  1471
```
```  1472 definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
```
```  1473 where
```
```  1474  "locally P S \<equiv>
```
```  1475         \<forall>w x. openin (top_of_set S) w \<and> x \<in> w
```
```  1476               \<longrightarrow> (\<exists>u v. openin (top_of_set S) u \<and> P v \<and>
```
```  1477                         x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
```
```  1478
```
```  1479 lemma locallyI:
```
```  1480   assumes "\<And>w x. \<lbrakk>openin (top_of_set S) w; x \<in> w\<rbrakk>
```
```  1481                   \<Longrightarrow> \<exists>u v. openin (top_of_set S) u \<and> P v \<and>
```
```  1482                         x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
```
```  1483     shows "locally P S"
```
```  1484 using assms by (force simp: locally_def)
```
```  1485
```
```  1486 lemma locallyE:
```
```  1487   assumes "locally P S" "openin (top_of_set S) w" "x \<in> w"
```
```  1488   obtains u v where "openin (top_of_set S) u"
```
```  1489                     "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
```
```  1490   using assms unfolding locally_def by meson
```
```  1491
```
```  1492 lemma locally_mono:
```
```  1493   assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t"
```
```  1494     shows "locally Q S"
```
```  1495 by (metis assms locally_def)
```
```  1496
```
```  1497 lemma locally_open_subset:
```
```  1498   assumes "locally P S" "openin (top_of_set S) t"
```
```  1499     shows "locally P t"
```
```  1500 using assms
```
```  1501 apply (simp add: locally_def)
```
```  1502 apply (erule all_forward)+
```
```  1503 apply (rule impI)
```
```  1504 apply (erule impCE)
```
```  1505  using openin_trans apply blast
```
```  1506 apply (erule ex_forward)
```
```  1507 by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
```
```  1508
```
```  1509 lemma locally_diff_closed:
```
```  1510     "\<lbrakk>locally P S; closedin (top_of_set S) t\<rbrakk> \<Longrightarrow> locally P (S - t)"
```
```  1511   using locally_open_subset closedin_def by fastforce
```
```  1512
```
```  1513 lemma locally_empty [iff]: "locally P {}"
```
```  1514   by (simp add: locally_def openin_subtopology)
```
```  1515
```
```  1516 lemma locally_singleton [iff]:
```
```  1517   fixes a :: "'a::metric_space"
```
```  1518   shows "locally P {a} \<longleftrightarrow> P {a}"
```
```  1519 apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
```
```  1520 using zero_less_one by blast
```
```  1521
```
```  1522 lemma locally_iff:
```
```  1523     "locally P S \<longleftrightarrow>
```
```  1524      (\<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)))"
```
```  1525 apply (simp add: le_inf_iff locally_def openin_open, safe)
```
```  1526 apply (metis IntE IntI le_inf_iff)
```
```  1527 apply (metis IntI Int_subset_iff)
```
```  1528 done
```
```  1529
```
```  1530 lemma locally_Int:
```
```  1531   assumes S: "locally P S" and t: "locally P t"
```
```  1532       and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
```
```  1533     shows "locally P (S \<inter> t)"
```
```  1534 using S t unfolding locally_iff
```
```  1535 apply clarify
```
```  1536 apply (drule_tac x=T in spec)+
```
```  1537 apply (drule_tac x=x in spec)+
```
```  1538 apply clarsimp
```
```  1539 apply (rename_tac U1 U2 V1 V2)
```
```  1540 apply (rule_tac x="U1 \<inter> U2" in exI)
```
```  1541 apply (simp add: open_Int)
```
```  1542 apply (rule_tac x="V1 \<inter> V2" in exI)
```
```  1543 apply (auto intro: P)
```
```  1544 done
```
```  1545
```
```  1546 lemma locally_Times:
```
```  1547   fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
```
```  1548   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)"
```
```  1549   shows "locally R (S \<times> T)"
```
```  1550     unfolding locally_def
```
```  1551 proof (clarify)
```
```  1552   fix W x y
```
```  1553   assume W: "openin (top_of_set (S \<times> T)) W" and xy: "(x, y) \<in> W"
```
```  1554   then obtain U V where "openin (top_of_set S) U" "x \<in> U"
```
```  1555                         "openin (top_of_set T) V" "y \<in> V" "U \<times> V \<subseteq> W"
```
```  1556     using Times_in_interior_subtopology by metis
```
```  1557   then obtain U1 U2 V1 V2
```
```  1558          where opeS: "openin (top_of_set S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
```
```  1559            and opeT: "openin (top_of_set T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
```
```  1560     by (meson PS QT locallyE)
```
```  1561   with \<open>U \<times> V \<subseteq> W\<close> 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"
```
```  1562     apply (rule_tac x="U1 \<times> V1" in exI)
```
```  1563     apply (rule_tac x="U2 \<times> V2" in exI)
```
```  1564     apply (auto simp: openin_Times R)
```
```  1565     done
```
```  1566 qed
```
```  1567
```
```  1568
```
```  1569 proposition homeomorphism_locally_imp:
```
```  1570   fixes S :: "'a::metric_space set" and t :: "'b::t2_space set"
```
```  1571   assumes S: "locally P S" and hom: "homeomorphism S t f g"
```
```  1572       and Q: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
```
```  1573     shows "locally Q t"
```
```  1574 proof (clarsimp simp: locally_def)
```
```  1575   fix W y
```
```  1576   assume "y \<in> W" and "openin (top_of_set t) W"
```
```  1577   then obtain T where T: "open T" "W = t \<inter> T"
```
```  1578     by (force simp: openin_open)
```
```  1579   then have "W \<subseteq> t" by auto
```
```  1580   have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
```
```  1581    and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
```
```  1582     using hom by (auto simp: homeomorphism_def)
```
```  1583   have gw: "g ` W = S \<inter> f -` W"
```
```  1584     using \<open>W \<subseteq> t\<close>
```
```  1585     apply auto
```
```  1586     using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
```
```  1587     using g \<open>W \<subseteq> t\<close> apply auto[1]
```
```  1588     by (simp add: f rev_image_eqI)
```
```  1589   have \<circ>: "openin (top_of_set S) (g ` W)"
```
```  1590   proof -
```
```  1591     have "continuous_on S f"
```
```  1592       using f(3) by blast
```
```  1593     then show "openin (top_of_set S) (g ` W)"
```
```  1594       by (simp add: gw Collect_conj_eq \<open>openin (top_of_set t) W\<close> continuous_on_open f(2))
```
```  1595   qed
```
```  1596   then obtain u v
```
```  1597     where osu: "openin (top_of_set S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
```
```  1598     using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
```
```  1599   have "v \<subseteq> S" using uv by (simp add: gw)
```
```  1600   have fv: "f ` v = t \<inter> {x. g x \<in> v}"
```
```  1601     using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto
```
```  1602   have "f ` v \<subseteq> W"
```
```  1603     using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
```
```  1604   have contvf: "continuous_on v f"
```
```  1605     using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast
```
```  1606   have contvg: "continuous_on (f ` v) g"
```
```  1607     using \<open>f ` v \<subseteq> W\<close> \<open>W \<subseteq> t\<close> continuous_on_subset [OF g(3)] by blast
```
```  1608   have homv: "homeomorphism v (f ` v) f g"
```
```  1609     using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
```
```  1610     apply (simp add: homeomorphism_def contvf contvg, auto)
```
```  1611     by (metis f(1) rev_image_eqI rev_subsetD)
```
```  1612   have 1: "openin (top_of_set t) (t \<inter> g -` u)"
```
```  1613     apply (rule continuous_on_open [THEN iffD1, rule_format])
```
```  1614     apply (rule \<open>continuous_on t g\<close>)
```
```  1615     using \<open>g ` t = S\<close> apply (simp add: osu)
```
```  1616     done
```
```  1617   have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
```
```  1618     apply (rule_tac x="f ` v" in exI)
```
```  1619     apply (intro conjI Q [OF \<open>P v\<close> homv])
```
```  1620     using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close>  \<open>f ` v \<subseteq> W\<close>  uv  apply (auto simp: fv)
```
```  1621     done
```
```  1622   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)"
```
```  1623     by (meson 1 2)
```
```  1624 qed
```
```  1625
```
```  1626 lemma homeomorphism_locally:
```
```  1627   fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
```
```  1628   assumes hom: "homeomorphism S t f g"
```
```  1629       and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)"
```
```  1630     shows "locally P S \<longleftrightarrow> locally Q t"
```
```  1631 apply (rule iffI)
```
```  1632 apply (erule homeomorphism_locally_imp [OF _ hom])
```
```  1633 apply (simp add: eq)
```
```  1634 apply (erule homeomorphism_locally_imp)
```
```  1635 using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
```
```  1636 done
```
```  1637
```
```  1638 lemma homeomorphic_locally:
```
```  1639   fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
```
```  1640   assumes hom: "S homeomorphic T"
```
```  1641           and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
```
```  1642     shows "locally P S \<longleftrightarrow> locally Q T"
```
```  1643 proof -
```
```  1644   obtain f g where hom: "homeomorphism S T f g"
```
```  1645     using assms by (force simp: homeomorphic_def)
```
```  1646   then show ?thesis
```
```  1647     using homeomorphic_def local.iff
```
```  1648     by (blast intro!: homeomorphism_locally)
```
```  1649 qed
```
```  1650
```
```  1651 lemma homeomorphic_local_compactness:
```
```  1652   fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
```
```  1653   shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
```
```  1654 by (simp add: homeomorphic_compactness homeomorphic_locally)
```
```  1655
```
```  1656 lemma locally_translation:
```
```  1657   fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
```
```  1658   shows
```
```  1659    "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
```
```  1660         \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
```
```  1661 apply (rule homeomorphism_locally [OF homeomorphism_translation])
```
```  1662 apply (simp add: homeomorphism_def)
```
```  1663 by metis
```
```  1664
```
```  1665 lemma locally_injective_linear_image:
```
```  1666   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1667   assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
```
```  1668     shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
```
```  1669 apply (rule linear_homeomorphism_image [OF f])
```
```  1670 apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
```
```  1671 by (metis iff homeomorphism_def)
```
```  1672
```
```  1673 lemma locally_open_map_image:
```
```  1674   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```  1675   assumes P: "locally P S"
```
```  1676       and f: "continuous_on S f"
```
```  1677       and oo: "\<And>t. openin (top_of_set S) t
```
```  1678                    \<Longrightarrow> openin (top_of_set (f ` S)) (f ` t)"
```
```  1679       and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
```
```  1680     shows "locally Q (f ` S)"
```
```  1681 proof (clarsimp simp add: locally_def)
```
```  1682   fix W y
```
```  1683   assume oiw: "openin (top_of_set (f ` S)) W" and "y \<in> W"
```
```  1684   then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
```
```  1685   have oivf: "openin (top_of_set S) (S \<inter> f -` W)"
```
```  1686     by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
```
```  1687   then obtain x where "x \<in> S" "f x = y"
```
```  1688     using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
```
```  1689   then obtain U V
```
```  1690     where "openin (top_of_set S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
```
```  1691     using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
```
```  1692     by auto
```
```  1693   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)"
```
```  1694     apply (rule_tac x="f ` U" in exI)
```
```  1695     apply (rule conjI, blast intro!: oo)
```
```  1696     apply (rule_tac x="f ` V" in exI)
```
```  1697     apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
```
```  1698     done
```
```  1699 qed
```
```  1700
```
```  1701
```
```  1702 subsection\<open>An induction principle for connected sets\<close>
```
```  1703
```
```  1704 proposition connected_induction:
```
```  1705   assumes "connected S"
```
```  1706       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"
```
```  1707       and opI: "\<And>a. a \<in> S
```
```  1708              \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
```
```  1709                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
```
```  1710       and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
```
```  1711     shows "Q b"
```
```  1712 proof -
```
```  1713   have 1: "openin (top_of_set S)
```
```  1714              {b. \<exists>T. openin (top_of_set S) T \<and>
```
```  1715                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
```
```  1716     apply (subst openin_subopen, clarify)
```
```  1717     apply (rule_tac x=T in exI, auto)
```
```  1718     done
```
```  1719   have 2: "openin (top_of_set S)
```
```  1720              {b. \<exists>T. openin (top_of_set S) T \<and>
```
```  1721                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
```
```  1722     apply (subst openin_subopen, clarify)
```
```  1723     apply (rule_tac x=T in exI, auto)
```
```  1724     done
```
```  1725   show ?thesis
```
```  1726     using \<open>connected S\<close>
```
```  1727     apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
```
```  1728     apply (elim disjE allE)
```
```  1729          apply (blast intro: 1)
```
```  1730         apply (blast intro: 2, simp_all)
```
```  1731        apply clarify apply (metis opI)
```
```  1732       using opD apply (blast intro: etc elim: dest:)
```
```  1733      using opI etc apply meson+
```
```  1734     done
```
```  1735 qed
```
```  1736
```
```  1737 lemma connected_equivalence_relation_gen:
```
```  1738   assumes "connected S"
```
```  1739       and etc: "a \<in> S" "b \<in> S" "P a" "P b"
```
```  1740       and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
```
```  1741       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"
```
```  1742       and opI: "\<And>a. a \<in> S
```
```  1743              \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
```
```  1744                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
```
```  1745     shows "R a b"
```
```  1746 proof -
```
```  1747   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"
```
```  1748     apply (rule connected_induction [OF \<open>connected S\<close> opD], simp_all)
```
```  1749     by (meson trans opI)
```
```  1750   then show ?thesis by (metis etc opI)
```
```  1751 qed
```
```  1752
```
```  1753 lemma connected_induction_simple:
```
```  1754   assumes "connected S"
```
```  1755       and etc: "a \<in> S" "b \<in> S" "P a"
```
```  1756       and opI: "\<And>a. a \<in> S
```
```  1757              \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and>
```
```  1758                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
```
```  1759     shows "P b"
```
```  1760 apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
```
```  1761 apply (frule opI)
```
```  1762 using etc apply simp_all
```
```  1763 done
```
```  1764
```
```  1765 lemma connected_equivalence_relation:
```
```  1766   assumes "connected S"
```
```  1767       and etc: "a \<in> S" "b \<in> S"
```
```  1768       and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
```
```  1769       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"
```
```  1770       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)"
```
```  1771     shows "R a b"
```
```  1772 proof -
```
```  1773   have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
```
```  1774     apply (rule connected_induction_simple [OF \<open>connected S\<close>], simp_all)
```
```  1775     by (meson local.sym local.trans opI openin_imp_subset subsetCE)
```
```  1776   then show ?thesis by (metis etc opI)
```
```  1777 qed
```
```  1778
```
```  1779 lemma locally_constant_imp_constant:
```
```  1780   assumes "connected S"
```
```  1781       and opI: "\<And>a. a \<in> S
```
```  1782              \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
```
```  1783     shows "f constant_on S"
```
```  1784 proof -
```
```  1785   have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
```
```  1786     apply (rule connected_equivalence_relation [OF \<open>connected S\<close>], simp_all)
```
```  1787     by (metis opI)
```
```  1788   then show ?thesis
```
```  1789     by (metis constant_on_def)
```
```  1790 qed
```
```  1791
```
```  1792 lemma locally_constant:
```
```  1793      "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
```
```  1794 apply (simp add: locally_def)
```
```  1795 apply (rule iffI)
```
```  1796  apply (rule locally_constant_imp_constant, assumption)
```
```  1797  apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
```
```  1798 by (meson constant_on_subset openin_imp_subset order_refl)
```
```  1799
```
```  1800
```
```  1801 subsection\<open>Basic properties of local compactness\<close>
```
```  1802
```
```  1803 proposition locally_compact:
```
```  1804   fixes s :: "'a :: metric_space set"
```
```  1805   shows
```
```  1806     "locally compact s \<longleftrightarrow>
```
```  1807      (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
```
```  1808                     openin (top_of_set s) u \<and> compact v)"
```
```  1809      (is "?lhs = ?rhs")
```
```  1810 proof
```
```  1811   assume ?lhs
```
```  1812   then show ?rhs
```
```  1813     apply clarify
```
```  1814     apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
```
```  1815     by auto
```
```  1816 next
```
```  1817   assume r [rule_format]: ?rhs
```
```  1818   have *: "\<exists>u v.
```
```  1819               openin (top_of_set s) u \<and>
```
```  1820               compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
```
```  1821           if "open T" "x \<in> s" "x \<in> T" for x T
```
```  1822   proof -
```
```  1823     obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (top_of_set s) u"
```
```  1824       using r [OF \<open>x \<in> s\<close>] by auto
```
```  1825     obtain e where "e>0" and e: "cball x e \<subseteq> T"
```
```  1826       using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
```
```  1827     show ?thesis
```
```  1828       apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
```
```  1829       apply (rule_tac x="cball x e \<inter> v" in exI)
```
```  1830       using that \<open>e > 0\<close> e uv
```
```  1831       apply auto
```
```  1832       done
```
```  1833   qed
```
```  1834   show ?lhs
```
```  1835     apply (rule locallyI)
```
```  1836     apply (subst (asm) openin_open)
```
```  1837     apply (blast intro: *)
```
```  1838     done
```
```  1839 qed
```
```  1840
```
```  1841 lemma locally_compactE:
```
```  1842   fixes s :: "'a :: metric_space set"
```
```  1843   assumes "locally compact s"
```
```  1844   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>
```
```  1845                              openin (top_of_set s) (u x) \<and> compact (v x)"
```
```  1846 using assms
```
```  1847 unfolding locally_compact by metis
```
```  1848
```
```  1849 lemma locally_compact_alt:
```
```  1850   fixes s :: "'a :: heine_borel set"
```
```  1851   shows "locally compact s \<longleftrightarrow>
```
```  1852          (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
```
```  1853                     openin (top_of_set s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
```
```  1854 apply (simp add: locally_compact)
```
```  1855 apply (intro ball_cong ex_cong refl iffI)
```
```  1856 apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
```
```  1857 by (meson closure_subset compact_closure)
```
```  1858
```
```  1859 lemma locally_compact_Int_cball:
```
```  1860   fixes s :: "'a :: heine_borel set"
```
```  1861   shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
```
```  1862         (is "?lhs = ?rhs")
```
```  1863 proof
```
```  1864   assume ?lhs
```
```  1865   then show ?rhs
```
```  1866     apply (simp add: locally_compact openin_contains_cball)
```
```  1867     apply (clarify | assumption | drule bspec)+
```
```  1868     by (metis (no_types, lifting)  compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
```
```  1869 next
```
```  1870   assume ?rhs
```
```  1871   then show ?lhs
```
```  1872     apply (simp add: locally_compact openin_contains_cball)
```
```  1873     apply (clarify | assumption | drule bspec)+
```
```  1874     apply (rule_tac x="ball x e \<inter> s" in exI, simp)
```
```  1875     apply (rule_tac x="cball x e \<inter> s" in exI)
```
```  1876     using compact_eq_bounded_closed
```
```  1877     apply auto
```
```  1878     apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
```
```  1879     done
```
```  1880 qed
```
```  1881
```
```  1882 lemma locally_compact_compact:
```
```  1883   fixes s :: "'a :: heine_borel set"
```
```  1884   shows "locally compact s \<longleftrightarrow>
```
```  1885          (\<forall>k. k \<subseteq> s \<and> compact k
```
```  1886               \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
```
```  1887                          openin (top_of_set s) u \<and> compact v))"
```
```  1888         (is "?lhs = ?rhs")
```
```  1889 proof
```
```  1890   assume ?lhs
```
```  1891   then obtain u v where
```
```  1892     uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
```
```  1893                              openin (top_of_set s) (u x) \<and> compact (v x)"
```
```  1894     by (metis locally_compactE)
```
```  1895   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"
```
```  1896           if "k \<subseteq> s" "compact k" for k
```
```  1897   proof -
```
```  1898     have "\<And>C. (\<forall>c\<in>C. openin (top_of_set k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
```
```  1899                     \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
```
```  1900       using that by (simp add: compact_eq_openin_cover)
```
```  1901     moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (top_of_set k) c"
```
```  1902       using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
```
```  1903     moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
```
```  1904       using that by clarsimp (meson subsetCE uv)
```
```  1905     ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D"
```
```  1906       by metis
```
```  1907     then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)"
```
```  1908       by (metis finite_subset_image)
```
```  1909     have Tuv: "\<Union>(u ` T) \<subseteq> \<Union>(v ` T)"
```
```  1910       using T that by (force simp: dest!: uv)
```
```  1911     show ?thesis
```
```  1912       apply (rule_tac x="\<Union>(u ` T)" in exI)
```
```  1913       apply (rule_tac x="\<Union>(v ` T)" in exI)
```
```  1914       apply (simp add: Tuv)
```
```  1915       using T that
```
```  1916       apply (auto simp: dest!: uv)
```
```  1917       done
```
```  1918   qed
```
```  1919   show ?rhs
```
```  1920     by (blast intro: *)
```
```  1921 next
```
```  1922   assume ?rhs
```
```  1923   then show ?lhs
```
```  1924     apply (clarsimp simp add: locally_compact)
```
```  1925     apply (drule_tac x="{x}" in spec, simp)
```
```  1926     done
```
```  1927 qed
```
```  1928
```
```  1929 lemma open_imp_locally_compact:
```
```  1930   fixes s :: "'a :: heine_borel set"
```
```  1931   assumes "open s"
```
```  1932     shows "locally compact s"
```
```  1933 proof -
```
```  1934   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"
```
```  1935           if "x \<in> s" for x
```
```  1936   proof -
```
```  1937     obtain e where "e>0" and e: "cball x e \<subseteq> s"
```
```  1938       using open_contains_cball assms \<open>x \<in> s\<close> by blast
```
```  1939     have ope: "openin (top_of_set s) (ball x e)"
```
```  1940       by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
```
```  1941     show ?thesis
```
```  1942       apply (rule_tac x="ball x e" in exI)
```
```  1943       apply (rule_tac x="cball x e" in exI)
```
```  1944       using \<open>e > 0\<close> e apply (auto simp: ope)
```
```  1945       done
```
```  1946   qed
```
```  1947   show ?thesis
```
```  1948     unfolding locally_compact
```
```  1949     by (blast intro: *)
```
```  1950 qed
```
```  1951
```
```  1952 lemma closed_imp_locally_compact:
```
```  1953   fixes s :: "'a :: heine_borel set"
```
```  1954   assumes "closed s"
```
```  1955     shows "locally compact s"
```
```  1956 proof -
```
```  1957   have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
```
```  1958                  openin (top_of_set s) u \<and> compact v"
```
```  1959           if "x \<in> s" for x
```
```  1960   proof -
```
```  1961     show ?thesis
```
```  1962       apply (rule_tac x = "s \<inter> ball x 1" in exI)
```
```  1963       apply (rule_tac x = "s \<inter> cball x 1" in exI)
```
```  1964       using \<open>x \<in> s\<close> assms apply auto
```
```  1965       done
```
```  1966   qed
```
```  1967   show ?thesis
```
```  1968     unfolding locally_compact
```
```  1969     by (blast intro: *)
```
```  1970 qed
```
```  1971
```
```  1972 lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
```
```  1973   by (simp add: closed_imp_locally_compact)
```
```  1974
```
```  1975 lemma locally_compact_Int:
```
```  1976   fixes s :: "'a :: t2_space set"
```
```  1977   shows "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
```
```  1978 by (simp add: compact_Int locally_Int)
```
```  1979
```
```  1980 lemma locally_compact_closedin:
```
```  1981   fixes s :: "'a :: heine_borel set"
```
```  1982   shows "\<lbrakk>closedin (top_of_set s) t; locally compact s\<rbrakk>
```
```  1983         \<Longrightarrow> locally compact t"
```
```  1984 unfolding closedin_closed
```
```  1985 using closed_imp_locally_compact locally_compact_Int by blast
```
```  1986
```
```  1987 lemma locally_compact_delete:
```
```  1988      fixes s :: "'a :: t1_space set"
```
```  1989      shows "locally compact s \<Longrightarrow> locally compact (s - {a})"
```
```  1990   by (auto simp: openin_delete locally_open_subset)
```
```  1991
```
```  1992 lemma locally_closed:
```
```  1993   fixes s :: "'a :: heine_borel set"
```
```  1994   shows "locally closed s \<longleftrightarrow> locally compact s"
```
```  1995         (is "?lhs = ?rhs")
```
```  1996 proof
```
```  1997   assume ?lhs
```
```  1998   then show ?rhs
```
```  1999     apply (simp only: locally_def)
```
```  2000     apply (erule all_forward imp_forward asm_rl exE)+
```
```  2001     apply (rule_tac x = "u \<inter> ball x 1" in exI)
```
```  2002     apply (rule_tac x = "v \<inter> cball x 1" in exI)
```
```  2003     apply (force intro: openin_trans)
```
```  2004     done
```
```  2005 next
```
```  2006   assume ?rhs then show ?lhs
```
```  2007     using compact_eq_bounded_closed locally_mono by blast
```
```  2008 qed
```
```  2009
```
```  2010 lemma locally_compact_openin_Un:
```
```  2011   fixes S :: "'a::euclidean_space set"
```
```  2012   assumes LCS: "locally compact S" and LCT:"locally compact T"
```
```  2013       and opS: "openin (top_of_set (S \<union> T)) S"
```
```  2014       and opT: "openin (top_of_set (S \<union> T)) T"
```
```  2015     shows "locally compact (S \<union> T)"
```
```  2016 proof -
```
```  2017   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
```
```  2018   proof -
```
```  2019     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
```
```  2020       using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
```
```  2021     moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> S"
```
```  2022       by (meson \<open>x \<in> S\<close> opS openin_contains_cball)
```
```  2023     then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> S"
```
```  2024       by force
```
```  2025     ultimately show ?thesis
```
```  2026       apply (rule_tac x="min e1 e2" in exI)
```
```  2027       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
```
```  2028       by (metis closed_Int closed_cball inf_left_commute)
```
```  2029   qed
```
```  2030   moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
```
```  2031   proof -
```
```  2032     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
```
```  2033       using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
```
```  2034     moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
```
```  2035       by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
```
```  2036     then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
```
```  2037       by force
```
```  2038     ultimately show ?thesis
```
```  2039       apply (rule_tac x="min e1 e2" in exI)
```
```  2040       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
```
```  2041       by (metis closed_Int closed_cball inf_left_commute)
```
```  2042   qed
```
```  2043   ultimately show ?thesis
```
```  2044     by (force simp: locally_compact_Int_cball)
```
```  2045 qed
```
```  2046
```
```  2047 lemma locally_compact_closedin_Un:
```
```  2048   fixes S :: "'a::euclidean_space set"
```
```  2049   assumes LCS: "locally compact S" and LCT:"locally compact T"
```
```  2050       and clS: "closedin (top_of_set (S \<union> T)) S"
```
```  2051       and clT: "closedin (top_of_set (S \<union> T)) T"
```
```  2052     shows "locally compact (S \<union> T)"
```
```  2053 proof -
```
```  2054   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
```
```  2055   proof -
```
```  2056     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
```
```  2057       using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
```
```  2058     moreover
```
```  2059     obtain e2 where "e2 > 0" and e2: "closed (cball x e2 \<inter> T)"
```
```  2060       using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
```
```  2061     ultimately show ?thesis
```
```  2062       apply (rule_tac x="min e1 e2" in exI)
```
```  2063       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
```
```  2064       by (metis closed_Int closed_Un closed_cball inf_left_commute)
```
```  2065   qed
```
```  2066   moreover
```
```  2067   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
```
```  2068   proof -
```
```  2069     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
```
```  2070       using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
```
```  2071     moreover
```
```  2072     obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
```
```  2073       using clT x by (fastforce simp: openin_contains_cball closedin_def)
```
```  2074     then have "closed (cball x e2 \<inter> T)"
```
```  2075     proof -
```
```  2076       have "{} = T - (T - cball x e2)"
```
```  2077         using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
```
```  2078       then show ?thesis
```
```  2079         by (simp add: Diff_Diff_Int inf_commute)
```
```  2080     qed
```
```  2081     ultimately show ?thesis
```
```  2082       apply (rule_tac x="min e1 e2" in exI)
```
```  2083       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
```
```  2084       by (metis closed_Int closed_Un closed_cball inf_left_commute)
```
```  2085   qed
```
```  2086   moreover
```
```  2087   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
```
```  2088   proof -
```
```  2089     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
```
```  2090       using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
```
```  2091     moreover
```
```  2092     obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
```
```  2093       using clS x by (fastforce simp: openin_contains_cball closedin_def)
```
```  2094     then have "closed (cball x e2 \<inter> S)"
```
```  2095       by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
```
```  2096     ultimately show ?thesis
```
```  2097       apply (rule_tac x="min e1 e2" in exI)
```
```  2098       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
```
```  2099       by (metis closed_Int closed_Un closed_cball inf_left_commute)
```
```  2100   qed
```
```  2101   ultimately show ?thesis
```
```  2102     by (auto simp: locally_compact_Int_cball)
```
```  2103 qed
```
```  2104
```
```  2105 lemma locally_compact_Times:
```
```  2106   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
```
```  2107   shows "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
```
```  2108   by (auto simp: compact_Times locally_Times)
```
```  2109
```
```  2110 lemma locally_compact_compact_subopen:
```
```  2111   fixes S :: "'a :: heine_borel set"
```
```  2112   shows
```
```  2113    "locally compact S \<longleftrightarrow>
```
```  2114     (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
```
```  2115           \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
```
```  2116                      openin (top_of_set S) U \<and> compact V))"
```
```  2117    (is "?lhs = ?rhs")
```
```  2118 proof
```
```  2119   assume L: ?lhs
```
```  2120   show ?rhs
```
```  2121   proof clarify
```
```  2122     fix K :: "'a set" and T :: "'a set"
```
```  2123     assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
```
```  2124     obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
```
```  2125                  and ope: "openin (top_of_set S) U"
```
```  2126       using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
```
```  2127     show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
```
```  2128                 openin (top_of_set S) U \<and> compact V"
```
```  2129     proof (intro exI conjI)
```
```  2130       show "K \<subseteq> U \<inter> T"
```
```  2131         by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
```
```  2132       show "U \<inter> T \<subseteq> closure(U \<inter> T)"
```
```  2133         by (rule closure_subset)
```
```  2134       show "closure (U \<inter> T) \<subseteq> S"
```
```  2135         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)
```
```  2136       show "openin (top_of_set S) (U \<inter> T)"
```
```  2137         by (simp add: \<open>open T\<close> ope openin_Int_open)
```
```  2138       show "compact (closure (U \<inter> T))"
```
```  2139         by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> bounded_subset compact_closure compact_eq_bounded_closed)
```
```  2140     qed auto
```
```  2141   qed
```
```  2142 next
```
```  2143   assume ?rhs then show ?lhs
```
```  2144     unfolding locally_compact_compact
```
```  2145     by (metis open_openin openin_topspace subtopology_superset top.extremum topspace_euclidean_subtopology)
```
```  2146 qed
```
```  2147
```
```  2148
```
```  2149 subsection\<open>Sura-Bura's results about compact components of sets\<close>
```
```  2150
```
```  2151 proposition Sura_Bura_compact:
```
```  2152   fixes S :: "'a::euclidean_space set"
```
```  2153   assumes "compact S" and C: "C \<in> components S"
```
```  2154   shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (top_of_set S) T \<and>
```
```  2155                            closedin (top_of_set S) T}"
```
```  2156          (is "C = \<Inter>?\<T>")
```
```  2157 proof
```
```  2158   obtain x where x: "C = connected_component_set S x" and "x \<in> S"
```
```  2159     using C by (auto simp: components_def)
```
```  2160   have "C \<subseteq> S"
```
```  2161     by (simp add: C in_components_subset)
```
```  2162   have "\<Inter>?\<T> \<subseteq> connected_component_set S x"
```
```  2163   proof (rule connected_component_maximal)
```
```  2164     have "x \<in> C"
```
```  2165       by (simp add: \<open>x \<in> S\<close> x)
```
```  2166     then show "x \<in> \<Inter>?\<T>"
```
```  2167       by blast
```
```  2168     have clo: "closed (\<Inter>?\<T>)"
```
```  2169       by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
```
```  2170     have False
```
```  2171       if K1: "closedin (top_of_set (\<Inter>?\<T>)) K1" and
```
```  2172          K2: "closedin (top_of_set (\<Inter>?\<T>)) K2" and
```
```  2173          K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
```
```  2174        for K1 K2
```
```  2175     proof -
```
```  2176       have "closed K1" "closed K2"
```
```  2177         using closedin_closed_trans clo K1 K2 by blast+
```
```  2178       then obtain V1 V2 where "open V1" "open V2" "K1 \<subseteq> V1" "K2 \<subseteq> V2" and V12: "V1 \<inter> V2 = {}"
```
```  2179         using separation_normal \<open>K1 \<inter> K2 = {}\<close> by metis
```
```  2180       have SV12_ne: "(S - (V1 \<union> V2)) \<inter> (\<Inter>?\<T>) \<noteq> {}"
```
```  2181       proof (rule compact_imp_fip)
```
```  2182         show "compact (S - (V1 \<union> V2))"
```
```  2183           by (simp add: \<open>open V1\<close> \<open>open V2\<close> \<open>compact S\<close> compact_diff open_Un)
```
```  2184         show clo\<T>: "closed T" if "T \<in> ?\<T>" for T
```
```  2185           using that \<open>compact S\<close>
```
```  2186           by (force intro: closedin_closed_trans simp add: compact_imp_closed)
```
```  2187         show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
```
```  2188         proof
```
```  2189           assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
```
```  2190           obtain D where opeD: "openin (top_of_set S) D"
```
```  2191                    and cloD: "closedin (top_of_set S) D"
```
```  2192                    and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
```
```  2193           proof (cases "\<F> = {}")
```
```  2194             case True
```
```  2195             with \<open>C \<subseteq> S\<close> djo that show ?thesis
```
```  2196               by force
```
```  2197           next
```
```  2198             case False show ?thesis
```
```  2199             proof
```
```  2200               show ope: "openin (top_of_set S) (\<Inter>\<F>)"
```
```  2201                 using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
```
```  2202               then show "closedin (top_of_set S) (\<Inter>\<F>)"
```
```  2203                 by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
```
```  2204               show "C \<subseteq> \<Inter>\<F>"
```
```  2205                 using \<F> by auto
```
```  2206               show "\<Inter>\<F> \<subseteq> V1 \<union> V2"
```
```  2207                 using ope djo openin_imp_subset by fastforce
```
```  2208             qed
```
```  2209           qed
```
```  2210           have "connected C"
```
```  2211             by (simp add: x)
```
```  2212           have "closed D"
```
```  2213             using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
```
```  2214           have cloV1: "closedin (top_of_set D) (D \<inter> closure V1)"
```
```  2215             and cloV2: "closedin (top_of_set D) (D \<inter> closure V2)"
```
```  2216             by (simp_all add: closedin_closed_Int)
```
```  2217           moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
```
```  2218             apply safe
```
```  2219             using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
```
```  2220                apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
```
```  2221             done
```
```  2222           ultimately have cloDV1: "closedin (top_of_set D) (D \<inter> V1)"
```
```  2223                       and cloDV2:  "closedin (top_of_set D) (D \<inter> V2)"
```
```  2224             by metis+
```
```  2225           then obtain U1 U2 where "closed U1" "closed U2"
```
```  2226                and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
```
```  2227             by (auto simp: closedin_closed)
```
```  2228           have "D \<inter> U1 \<inter> C \<noteq> {}"
```
```  2229           proof
```
```  2230             assume "D \<inter> U1 \<inter> C = {}"
```
```  2231             then have *: "C \<subseteq> D \<inter> V2"
```
```  2232               using D1 DV12 \<open>C \<subseteq> D\<close> by auto
```
```  2233             have "\<Inter>?\<T> \<subseteq> D \<inter> V2"
```
```  2234               apply (rule Inter_lower)
```
```  2235               using * apply simp
```
```  2236               by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
```
```  2237             then show False
```
```  2238               using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
```
```  2239           qed
```
```  2240           moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
```
```  2241           proof
```
```  2242             assume "D \<inter> U2 \<inter> C = {}"
```
```  2243             then have *: "C \<subseteq> D \<inter> V1"
```
```  2244               using D2 DV12 \<open>C \<subseteq> D\<close> by auto
```
```  2245             have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
```
```  2246               apply (rule Inter_lower)
```
```  2247               using * apply simp
```
```  2248               by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
```
```  2249             then show False
```
```  2250               using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
```
```  2251           qed
```
```  2252           ultimately show False
```
```  2253             using \<open>connected C\<close> unfolding connected_closed
```
```  2254             apply (simp only: not_ex)
```
```  2255             apply (drule_tac x="D \<inter> U1" in spec)
```
```  2256             apply (drule_tac x="D \<inter> U2" in spec)
```
```  2257             using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
```
```  2258             by blast
```
```  2259         qed
```
```  2260       qed
```
```  2261       show False
```
```  2262         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)
```
```  2263     qed
```
```  2264     then show "connected (\<Inter>?\<T>)"
```
```  2265       by (auto simp: connected_closedin_eq)
```
```  2266     show "\<Inter>?\<T> \<subseteq> S"
```
```  2267       by (fastforce simp: C in_components_subset)
```
```  2268   qed
```
```  2269   with x show "\<Inter>?\<T> \<subseteq> C" by simp
```
```  2270 qed auto
```
```  2271
```
```  2272
```
```  2273 corollary Sura_Bura_clopen_subset:
```
```  2274   fixes S :: "'a::euclidean_space set"
```
```  2275   assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
```
```  2276       and U: "open U" "C \<subseteq> U"
```
```  2277   obtains K where "openin (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
```
```  2278 proof (rule ccontr)
```
```  2279   assume "\<not> thesis"
```
```  2280   with that have neg: "\<nexists>K. openin (top_of_set S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
```
```  2281     by metis
```
```  2282   obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
```
```  2283                and opeSV: "openin (top_of_set S) V"
```
```  2284     using S U \<open>compact C\<close>
```
```  2285     apply (simp add: locally_compact_compact_subopen)
```
```  2286     by (meson C in_components_subset)
```
```  2287   let ?\<T> = "{T. C \<subseteq> T \<and> openin (top_of_set K) T \<and> compact T \<and> T \<subseteq> K}"
```
```  2288   have CK: "C \<in> components K"
```
```  2289     by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
```
```  2290   with \<open>compact K\<close>
```
```  2291   have "C = \<Inter>{T. C \<subseteq> T \<and> openin (top_of_set K) T \<and> closedin (top_of_set K) T}"
```
```  2292     by (simp add: Sura_Bura_compact)
```
```  2293   then have Ceq: "C = \<Inter>?\<T>"
```
```  2294     by (simp add: closedin_compact_eq \<open>compact K\<close>)
```
```  2295   obtain W where "open W" and W: "V = S \<inter> W"
```
```  2296     using opeSV by (auto simp: openin_open)
```
```  2297   have "-(U \<inter> W) \<inter> \<Inter>?\<T> \<noteq> {}"
```
```  2298   proof (rule closed_imp_fip_compact)
```
```  2299     show "- (U \<inter> W) \<inter> \<Inter>\<F> \<noteq> {}"
```
```  2300       if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
```
```  2301     proof (cases "\<F> = {}")
```
```  2302       case True
```
```  2303       have False if "U = UNIV" "W = UNIV"
```
```  2304       proof -
```
```  2305         have "V = S"
```
```  2306           by (simp add: W \<open>W = UNIV\<close>)
```
```  2307         with neg show False
```
```  2308           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
```
```  2309       qed
```
```  2310       with True show ?thesis
```
```  2311         by auto
```
```  2312     next
```
```  2313       case False
```
```  2314       show ?thesis
```
```  2315       proof
```
```  2316         assume "- (U \<inter> W) \<inter> \<Inter>\<F> = {}"
```
```  2317         then have FUW: "\<Inter>\<F> \<subseteq> U \<inter> W"
```
```  2318           by blast
```
```  2319         have "C \<subseteq> \<Inter>\<F>"
```
```  2320           using \<F> by auto
```
```  2321         moreover have "compact (\<Inter>\<F>)"
```
```  2322           by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
```
```  2323         moreover have "\<Inter>\<F> \<subseteq> K"
```
```  2324           using False that(2) by fastforce
```
```  2325         moreover have opeKF: "openin (top_of_set K) (\<Inter>\<F>)"
```
```  2326           using False \<F> \<open>finite \<F>\<close> by blast
```
```  2327         then have opeVF: "openin (top_of_set V) (\<Inter>\<F>)"
```
```  2328           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
```
```  2329         then have "openin (top_of_set S) (\<Inter>\<F>)"
```
```  2330           by (metis opeSV openin_trans)
```
```  2331         moreover have "\<Inter>\<F> \<subseteq> U"
```
```  2332           by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
```
```  2333         ultimately show False
```
```  2334           using neg by blast
```
```  2335       qed
```
```  2336     qed
```
```  2337   qed (use \<open>open W\<close> \<open>open U\<close> in auto)
```
```  2338   with W Ceq \<open>C \<subseteq> V\<close> \<open>C \<subseteq> U\<close> show False
```
```  2339     by auto
```
```  2340 qed
```
```  2341
```
```  2342
```
```  2343 corollary Sura_Bura_clopen_subset_alt:
```
```  2344   fixes S :: "'a::euclidean_space set"
```
```  2345   assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
```
```  2346       and opeSU: "openin (top_of_set S) U" and "C \<subseteq> U"
```
```  2347   obtains K where "openin (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
```
```  2348 proof -
```
```  2349   obtain V where "open V" "U = S \<inter> V"
```
```  2350     using opeSU by (auto simp: openin_open)
```
```  2351   with \<open>C \<subseteq> U\<close> have "C \<subseteq> V"
```
```  2352     by auto
```
```  2353   then show ?thesis
```
```  2354     using Sura_Bura_clopen_subset [OF S C \<open>compact C\<close> \<open>open V\<close>]
```
```  2355     by (metis \<open>U = S \<inter> V\<close> inf.bounded_iff openin_imp_subset that)
```
```  2356 qed
```
```  2357
```
```  2358 corollary Sura_Bura:
```
```  2359   fixes S :: "'a::euclidean_space set"
```
```  2360   assumes "locally compact S" "C \<in> components S" "compact C"
```
```  2361   shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (top_of_set S) K}"
```
```  2362          (is "C = ?rhs")
```
```  2363 proof
```
```  2364   show "?rhs \<subseteq> C"
```
```  2365   proof (clarsimp, rule ccontr)
```
```  2366     fix x
```
```  2367     assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (top_of_set S) X \<longrightarrow> x \<in> X"
```
```  2368       and "x \<notin> C"
```
```  2369     obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
```
```  2370       using separation_normal [of "{x}" C]
```
```  2371       by (metis Int_empty_left \<open>x \<notin> C\<close> \<open>compact C\<close> closed_empty closed_insert compact_imp_closed insert_disjoint(1))
```
```  2372     have "x \<notin> V"
```
```  2373       using \<open>U \<inter> V = {}\<close> \<open>{x} \<subseteq> U\<close> by blast
```
```  2374     then show False
```
```  2375       by (meson "*" Sura_Bura_clopen_subset \<open>C \<subseteq> V\<close> \<open>open V\<close> assms(1) assms(2) assms(3) subsetCE)
```
```  2376   qed
```
```  2377 qed blast
```
```  2378
```
```  2379
```
```  2380 subsection\<open>Special cases of local connectedness and path connectedness\<close>
```
```  2381
```
```  2382 lemma locally_connected_1:
```
```  2383   assumes
```
```  2384     "\<And>v x. \<lbrakk>openin (top_of_set S) v; x \<in> v\<rbrakk>
```
```  2385               \<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and>
```
```  2386                       connected u \<and> x \<in> u \<and> u \<subseteq> v"
```
```  2387    shows "locally connected S"
```
```  2388 apply (clarsimp simp add: locally_def)
```
```  2389 apply (drule assms; blast)
```
```  2390 done
```
```  2391
```
```  2392 lemma locally_connected_2:
```
```  2393   assumes "locally connected S"
```
```  2394           "openin (top_of_set S) t"
```
```  2395           "x \<in> t"
```
```  2396    shows "openin (top_of_set S) (connected_component_set t x)"
```
```  2397 proof -
```
```  2398   { fix y :: 'a
```
```  2399     let ?SS = "top_of_set S"
```
```  2400     assume 1: "openin ?SS t"
```
```  2401               "\<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))"
```
```  2402     and "connected_component t x y"
```
```  2403     then have "y \<in> t" and y: "y \<in> connected_component_set t x"
```
```  2404       using connected_component_subset by blast+
```
```  2405     obtain F where
```
```  2406       "\<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))"
```
```  2407       by moura
```
```  2408     then obtain G where
```
```  2409        "\<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)"
```
```  2410       by moura
```
```  2411     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"
```
```  2412       using 1 \<open>y \<in> t\<close> by presburger
```
```  2413     have "G y t \<subseteq> connected_component_set t y"
```
```  2414       by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
```
```  2415     then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> connected_component_set t x"
```
```  2416       by (metis (no_types) * connected_component_eq dual_order.trans y)
```
```  2417   }
```
```  2418   then show ?thesis
```
```  2419     using assms openin_subopen by (force simp: locally_def)
```
```  2420 qed
```
```  2421
```
```  2422 lemma locally_connected_3:
```
```  2423   assumes "\<And>t x. \<lbrakk>openin (top_of_set S) t; x \<in> t\<rbrakk>
```
```  2424               \<Longrightarrow> openin (top_of_set S)
```
```  2425                           (connected_component_set t x)"
```
```  2426           "openin (top_of_set S) v" "x \<in> v"
```
```  2427    shows  "\<exists>u. openin (top_of_set S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
```
```  2428 using assms connected_component_subset by fastforce
```
```  2429
```
```  2430 lemma locally_connected:
```
```  2431   "locally connected S \<longleftrightarrow>
```
```  2432    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
```
```  2433           \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
```
```  2434 by (metis locally_connected_1 locally_connected_2 locally_connected_3)
```
```  2435
```
```  2436 lemma locally_connected_open_connected_component:
```
```  2437   "locally connected S \<longleftrightarrow>
```
```  2438    (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t
```
```  2439           \<longrightarrow> openin (top_of_set S) (connected_component_set t x))"
```
```  2440 by (metis locally_connected_1 locally_connected_2 locally_connected_3)
```
```  2441
```
```  2442 lemma locally_path_connected_1:
```
```  2443   assumes
```
```  2444     "\<And>v x. \<lbrakk>openin (top_of_set S) v; x \<in> v\<rbrakk>
```
```  2445               \<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
```
```  2446    shows "locally path_connected S"
```
```  2447 apply (clarsimp simp add: locally_def)
```
```  2448 apply (drule assms; blast)
```
```  2449 done
```
```  2450
```
```  2451 lemma locally_path_connected_2:
```
```  2452   assumes "locally path_connected S"
```
```  2453           "openin (top_of_set S) t"
```
```  2454           "x \<in> t"
```
```  2455    shows "openin (top_of_set S) (path_component_set t x)"
```
```  2456 proof -
```
```  2457   { fix y :: 'a
```
```  2458     let ?SS = "top_of_set S"
```
```  2459     assume 1: "openin ?SS t"
```
```  2460               "\<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))"
```
```  2461     and "path_component t x y"
```
```  2462     then have "y \<in> t" and y: "y \<in> path_component_set t x"
```
```  2463       using path_component_mem(2) by blast+
```
```  2464     obtain F where
```
```  2465       "\<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))"
```
```  2466       by moura
```
```  2467     then obtain G where
```
```  2468        "\<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)"
```
```  2469       by moura
```
```  2470     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"
```
```  2471       using 1 \<open>y \<in> t\<close> by presburger
```
```  2472     have "G y t \<subseteq> path_component_set t y"
```
```  2473       using * path_component_maximal rev_subsetD by blast
```
```  2474     then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x"
```
```  2475       by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> dual_order.trans path_component_eq y)
```
```  2476   }
```
```  2477   then show ?thesis
```
```  2478     using assms openin_subopen by (force simp: locally_def)
```
```  2479 qed
```
```  2480
```
```  2481 lemma locally_path_connected_3:
```
```  2482   assumes "\<And>t x. \<lbrakk>openin (top_of_set S) t; x \<in> t\<rbrakk>
```
```  2483               \<Longrightarrow> openin (top_of_set S) (path_component_set t x)"
```
```  2484           "openin (top_of_set S) v" "x \<in> v"
```
```  2485    shows  "\<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
```
```  2486 proof -
```
```  2487   have "path_component v x x"
```
```  2488     by (meson assms(3) path_component_refl)
```
```  2489   then show ?thesis
```
```  2490     by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
```
```  2491 qed
```
```  2492
```
```  2493 proposition locally_path_connected:
```
```  2494   "locally path_connected S \<longleftrightarrow>
```
```  2495    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
```
```  2496           \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))"
```
```  2497   by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
```
```  2498
```
```  2499 proposition locally_path_connected_open_path_component:
```
```  2500   "locally path_connected S \<longleftrightarrow>
```
```  2501    (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t
```
```  2502           \<longrightarrow> openin (top_of_set S) (path_component_set t x))"
```
```  2503   by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
```
```  2504
```
```  2505 lemma locally_connected_open_component:
```
```  2506   "locally connected S \<longleftrightarrow>
```
```  2507    (\<forall>t c. openin (top_of_set S) t \<and> c \<in> components t
```
```  2508           \<longrightarrow> openin (top_of_set S) c)"
```
```  2509 by (metis components_iff locally_connected_open_connected_component)
```
```  2510
```
```  2511 proposition locally_connected_im_kleinen:
```
```  2512   "locally connected S \<longleftrightarrow>
```
```  2513    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
```
```  2514        \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and>
```
```  2515                 x \<in> u \<and> u \<subseteq> v \<and>
```
```  2516                 (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
```
```  2517    (is "?lhs = ?rhs")
```
```  2518 proof
```
```  2519   assume ?lhs
```
```  2520   then show ?rhs
```
```  2521     by (fastforce simp add: locally_connected)
```
```  2522 next
```
```  2523   assume ?rhs
```
```  2524   have *: "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> c"
```
```  2525        if "openin (top_of_set S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
```
```  2526   proof -
```
```  2527     from that \<open>?rhs\<close> [rule_format, of t x]
```
```  2528     obtain u where u:
```
```  2529       "openin (top_of_set S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
```
```  2530        (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
```
```  2531       using in_components_subset by auto
```
```  2532     obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
```
```  2533       "\<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))"
```
```  2534       by moura
```
```  2535     then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)"
```
```  2536       by (meson components_iff c)
```
```  2537     obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
```
```  2538         G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)"
```
```  2539       by moura
```
```  2540      have "G c u \<notin> u \<or> G c u \<in> c"
```
```  2541       using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3))
```
```  2542     then show ?thesis
```
```  2543       using G u by auto
```
```  2544   qed
```
```  2545   show ?lhs
```
```  2546     apply (clarsimp simp add: locally_connected_open_component)
```
```  2547     apply (subst openin_subopen)
```
```  2548     apply (blast intro: *)
```
```  2549     done
```
```  2550 qed
```
```  2551
```
```  2552 proposition locally_path_connected_im_kleinen:
```
```  2553   "locally path_connected S \<longleftrightarrow>
```
```  2554    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
```
```  2555        \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and>
```
```  2556                 x \<in> u \<and> u \<subseteq> v \<and>
```
```  2557                 (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
```
```  2558                                 pathstart p = x \<and> pathfinish p = y))))"
```
```  2559    (is "?lhs = ?rhs")
```
```  2560 proof
```
```  2561   assume ?lhs
```
```  2562   then show ?rhs
```
```  2563     apply (simp add: locally_path_connected path_connected_def)
```
```  2564     apply (erule all_forward ex_forward imp_forward conjE | simp)+
```
```  2565     by (meson dual_order.trans)
```
```  2566 next
```
```  2567   assume ?rhs
```
```  2568   have *: "\<exists>T. openin (top_of_set S) T \<and>
```
```  2569                x \<in> T \<and> T \<subseteq> path_component_set u z"
```
```  2570        if "openin (top_of_set S) u" and "z \<in> u" and c: "path_component u z x" for u z x
```
```  2571   proof -
```
```  2572     have "x \<in> u"
```
```  2573       by (meson c path_component_mem(2))
```
```  2574     with that \<open>?rhs\<close> [rule_format, of u x]
```
```  2575     obtain U where U:
```
```  2576       "openin (top_of_set S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
```
```  2577        (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
```
```  2578        by blast
```
```  2579     show ?thesis
```
```  2580       apply (rule_tac x=U in exI)
```
```  2581       apply (auto simp: U)
```
```  2582       apply (metis U c path_component_trans path_component_def)
```
```  2583       done
```
```  2584   qed
```
```  2585   show ?lhs
```
```  2586     apply (clarsimp simp add: locally_path_connected_open_path_component)
```
```  2587     apply (subst openin_subopen)
```
```  2588     apply (blast intro: *)
```
```  2589     done
```
```  2590 qed
```
```  2591
```
```  2592 lemma locally_path_connected_imp_locally_connected:
```
```  2593   "locally path_connected S \<Longrightarrow> locally connected S"
```
```  2594 using locally_mono path_connected_imp_connected by blast
```
```  2595
```
```  2596 lemma locally_connected_components:
```
```  2597   "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
```
```  2598 by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
```
```  2599
```
```  2600 lemma locally_path_connected_components:
```
```  2601   "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally path_connected c"
```
```  2602 by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)
```
```  2603
```
```  2604 lemma locally_path_connected_connected_component:
```
```  2605   "locally path_connected S \<Longrightarrow> locally path_connected (connected_component_set S x)"
```
```  2606 by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)
```
```  2607
```
```  2608 lemma open_imp_locally_path_connected:
```
```  2609   fixes S :: "'a :: real_normed_vector set"
```
```  2610   shows "open S \<Longrightarrow> locally path_connected S"
```
```  2611 apply (rule locally_mono [of convex])
```
```  2612 apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
```
```  2613 apply (meson open_ball centre_in_ball convex_ball openE order_trans)
```
```  2614 done
```
```  2615
```
```  2616 lemma open_imp_locally_connected:
```
```  2617   fixes S :: "'a :: real_normed_vector set"
```
```  2618   shows "open S \<Longrightarrow> locally connected S"
```
```  2619 by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)
```
```  2620
```
```  2621 lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
```
```  2622   by (simp add: open_imp_locally_path_connected)
```
```  2623
```
```  2624 lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
```
```  2625   by (simp add: open_imp_locally_connected)
```
```  2626
```
```  2627 lemma openin_connected_component_locally_connected:
```
```  2628     "locally connected S
```
```  2629      \<Longrightarrow> openin (top_of_set S) (connected_component_set S x)"
```
```  2630 apply (simp add: locally_connected_open_connected_component)
```
```  2631 by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
```
```  2632
```
```  2633 lemma openin_components_locally_connected:
```
```  2634     "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (top_of_set S) c"
```
```  2635   using locally_connected_open_component openin_subtopology_self by blast
```
```  2636
```
```  2637 lemma openin_path_component_locally_path_connected:
```
```  2638   "locally path_connected S
```
```  2639         \<Longrightarrow> openin (top_of_set S) (path_component_set S x)"
```
```  2640 by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
```
```  2641
```
```  2642 lemma closedin_path_component_locally_path_connected:
```
```  2643     "locally path_connected S
```
```  2644         \<Longrightarrow> closedin (top_of_set S) (path_component_set S x)"
```
```  2645 apply  (simp add: closedin_def path_component_subset complement_path_component_Union)
```
```  2646 apply (rule openin_Union)
```
```  2647 using openin_path_component_locally_path_connected by auto
```
```  2648
```
```  2649 lemma convex_imp_locally_path_connected:
```
```  2650   fixes S :: "'a:: real_normed_vector set"
```
```  2651   shows "convex S \<Longrightarrow> locally path_connected S"
```
```  2652 apply (clarsimp simp add: locally_path_connected)
```
```  2653 apply (subst (asm) openin_open)
```
```  2654 apply clarify
```
```  2655 apply (erule (1) openE)
```
```  2656 apply (rule_tac x = "S \<inter> ball x e" in exI)
```
```  2657 apply (force simp: convex_Int convex_imp_path_connected)
```
```  2658 done
```
```  2659
```
```  2660 lemma convex_imp_locally_connected:
```
```  2661   fixes S :: "'a:: real_normed_vector set"
```
```  2662   shows "convex S \<Longrightarrow> locally connected S"
```
```  2663   by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
```
```  2664
```
```  2665
```
```  2666 subsection\<open>Relations between components and path components\<close>
```
```  2667
```
```  2668 lemma path_component_eq_connected_component:
```
```  2669   assumes "locally path_connected S"
```
```  2670     shows "(path_component S x = connected_component S x)"
```
```  2671 proof (cases "x \<in> S")
```
```  2672   case True
```
```  2673   have "openin (top_of_set (connected_component_set S x)) (path_component_set S x)"
```
```  2674     apply (rule openin_subset_trans [of S])
```
```  2675     apply (intro conjI openin_path_component_locally_path_connected [OF assms])
```
```  2676     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
```
```  2677     done
```
```  2678   moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
```
```  2679     apply (rule closedin_subset_trans [of S])
```
```  2680     apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
```
```  2681     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
```
```  2682     done
```
```  2683   ultimately have *: "path_component_set S x = connected_component_set S x"
```
```  2684     by (metis connected_connected_component connected_clopen True path_component_eq_empty)
```
```  2685   then show ?thesis
```
```  2686     by blast
```
```  2687 next
```
```  2688   case False then show ?thesis
```
```  2689     by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
```
```  2690 qed
```
```  2691
```
```  2692 lemma path_component_eq_connected_component_set:
```
```  2693      "locally path_connected S \<Longrightarrow> (path_component_set S x = connected_component_set S x)"
```
```  2694 by (simp add: path_component_eq_connected_component)
```
```  2695
```
```  2696 lemma locally_path_connected_path_component:
```
```  2697      "locally path_connected S \<Longrightarrow> locally path_connected (path_component_set S x)"
```
```  2698 using locally_path_connected_connected_component path_component_eq_connected_component by fastforce
```
```  2699
```
```  2700 lemma open_path_connected_component:
```
```  2701   fixes S :: "'a :: real_normed_vector set"
```
```  2702   shows "open S \<Longrightarrow> path_component S x = connected_component S x"
```
```  2703 by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)
```
```  2704
```
```  2705 lemma open_path_connected_component_set:
```
```  2706   fixes S :: "'a :: real_normed_vector set"
```
```  2707   shows "open S \<Longrightarrow> path_component_set S x = connected_component_set S x"
```
```  2708 by (simp add: open_path_connected_component)
```
```  2709
```
```  2710 proposition locally_connected_quotient_image:
```
```  2711   assumes lcS: "locally connected S"
```
```  2712       and oo: "\<And>T. T \<subseteq> f ` S
```
```  2713                 \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
```
```  2714                     openin (top_of_set (f ` S)) T"
```
```  2715     shows "locally connected (f ` S)"
```
```  2716 proof (clarsimp simp: locally_connected_open_component)
```
```  2717   fix U C
```
```  2718   assume opefSU: "openin (top_of_set (f ` S)) U" and "C \<in> components U"
```
```  2719   then have "C \<subseteq> U" "U \<subseteq> f ` S"
```
```  2720     by (meson in_components_subset openin_imp_subset)+
```
```  2721   then have "openin (top_of_set (f ` S)) C \<longleftrightarrow>
```
```  2722              openin (top_of_set S) (S \<inter> f -` C)"
```
```  2723     by (auto simp: oo)
```
```  2724   moreover have "openin (top_of_set S) (S \<inter> f -` C)"
```
```  2725   proof (subst openin_subopen, clarify)
```
```  2726     fix x
```
```  2727     assume "x \<in> S" "f x \<in> C"
```
```  2728     show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
```
```  2729     proof (intro conjI exI)
```
```  2730       show "openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
```
```  2731       proof (rule ccontr)
```
```  2732         assume **: "\<not> openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
```
```  2733         then have "x \<notin> (S \<inter> f -` U)"
```
```  2734           using \<open>U \<subseteq> f ` S\<close> opefSU lcS locally_connected_2 oo by blast
```
```  2735         with ** show False
```
```  2736           by (metis (no_types) connected_component_eq_empty empty_iff openin_subopen)
```
```  2737       qed
```
```  2738     next
```
```  2739       show "x \<in> connected_component_set (S \<inter> f -` U) x"
```
```  2740         using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by auto
```
```  2741     next
```
```  2742       have contf: "continuous_on S f"
```
```  2743         by (simp add: continuous_on_open oo openin_imp_subset)
```
```  2744       then have "continuous_on (connected_component_set (S \<inter> f -` U) x) f"
```
```  2745         apply (rule continuous_on_subset)
```
```  2746         using connected_component_subset apply blast
```
```  2747         done
```
```  2748       then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
```
```  2749         by (rule connected_continuous_image [OF _ connected_connected_component])
```
```  2750       moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
```
```  2751         using connected_component_in by blast
```
```  2752       moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
```
```  2753         using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
```
```  2754       ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
```
```  2755         by (rule components_maximal [OF \<open>C \<in> components U\<close>])
```
```  2756       have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
```
```  2757         using connected_component_subset fC by blast
```
```  2758       have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
```
```  2759       proof -
```
```  2760         { assume "x \<in> connected_component_set (S \<inter> f -` U) x"
```
```  2761           then have ?thesis
```
```  2762             using cUC connected_component_idemp connected_component_mono by blast }
```
```  2763         then show ?thesis
```
```  2764           using connected_component_eq_empty by auto
```
```  2765       qed
```
```  2766       also have "\<dots> \<subseteq> (S \<inter> f -` C)"
```
```  2767         by (rule connected_component_subset)
```
```  2768       finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
```
```  2769     qed
```
```  2770   qed
```
```  2771   ultimately show "openin (top_of_set (f ` S)) C"
```
```  2772     by metis
```
```  2773 qed
```
```  2774
```
```  2775 text\<open>The proof resembles that above but is not identical!\<close>
```
```  2776 proposition locally_path_connected_quotient_image:
```
```  2777   assumes lcS: "locally path_connected S"
```
```  2778       and oo: "\<And>T. T \<subseteq> f ` S
```
```  2779                 \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow> openin (top_of_set (f ` S)) T"
```
```  2780     shows "locally path_connected (f ` S)"
```
```  2781 proof (clarsimp simp: locally_path_connected_open_path_component)
```
```  2782   fix U y
```
```  2783   assume opefSU: "openin (top_of_set (f ` S)) U" and "y \<in> U"
```
```  2784   then have "path_component_set U y \<subseteq> U" "U \<subseteq> f ` S"
```
```  2785     by (meson path_component_subset openin_imp_subset)+
```
```  2786   then have "openin (top_of_set (f ` S)) (path_component_set U y) \<longleftrightarrow>
```
```  2787              openin (top_of_set S) (S \<inter> f -` path_component_set U y)"
```
```  2788   proof -
```
```  2789     have "path_component_set U y \<subseteq> f ` S"
```
```  2790       using \<open>U \<subseteq> f ` S\<close> \<open>path_component_set U y \<subseteq> U\<close> by blast
```
```  2791     then show ?thesis
```
```  2792       using oo by blast
```
```  2793   qed
```
```  2794   moreover have "openin (top_of_set S) (S \<inter> f -` path_component_set U y)"
```
```  2795   proof (subst openin_subopen, clarify)
```
```  2796     fix x
```
```  2797     assume "x \<in> S" and Uyfx: "path_component U y (f x)"
```
```  2798     then have "f x \<in> U"
```
```  2799       using path_component_mem by blast
```
```  2800     show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
```
```  2801     proof (intro conjI exI)
```
```  2802       show "openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
```
```  2803       proof (rule ccontr)
```
```  2804         assume **: "\<not> openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
```
```  2805         then have "x \<notin> (S \<inter> f -` U)"
```
```  2806           by (metis (no_types, lifting) \<open>U \<subseteq> f ` S\<close> opefSU lcS oo locally_path_connected_open_path_component)
```
```  2807         then show False
```
```  2808           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
```
```  2809       qed
```
```  2810     next
```
```  2811       show "x \<in> path_component_set (S \<inter> f -` U) x"
```
```  2812         by (simp add: \<open>f x \<in> U\<close> \<open>x \<in> S\<close> path_component_refl)
```
```  2813     next
```
```  2814       have contf: "continuous_on S f"
```
```  2815         by (simp add: continuous_on_open oo openin_imp_subset)
```
```  2816       then have "continuous_on (path_component_set (S \<inter> f -` U) x) f"
```
```  2817         apply (rule continuous_on_subset)
```
```  2818         using path_component_subset apply blast
```
```  2819         done
```
```  2820       then have "path_connected (f ` path_component_set (S \<inter> f -` U) x)"
```
```  2821         by (simp add: path_connected_continuous_image)
```
```  2822       moreover have "f ` path_component_set (S \<inter> f -` U) x \<subseteq> U"
```
```  2823         using path_component_mem by fastforce
```
```  2824       moreover have "f x \<in> f ` path_component_set (S \<inter> f -` U) x"
```
```  2825         by (force simp: \<open>x \<in> S\<close> \<open>f x \<in> U\<close> path_component_refl_eq)
```
```  2826       ultimately have "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U (f x)"
```
```  2827         by (meson path_component_maximal)
```
```  2828        also have  "\<dots> \<subseteq> path_component_set U y"
```
```  2829         by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
```
```  2830       finally have fC: "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U y" .
```
```  2831       have cUC: "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)"
```
```  2832         using path_component_subset fC by blast
```
```  2833       have "path_component_set (S \<inter> f -` U) x \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) x"
```
```  2834       proof -
```
```  2835         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"
```
```  2836           using cUC path_component_mono by blast
```
```  2837         then show ?thesis
```
```  2838           using path_component_path_component by blast
```
```  2839       qed
```
```  2840       also have "\<dots> \<subseteq> (S \<inter> f -` path_component_set U y)"
```
```  2841         by (rule path_component_subset)
```
```  2842       finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
```
```  2843     qed
```
```  2844   qed
```
```  2845   ultimately show "openin (top_of_set (f ` S)) (path_component_set U y)"
```
```  2846     by metis
```
```  2847 qed
```
```  2848
```
```  2849 subsection%unimportant\<open>Components, continuity, openin, closedin\<close>
```
```  2850
```
```  2851 lemma continuous_on_components_gen:
```
```  2852  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  2853   assumes "\<And>c. c \<in> components S \<Longrightarrow>
```
```  2854               openin (top_of_set S) c \<and> continuous_on c f"
```
```  2855     shows "continuous_on S f"
```
```  2856 proof (clarsimp simp: continuous_openin_preimage_eq)
```
```  2857   fix t :: "'b set"
```
```  2858   assume "open t"
```
```  2859   have *: "S \<inter> f -` t = (\<Union>c \<in> components S. c \<inter> f -` t)"
```
```  2860     by auto
```
```  2861   show "openin (top_of_set S) (S \<inter> f -` t)"
```
```  2862     unfolding * using \<open>open t\<close> assms continuous_openin_preimage_gen openin_trans openin_Union by blast
```
```  2863 qed
```
```  2864
```
```  2865 lemma continuous_on_components:
```
```  2866  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  2867   assumes "locally connected S "
```
```  2868           "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
```
```  2869     shows "continuous_on S f"
```
```  2870 apply (rule continuous_on_components_gen)
```
```  2871 apply (auto simp: assms intro: openin_components_locally_connected)
```
```  2872 done
```
```  2873
```
```  2874 lemma continuous_on_components_eq:
```
```  2875     "locally connected S
```
```  2876      \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
```
```  2877 by (meson continuous_on_components continuous_on_subset in_components_subset)
```
```  2878
```
```  2879 lemma continuous_on_components_open:
```
```  2880  fixes S :: "'a::real_normed_vector set"
```
```  2881   assumes "open S "
```
```  2882           "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
```
```  2883     shows "continuous_on S f"
```
```  2884 using continuous_on_components open_imp_locally_connected assms by blast
```
```  2885
```
```  2886 lemma continuous_on_components_open_eq:
```
```  2887   fixes S :: "'a::real_normed_vector set"
```
```  2888   shows "open S \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
```
```  2889 using continuous_on_subset in_components_subset
```
```  2890 by (blast intro: continuous_on_components_open)
```
```  2891
```
```  2892 lemma closedin_union_complement_components:
```
```  2893   assumes u: "locally connected u"
```
```  2894       and S: "closedin (top_of_set u) S"
```
```  2895       and cuS: "c \<subseteq> components(u - S)"
```
```  2896     shows "closedin (top_of_set u) (S \<union> \<Union>c)"
```
```  2897 proof -
```
```  2898   have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c'
```
```  2899     by (simp add: disjnt_def) blast
```
```  2900   have "S \<subseteq> u"
```
```  2901     using S closedin_imp_subset by blast
```
```  2902   moreover have "u - S = \<Union>c \<union> \<Union>(components (u - S) - c)"
```
```  2903     by (metis Diff_partition Union_components Union_Un_distrib assms(3))
```
```  2904   moreover have "disjnt (\<Union>c) (\<Union>(components (u - S) - c))"
```
```  2905     apply (rule di)
```
```  2906     by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
```
```  2907   ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))"
```
```  2908     by (auto simp: disjnt_def)
```
```  2909   have *: "openin (top_of_set u) (\<Union>(components (u - S) - c))"
```
```  2910     apply (rule openin_Union)
```
```  2911     apply (rule openin_trans [of "u - S"])
```
```  2912     apply (simp add: u S locally_diff_closed openin_components_locally_connected)
```
```  2913     apply (simp add: openin_diff S)
```
```  2914     done
```
```  2915   have "openin (top_of_set u) (u - (u - \<Union>(components (u - S) - c)))"
```
```  2916     apply (rule openin_diff, simp)
```
```  2917     apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
```
```  2918     done
```
```  2919   then show ?thesis
```
```  2920     by (force simp: eq closedin_def)
```
```  2921 qed
```
```  2922
```
```  2923 lemma closed_union_complement_components:
```
```  2924   fixes S :: "'a::real_normed_vector set"
```
```  2925   assumes S: "closed S" and c: "c \<subseteq> components(- S)"
```
```  2926     shows "closed(S \<union> \<Union> c)"
```
```  2927 proof -
```
```  2928   have "closedin (top_of_set UNIV) (S \<union> \<Union>c)"
```
```  2929     apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
```
```  2930     using S c apply (simp_all add: Compl_eq_Diff_UNIV)
```
```  2931     done
```
```  2932   then show ?thesis by simp
```
```  2933 qed
```
```  2934
```
```  2935 lemma closedin_Un_complement_component:
```
```  2936   fixes S :: "'a::real_normed_vector set"
```
```  2937   assumes u: "locally connected u"
```
```  2938       and S: "closedin (top_of_set u) S"
```
```  2939       and c: " c \<in> components(u - S)"
```
```  2940     shows "closedin (top_of_set u) (S \<union> c)"
```
```  2941 proof -
```
```  2942   have "closedin (top_of_set u) (S \<union> \<Union>{c})"
```
```  2943     using c by (blast intro: closedin_union_complement_components [OF u S])
```
```  2944   then show ?thesis
```
```  2945     by simp
```
```  2946 qed
```
```  2947
```
```  2948 lemma closed_Un_complement_component:
```
```  2949   fixes S :: "'a::real_normed_vector set"
```
```  2950   assumes S: "closed S" and c: " c \<in> components(-S)"
```
```  2951     shows "closed (S \<union> c)"
```
```  2952   by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
```
```  2953       locally_connected_UNIV subtopology_UNIV)
```
```  2954
```
```  2955
```
```  2956 subsection\<open>Existence of isometry between subspaces of same dimension\<close>
```
```  2957
```
```  2958 lemma isometry_subset_subspace:
```
```  2959   fixes S :: "'a::euclidean_space set"
```
```  2960     and T :: "'b::euclidean_space set"
```
```  2961   assumes S: "subspace S"
```
```  2962       and T: "subspace T"
```
```  2963       and d: "dim S \<le> dim T"
```
```  2964   obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
```
```  2965 proof -
```
```  2966   obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
```
```  2967              and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
```
```  2968              and "independent B" "finite B" "card B = dim S" "span B = S"
```
```  2969     by (metis orthonormal_basis_subspace [OF S] independent_finite)
```
```  2970   obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
```
```  2971              and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
```
```  2972              and "independent C" "finite C" "card C = dim T" "span C = T"
```
```  2973     by (metis orthonormal_basis_subspace [OF T] independent_finite)
```
```  2974   obtain fb where "fb ` B \<subseteq> C" "inj_on fb B"
```
```  2975     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)
```
```  2976   then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
```
```  2977     using Corth
```
```  2978     apply (auto simp: pairwise_def orthogonal_clauses)
```
```  2979     by (meson subsetD image_eqI inj_on_def)
```
```  2980   obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
```
```  2981     using linear_independent_extend \<open>independent B\<close> by fastforce
```
```  2982   have "span (f ` B) \<subseteq> span C"
```
```  2983     by (metis \<open>fb ` B \<subseteq> C\<close> ffb image_cong span_mono)
```
```  2984   then have "f ` S \<subseteq> T"
```
```  2985     unfolding \<open>span B = S\<close> \<open>span C = T\<close> span_linear_image[OF \<open>linear f\<close>] .
```
```  2986   have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
```
```  2987     using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto
```
```  2988   have "norm (f x) = norm x" if "x \<in> S" for x
```
```  2989   proof -
```
```  2990     interpret linear f by fact
```
```  2991     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
```
```  2992       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
```
```  2993     have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by (simp add: sum scale ffb x)
```
```  2994     also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
```
```  2995       apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
```
```  2996       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
```
```  2997       done
```
```  2998     also have "\<dots> = norm x ^2"
```
```  2999       by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
```
```  3000     finally show ?thesis
```
```  3001       by (simp add: norm_eq_sqrt_inner)
```
```  3002   qed
```
```  3003   then show ?thesis
```
```  3004     by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
```
```  3005 qed
```
```  3006
```
```  3007 proposition isometries_subspaces:
```
```  3008   fixes S :: "'a::euclidean_space set"
```
```  3009     and T :: "'b::euclidean_space set"
```
```  3010   assumes S: "subspace S"
```
```  3011       and T: "subspace T"
```
```  3012       and d: "dim S = dim T"
```
```  3013   obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
```
```  3014                     "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
```
```  3015                     "\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x"
```
```  3016                     "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
```
```  3017                     "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
```
```  3018 proof -
```
```  3019   obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
```
```  3020              and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
```
```  3021              and "independent B" "finite B" "card B = dim S" "span B = S"
```
```  3022     by (metis orthonormal_basis_subspace [OF S] independent_finite)
```
```  3023   obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
```
```  3024              and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
```
```  3025              and "independent C" "finite C" "card C = dim T" "span C = T"
```
```  3026     by (metis orthonormal_basis_subspace [OF T] independent_finite)
```
```  3027   obtain fb where "bij_betw fb B C"
```
```  3028     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)
```
```  3029   then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
```
```  3030     using Corth
```
```  3031     apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
```
```  3032     by (meson subsetD image_eqI inj_on_def)
```
```  3033   obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
```
```  3034     using linear_independent_extend \<open>independent B\<close> by fastforce
```
```  3035   interpret f: linear f by fact
```
```  3036   define gb where "gb \<equiv> inv_into B fb"
```
```  3037   then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C"
```
```  3038     using Borth
```
```  3039     apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
```
```  3040     by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
```
```  3041   obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x"
```
```  3042     using linear_independent_extend \<open>independent C\<close> by fastforce
```
```  3043   interpret g: linear g by fact
```
```  3044   have "span (f ` B) \<subseteq> span C"
```
```  3045     by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb image_cong)
```
```  3046   then have "f ` S \<subseteq> T"
```
```  3047     unfolding \<open>span B = S\<close> \<open>span C = T\<close>
```
```  3048       span_linear_image[OF \<open>linear f\<close>] .
```
```  3049   have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
```
```  3050     using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce
```
```  3051   have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x
```
```  3052   proof -
```
```  3053     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
```
```  3054       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
```
```  3055     have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
```
```  3056       using linear_sum [OF \<open>linear f\<close>] x by auto
```
```  3057     also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
```
```  3058       by (simp add: f.sum f.scale)
```
```  3059     also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
```
```  3060       by (simp add: ffb cong: sum.cong)
```
```  3061     finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
```
```  3062     then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
```
```  3063     also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
```
```  3064       apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
```
```  3065       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
```
```  3066       done
```
```  3067     also have "\<dots> = (norm x)\<^sup>2"
```
```  3068       by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
```
```  3069     finally show "norm (f x) = norm x"
```
```  3070       by (simp add: norm_eq_sqrt_inner)
```
```  3071     have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
```
```  3072     also have "\<dots> = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
```
```  3073       by (simp add: g.sum g.scale)
```
```  3074     also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
```
```  3075       by (simp add: g.scale)
```
```  3076     also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R v)"
```
```  3077       apply (rule sum.cong [OF refl])
```
```  3078       using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
```
```  3079     also have "\<dots> = x"
```
```  3080       using x by blast
```
```  3081     finally show "g (f x) = x" .
```
```  3082   qed
```
```  3083   have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x"
```
```  3084     by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into)
```
```  3085   have g [simp]: "f (g x) = x" if "x \<in> T" for x
```
```  3086   proof -
```
```  3087     obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
```
```  3088       using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
```
```  3089     have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
```
```  3090       by (simp add: x g.sum)
```
```  3091     also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
```
```  3092       by (simp add: g.scale)
```
```  3093     also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
```
```  3094       by (simp add: ggb cong: sum.cong)
```
```  3095     finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
```
```  3096     also have "\<dots> = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
```
```  3097       by (simp add: f.scale f.sum)
```
```  3098     also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
```
```  3099       by (simp add: f.scale f.sum)
```
```  3100     also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R v)"
```
```  3101       using \<open>bij_betw fb B C\<close>
```
```  3102       by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
```
```  3103     also have "\<dots> = x"
```
```  3104       using x by blast
```
```  3105     finally show "f (g x) = x" .
```
```  3106   qed
```
```  3107   have gim: "g ` T = S"
```
```  3108     by (metis (full_types) S T \<open>f ` S \<subseteq> T\<close> d dim_eq_span dim_image_le f(2) g.linear_axioms
```
```  3109         image_iff linear_subspace_image span_eq_iff subset_iff)
```
```  3110   have fim: "f ` S = T"
```
```  3111     using \<open>g ` T = S\<close> image_iff by fastforce
```
```  3112   have [simp]: "norm (g x) = norm x" if "x \<in> T" for x
```
```  3113     using fim that by auto
```
```  3114   show ?thesis
```
```  3115     apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>])
```
```  3116     apply (simp_all add: fim gim)
```
```  3117     done
```
```  3118 qed
```
```  3119
```
```  3120 corollary isometry_subspaces:
```
```  3121   fixes S :: "'a::euclidean_space set"
```
```  3122     and T :: "'b::euclidean_space set"
```
```  3123   assumes S: "subspace S"
```
```  3124       and T: "subspace T"
```
```  3125       and d: "dim S = dim T"
```
```  3126   obtains f where "linear f" "f ` S = T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
```
```  3127 using isometries_subspaces [OF assms]
```
```  3128 by metis
```
```  3129
```
```  3130 corollary isomorphisms_UNIV_UNIV:
```
```  3131   assumes "DIM('M) = DIM('N)"
```
```  3132   obtains f::"'M::euclidean_space \<Rightarrow>'N::euclidean_space" and g
```
```  3133   where "linear f" "linear g"
```
```  3134                     "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y"
```
```  3135                     "\<And>x. g (f x) = x" "\<And>y. f(g y) = y"
```
```  3136   using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])
```
```  3137
```
```  3138 lemma homeomorphic_subspaces:
```
```  3139   fixes S :: "'a::euclidean_space set"
```
```  3140     and T :: "'b::euclidean_space set"
```
```  3141   assumes S: "subspace S"
```
```  3142       and T: "subspace T"
```
```  3143       and d: "dim S = dim T"
```
```  3144     shows "S homeomorphic T"
```
```  3145 proof -
```
```  3146   obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
```
```  3147                    "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
```
```  3148     by (blast intro: isometries_subspaces [OF assms])
```
```  3149   then show ?thesis
```
```  3150     apply (simp add: homeomorphic_def homeomorphism_def)
```
```  3151     apply (rule_tac x=f in exI)
```
```  3152     apply (rule_tac x=g in exI)
```
```  3153     apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
```
```  3154     done
```
```  3155 qed
```
```  3156
```
```  3157 lemma homeomorphic_affine_sets:
```
```  3158   assumes "affine S" "affine T" "aff_dim S = aff_dim T"
```
```  3159     shows "S homeomorphic T"
```
```  3160 proof (cases "S = {} \<or> T = {}")
```
```  3161   case True  with assms aff_dim_empty homeomorphic_empty show ?thesis
```
```  3162     by metis
```
```  3163 next
```
```  3164   case False
```
```  3165   then obtain a b where ab: "a \<in> S" "b \<in> T" by auto
```
```  3166   then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
```
```  3167     using affine_diffs_subspace assms by blast+
```
```  3168   have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
```
```  3169     using assms ab  by (simp add: aff_dim_eq_dim  [OF hull_inc] image_def)
```
```  3170   have "S homeomorphic ((+) (- a) ` S)"
```
```  3171     by (fact homeomorphic_translation)
```
```  3172   also have "\<dots> homeomorphic ((+) (- b) ` T)"
```
```  3173     by (rule homeomorphic_subspaces [OF ss dd])
```
```  3174   also have "\<dots> homeomorphic T"
```
```  3175     using homeomorphic_translation [of T "- b"] by (simp add: homeomorphic_sym [of T])
```
```  3176   finally show ?thesis .
```
```  3177 qed
```
```  3178
```
```  3179
```
```  3180 subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
```
```  3181
```
```  3182 locale%important Retracts =
```
```  3183   fixes s h t k
```
```  3184   assumes conth: "continuous_on s h"
```
```  3185       and imh: "h ` s = t"
```
```  3186       and contk: "continuous_on t k"
```
```  3187       and imk: "k ` t \<subseteq> s"
```
```  3188       and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
```
```  3189
```
```  3190 begin
```
```  3191
```
```  3192 lemma homotopically_trivial_retraction_gen:
```
```  3193   assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
```
```  3194       and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
```
```  3195       and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
```
```  3196       and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f;
```
```  3197                        continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk>
```
```  3198                        \<Longrightarrow> homotopic_with P u s f g"
```
```  3199       and contf: "continuous_on u f" and imf: "f ` u \<subseteq> t" and Qf: "Q f"
```
```  3200       and contg: "continuous_on u g" and img: "g ` u \<subseteq> t" and Qg: "Q g"
```
```  3201     shows "homotopic_with Q u t f g"
```
```  3202 proof -
```
```  3203   have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
```
```  3204   have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto
```
```  3205   have "continuous_on u (k \<circ> f)"
```
```  3206     using contf continuous_on_compose continuous_on_subset contk imf by blast
```
```  3207   moreover have "(k \<circ> f) ` u \<subseteq> s"
```
```  3208     using imf imk by fastforce
```
```  3209   moreover have "P (k \<circ> f)"
```
```  3210     by (simp add: P Qf contf imf)
```
```  3211   moreover have "continuous_on u (k \<circ> g)"
```
```  3212     using contg continuous_on_compose continuous_on_subset contk img by blast
```
```  3213   moreover have "(k \<circ> g) ` u \<subseteq> s"
```
```  3214     using img imk by fastforce
```
```  3215   moreover have "P (k \<circ> g)"
```
```  3216     by (simp add: P Qg contg img)
```
```  3217   ultimately have "homotopic_with P u s (k \<circ> f) (k \<circ> g)"
```
```  3218     by (rule hom)
```
```  3219   then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
```
```  3220     apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
```
```  3221     using Q by (auto simp: conth imh)
```
```  3222   then show ?thesis
```
```  3223     apply (rule homotopic_with_eq)
```
```  3224     apply (metis feq)
```
```  3225     apply (metis geq)
```
```  3226     apply (metis Qeq)
```
```  3227     done
```
```  3228 qed
```
```  3229
```
```  3230 lemma homotopically_trivial_retraction_null_gen:
```
```  3231   assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
```
```  3232       and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
```
```  3233       and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
```
```  3234       and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk>
```
```  3235                      \<Longrightarrow> \<exists>c. homotopic_with P u s f (\<lambda>x. c)"
```
```  3236       and contf: "continuous_on u f" and imf:"f ` u \<subseteq> t" and Qf: "Q f"
```
```  3237   obtains c where "homotopic_with Q u t f (\<lambda>x. c)"
```
```  3238 proof -
```
```  3239   have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
```
```  3240   have "continuous_on u (k \<circ> f)"
```
```  3241     using contf continuous_on_compose continuous_on_subset contk imf by blast
```
```  3242   moreover have "(k \<circ> f) ` u \<subseteq> s"
```
```  3243     using imf imk by fastforce
```
```  3244   moreover have "P (k \<circ> f)"
```
```  3245     by (simp add: P Qf contf imf)
```
```  3246   ultimately obtain c where "homotopic_with P u s (k \<circ> f) (\<lambda>x. c)"
```
```  3247     by (metis hom)
```
```  3248   then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
```
```  3249     apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
```
```  3250     using Q by (auto simp: conth imh)
```
```  3251   then show ?thesis
```
```  3252     apply (rule_tac c = "h c" in that)
```
```  3253     apply (erule homotopic_with_eq)
```
```  3254     apply (metis feq, simp)
```
```  3255     apply (metis Qeq)
```
```  3256     done
```
```  3257 qed
```
```  3258
```
```  3259 lemma cohomotopically_trivial_retraction_gen:
```
```  3260   assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
```
```  3261       and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
```
```  3262       and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
```
```  3263       and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f;
```
```  3264                        continuous_on s g; g ` s \<subseteq> u; P g\<rbrakk>
```
```  3265                        \<Longrightarrow> homotopic_with P s u f g"
```
```  3266       and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
```
```  3267       and contg: "continuous_on t g" and img: "g ` t \<subseteq> u" and Qg: "Q g"
```
```  3268     shows "homotopic_with Q t u f g"
```
```  3269 proof -
```
```  3270   have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
```
```  3271   have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto
```
```  3272   have "continuous_on s (f \<circ> h)"
```
```  3273     using contf conth continuous_on_compose imh by blast
```
```  3274   moreover have "(f \<circ> h) ` s \<subseteq> u"
```
```  3275     using imf imh by fastforce
```
```  3276   moreover have "P (f \<circ> h)"
```
```  3277     by (simp add: P Qf contf imf)
```
```  3278   moreover have "continuous_on s (g \<circ> h)"
```
```  3279     using contg continuous_on_compose continuous_on_subset conth imh by blast
```
```  3280   moreover have "(g \<circ> h) ` s \<subseteq> u"
```
```  3281     using img imh by fastforce
```
```  3282   moreover have "P (g \<circ> h)"
```
```  3283     by (simp add: P Qg contg img)
```
```  3284   ultimately have "homotopic_with P s u (f \<circ> h) (g \<circ> h)"
```
```  3285     by (rule hom)
```
```  3286   then have "homotopic_with Q t u (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
```
```  3287     apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
```
```  3288     using Q by (auto simp: contk imk)
```
```  3289   then show ?thesis
```
```  3290     apply (rule homotopic_with_eq)
```
```  3291     apply (metis feq)
```
```  3292     apply (metis geq)
```
```  3293     apply (metis Qeq)
```
```  3294     done
```
```  3295 qed
```
```  3296
```
```  3297 lemma cohomotopically_trivial_retraction_null_gen:
```
```  3298   assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
```
```  3299       and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
```
```  3300       and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
```
```  3301       and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk>
```
```  3302                        \<Longrightarrow> \<exists>c. homotopic_with P s u f (\<lambda>x. c)"
```
```  3303       and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
```
```  3304   obtains c where "homotopic_with Q t u f (\<lambda>x. c)"
```
```  3305 proof -
```
```  3306   have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
```
```  3307   have "continuous_on s (f \<circ> h)"
```
```  3308     using contf conth continuous_on_compose imh by blast
```
```  3309   moreover have "(f \<circ> h) ` s \<subseteq> u"
```
```  3310     using imf imh by fastforce
```
```  3311   moreover have "P (f \<circ> h)"
```
```  3312     by (simp add: P Qf contf imf)
```
```  3313   ultimately obtain c where "homotopic_with P s u (f \<circ> h) (\<lambda>x. c)"
```
```  3314     by (metis hom)
```
```  3315   then have "homotopic_with Q t u (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
```
```  3316     apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
```
```  3317     using Q by (auto simp: contk imk)
```
```  3318   then show ?thesis
```
```  3319     apply (rule_tac c = c in that)
```
```  3320     apply (erule homotopic_with_eq)
```
```  3321     apply (metis feq, simp)
```
```  3322     apply (metis Qeq)
```
```  3323     done
```
```  3324 qed
```
```  3325
```
```  3326 end
```
```  3327
```
```  3328 lemma simply_connected_retraction_gen:
```
```  3329   shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T;
```
```  3330           continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
```
```  3331         \<Longrightarrow> simply_connected T"
```
```  3332 apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
```
```  3333 apply (rule Retracts.homotopically_trivial_retraction_gen
```
```  3334         [of S h _ k _ "\<lambda>p. pathfinish p = pathstart p"  "\<lambda>p. pathfinish p = pathstart p"])
```
```  3335 apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
```
```  3336 done
```
```  3337
```
```  3338 lemma homeomorphic_simply_connected:
```
```  3339     "\<lbrakk>S homeomorphic T; simply_connected S\<rbrakk> \<Longrightarrow> simply_connected T"
```
```  3340   by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)
```
```  3341
```
```  3342 lemma homeomorphic_simply_connected_eq:
```
```  3343     "S homeomorphic T \<Longrightarrow> (simply_connected S \<longleftrightarrow> simply_connected T)"
```
```  3344   by (metis homeomorphic_simply_connected homeomorphic_sym)
```
```  3345
```
```  3346
```
```  3347 subsection\<open>Homotopy equivalence\<close>
```
```  3348
```
```  3349 definition%important homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
```
```  3350              (infix "homotopy'_eqv" 50)
```
```  3351   where "S homotopy_eqv T \<equiv>
```
```  3352         \<exists>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
```
```  3353               continuous_on T g \<and> g ` T \<subseteq> S \<and>
```
```  3354               homotopic_with (\<lambda>x. True) S S (g \<circ> f) id \<and>
```
```  3355               homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
```
```  3356
```
```  3357 lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
```
```  3358   unfolding homeomorphic_def homotopy_eqv_def homeomorphism_def
```
```  3359   by (fastforce intro!: homotopic_with_equal continuous_on_compose)
```
```  3360
```
```  3361 lemma homotopy_eqv_refl: "S homotopy_eqv S"
```
```  3362   by (rule homeomorphic_imp_homotopy_eqv homeomorphic_refl)+
```
```  3363
```
```  3364 lemma homotopy_eqv_sym: "S homotopy_eqv T \<longleftrightarrow> T homotopy_eqv S"
```
```  3365   by (auto simp: homotopy_eqv_def)
```
```  3366
```
```  3367 lemma homotopy_eqv_trans [trans]:
```
```  3368     fixes S :: "'a::real_normed_vector set" and U :: "'c::real_normed_vector set"
```
```  3369   assumes ST: "S homotopy_eqv T" and TU: "T homotopy_eqv U"
```
```  3370     shows "S homotopy_eqv U"
```
```  3371 proof -
```
```  3372   obtain f1 g1 where f1: "continuous_on S f1" "f1 ` S \<subseteq> T"
```
```  3373                  and g1: "continuous_on T g1" "g1 ` T \<subseteq> S"
```
```  3374                  and hom1: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> f1) id"
```
```  3375                            "homotopic_with (\<lambda>x. True) T T (f1 \<circ> g1) id"
```
```  3376     using ST by (auto simp: homotopy_eqv_def)
```
```  3377   obtain f2 g2 where f2: "continuous_on T f2" "f2 ` T \<subseteq> U"
```
```  3378                  and g2: "continuous_on U g2" "g2 ` U \<subseteq> T"
```
```  3379                  and hom2: "homotopic_with (\<lambda>x. True) T T (g2 \<circ> f2) id"
```
```  3380                            "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
```
```  3381     using TU by (auto simp: homotopy_eqv_def)
```
```  3382   have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
```
```  3383     by (rule homotopic_with_compose_continuous_right hom2 f1)+
```
```  3384   then have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
```
```  3385     by (simp add: o_assoc)
```
```  3386   then have "homotopic_with (\<lambda>x. True) S S
```
```  3387          (g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
```
```  3388     by (simp add: g1 homotopic_with_compose_continuous_left)
```
```  3389   moreover have "homotopic_with (\<lambda>x. True) S S (g1 \<circ> id \<circ> f1) id"
```
```  3390     using hom1 by simp
```
```  3391   ultimately have SS: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
```
```  3392     apply (simp add: o_assoc)
```
```  3393     apply (blast intro: homotopic_with_trans)
```
```  3394     done
```
```  3395   have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
```
```  3396     by (rule homotopic_with_compose_continuous_right hom1 g2)+
```
```  3397   then have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> (g1 \<circ> g2)) (id \<circ> g2)"
```
```  3398     by (simp add: o_assoc)
```
```  3399   then have "homotopic_with (\<lambda>x. True) U U
```
```  3400          (f2 \<circ> (f1 \<circ> (g1 \<circ> g2))) (f2 \<circ> (id \<circ> g2))"
```
```  3401     by (simp add: f2 homotopic_with_compose_continuous_left)
```
```  3402   moreover have "homotopic_with (\<lambda>x. True) U U (f2 \<circ> id \<circ> g2) id"
```
```  3403     using hom2 by simp
```
```  3404   ultimately have UU: "homotopic_with (\<lambda>x. True) U U (f2 \<circ> f1 \<circ> (g1 \<circ> g2)) id"
```
```  3405     apply (simp add: o_assoc)
```
```  3406     apply (blast intro: homotopic_with_trans)
```
```  3407     done
```
```  3408   show ?thesis
```
```  3409     unfolding homotopy_eqv_def
```
```  3410     apply (rule_tac x = "f2 \<circ> f1" in exI)
```
```  3411     apply (rule_tac x = "g1 \<circ> g2" in exI)
```
```  3412     apply (intro conjI continuous_on_compose SS UU)
```
```  3413     using f1 f2 g1 g2  apply (force simp: elim!: continuous_on_subset)+
```
```  3414     done
```
```  3415 qed
```
```  3416
```
```  3417 lemma homotopy_eqv_inj_linear_image:
```
```  3418   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3419   assumes "linear f" "inj f"
```
```  3420     shows "(f ` S) homotopy_eqv S"
```
```  3421 apply (rule homeomorphic_imp_homotopy_eqv)
```
```  3422 using assms homeomorphic_sym linear_homeomorphic_image by auto
```
```  3423
```
```  3424 lemma homotopy_eqv_translation:
```
```  3425     fixes S :: "'a::real_normed_vector set"
```
```  3426     shows "(+) a ` S homotopy_eqv S"
```
```  3427   apply (rule homeomorphic_imp_homotopy_eqv)
```
```  3428   using homeomorphic_translation homeomorphic_sym by blast
```
```  3429
```
```  3430 lemma homotopy_eqv_homotopic_triviality_imp:
```
```  3431   fixes S :: "'a::real_normed_vector set"
```
```  3432     and T :: "'b::real_normed_vector set"
```
```  3433     and U :: "'c::real_normed_vector set"
```
```  3434   assumes "S homotopy_eqv T"
```
```  3435       and f: "continuous_on U f" "f ` U \<subseteq> T"
```
```  3436       and g: "continuous_on U g" "g ` U \<subseteq> T"
```
```  3437       and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
```
```  3438                          continuous_on U g; g ` U \<subseteq> S\<rbrakk>
```
```  3439                          \<Longrightarrow> homotopic_with (\<lambda>x. True) U S f g"
```
```  3440     shows "homotopic_with (\<lambda>x. True) U T f g"
```
```  3441 proof -
```
```  3442   obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
```
```  3443                and k: "continuous_on T k" "k ` T \<subseteq> S"
```
```  3444                and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
```
```  3445                         "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
```
```  3446     using assms by (auto simp: homotopy_eqv_def)
```
```  3447   have "homotopic_with (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
```
```  3448     apply (rule homUS)
```
```  3449     using f g k
```
```  3450     apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
```
```  3451     apply (force simp: o_def)+
```
```  3452     done
```
```  3453   then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
```
```  3454     apply (rule homotopic_with_compose_continuous_left)
```
```  3455     apply (simp_all add: h)
```
```  3456     done
```
```  3457   moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
```
```  3458     apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
```
```  3459     apply (auto simp: hom f)
```
```  3460     done
```
```  3461   moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
```
```  3462     apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
```
```  3463     apply (auto simp: hom g)
```
```  3464     done
```
```  3465   ultimately show "homotopic_with (\<lambda>x. True) U T f g"
```
```  3466     apply (simp add: o_assoc)
```
```  3467     using homotopic_with_trans homotopic_with_sym by blast
```
```  3468 qed
```
```  3469
```
```  3470 lemma homotopy_eqv_homotopic_triviality:
```
```  3471   fixes S :: "'a::real_normed_vector set"
```
```  3472     and T :: "'b::real_normed_vector set"
```
```  3473     and U :: "'c::real_normed_vector set"
```
```  3474   assumes "S homotopy_eqv T"
```
```  3475     shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
```
```  3476                    continuous_on U g \<and> g ` U \<subseteq> S
```
```  3477                    \<longrightarrow> homotopic_with (\<lambda>x. True) U S f g) \<longleftrightarrow>
```
```  3478            (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and>
```
```  3479                   continuous_on U g \<and> g ` U \<subseteq> T
```
```  3480                   \<longrightarrow> homotopic_with (\<lambda>x. True) U T f g)"
```
```  3481 apply (rule iffI)
```
```  3482 apply (metis assms homotopy_eqv_homotopic_triviality_imp)
```
```  3483 by (metis (no_types) assms homotopy_eqv_homotopic_triviality_imp homotopy_eqv_sym)
```
```  3484
```
```  3485 lemma homotopy_eqv_cohomotopic_triviality_null_imp:
```
```  3486   fixes S :: "'a::real_normed_vector set"
```
```  3487     and T :: "'b::real_normed_vector set"
```
```  3488     and U :: "'c::real_normed_vector set"
```
```  3489   assumes "S homotopy_eqv T"
```
```  3490       and f: "continuous_on T f" "f ` T \<subseteq> U"
```
```  3491       and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
```
```  3492                       \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c)"
```
```  3493   obtains c where "homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)"
```
```  3494 proof -
```
```  3495   obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
```
```  3496                and k: "continuous_on T k" "k ` T \<subseteq> S"
```
```  3497                and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
```
```  3498                         "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
```
```  3499     using assms by (auto simp: homotopy_eqv_def)
```
```  3500   obtain c where "homotopic_with (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
```
```  3501     apply (rule exE [OF homSU [of "f \<circ> h"]])
```
```  3502     apply (intro continuous_on_compose h)
```
```  3503     using h f  apply (force elim!: continuous_on_subset)+
```
```  3504     done
```
```  3505   then have "homotopic_with (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
```
```  3506     apply (rule homotopic_with_compose_continuous_right [where X=S])
```
```  3507     using k by auto
```
```  3508   moreover have "homotopic_with (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
```
```  3509     apply (rule homotopic_with_compose_continuous_left [where Y=T])
```
```  3510       apply (simp add: hom homotopic_with_symD)
```
```  3511      using f apply auto
```
```  3512     done
```
```  3513   ultimately show ?thesis
```
```  3514     apply (rule_tac c=c in that)
```
```  3515     apply (simp add: o_def)
```
```  3516     using homotopic_with_trans by blast
```
```  3517 qed
```
```  3518
```
```  3519 lemma homotopy_eqv_cohomotopic_triviality_null:
```
```  3520   fixes S :: "'a::real_normed_vector set"
```
```  3521     and T :: "'b::real_normed_vector set"
```
```  3522     and U :: "'c::real_normed_vector set"
```
```  3523   assumes "S homotopy_eqv T"
```
```  3524     shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
```
```  3525                 \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c))) \<longleftrightarrow>
```
```  3526            (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U
```
```  3527                 \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)))"
```
```  3528 apply (rule iffI)
```
```  3529 apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
```
```  3530 by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
```
```  3531
```
```  3532 lemma homotopy_eqv_homotopic_triviality_null_imp:
```
```  3533   fixes S :: "'a::real_normed_vector set"
```
```  3534     and T :: "'b::real_normed_vector set"
```
```  3535     and U :: "'c::real_normed_vector set"
```
```  3536   assumes "S homotopy_eqv T"
```
```  3537       and f: "continuous_on U f" "f ` U \<subseteq> T"
```
```  3538       and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
```
```  3539                       \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
```
```  3540     shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
```
```  3541 proof -
```
```  3542   obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
```
```  3543                and k: "continuous_on T k" "k ` T \<subseteq> S"
```
```  3544                and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
```
```  3545                         "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
```
```  3546     using assms by (auto simp: homotopy_eqv_def)
```
```  3547   obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
```
```  3548     apply (rule exE [OF homSU [of "k \<circ> f"]])
```
```  3549     apply (intro continuous_on_compose h)
```
```  3550     using k f  apply (force elim!: continuous_on_subset)+
```
```  3551     done
```
```  3552   then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
```
```  3553     apply (rule homotopic_with_compose_continuous_left [where Y=S])
```
```  3554     using h by auto
```
```  3555   moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
```
```  3556     apply (rule homotopic_with_compose_continuous_right [where X=T])
```
```  3557       apply (simp add: hom homotopic_with_symD)
```
```  3558      using f apply auto
```
```  3559     done
```
```  3560   ultimately show ?thesis
```
```  3561     using homotopic_with_trans by (fastforce simp add: o_def)
```
```  3562 qed
```
```  3563
```
```  3564 lemma homotopy_eqv_homotopic_triviality_null:
```
```  3565   fixes S :: "'a::real_normed_vector set"
```
```  3566     and T :: "'b::real_normed_vector set"
```
```  3567     and U :: "'c::real_normed_vector set"
```
```  3568   assumes "S homotopy_eqv T"
```
```  3569     shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
```
```  3570                   \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
```
```  3571            (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
```
```  3572                   \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
```
```  3573 apply (rule iffI)
```
```  3574 apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
```
```  3575 by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
```
```  3576
```
```  3577 lemma homotopy_eqv_contractible_sets:
```
```  3578   fixes S :: "'a::real_normed_vector set"
```
```  3579     and T :: "'b::real_normed_vector set"
```
```  3580   assumes "contractible S" "contractible T" "S = {} \<longleftrightarrow> T = {}"
```
```  3581     shows "S homotopy_eqv T"
```
```  3582 proof (cases "S = {}")
```
```  3583   case True with assms show ?thesis
```
```  3584     by (simp add: homeomorphic_imp_homotopy_eqv)
```
```  3585 next
```
```  3586   case False
```
```  3587   with assms obtain a b where "a \<in> S" "b \<in> T"
```
```  3588     by auto
```
```  3589   then show ?thesis
```
```  3590     unfolding homotopy_eqv_def
```
```  3591     apply (rule_tac x="\<lambda>x. b" in exI)
```
```  3592     apply (rule_tac x="\<lambda>x. a" in exI)
```
```  3593     apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
```
```  3594     apply (auto simp: o_def continuous_on_const)
```
```  3595     done
```
```  3596 qed
```
```  3597
```
```  3598 lemma homotopy_eqv_empty1 [simp]:
```
```  3599   fixes S :: "'a::real_normed_vector set"
```
```  3600   shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}"
```
```  3601 apply (rule iffI)
```
```  3602 using homotopy_eqv_def apply fastforce
```
```  3603 by (simp add: homotopy_eqv_contractible_sets)
```
```  3604
```
```  3605 lemma homotopy_eqv_empty2 [simp]:
```
```  3606   fixes S :: "'a::real_normed_vector set"
```
```  3607   shows "({}::'b::real_normed_vector set) homotopy_eqv S \<longleftrightarrow> S = {}"
```
```  3608 by (metis homotopy_eqv_empty1 homotopy_eqv_sym)
```
```  3609
```
```  3610 lemma homotopy_eqv_contractibility:
```
```  3611   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
```
```  3612   shows "S homotopy_eqv T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
```
```  3613 unfolding homotopy_eqv_def
```
```  3614 by (blast intro: homotopy_dominated_contractibility)
```
```  3615
```
```  3616 lemma homotopy_eqv_sing:
```
```  3617   fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
```
```  3618   shows "S homotopy_eqv {a} \<longleftrightarrow> S \<noteq> {} \<and> contractible S"
```
```  3619 proof (cases "S = {}")
```
```  3620   case True then show ?thesis
```
```  3621     by simp
```
```  3622 next
```
```  3623   case False then show ?thesis
```
```  3624     by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
```
```  3625 qed
```
```  3626
```
```  3627 lemma homeomorphic_contractible_eq:
```
```  3628   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
```
```  3629   shows "S homeomorphic T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
```
```  3630 by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)
```
```  3631
```
```  3632 lemma homeomorphic_contractible:
```
```  3633   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
```
```  3634   shows "\<lbrakk>contractible S; S homeomorphic T\<rbrakk> \<Longrightarrow> contractible T"
```
```  3635   by (metis homeomorphic_contractible_eq)
```
```  3636
```
```  3637
```
```  3638 subsection%unimportant\<open>Misc other results\<close>
```
```  3639
```
```  3640 lemma bounded_connected_Compl_real:
```
```  3641   fixes S :: "real set"
```
```  3642   assumes "bounded S" and conn: "connected(- S)"
```
```  3643     shows "S = {}"
```
```  3644 proof -
```
```  3645   obtain a b where "S \<subseteq> box a b"
```
```  3646     by (meson assms bounded_subset_box_symmetric)
```
```  3647   then have "a \<notin> S" "b \<notin> S"
```
```  3648     by auto
```
```  3649   then have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> - S"
```
```  3650     by (meson Compl_iff conn connected_iff_interval)
```
```  3651   then show ?thesis
```
```  3652     using \<open>S \<subseteq> box a b\<close> by auto
```
```  3653 qed
```
```  3654
```
```  3655 corollary bounded_path_connected_Compl_real:
```
```  3656   fixes S :: "real set"
```
```  3657   assumes "bounded S" "path_connected(- S)" shows "S = {}"
```
```  3658   by (simp add: assms bounded_connected_Compl_real path_connected_imp_connected)
```
```  3659
```
```  3660 lemma bounded_connected_Compl_1:
```
```  3661   fixes S :: "'a::{euclidean_space} set"
```
```  3662   assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
```
```  3663     shows "S = {}"
```
```  3664 proof -
```
```  3665   have "DIM('a) = DIM(real)"
```
```  3666     by (simp add: "1")
```
```  3667   then obtain f::"'a \<Rightarrow> real" and g
```
```  3668   where "linear f" "\<And>x. norm(f x) = norm x" "\<And>x. g(f x) = x" "\<And>y. f(g y) = y"
```
```  3669     by (rule isomorphisms_UNIV_UNIV) blast
```
```  3670   with \<open>bounded S\<close> have "bounded (f ` S)"
```
```  3671     using bounded_linear_image linear_linear by blast
```
```  3672   have "connected (f ` (-S))"
```
```  3673     using connected_linear_image assms \<open>linear f\<close> by blast
```
```  3674   moreover have "f ` (-S) = - (f ` S)"
```
```  3675     apply (rule bij_image_Compl_eq)
```
```  3676     apply (auto simp: bij_def)
```
```  3677      apply (metis \<open>\<And>x. g (f x) = x\<close> injI)
```
```  3678     by (metis UNIV_I \<open>\<And>y. f (g y) = y\<close> image_iff)
```
```  3679   finally have "connected (- (f ` S))"
```
```  3680     by simp
```
```  3681   then have "f ` S = {}"
```
```  3682     using \<open>bounded (f ` S)\<close> bounded_connected_Compl_real by blast
```
```  3683   then show ?thesis
```
```  3684     by blast
```
```  3685 qed
```
```  3686
```
```  3687
```
```  3688 subsection%unimportant\<open>Some Uncountable Sets\<close>
```
```  3689
```
```  3690 lemma uncountable_closed_segment:
```
```  3691   fixes a :: "'a::real_normed_vector"
```
```  3692   assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
```
```  3693 unfolding path_image_linepath [symmetric] path_image_def
```
```  3694   using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
```
```  3695         countable_image_inj_on by auto
```
```  3696
```
```  3697 lemma uncountable_open_segment:
```
```  3698   fixes a :: "'a::real_normed_vector"
```
```  3699   assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
```
```  3700   by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
```
```  3701
```
```  3702 lemma uncountable_convex:
```
```  3703   fixes a :: "'a::real_normed_vector"
```
```  3704   assumes "convex S" "a \<in> S" "b \<in> S" "a \<noteq> b"
```
```  3705     shows "uncountable S"
```
```  3706 proof -
```
```  3707   have "uncountable (closed_segment a b)"
```
```  3708     by (simp add: uncountable_closed_segment assms)
```
```  3709   then show ?thesis
```
```  3710     by (meson assms convex_contains_segment countable_subset)
```
```  3711 qed
```
```  3712
```
```  3713 lemma uncountable_ball:
```
```  3714   fixes a :: "'a::euclidean_space"
```
```  3715   assumes "r > 0"
```
```  3716     shows "uncountable (ball a r)"
```
```  3717 proof -
```
```  3718   have "uncountable (open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)))"
```
```  3719     by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
```
```  3720   moreover have "open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)) \<subseteq> ball a r"
```
```  3721     using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
```
```  3722   ultimately show ?thesis
```
```  3723     by (metis countable_subset)
```
```  3724 qed
```
```  3725
```
```  3726 lemma ball_minus_countable_nonempty:
```
```  3727   assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
```
```  3728   shows   "ball z r - A \<noteq> {}"
```
```  3729 proof
```
```  3730   assume *: "ball z r - A = {}"
```
```  3731   have "uncountable (ball z r - A)"
```
```  3732     by (intro uncountable_minus_countable assms uncountable_ball)
```
```  3733   thus False by (subst (asm) *) auto
```
```  3734 qed
```
```  3735
```
```  3736 lemma uncountable_cball:
```
```  3737   fixes a :: "'a::euclidean_space"
```
```  3738   assumes "r > 0"
```
```  3739   shows "uncountable (cball a r)"
```
```  3740   using assms countable_subset uncountable_ball by auto
```
```  3741
```
```  3742 lemma pairwise_disjnt_countable:
```
```  3743   fixes \<N> :: "nat set set"
```
```  3744   assumes "pairwise disjnt \<N>"
```
```  3745     shows "countable \<N>"
```
```  3746 proof -
```
```  3747   have "inj_on (\<lambda>X. SOME n. n \<in> X) (\<N> - {{}})"
```
```  3748     apply (clarsimp simp add: inj_on_def)
```
```  3749     by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
```
```  3750   then show ?thesis
```
```  3751     by (metis countable_Diff_eq countable_def)
```
```  3752 qed
```
```  3753
```
```  3754 lemma pairwise_disjnt_countable_Union:
```
```  3755     assumes "countable (\<Union>\<N>)" and pwd: "pairwise disjnt \<N>"
```
```  3756     shows "countable \<N>"
```
```  3757 proof -
```
```  3758   obtain f :: "_ \<Rightarrow> nat" where f: "inj_on f (\<Union>\<N>)"
```
```  3759     using assms by blast
```
```  3760   then have "pairwise disjnt (\<Union> X \<in> \<N>. {f ` X})"
```
```  3761     using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
```
```  3762   then have "countable (\<Union> X \<in> \<N>. {f ` X})"
```
```  3763     using pairwise_disjnt_countable by blast
```
```  3764   then show ?thesis
```
```  3765     by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
```
```  3766 qed
```
```  3767
```
```  3768 lemma connected_uncountable:
```
```  3769   fixes S :: "'a::metric_space set"
```
```  3770   assumes "connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
```
```  3771 proof -
```
```  3772   have "continuous_on S (dist a)"
```
```  3773     by (intro continuous_intros)
```
```  3774   then have "connected (dist a ` S)"
```
```  3775     by (metis connected_continuous_image \<open>connected S\<close>)
```
```  3776   then have "closed_segment 0 (dist a b) \<subseteq> (dist a ` S)"
```
```  3777     by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
```
```  3778   then have "uncountable (dist a ` S)"
```
```  3779     by (metis \<open>a \<noteq> b\<close> countable_subset dist_eq_0_iff uncountable_closed_segment)
```
```  3780   then show ?thesis
```
```  3781     by blast
```
```  3782 qed
```
```  3783
```
```  3784 lemma path_connected_uncountable:
```
```  3785   fixes S :: "'a::metric_space set"
```
```  3786   assumes "path_connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
```
```  3787   using path_connected_imp_connected assms connected_uncountable by metis
```
```  3788
```
```  3789 lemma connected_finite_iff_sing:
```
```  3790   fixes S :: "'a::metric_space set"
```
```  3791   assumes "connected S"
```
```  3792   shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"  (is "_ = ?rhs")
```
```  3793 proof -
```
```  3794   have "uncountable S" if "\<not> ?rhs"
```
```  3795     using connected_uncountable assms that by blast
```
```  3796   then show ?thesis
```
```  3797     using uncountable_infinite by auto
```
```  3798 qed
```
```  3799
```
```  3800 lemma connected_card_eq_iff_nontrivial:
```
```  3801   fixes S :: "'a::metric_space set"
```
```  3802   shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
```
```  3803   apply (auto simp: countable_finite finite_subset)
```
```  3804   by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
```
```  3805
```
```  3806 lemma simple_path_image_uncountable:
```
```  3807   fixes g :: "real \<Rightarrow> 'a::metric_space"
```
```  3808   assumes "simple_path g"
```
```  3809   shows "uncountable (path_image g)"
```
```  3810 proof -
```
```  3811   have "g 0 \<in> path_image g" "g (1/2) \<in> path_image g"
```
```  3812     by (simp_all add: path_defs)
```
```  3813   moreover have "g 0 \<noteq> g (1/2)"
```
```  3814     using assms by (fastforce simp add: simple_path_def)
```
```  3815   ultimately show ?thesis
```
```  3816     apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
```
```  3817     by blast
```
```  3818 qed
```
```  3819
```
```  3820 lemma arc_image_uncountable:
```
```  3821   fixes g :: "real \<Rightarrow> 'a::metric_space"
```
```  3822   assumes "arc g"
```
```  3823   shows "uncountable (path_image g)"
```
```  3824   by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
```
```  3825
```
```  3826
```
```  3827 subsection%unimportant\<open> Some simple positive connection theorems\<close>
```
```  3828
```
```  3829 proposition path_connected_convex_diff_countable:
```
```  3830   fixes U :: "'a::euclidean_space set"
```
```  3831   assumes "convex U" "\<not> collinear U" "countable S"
```
```  3832     shows "path_connected(U - S)"
```
```  3833 proof (clarsimp simp add: path_connected_def)
```
```  3834   fix a b
```
```  3835   assume "a \<in> U" "a \<notin> S" "b \<in> U" "b \<notin> S"
```
```  3836   let ?m = "midpoint a b"
```
```  3837   show "\<exists>g. path g \<and> path_image g \<subseteq> U - S \<and> pathstart g = a \<and> pathfinish g = b"
```
```  3838   proof (cases "a = b")
```
```  3839     case True
```
```  3840     then show ?thesis
```
```  3841       by (metis DiffI \<open>a \<in> U\<close> \<open>a \<notin> S\<close> path_component_def path_component_refl)
```
```  3842   next
```
```  3843     case False
```
```  3844     then have "a \<noteq> ?m" "b \<noteq> ?m"
```
```  3845       using midpoint_eq_endpoint by fastforce+
```
```  3846     have "?m \<in> U"
```
```  3847       using \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>convex U\<close> convex_contains_segment by force
```
```  3848     obtain c where "c \<in> U" and nc_abc: "\<not> collinear {a,b,c}"
```
```  3849       by (metis False \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>\<not> collinear U\<close> collinear_triples insert_absorb)
```
```  3850     have ncoll_mca: "\<not> collinear {?m,c,a}"
```
```  3851       by (metis (full_types) \<open>a \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
```
```  3852     have ncoll_mcb: "\<not> collinear {?m,c,b}"
```
```  3853       by (metis (full_types) \<open>b \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
```
```  3854     have "c \<noteq> ?m"
```
```  3855       by (metis collinear_midpoint insert_commute nc_abc)
```
```  3856     then have "closed_segment ?m c \<subseteq> U"
```
```  3857       by (simp add: \<open>c \<in> U\<close> \<open>?m \<in> U\<close> \<open>convex U\<close> closed_segment_subset)
```
```  3858     then obtain z where z: "z \<in> closed_segment ?m c"
```
```  3859                     and disjS: "(closed_segment a z \<union> closed_segment z b) \<inter> S = {}"
```
```  3860     proof -
```
```  3861       have False if "closed_segment ?m c \<subseteq> {z. (closed_segment a z \<union> closed_segment z b) \<inter> S \<noteq> {}}"
```
```  3862       proof -
```
```  3863         have closb: "closed_segment ?m c \<subseteq>
```
```  3864                  {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> {}}"
```
```  3865           using that by blast
```
```  3866         have *: "countable {z \<in> closed_segment ?m c. closed_segment z u \<inter> S \<noteq> {}}"
```
```  3867           if "u \<in> U" "u \<notin> S" and ncoll: "\<not> collinear {?m, c, u}" for u
```
```  3868         proof -
```
```  3869           have **: False if x1: "x1 \<in> closed_segment ?m c" and x2: "x2 \<in> closed_segment ?m c"
```
```  3870                             and "x1 \<noteq> x2" "x1 \<noteq> u"
```
```  3871                             and w: "w \<in> closed_segment x1 u" "w \<in> closed_segment x2 u"
```
```  3872                             and "w \<in> S" for x1 x2 w
```
```  3873           proof -
```
```  3874             have "x1 \<in> affine hull {?m,c}" "x2 \<in> affine hull {?m,c}"
```
```  3875               using segment_as_ball x1 x2 by auto
```
```  3876             then have coll_x1: "collinear {x1, ?m, c}" and coll_x2: "collinear {?m, c, x2}"
```
```  3877               by (simp_all add: affine_hull_3_imp_collinear) (metis affine_hull_3_imp_collinear insert_commute)
```
```  3878             have "\<not> collinear {x1, u, x2}"
```
```  3879             proof
```
```  3880               assume "collinear {x1, u, x2}"
```
```  3881               then have "collinear {?m, c, u}"
```
```  3882                 by (metis (full_types) \<open>c \<noteq> ?m\<close> coll_x1 coll_x2 collinear_3_trans insert_commute ncoll \<open>x1 \<noteq> x2\<close>)
```
```  3883               with ncoll show False ..
```
```  3884             qed
```
```  3885             then have "closed_segment x1 u \<inter> closed_segment u x2 = {u}"
```
```  3886               by (blast intro!: Int_closed_segment)
```
```  3887             then have "w = u"
```
```  3888               using closed_segment_commute w by auto
```
```  3889             show ?thesis
```
```  3890               using \<open>u \<notin> S\<close> \<open>w = u\<close> that(7) by auto
```
```  3891           qed
```
```  3892           then have disj: "disjoint ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}))"
```
```  3893             by (fastforce simp: pairwise_def disjnt_def)
```
```  3894           have cou: "countable ((\<Union>z \<in> closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
```
```  3895             apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
```
```  3896              apply (rule countable_subset [OF _ \<open>countable S\<close>], auto)
```
```  3897             done
```
```  3898           define f where "f \<equiv> \<lambda>X. (THE z. z \<in> closed_segment ?m c \<and> X = closed_segment z u \<inter> S)"
```
```  3899           show ?thesis
```
```  3900           proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
```
```  3901             fix x
```
```  3902             assume x: "x \<in> closed_segment ?m c" "closed_segment x u \<inter> S \<noteq> {}"
```
```  3903             show "x \<in> f ` ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
```
```  3904             proof (rule_tac x="closed_segment x u \<inter> S" in image_eqI)
```
```  3905               show "x = f (closed_segment x u \<inter> S)"
```
```  3906                 unfolding f_def
```
```  3907                 apply (rule the_equality [symmetric])
```
```  3908                 using x  apply (auto simp: dest: **)
```
```  3909                 done
```
```  3910             qed (use x in auto)
```
```  3911           qed
```
```  3912         qed
```
```  3913         have "uncountable (closed_segment ?m c)"
```
```  3914           by (metis \<open>c \<noteq> ?m\<close> uncountable_closed_segment)
```
```  3915         then show False
```
```  3916           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]
```
```  3917           apply (simp add: closed_segment_commute)
```
```  3918           by (simp add: countable_subset)
```
```  3919       qed
```
```  3920       then show ?thesis
```
```  3921         by (force intro: that)
```
```  3922     qed
```
```  3923     show ?thesis
```
```  3924     proof (intro exI conjI)
```
```  3925       have "path_image (linepath a z +++ linepath z b) \<subseteq> U"
```
```  3926         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)
```
```  3927       with disjS show "path_image (linepath a z +++ linepath z b) \<subseteq> U - S"
```
```  3928         by (force simp: path_image_join)
```
```  3929     qed auto
```
```  3930   qed
```
```  3931 qed
```
```  3932
```
```  3933
```
```  3934 corollary connected_convex_diff_countable:
```
```  3935   fixes U :: "'a::euclidean_space set"
```
```  3936   assumes "convex U" "\<not> collinear U" "countable S"
```
```  3937   shows "connected(U - S)"
```
```  3938   by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)
```
```  3939
```
```  3940 lemma path_connected_punctured_convex:
```
```  3941   assumes "convex S" and aff: "aff_dim S \<noteq> 1"
```
```  3942     shows "path_connected(S - {a})"
```
```  3943 proof -
```
```  3944   consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S \<ge> 2"
```
```  3945     using assms aff_dim_geq [of S] by linarith
```
```  3946   then show ?thesis
```
```  3947   proof cases
```
```  3948     assume "aff_dim S = -1"
```
```  3949     then show ?thesis
```
```  3950       by (metis aff_dim_empty empty_Diff path_connected_empty)
```
```  3951   next
```
```  3952     assume "aff_dim S = 0"
```
```  3953     then show ?thesis
```
```  3954       by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
```
```  3955   next
```
```  3956     assume ge2: "aff_dim S \<ge> 2"
```
```  3957     then have "\<not> collinear S"
```
```  3958     proof (clarsimp simp add: collinear_affine_hull)
```
```  3959       fix u v
```
```  3960       assume "S \<subseteq> affine hull {u, v}"
```
```  3961       then have "aff_dim S \<le> aff_dim {u, v}"
```
```  3962         by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
```
```  3963       with ge2 show False
```
```  3964         by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
```
```  3965     qed
```
```  3966     then show ?thesis
```
```  3967       apply (rule path_connected_convex_diff_countable [OF \<open>convex S\<close>])
```
```  3968       by simp
```
```  3969   qed
```
```  3970 qed
```
```  3971
```
```  3972 lemma connected_punctured_convex:
```
```  3973   shows "\<lbrakk>convex S; aff_dim S \<noteq> 1\<rbrakk> \<Longrightarrow> connected(S - {a})"
```
```  3974   using path_connected_imp_connected path_connected_punctured_convex by blast
```
```  3975
```
```  3976 lemma path_connected_complement_countable:
```
```  3977   fixes S :: "'a::euclidean_space set"
```
```  3978   assumes "2 \<le> DIM('a)" "countable S"
```
```  3979   shows "path_connected(- S)"
```
```  3980 proof -
```
```  3981   have "path_connected(UNIV - S)"
```
```  3982     apply (rule path_connected_convex_diff_countable)
```
```  3983     using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
```
```  3984   then show ?thesis
```
```  3985     by (simp add: Compl_eq_Diff_UNIV)
```
```  3986 qed
```
```  3987
```
```  3988 proposition path_connected_openin_diff_countable:
```
```  3989   fixes S :: "'a::euclidean_space set"
```
```  3990   assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
```
```  3991       and "\<not> collinear S" "countable T"
```
```  3992     shows "path_connected(S - T)"
```
```  3993 proof (clarsimp simp add: path_connected_component)
```
```  3994   fix x y
```
```  3995   assume xy: "x \<in> S" "x \<notin> T" "y \<in> S" "y \<notin> T"
```
```  3996   show "path_component (S - T) x y"
```
```  3997   proof (rule connected_equivalence_relation_gen [OF \<open>connected S\<close>, where P = "\<lambda>x. x \<notin> T"])
```
```  3998     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
```
```  3999     proof -
```
```  4000       have "openin (top_of_set (affine hull S)) U"
```
```  4001         using opeU ope openin_trans by blast
```
```  4002       with \<open>x \<in> U\<close> obtain r where Usub: "U \<subseteq> affine hull S" and "r > 0"
```
```  4003                               and subU: "ball x r \<inter> affine hull S \<subseteq> U"
```
```  4004         by (auto simp: openin_contains_ball)
```
```  4005       with \<open>x \<in> U\<close> have x: "x \<in> ball x r \<inter> affine hull S"
```
```  4006         by auto
```
```  4007       have "\<not> S \<subseteq> {x}"
```
```  4008         using \<open>\<not> collinear S\<close>  collinear_subset by blast
```
```  4009       then obtain x' where "x' \<noteq> x" "x' \<in> S"
```
```  4010         by blast
```
```  4011       obtain y where y: "y \<noteq> x" "y \<in> ball x r \<inter> affine hull S"
```
```  4012       proof
```
```  4013         show "x + (r / 2 / norm(x' - x)) *\<^sub>R (x' - x) \<noteq> x"
```
```  4014           using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> by auto
```
```  4015         show "x + (r / 2 / norm (x' - x)) *\<^sub>R (x' - x) \<in> ball x r \<inter> affine hull S"
```
```  4016           using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> \<open>x' \<in> S\<close> x
```
```  4017           by (simp add: dist_norm mem_affine_3_minus hull_inc)
```
```  4018       qed
```
```  4019       have "convex (ball x r \<inter> affine hull S)"
```
```  4020         by (simp add: affine_imp_convex convex_Int)
```
```  4021       with x y subU have "uncountable U"
```
```  4022         by (meson countable_subset uncountable_convex)
```
```  4023       then have "\<not> U \<subseteq> T"
```
```  4024         using \<open>countable T\<close> countable_subset by blast
```
```  4025       then show ?thesis by blast
```
```  4026     qed
```
```  4027     show "\<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and>
```
```  4028               (\<forall>x\<in>U. \<forall>y\<in>U. x \<notin> T \<and> y \<notin> T \<longrightarrow> path_component (S - T) x y)"
```
```  4029           if "x \<in> S" for x
```
```  4030     proof -
```
```  4031       obtain r where Ssub: "S \<subseteq> affine hull S" and "r > 0"
```
```  4032                  and subS: "ball x r \<inter> affine hull S \<subseteq> S"
```
```  4033         using ope \<open>x \<in> S\<close> by (auto simp: openin_contains_ball)
```
```  4034       then have conv: "convex (ball x r \<inter> affine hull S)"
```
```  4035         by (simp add: affine_imp_convex convex_Int)
```
```  4036       have "\<not> aff_dim (affine hull S) \<le> 1"
```
```  4037         using \<open>\<not> collinear S\<close> collinear_aff_dim by auto
```
```  4038       then have "\<not> collinear (ball x r \<inter> affine hull S)"
```
```  4039         apply (simp add: collinear_aff_dim)
```
```  4040         by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball \<open>0 < r\<close> aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
```
```  4041       then have *: "path_connected ((ball x r \<inter> affine hull S) - T)"
```
```  4042         by (rule path_connected_convex_diff_countable [OF conv _ \<open>countable T\<close>])
```
```  4043       have ST: "ball x r \<inter> affine hull S - T \<subseteq> S - T"
```
```  4044         using subS by auto
```
```  4045       show ?thesis
```
```  4046       proof (intro exI conjI)
```
```  4047         show "x \<in> ball x r \<inter> affine hull S"
```
```  4048           using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
```
```  4049         have "openin (top_of_set (affine hull S)) (ball x r \<inter> affine hull S)"
```
```  4050           by (subst inf.commute) (simp add: openin_Int_open)
```
```  4051         then show "openin (top_of_set S) (ball x r \<inter> affine hull S)"
```
```  4052           by (rule openin_subset_trans [OF _ subS Ssub])
```
```  4053       qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
```
```  4054     qed
```
```  4055   qed (use xy path_component_trans in auto)
```
```  4056 qed
```
```  4057
```
```  4058 corollary connected_openin_diff_countable:
```
```  4059   fixes S :: "'a::euclidean_space set"
```
```  4060   assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
```
```  4061       and "\<not> collinear S" "countable T"
```
```  4062     shows "connected(S - T)"
```
```  4063   by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])
```
```  4064
```
```  4065 corollary path_connected_open_diff_countable:
```
```  4066   fixes S :: "'a::euclidean_space set"
```
```  4067   assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
```
```  4068   shows "path_connected(S - T)"
```
```  4069 proof (cases "S = {}")
```
```  4070   case True
```
```  4071   then show ?thesis
```
```  4072     by (simp add: path_connected_empty)
```
```  4073 next
```
```  4074   case False
```
```  4075   show ?thesis
```
```  4076   proof (rule path_connected_openin_diff_countable)
```
```  4077     show "openin (top_of_set (affine hull S)) S"
```
```  4078       by (simp add: assms hull_subset open_subset)
```
```  4079     show "\<not> collinear S"
```
```  4080       using assms False by (simp add: collinear_aff_dim aff_dim_open)
```
```  4081   qed (simp_all add: assms)
```
```  4082 qed
```
```  4083
```
```  4084 corollary connected_open_diff_countable:
```
```  4085   fixes S :: "'a::euclidean_space set"
```
```  4086   assumes "2 \<le> DIM('a)" "open S" "connected S" "countable T"
```
```  4087   shows "connected(S - T)"
```
```  4088 by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)
```
```  4089
```
```  4090
```
```  4091
```
```  4092 subsection%unimportant \<open>Self-homeomorphisms shuffling points about\<close>
```
```  4093
```
```  4094 subsubsection%unimportant\<open>The theorem \<open>homeomorphism_moving_points_exists\<close>\<close>
```
```  4095
```
```  4096 lemma homeomorphism_moving_point_1:
```
```  4097   fixes a :: "'a::euclidean_space"
```
```  4098   assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T"
```
```  4099   obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
```
```  4100                     "f a = u" "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = x"
```
```  4101 proof -
```
```  4102   have nou: "norm (u - a) < r" and "u \<in> T"
```
```  4103     using u by (auto simp: dist_norm norm_minus_commute)
```
```  4104   then have "0 < r"
```
```  4105     by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
```
```  4106   define f where "f \<equiv> \<lambda>x. (1 - norm(x - a) / r) *\<^sub>R (u - a) + x"
```
```  4107   have *: "False" if eq: "x + (norm y / r) *\<^sub>R u = y + (norm x / r) *\<^sub>R u"
```
```  4108                   and nou: "norm u < r" and yx: "norm y < norm x" for x y and u::'a
```
```  4109   proof -
```
```  4110     have "x = y + (norm x / r - (norm y / r)) *\<^sub>R u"
```
```  4111       using eq by (simp add: algebra_simps)
```
```  4112     then have "norm x = norm (y + ((norm x - norm y) / r) *\<^sub>R u)"
```
```  4113       by (metis diff_divide_distrib)
```
```  4114     also have "\<dots> \<le> norm y + norm(((norm x - norm y) / r) *\<^sub>R u)"
```
```  4115       using norm_triangle_ineq by blast
```
```  4116     also have "\<dots> = norm y + (norm x - norm y) * (norm u / r)"
```
```  4117       using yx \<open>r > 0\<close>
```
```  4118       by (simp add: divide_simps)
```
```  4119     also have "\<dots> < norm y + (norm x - norm y) * 1"
```
```  4120       apply (subst add_less_cancel_left)
```
```  4121       apply (rule mult_strict_left_mono)
```
```  4122       using nou \<open>0 < r\<close> yx
```
```  4123        apply (simp_all add: field_simps)
```
```  4124       done
```
```  4125     also have "\<dots> = norm x"
```
```  4126       by simp
```
```  4127     finally show False by simp
```
```  4128   qed
```
```  4129   have "inj f"
```
```  4130     unfolding f_def
```
```  4131   proof (clarsimp simp: inj_on_def)
```
```  4132     fix x y
```
```  4133     assume "(1 - norm (x - a) / r) *\<^sub>R (u - a) + x =
```
```  4134             (1 - norm (y - a) / r) *\<^sub>R (u - a) + y"
```
```  4135     then have eq: "(x - a) + (norm (y - a) / r) *\<^sub>R (u - a) = (y - a) + (norm (x - a) / r) *\<^sub>R (u - a)"
```
```  4136       by (auto simp: algebra_simps)
```
```  4137     show "x=y"
```
```  4138     proof (cases "norm (x - a) = norm (y - a)")
```
```  4139       case True
```
```  4140       then show ?thesis
```
```  4141         using eq by auto
```
```  4142     next
```
```  4143       case False
```
```  4144       then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
```
```  4145         by linarith
```
```  4146       then have "False"
```
```  4147       proof cases
```
```  4148         case 1 show False
```
```  4149           using * [OF _ nou 1] eq by simp
```
```  4150       next
```
```  4151         case 2 with * [OF eq nou] show False
```
```  4152           by auto
```
```  4153       qed
```
```  4154       then show "x=y" ..
```
```  4155     qed
```
```  4156   qed
```
```  4157   then have inj_onf: "inj_on f (cball a r \<inter> T)"
```
```  4158     using inj_on_Int by fastforce
```
```  4159   have contf: "continuous_on (cball a r \<inter> T) f"
```
```  4160     unfolding f_def using \<open>0 < r\<close>  by (intro continuous_intros) blast
```
```  4161   have fim: "f ` (cball a r \<inter> T) = cball a r \<inter> T"
```
```  4162   proof
```
```  4163     have *: "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> r" if "norm y \<le> r" "norm u < r" for y u::'a
```
```  4164     proof -
```
```  4165       have "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> norm y + norm((1 - norm y / r) *\<^sub>R u)"
```
```  4166         using norm_triangle_ineq by blast
```
```  4167       also have "\<dots> = norm y + abs(1 - norm y / r) * norm u"
```
```  4168         by simp
```
```  4169       also have "\<dots> \<le> r"
```
```  4170       proof -
```
```  4171         have "(r - norm u) * (r - norm y) \<ge> 0"
```
```  4172           using that by auto
```
```  4173         then have "r * norm u + r * norm y \<le> r * r + norm u * norm y"
```
```  4174           by (simp add: algebra_simps)
```
```  4175         then show ?thesis
```
```  4176         using that \<open>0 < r\<close> by (simp add: abs_if field_simps)
```
```  4177       qed
```
```  4178       finally show ?thesis .
```
```  4179     qed
```
```  4180     have "f ` (cball a r) \<subseteq> cball a r"
```
```  4181       apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
```
```  4182       using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
```
```  4183     moreover have "f ` T \<subseteq> T"
```
```  4184       unfolding f_def using \<open>affine T\<close> \<open>a \<in> T\<close> \<open>u \<in> T\<close>
```
```  4185       by (force simp: add.commute mem_affine_3_minus)
```
```  4186     ultimately show "f ` (cball a r \<inter> T) \<subseteq> cball a r \<inter> T"
```
```  4187       by blast
```
```  4188   next
```
```  4189     show "cball a r \<inter> T \<subseteq> f ` (cball a r \<inter> T)"
```
```  4190     proof (clarsimp simp add: dist_norm norm_minus_commute)
```
```  4191       fix x
```
```  4192       assume x: "norm (x - a) \<le> r" and "x \<in> T"
```
```  4193       have "\<exists>v \<in> {0..1}. ((1 - v) * r - norm ((x - a) - v *\<^sub>R (u - a))) \<bullet> 1 = 0"
```
```  4194         by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
```
```  4195       then obtain v where "0\<le>v" "v\<le>1" and v: "(1 - v) * r = norm ((x - a) - v *\<^sub>R (u - a))"
```
```  4196         by auto
```
```  4197       show "x \<in> f ` (cball a r \<inter> T)"
```
```  4198       proof (rule image_eqI)
```
```  4199         show "x = f (x - v *\<^sub>R (u - a))"
```
```  4200           using \<open>r > 0\<close> v by (simp add: f_def field_simps)
```
```  4201         have "x - v *\<^sub>R (u - a) \<in> cball a r"
```
```  4202           using \<open>r > 0\<close> v \<open>0 \<le> v\<close>
```
```  4203           apply (simp add: field_simps dist_norm norm_minus_commute)
```
```  4204           by (metis le_add_same_cancel2 order.order_iff_strict zero_le_mult_iff)
```
```  4205         moreover have "x - v *\<^sub>R (u - a) \<in> T"
```
```  4206           by (simp add: f_def \<open>affine T\<close> \<open>u \<in> T\<close> \<open>x \<in> T\<close> assms mem_affine_3_minus2)
```
```  4207         ultimately show "x - v *\<^sub>R (u - a) \<in> cball a r \<inter> T"
```
```  4208           by blast
```
```  4209       qed
```
```  4210     qed
```
```  4211   qed
```
```  4212   have "\<exists>g. homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
```
```  4213     apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
```
```  4214     apply (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
```
```  4215     done
```
```  4216   then show ?thesis
```
```  4217     apply (rule exE)
```
```  4218     apply (erule_tac f=f in that)
```
```  4219     using \<open>r > 0\<close>
```
```  4220      apply (simp_all add: f_def dist_norm norm_minus_commute)
```
```  4221     done
```
```  4222 qed
```
```  4223
```
```  4224 corollary%unimportant homeomorphism_moving_point_2:
```
```  4225   fixes a :: "'a::euclidean_space"
```
```  4226   assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
```
```  4227   obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
```
```  4228                     "f u = v" "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
```
```  4229 proof -
```
```  4230   have "0 < r"
```
```  4231     by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
```
```  4232   obtain f1 g1 where hom1: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f1 g1"
```
```  4233                  and "f1 a = u" and f1: "\<And>x. x \<in> sphere a r \<Longrightarrow> f1 x = x"
```
```  4234     using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u] by blast
```
```  4235   obtain f2 g2 where hom2: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f2 g2"
```
```  4236                  and "f2 a = v" and f2: "\<And>x. x \<in> sphere a r \<Longrightarrow> f2 x = x"
```
```  4237     using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> v] by blast
```
```  4238   show ?thesis
```
```  4239   proof
```
```  4240     show "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) (f2 \<circ> g1) (f1 \<circ> g2)"
```
```  4241       by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
```
```  4242     have "g1 u = a"
```
```  4243       using \<open>0 < r\<close> \<open>f1 a = u\<close> assms hom1 homeomorphism_apply1 by fastforce
```
```  4244     then show "(f2 \<circ> g1) u = v"
```
```  4245       by (simp add: \<open>f2 a = v\<close>)
```
```  4246     show "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> (f2 \<circ> g1) x = x"
```
```  4247       using f1 f2 hom1 homeomorphism_apply1 by fastforce
```
```  4248   qed
```
```  4249 qed
```
```  4250
```
```  4251
```
```  4252 corollary%unimportant homeomorphism_moving_point_3:
```
```  4253   fixes a :: "'a::euclidean_space"
```
```  4254   assumes "affine T" "a \<in> T" and ST: "ball a r \<inter> T \<subseteq> S" "S \<subseteq> T"
```
```  4255       and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
```
```  4256   obtains f g where "homeomorphism S S f g"
```
```  4257                     "f u = v" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> ball a r \<inter> T"
```
```  4258 proof -
```
```  4259   obtain f g where hom: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
```
```  4260                and "f u = v" and fid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
```
```  4261     using homeomorphism_moving_point_2 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u v] by blast
```
```  4262   have gid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> g x = x"
```
```  4263     using fid hom homeomorphism_apply1 by fastforce
```
```  4264   define ff where "ff \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then f x else x"
```
```  4265   define gg where "gg \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then g x else x"
```
```  4266   show ?thesis
```
```  4267   proof
```
```  4268     show "homeomorphism S S ff gg"
```
```  4269     proof (rule homeomorphismI)
```
```  4270       have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) ff"
```
```  4271         apply (simp add: ff_def)
```
```  4272         apply (rule continuous_on_cases)
```
```  4273         using homeomorphism_cont1 [OF hom]
```
```  4274             apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id fid)
```
```  4275         done
```
```  4276       then show "continuous_on S ff"
```
```  4277         apply (rule continuous_on_subset)
```
```  4278         using ST by auto
```
```  4279       have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
```
```  4280         apply (simp add: gg_def)
```
```  4281         apply (rule continuous_on_cases)
```
```  4282         using homeomorphism_cont2 [OF hom]
```
```  4283             apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id gid)
```
```  4284         done
```
```  4285       then show "continuous_on S gg"
```
```  4286         apply (rule continuous_on_subset)
```
```  4287         using ST by auto
```
```  4288       show "ff ` S \<subseteq> S"
```
```  4289       proof (clarsimp simp add: ff_def)
```
```  4290         fix x
```
```  4291         assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
```
```  4292         then have "f x \<in> cball a r \<inter> T"
```
```  4293           using homeomorphism_image1 [OF hom] by force
```
```  4294         then show "f x \<in> S"
```
```  4295           using ST(1) \<open>x \<in> T\<close> gid hom homeomorphism_def x by fastforce
```
```  4296       qed
```
```  4297       show "gg ` S \<subseteq> S"
```
```  4298       proof (clarsimp simp add: gg_def)
```
```  4299         fix x
```
```  4300         assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
```
```  4301         then have "g x \<in> cball a r \<inter> T"
```
```  4302           using homeomorphism_image2 [OF hom] by force
```
```  4303         then have "g x \<in> ball a r"
```
```  4304           using homeomorphism_apply2 [OF hom]
```
```  4305             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)
```
```  4306         then show "g x \<in> S"
```
```  4307           using ST(1) \<open>g x \<in> cball a r \<inter> T\<close> by force
```
```  4308         qed
```
```  4309       show "\<And>x. x \<in> S \<Longrightarrow> gg (ff x) = x"
```
```  4310         apply (simp add: ff_def gg_def)
```
```  4311         using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
```
```  4312         apply auto
```
```  4313         apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
```
```  4314         done
```
```  4315       show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
```
```  4316         apply (simp add: ff_def gg_def)
```
```  4317         using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
```
```  4318         apply auto
```
```  4319         apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
```
```  4320         done
```
```  4321     qed
```
```  4322     show "ff u = v"
```
```  4323       using u by (auto simp: ff_def \<open>f u = v\<close>)
```
```  4324     show "{x. \<not> (ff x = x \<and> gg x = x)} \<subseteq> ball a r \<inter> T"
```
```  4325       by (auto simp: ff_def gg_def)
```
```  4326   qed
```
```  4327 qed
```
```  4328
```
```  4329
```
```  4330 proposition%unimportant homeomorphism_moving_point:
```
```  4331   fixes a :: "'a::euclidean_space"
```
```  4332   assumes ope: "openin (top_of_set (affine hull S)) S"
```
```  4333       and "S \<subseteq> T"
```
```  4334       and TS: "T \<subseteq> affine hull S"
```
```  4335       and S: "connected S" "a \<in> S" "b \<in> S"
```
```  4336   obtains f g where "homeomorphism T T f g" "f a = b"
```
```  4337                     "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
```
```  4338                     "bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4339 proof -
```
```  4340   have 1: "\<exists>h k. homeomorphism T T h k \<and> h (f d) = d \<and>
```
```  4341               {x. \<not> (h x = x \<and> k x = x)} \<subseteq> S \<and> bounded {x. \<not> (h x = x \<and> k x = x)}"
```
```  4342         if "d \<in> S" "f d \<in> S" and homfg: "homeomorphism T T f g"
```
```  4343         and S: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
```
```  4344         and bo: "bounded {x. \<not> (f x = x \<and> g x = x)}" for d f g
```
```  4345   proof (intro exI conjI)
```
```  4346     show homgf: "homeomorphism T T g f"
```
```  4347       by (metis homeomorphism_symD homfg)
```
```  4348     then show "g (f d) = d"
```
```  4349       by (meson \<open>S \<subseteq> T\<close> homeomorphism_def subsetD \<open>d \<in> S\<close>)
```
```  4350     show "{x. \<not> (g x = x \<and> f x = x)} \<subseteq> S"
```
```  4351       using S by blast
```
```  4352     show "bounded {x. \<not> (g x = x \<and> f x = x)}"
```
```  4353       using bo by (simp add: conj_commute)
```
```  4354   qed
```
```  4355   have 2: "\<exists>f g. homeomorphism T T f g \<and> f x = f2 (f1 x) \<and>
```
```  4356                  {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4357              if "x \<in> S" "f1 x \<in> S" "f2 (f1 x) \<in> S"
```
```  4358                 and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
```
```  4359                 and sub: "{x. \<not> (f1 x = x \<and> g1 x = x)} \<subseteq> S"   "{x. \<not> (f2 x = x \<and> g2 x = x)} \<subseteq> S"
```
```  4360                 and bo: "bounded {x. \<not> (f1 x = x \<and> g1 x = x)}"  "bounded {x. \<not> (f2 x = x \<and> g2 x = x)}"
```
```  4361              for x f1 f2 g1 g2
```
```  4362   proof (intro exI conjI)
```
```  4363     show homgf: "homeomorphism T T (f2 \<circ> f1) (g1 \<circ> g2)"
```
```  4364       by (metis homeomorphism_compose hom)
```
```  4365     then show "(f2 \<circ> f1) x = f2 (f1 x)"
```
```  4366       by force
```
```  4367     show "{x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)} \<subseteq> S"
```
```  4368       using sub by force
```
```  4369     have "bounded ({x. \<not>(f1 x = x \<and> g1 x = x)} \<union> {x. \<not>(f2 x = x \<and> g2 x = x)})"
```
```  4370       using bo by simp
```
```  4371     then show "bounded {x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)}"
```
```  4372       by (rule bounded_subset) auto
```
```  4373   qed
```
```  4374   have 3: "\<exists>U. openin (top_of_set S) U \<and>
```
```  4375               d \<in> U \<and>
```
```  4376               (\<forall>x\<in>U.
```
```  4377                   \<exists>f g. homeomorphism T T f g \<and> f d = x \<and>
```
```  4378                         {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
```
```  4379                         bounded {x. \<not> (f x = x \<and> g x = x)})"
```
```  4380            if "d \<in> S" for d
```
```  4381   proof -
```
```  4382     obtain r where "r > 0" and r: "ball d r \<inter> affine hull S \<subseteq> S"
```
```  4383       by (metis \<open>d \<in> S\<close> ope openin_contains_ball)
```
```  4384     have *: "\<exists>f g. homeomorphism T T f g \<and> f d = e \<and>
```
```  4385                    {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
```
```  4386                    bounded {x. \<not> (f x = x \<and> g x = x)}" if "e \<in> S" "e \<in> ball d r" for e
```
```  4387       apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
```
```  4388       using r \<open>S \<subseteq> T\<close> TS that
```
```  4389             apply (auto simp: \<open>d \<in> S\<close> \<open>0 < r\<close> hull_inc)
```
```  4390       using bounded_subset by blast
```
```  4391     show ?thesis
```
```  4392       apply (rule_tac x="S \<inter> ball d r" in exI)
```
```  4393       apply (intro conjI)
```
```  4394         apply (simp add: openin_open_Int)
```
```  4395        apply (simp add: \<open>0 < r\<close> that)
```
```  4396       apply (blast intro: *)
```
```  4397       done
```
```  4398   qed
```
```  4399   have "\<exists>f g. homeomorphism T T f g \<and> f a = b \<and>
```
```  4400               {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4401     apply (rule connected_equivalence_relation [OF S], safe)
```
```  4402       apply (blast intro: 1 2 3)+
```
```  4403     done
```
```  4404   then show ?thesis
```
```  4405     using that by auto
```
```  4406 qed
```
```  4407
```
```  4408
```
```  4409 lemma homeomorphism_moving_points_exists_gen:
```
```  4410   assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
```
```  4411              "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
```
```  4412       and "2 \<le> aff_dim S"
```
```  4413       and ope: "openin (top_of_set (affine hull S)) S"
```
```  4414       and "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
```
```  4415   shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
```
```  4416                {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4417   using assms
```
```  4418 proof (induction K)
```
```  4419   case empty
```
```  4420   then show ?case
```
```  4421     by (force simp: homeomorphism_ident)
```
```  4422 next
```
```  4423   case (insert i K)
```
```  4424   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"
```
```  4425        and pw: "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"
```
```  4426        and "x i \<in> S" "y i \<in> S"
```
```  4427        and xyS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
```
```  4428     by (simp_all add: pairwise_insert)
```
```  4429   obtain f g where homfg: "homeomorphism T T f g" and feq: "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
```
```  4430                and fg_sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
```
```  4431                and bo_fg: "bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4432     using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
```
```  4433   then have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
```
```  4434                    {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4435     using insert by blast
```
```  4436   have aff_eq: "affine hull (S - y ` K) = affine hull S"
```
```  4437     apply (rule affine_hull_Diff)
```
```  4438     apply (auto simp: insert)
```
```  4439     using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
```
```  4440   have f_in_S: "f x \<in> S" if "x \<in> S" for x
```
```  4441     using homfg fg_sub homeomorphism_apply1 \<open>S \<subseteq> T\<close>
```
```  4442   proof -
```
```  4443     have "(f (f x) \<noteq> f x \<or> g (f x) \<noteq> f x) \<or> f x \<in> S"
```
```  4444       by (metis \<open>S \<subseteq> T\<close> homfg subsetD homeomorphism_apply1 that)
```
```  4445     then show ?thesis
```
```  4446       using fg_sub by force
```
```  4447   qed
```
```  4448   obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
```
```  4449                and hk_sub: "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
```
```  4450                and bo_hk:  "bounded {x. \<not> (h x = x \<and> k x = x)}"
```
```  4451   proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
```
```  4452     show "openin (top_of_set (affine hull (S - y ` K))) (S - y ` K)"
```
```  4453       by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
```
```  4454     show "S - y ` K \<subseteq> T"
```
```  4455       using \<open>S \<subseteq> T\<close> by auto
```
```  4456     show "T \<subseteq> affine hull (S - y ` K)"
```
```  4457       using insert by (simp add: aff_eq)
```
```  4458     show "connected (S - y ` K)"
```
```  4459     proof (rule connected_openin_diff_countable [OF \<open>connected S\<close> ope])
```
```  4460       show "\<not> collinear S"
```
```  4461         using collinear_aff_dim \<open>2 \<le> aff_dim S\<close> by force
```
```  4462       show "countable (y ` K)"
```
```  4463         using countable_finite insert.hyps(1) by blast
```
```  4464     qed
```
```  4465     show "f (x i) \<in> S - y ` K"
```
```  4466       apply (auto simp: f_in_S \<open>x i \<in> S\<close>)
```
```  4467         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)
```
```  4468     show "y i \<in> S - y ` K"
```
```  4469       using insert.hyps xney by (auto simp: \<open>y i \<in> S\<close>)
```
```  4470   qed blast
```
```  4471   show ?case
```
```  4472   proof (intro exI conjI)
```
```  4473     show "homeomorphism T T (h \<circ> f) (g \<circ> k)"
```
```  4474       using homfg homhk homeomorphism_compose by blast
```
```  4475     show "\<forall>i \<in> insert i K. (h \<circ> f) (x i) = y i"
```
```  4476       using feq hk_sub by (auto simp: heq)
```
```  4477     show "{x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)} \<subseteq> S"
```
```  4478       using fg_sub hk_sub by force
```
```  4479     have "bounded ({x. \<not>(f x = x \<and> g x = x)} \<union> {x. \<not>(h x = x \<and> k x = x)})"
```
```  4480       using bo_fg bo_hk bounded_Un by blast
```
```  4481     then show "bounded {x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)}"
```
```  4482       by (rule bounded_subset) auto
```
```  4483   qed
```
```  4484 qed
```
```  4485
```
```  4486 proposition%unimportant homeomorphism_moving_points_exists:
```
```  4487   fixes S :: "'a::euclidean_space set"
```
```  4488   assumes 2: "2 \<le> DIM('a)" "open S" "connected S" "S \<subseteq> T" "finite K"
```
```  4489       and KS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
```
```  4490       and pw: "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
```
```  4491       and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
```
```  4492   obtains f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
```
```  4493                     "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. (\<not> (f x = x \<and> g x = x))}"
```
```  4494 proof (cases "S = {}")
```
```  4495   case True
```
```  4496   then show ?thesis
```
```  4497     using KS homeomorphism_ident that by fastforce
```
```  4498 next
```
```  4499   case False
```
```  4500   then have affS: "affine hull S = UNIV"
```
```  4501     by (simp add: affine_hull_open \<open>open S\<close>)
```
```  4502   then have ope: "openin (top_of_set (affine hull S)) S"
```
```  4503     using \<open>open S\<close> open_openin by auto
```
```  4504   have "2 \<le> DIM('a)" by (rule 2)
```
```  4505   also have "\<dots> = aff_dim (UNIV :: 'a set)"
```
```  4506     by simp
```
```  4507   also have "\<dots> \<le> aff_dim S"
```
```  4508     by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
```
```  4509   finally have "2 \<le> aff_dim S"
```
```  4510     by linarith
```
```  4511   then show ?thesis
```
```  4512     using homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> KS pw _ ope S] that by fastforce
```
```  4513 qed
```
```  4514
```
```  4515 subsubsection%unimportant\<open>The theorem \<open>homeomorphism_grouping_points_exists\<close>\<close>
```
```  4516
```
```  4517 lemma homeomorphism_grouping_point_1:
```
```  4518   fixes a::real and c::real
```
```  4519   assumes "a < b" "c < d"
```
```  4520   obtains f g where "homeomorphism (cbox a b) (cbox c d) f g" "f a = c" "f b = d"
```
```  4521 proof -
```
```  4522   define f where "f \<equiv> \<lambda>x. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
```
```  4523   have "\<exists>g. homeomorphism (cbox a b) (cbox c d) f g"
```
```  4524   proof (rule homeomorphism_compact)
```
```  4525     show "continuous_on (cbox a b) f"
```
```  4526       apply (simp add: f_def)
```
```  4527       apply (intro continuous_intros)
```
```  4528       using assms by auto
```
```  4529     have "f ` {a..b} = {c..d}"
```
```  4530       unfolding f_def image_affinity_atLeastAtMost
```
```  4531       using assms sum_sqs_eq by (auto simp: divide_simps algebra_simps)
```
```  4532     then show "f ` cbox a b = cbox c d"
```
```  4533       by auto
```
```  4534     show "inj_on f (cbox a b)"
```
```  4535       unfolding f_def inj_on_def using assms by auto
```
```  4536   qed auto
```
```  4537   then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
```
```  4538   then show ?thesis
```
```  4539   proof
```
```  4540     show "f a = c"
```
```  4541       by (simp add: f_def)
```
```  4542     show "f b = d"
```
```  4543       using assms sum_sqs_eq [of a b] by (auto simp: f_def divide_simps algebra_simps)
```
```  4544   qed
```
```  4545 qed
```
```  4546
```
```  4547 lemma homeomorphism_grouping_point_2:
```
```  4548   fixes a::real and w::real
```
```  4549   assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
```
```  4550       and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
```
```  4551       and "b \<in> cbox a c" "v \<in> cbox u w"
```
```  4552       and eq: "f1 a = u" "f1 b = v" "f2 b = v" "f2 c = w"
```
```  4553  obtains f g where "homeomorphism (cbox a c) (cbox u w) f g" "f a = u" "f c = w"
```
```  4554                    "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = f1 x" "\<And>x. x \<in> cbox b c \<Longrightarrow> f x = f2 x"
```
```  4555 proof -
```
```  4556   have le: "a \<le> b" "b \<le> c" "u \<le> v" "v \<le> w"
```
```  4557     using assms by simp_all
```
```  4558   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"
```
```  4559     by auto
```
```  4560   define f where "f \<equiv> \<lambda>x. if x \<le> b then f1 x else f2 x"
```
```  4561   have "\<exists>g. homeomorphism (cbox a c) (cbox u w) f g"
```
```  4562   proof (rule homeomorphism_compact)
```
```  4563     have cf1: "continuous_on (cbox a b) f1"
```
```  4564       using hom_ab homeomorphism_cont1 by blast
```
```  4565     have cf2: "continuous_on (cbox b c) f2"
```
```  4566       using hom_bc homeomorphism_cont1 by blast
```
```  4567     show "continuous_on (cbox a c) f"
```
```  4568       apply (simp add: f_def)
```
```  4569       apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
```
```  4570       using le eq apply (force simp: continuous_on_id)+
```
```  4571       done
```
```  4572     have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
```
```  4573       unfolding f_def using eq by force+
```
```  4574     then show "f ` cbox a c = cbox u w"
```
```  4575       apply (simp only: ac uw image_Un)
```
```  4576       by (metis hom_ab hom_bc homeomorphism_def)
```
```  4577     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
```
```  4578     proof -
```
```  4579       have "f1 x \<in> cbox u v"
```
```  4580         by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
```
```  4581       moreover have "f2 y \<in> cbox v w"
```
```  4582         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)
```
```  4583       moreover have "f2 y \<noteq> f2 b"
```
```  4584         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)
```
```  4585       ultimately show ?thesis
```
```  4586         using le eq by simp
```
```  4587     qed
```
```  4588     have "inj_on f1 (cbox a b)"
```
```  4589       by (metis (full_types) hom_ab homeomorphism_def inj_onI)
```
```  4590     moreover have "inj_on f2 (cbox b c)"
```
```  4591       by (metis (full_types) hom_bc homeomorphism_def inj_onI)
```
```  4592     ultimately show "inj_on f (cbox a c)"
```
```  4593       apply (simp (no_asm) add: inj_on_def)
```
```  4594       apply (simp add: f_def inj_on_eq_iff)
```
```  4595       using neq12  apply force
```
```  4596       done
```
```  4597   qed auto
```
```  4598   then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
```
```  4599   then show ?thesis
```
```  4600     apply (rule that)
```
```  4601     using eq le by (auto simp: f_def)
```
```  4602 qed
```
```  4603
```
```  4604 lemma homeomorphism_grouping_point_3:
```
```  4605   fixes a::real
```
```  4606   assumes cbox_sub: "cbox c d \<subseteq> box a b" "cbox u v \<subseteq> box a b"
```
```  4607       and box_ne: "box c d \<noteq> {}" "box u v \<noteq> {}"
```
```  4608   obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
```
```  4609                     "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v"
```
```  4610 proof -
```
```  4611   have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d \<noteq> {}"
```
```  4612     using assms
```
```  4613     by (simp_all add: cbox_sub subset_eq)
```
```  4614   obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
```
```  4615                    and f1_eq: "f1 a = a" "f1 c = u"
```
```  4616     using homeomorphism_grouping_point_1 [OF \<open>a < c\<close> \<open>a < u\<close>] .
```
```  4617   obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
```
```  4618                    and f2_eq: "f2 c = u" "f2 d = v"
```
```  4619     using homeomorphism_grouping_point_1 [OF \<open>c < d\<close> \<open>u < v\<close>] .
```
```  4620   obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
```
```  4621                    and f3_eq: "f3 d = v" "f3 b = b"
```
```  4622     using homeomorphism_grouping_point_1 [OF \<open>d < b\<close> \<open>v < b\<close>] .
```
```  4623   obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
```
```  4624                  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"
```
```  4625     using homeomorphism_grouping_point_2 [OF 1 2] less  by (auto simp: f1_eq f2_eq)
```
```  4626   obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
```
```  4627                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"
```
```  4628     using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
```
```  4629   show ?thesis
```
```  4630     apply (rule that [OF fg])
```
```  4631     using f4_eq f_eq homeomorphism_image1 [OF 2]
```
```  4632     apply simp
```
```  4633     by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
```
```  4634 qed
```
```  4635
```
```  4636
```
```  4637 lemma homeomorphism_grouping_point_4:
```
```  4638   fixes T :: "real set"
```
```  4639   assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
```
```  4640   obtains f g where "homeomorphism T T f g"
```
```  4641                     "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
```
```  4642                     "bounded {x. (\<not> (f x = x \<and> g x = x))}"
```
```  4643 proof -
```
```  4644   obtain c d where "box c d \<noteq> {}" "cbox c d \<subseteq> U"
```
```  4645   proof -
```
```  4646     obtain u where "u \<in> U"
```
```  4647       using \<open>U \<noteq> {}\<close> by blast
```
```  4648     then obtain e where "e > 0" "cball u e \<subseteq> U"
```
```  4649       using \<open>open U\<close> open_contains_cball by blast
```
```  4650     then show ?thesis
```
```  4651       by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
```
```  4652   qed
```
```  4653   have "compact K"
```
```  4654     by (simp add: \<open>finite K\<close> finite_imp_compact)
```
```  4655   obtain a b where "box a b \<noteq> {}" "K \<subseteq> cbox a b" "cbox a b \<subseteq> S"
```
```  4656   proof (cases "K = {}")
```
```  4657     case True then show ?thesis
```
```  4658       using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> that by blast
```
```  4659   next
```
```  4660     case False
```
```  4661     then obtain a b where "a \<in> K" "b \<in> K"
```
```  4662             and a: "\<And>x. x \<in> K \<Longrightarrow> a \<le> x" and b: "\<And>x. x \<in> K \<Longrightarrow> x \<le> b"
```
```  4663       using compact_attains_inf compact_attains_sup by (metis \<open>compact K\<close>)+
```
```  4664     obtain e where "e > 0" "cball b e \<subseteq> S"
```
```  4665       using \<open>open S\<close> open_contains_cball
```
```  4666       by (metis \<open>b \<in> K\<close> \<open>K \<subseteq> S\<close> subsetD)
```
```  4667     show ?thesis
```
```  4668     proof
```
```  4669       show "box a (b + e) \<noteq> {}"
```
```  4670         using \<open>0 < e\<close> \<open>b \<in> K\<close> a by force
```
```  4671       show "K \<subseteq> cbox a (b + e)"
```
```  4672         using \<open>0 < e\<close> a b by fastforce
```
```  4673       have "a \<in> S"
```
```  4674         using \<open>a \<in> K\<close> assms(6) by blast
```
```  4675       have "b + e \<in> S"
```
```  4676         using \<open>0 < e\<close> \<open>cball b e \<subseteq> S\<close>  by (force simp: dist_norm)
```
```  4677       show "cbox a (b + e) \<subseteq> S"
```
```  4678         using \<open>a \<in> S\<close> \<open>b + e \<in> S\<close> \<open>connected S\<close> connected_contains_Icc by auto
```
```  4679     qed
```
```  4680   qed
```
```  4681   obtain w z where "cbox w z \<subseteq> S" and sub_wz: "cbox a b \<union> cbox c d \<subseteq> box w z"
```
```  4682   proof -
```
```  4683     have "a \<in> S" "b \<in> S"
```
```  4684       using \<open>box a b \<noteq> {}\<close> \<open>cbox a b \<subseteq> S\<close> by auto
```
```  4685     moreover have "c \<in> S" "d \<in> S"
```
```  4686       using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> by force+
```
```  4687     ultimately have "min a c \<in> S" "max b d \<in> S"
```
```  4688       by linarith+
```
```  4689     then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 \<subseteq> S" "e2 > 0" "cball (max b d) e2 \<subseteq> S"
```
```  4690       using \<open>open S\<close> open_contains_cball by metis
```
```  4691     then have *: "min a c - e1 \<in> S" "max b d + e2 \<in> S"
```
```  4692       by (auto simp: dist_norm)
```
```  4693     show ?thesis
```
```  4694     proof
```
```  4695       show "cbox (min a c - e1) (max b d+ e2) \<subseteq> S"
```
```  4696         using * \<open>connected S\<close> connected_contains_Icc by auto
```
```  4697       show "cbox a b \<union> cbox c d \<subseteq> box (min a c - e1) (max b d + e2)"
```
```  4698         using \<open>0 < e1\<close> \<open>0 < e2\<close> by auto
```
```  4699     qed
```
```  4700   qed
```
```  4701   then
```
```  4702   obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
```
```  4703                and "f w = w" "f z = z"
```
```  4704                and fin: "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<in> cbox c d"
```
```  4705     using homeomorphism_grouping_point_3 [of a b w z c d]
```
```  4706     using \<open>box a b \<noteq> {}\<close> \<open>box c d \<noteq> {}\<close> by blast
```
```  4707   have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
```
```  4708     using hom homeomorphism_def by blast+
```
```  4709   define f' where "f' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
```
```  4710   define g' where "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
```
```  4711   show ?thesis
```
```  4712   proof
```
```  4713     have T: "cbox w z \<union> (T - box w z) = T"
```
```  4714       using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
```
```  4715     show "homeomorphism T T f' g'"
```
```  4716     proof
```
```  4717       have clo: "closedin (top_of_set (cbox w z \<union> (T - box w z))) (T - box w z)"
```
```  4718         by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
```
```  4719       have "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
```
```  4720         unfolding f'_def g'_def
```
```  4721          apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
```
```  4722          apply (simp_all add: closed_subset)
```
```  4723         using \<open>f w = w\<close> \<open>f z = z\<close> apply force
```
```  4724         by (metis \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_def less_eq_real_def mem_box_real(2))
```
```  4725       then show "continuous_on T f'" "continuous_on T g'"
```
```  4726         by (simp_all only: T)
```
```  4727       show "f' ` T \<subseteq> T"
```
```  4728         unfolding f'_def
```
```  4729         by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
```
```  4730       show "g' ` T \<subseteq> T"
```
```  4731         unfolding g'_def
```
```  4732         by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
```
```  4733       show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
```
```  4734         unfolding f'_def g'_def
```
```  4735         using homeomorphism_apply1 [OF hom]  homeomorphism_image1 [OF hom] by fastforce
```
```  4736       show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
```
```  4737         unfolding f'_def g'_def
```
```  4738         using homeomorphism_apply2 [OF hom]  homeomorphism_image2 [OF hom] by fastforce
```
```  4739     qed
```
```  4740     show "\<And>x. x \<in> K \<Longrightarrow> f' x \<in> U"
```
```  4741       using fin sub_wz \<open>K \<subseteq> cbox a b\<close> \<open>cbox c d \<subseteq> U\<close> by (force simp: f'_def)
```
```  4742     show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
```
```  4743       using \<open>cbox w z \<subseteq> S\<close> by (auto simp: f'_def g'_def)
```
```  4744     show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
```
```  4745       apply (rule bounded_subset [of "cbox w z"])
```
```  4746       using bounded_cbox apply blast
```
```  4747       apply (auto simp: f'_def g'_def)
```
```  4748       done
```
```  4749   qed
```
```  4750 qed
```
```  4751
```
```  4752 proposition%unimportant homeomorphism_grouping_points_exists:
```
```  4753   fixes S :: "'a::euclidean_space set"
```
```  4754   assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
```
```  4755   obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
```
```  4756                     "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
```
```  4757 proof (cases "2 \<le> DIM('a)")
```
```  4758   case True
```
```  4759   have TS: "T \<subseteq> affine hull S"
```
```  4760     using affine_hull_open assms by blast
```
```  4761   have "infinite U"
```
```  4762     using \<open>open U\<close> \<open>U \<noteq> {}\<close> finite_imp_not_open by blast
```
```  4763   then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
```
```  4764     using infinite_arbitrarily_large by metis
```
```  4765   then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
```
```  4766     using \<open>finite K\<close> finite_same_card_bij by blast
```
```  4767   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)}"
```
```  4768   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>])
```
```  4769     show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
```
```  4770       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
```
```  4771     show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
```
```  4772       using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
```
```  4773   qed (use affine_hull_open assms that in auto)
```
```  4774   then show ?thesis
```
```  4775     using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
```
```  4776 next
```
```  4777   case False
```
```  4778   with DIM_positive have "DIM('a) = 1"
```
```  4779     by (simp add: dual_order.antisym)
```
```  4780   then obtain h::"'a \<Rightarrow>real" and j
```
```  4781   where "linear h" "linear j"
```
```  4782     and noh: "\<And>x. norm(h x) = norm x" and noj: "\<And>y. norm(j y) = norm y"
```
```  4783     and hj:  "\<And>x. j(h x) = x" "\<And>y. h(j y) = y"
```
```  4784     and ranh: "surj h"
```
```  4785     using isomorphisms_UNIV_UNIV
```
```  4786     by (metis (mono_tags, hide_lams) DIM_real UNIV_eq_I range_eqI)
```
```  4787   obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
```
```  4788                and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
```
```  4789                and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
```
```  4790                and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4791     apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
```
```  4792     by (simp_all add: assms image_mono  \<open>linear h\<close> open_surjective_linear_image connected_linear_image ranh)
```
```  4793   have jf: "j (f (h x)) = x \<longleftrightarrow> f (h x) = h x" for x
```
```  4794     by (metis hj)
```
```  4795   have jg: "j (g (h x)) = x \<longleftrightarrow> g (h x) = h x" for x
```
```  4796     by (metis hj)
```
```  4797   have cont_hj: "continuous_on X h"  "continuous_on Y j" for X Y
```
```  4798     by (simp_all add: \<open>linear h\<close> \<open>linear j\<close> linear_linear linear_continuous_on)
```
```  4799   show ?thesis
```
```  4800   proof
```
```  4801     show "homeomorphism T T (j \<circ> f \<circ> h) (j \<circ> g \<circ> h)"
```
```  4802     proof
```
```  4803       show "continuous_on T (j \<circ> f \<circ> h)" "continuous_on T (j \<circ> g \<circ> h)"
```
```  4804         using hom homeomorphism_def
```
```  4805         by (blast intro: continuous_on_compose cont_hj)+
```
```  4806       show "(j \<circ> f \<circ> h) ` T \<subseteq> T" "(j \<circ> g \<circ> h) ` T \<subseteq> T"
```
```  4807         by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
```
```  4808       show "\<And>x. x \<in> T \<Longrightarrow> (j \<circ> g \<circ> h) ((j \<circ> f \<circ> h) x) = x"
```
```  4809         using hj hom homeomorphism_apply1 by fastforce
```
```  4810       show "\<And>y. y \<in> T \<Longrightarrow> (j \<circ> f \<circ> h) ((j \<circ> g \<circ> h) y) = y"
```
```  4811         using hj hom homeomorphism_apply2 by fastforce
```
```  4812     qed
```
```  4813     show "{x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} \<subseteq> S"
```
```  4814       apply (clarsimp simp: jf jg hj)
```
```  4815       using sub hj
```
```  4816       apply (drule_tac c="h x" in subsetD, force)
```
```  4817       by (metis imageE)
```
```  4818     have "bounded (j ` {x. (\<not> (f x = x \<and> g x = x))})"
```
```  4819       by (rule bounded_linear_image [OF bou]) (use \<open>linear j\<close> linear_conv_bounded_linear in auto)
```
```  4820     moreover
```
```  4821     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))}"
```
```  4822       using hj by (auto simp: jf jg image_iff, metis+)
```
```  4823     ultimately show "bounded {x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)}"
```
```  4824       by metis
```
```  4825     show "\<And>x. x \<in> K \<Longrightarrow> (j \<circ> f \<circ> h) x \<in> U"
```
```  4826       using f hj by fastforce
```
```  4827   qed
```
```  4828 qed
```
```  4829
```
```  4830
```
```  4831 proposition%unimportant homeomorphism_grouping_points_exists_gen:
```
```  4832   fixes S :: "'a::euclidean_space set"
```
```  4833   assumes opeU: "openin (top_of_set S) U"
```
```  4834       and opeS: "openin (top_of_set (affine hull S)) S"
```
```  4835       and "U \<noteq> {}" "finite K" "K \<subseteq> S" and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
```
```  4836   obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
```
```  4837                     "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
```
```  4838 proof (cases "2 \<le> aff_dim S")
```
```  4839   case True
```
```  4840   have opeU': "openin (top_of_set (affine hull S)) U"
```
```  4841     using opeS opeU openin_trans by blast
```
```  4842   obtain u where "u \<in> U" "u \<in> S"
```
```  4843     using \<open>U \<noteq> {}\<close> opeU openin_imp_subset by fastforce+
```
```  4844   have "infinite U"
```
```  4845     apply (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
```
```  4846     apply (rule connected_imp_perfect_aff_dim [OF \<open>connected S\<close> _ \<open>u \<in> S\<close>])
```
```  4847     using True apply simp
```
```  4848     done
```
```  4849   then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
```
```  4850     using infinite_arbitrarily_large by metis
```
```  4851   then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
```
```  4852     using \<open>finite K\<close> finite_same_card_bij by blast
```
```  4853   have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(id i) = \<gamma> i) \<and>
```
```  4854                {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4855   proof (rule homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> _ _ True opeS S])
```
```  4856     show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
```
```  4857       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)
```
```  4858     show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
```
```  4859       using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
```
```  4860   qed
```
```  4861   then show ?thesis
```
```  4862     using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
```
```  4863 next
```
```  4864   case False
```
```  4865   with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
```
```  4866   then show ?thesis
```
```  4867   proof cases
```
```  4868     assume "aff_dim S = -1"
```
```  4869     then have "S = {}"
```
```  4870       using aff_dim_empty by blast
```
```  4871     then have "False"
```
```  4872       using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
```
```  4873     then show ?thesis ..
```
```  4874   next
```
```  4875     assume "aff_dim S = 0"
```
```  4876     then obtain a where "S = {a}"
```
```  4877       using aff_dim_eq_0 by blast
```
```  4878     then have "K \<subseteq> U"
```
```  4879       using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
```
```  4880     show ?thesis
```
```  4881       apply (rule that [of id id])
```
```  4882       using \<open>K \<subseteq> U\<close> by (auto simp: continuous_on_id intro: homeomorphismI)
```
```  4883   next
```
```  4884     assume "aff_dim S = 1"
```
```  4885     then have "affine hull S homeomorphic (UNIV :: real set)"
```
```  4886       by (auto simp: homeomorphic_affine_sets)
```
```  4887     then obtain h::"'a\<Rightarrow>real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
```
```  4888       using homeomorphic_def by blast
```
```  4889     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"
```
```  4890       by (auto simp: homeomorphism_def)
```
```  4891     have connh: "connected (h ` S)"
```
```  4892       by (meson Topological_Spaces.connected_continuous_image \<open>connected S\<close> homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
```
```  4893     have hUS: "h ` U \<subseteq> h ` S"
```
```  4894       by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
```
```  4895     have opn: "openin (top_of_set (affine hull S)) U \<Longrightarrow> open (h ` U)" for U
```
```  4896       using homeomorphism_imp_open_map [OF homhj]  by simp
```
```  4897     have "open (h ` U)" "open (h ` S)"
```
```  4898       by (auto intro: opeS opeU openin_trans opn)
```
```  4899     then obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
```
```  4900                  and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
```
```  4901                  and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
```
```  4902                  and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
```
```  4903       apply (rule homeomorphism_grouping_points_exists [of "h ` U" "h ` S" "h ` K" "h ` T"])
```
```  4904       using assms by (auto simp: connh hUS)
```
```  4905     have jf: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (f (h x)) = x \<longleftrightarrow> f (h x) = h x"
```
```  4906       by (metis h j)
```
```  4907     have jg: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (g (h x)) = x \<longleftrightarrow> g (h x) = h x"
```
```  4908       by (metis h j)
```
```  4909     have cont_hj: "continuous_on T h"  "continuous_on Y j" for Y
```
```  4910       apply (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
```
```  4911       using homeomorphism_def homhj apply blast
```
```  4912       by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
```
```  4913     define f' where "f' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> f \<circ> h) x else x"
```
```  4914     define g' where "g' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> g \<circ> h) x else x"
```
```  4915     show ?thesis
```
```  4916     proof
```
```  4917       show "homeomorphism T T f' g'"
```
```  4918       proof
```
```  4919         have "continuous_on T (j \<circ> f \<circ> h)"
```
```  4920           apply (intro continuous_on_compose cont_hj)
```
```  4921           using hom homeomorphism_def by blast
```
```  4922         then show "continuous_on T f'"
```
```  4923           apply (rule continuous_on_eq)
```
```  4924           using \<open>T \<subseteq> affine hull S\<close> f'_def by auto
```
```  4925         have "continuous_on T (j \<circ> g \<circ> h)"
```
```  4926           apply (intro continuous_on_compose cont_hj)
```
```  4927           using hom homeomorphism_def by blast
```
```  4928         then show "continuous_on T g'"
```
```  4929           apply (rule continuous_on_eq)
```
```  4930           using \<open>T \<subseteq> affine hull S\<close> g'_def by auto
```
```  4931         show "f' ` T \<subseteq> T"
```
```  4932         proof (clarsimp simp: f'_def)
```
```  4933           fix x assume "x \<in> T"
```
```  4934           then have "f (h x) \<in> h ` T"
```
```  4935             by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
```
```  4936           then show "j (f (h x)) \<in> T"
```
```  4937             using \<open>T \<subseteq> affine hull S\<close> h by auto
```
```  4938         qed
```
```  4939         show "g' ` T \<subseteq> T"
```
```  4940         proof (clarsimp simp: g'_def)
```
```  4941           fix x assume "x \<in> T"
```
```  4942           then have "g (h x) \<in> h ` T"
```
```  4943             by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
```
```  4944           then show "j (g (h x)) \<in> T"
```
```  4945             using \<open>T \<subseteq> affine hull S\<close> h by auto
```
```  4946         qed
```
```  4947         show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
```
```  4948           using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
```
```  4949         show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
```
```  4950           using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
```
```  4951       qed
```
```  4952     next
```
```  4953       show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
```
```  4954         apply (clarsimp simp: f'_def g'_def jf jg)
```
```  4955         apply (rule imageE [OF subsetD [OF sub]], force)
```
```  4956         by (metis h hull_inc)
```
```  4957     next
```
```  4958       have "compact (j ` closure {x. \<not> (f x = x \<and> g x = x)})"
```
```  4959         using bou by (auto simp: compact_continuous_image cont_hj)
```
```  4960       then have "bounded (j ` {x. \<not> (f x = x \<and> g x = x)})"
```
```  4961         by (rule bounded_closure_image [OF compact_imp_bounded])
```
```  4962       moreover
```
```  4963       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))}"
```
```  4964         using h j by (auto simp: image_iff; metis)
```
```  4965       ultimately have "bounded {x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x}"
```
```  4966         by metis
```
```  4967       then show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
```
```  4968         by (simp add: f'_def g'_def Collect_mono bounded_subset)
```
```  4969     next
```
```  4970       show "f' x \<in> U" if "x \<in> K" for x
```
```  4971       proof -
```
```  4972         have "U \<subseteq> S"
```
```  4973           using opeU openin_imp_subset by blast
```
```  4974         then have "j (f (h x)) \<in> U"
```
```  4975           using f h hull_subset that by fastforce
```
```  4976         then show "f' x \<in> U"
```
```  4977           using \<open>K \<subseteq> S\<close> S f'_def that by auto
```
```  4978       qed
```
```  4979     qed
```
```  4980   qed
```
```  4981 qed
```
```  4982
```
```  4983
```
```  4984 subsection\<open>Nullhomotopic mappings\<close>
```
```  4985
```
```  4986 text\<open> A mapping out of a sphere is nullhomotopic iff it extends to the ball.
```
```  4987 This even works out in the degenerate cases when the radius is \<open>\<le>\<close> 0, and
```
```  4988 we also don't need to explicitly assume continuity since it's already implicit
```
```  4989 in both sides of the equivalence.\<close>
```
```  4990
```
```  4991 lemma nullhomotopic_from_lemma:
```
```  4992   assumes contg: "continuous_on (cball a r - {a}) g"
```
```  4993       and fa: "\<And>e. 0 < e
```
```  4994                \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>x. x \<noteq> a \<and> norm(x - a) < d \<longrightarrow> norm(g x - f a) < e)"
```
```  4995       and r: "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> f x = g x"
```
```  4996     shows "continuous_on (cball a r) f"
```
```  4997 proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
```
```  4998   fix x
```
```  4999   assume x: "dist a x \<le> r"
```
```  5000   show "continuous (at x within cball a r) f"
```
```  5001   proof (cases "x=a")
```
```  5002     case True
```
```  5003     then show ?thesis
```
```  5004       by (metis continuous_within_eps_delta fa dist_norm dist_self r)
```
```  5005   next
```
```  5006     case False
```
```  5007     show ?thesis
```
```  5008     proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
```
```  5009       have "\<exists>d>0. \<forall>x'\<in>cball a r.
```
```  5010                       dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
```
```  5011       proof -
```
```  5012         obtain d where "d > 0"
```
```  5013            and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
```
```  5014                                  dist (g x') (g x) < e"
```
```  5015           using contg False x \<open>e>0\<close>
```
```  5016           unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
```
```  5017         show ?thesis
```
```  5018           using \<open>d > 0\<close> \<open>x \<noteq> a\<close>
```
```  5019           by (rule_tac x="min d (norm(x - a))" in exI)
```
```  5020              (auto simp: dist_commute dist_norm [symmetric]  intro!: d)
```
```  5021       qed
```
```  5022       then show "continuous (at x within cball a r) g"
```
```  5023         using contg False by (auto simp: continuous_within_eps_delta)
```
```  5024       show "0 < norm (x - a)"
```
```  5025         using False by force
```
```  5026       show "x \<in> cball a r"
```
```  5027         by (simp add: x)
```
```  5028       show "\<And>x'. \<lbrakk>x' \<in> cball a r; dist x' x < norm (x - a)\<rbrakk>
```
```  5029         \<Longrightarrow> g x' = f x'"
```
```  5030         by (metis dist_commute dist_norm less_le r)
```
```  5031     qed
```
```  5032   qed
```
```  5033 qed
```
```  5034
```
```  5035 proposition nullhomotopic_from_sphere_extension:
```
```  5036   fixes f :: "'M::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  5037   shows  "(\<exists>c. homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) \<longleftrightarrow>
```
```  5038           (\<exists>g. continuous_on (cball a r) g \<and> g ` (cball a r) \<subseteq> S \<and>
```
```  5039                (\<forall>x \<in> sphere a r. g x = f x))"
```
```  5040          (is "?lhs = ?rhs")
```
```  5041 proof (cases r "0::real" rule: linorder_cases)
```
```  5042   case equal
```
```  5043   then show ?thesis
```
```  5044     apply (auto simp: homotopic_with)
```
```  5045     apply (rule_tac x="\<lambda>x. h (0, a)" in exI)
```
```  5046      apply (fastforce simp add:)
```
```  5047     using continuous_on_const by blast
```
```  5048 next
```
```  5049   case greater
```
```  5050   let ?P = "continuous_on {x. norm(x - a) = r} f \<and> f ` {x. norm(x - a) = r} \<subseteq> S"
```
```  5051   have ?P if ?lhs using that
```
```  5052   proof
```
```  5053     fix c
```
```  5054     assume c: "homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
```
```  5055     then have contf: "continuous_on (sphere a r) f" and fim: "f ` sphere a r \<subseteq> S"
```
```  5056       by (auto simp: homotopic_with_imp_subset1 homotopic_with_imp_continuous)
```
```  5057     show ?P
```
```  5058       using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
```
```  5059   qed
```
```  5060   moreover have ?P if ?rhs using that
```
```  5061   proof
```
```  5062     fix g
```
```  5063     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)"
```
```  5064     then
```
```  5065     show ?P
```
```  5066       apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
```
```  5067       apply (auto simp: dist_norm norm_minus_commute)
```
```  5068       by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
```
```  5069   qed
```
```  5070   moreover have ?thesis if ?P
```
```  5071   proof
```
```  5072     assume ?lhs
```
```  5073     then obtain c where "homotopic_with (\<lambda>x. True) (sphere a r) S (\<lambda>x. c) f"
```
```  5074       using homotopic_with_sym by blast
```
```  5075     then obtain h where conth: "continuous_on ({0..1::real} \<times> sphere a r) h"
```
```  5076                     and him: "h ` ({0..1} \<times> sphere a r) \<subseteq> S"
```
```  5077                     and h: "\<And>x. h(0, x) = c" "\<And>x. h(1, x) = f x"
```
```  5078       by (auto simp: homotopic_with_def)
```
```  5079     obtain b1::'M where "b1 \<in> Basis"
```
```  5080       using SOME_Basis by auto
```
```  5081     have "c \<in> S"
```
```  5082       apply (rule him [THEN subsetD])
```
```  5083       apply (rule_tac x = "(0, a + r *\<^sub>R b1)" in image_eqI)
```
```  5084       using h greater \<open>b1 \<in> Basis\<close>
```
```  5085        apply (auto simp: dist_norm)
```
```  5086       done
```
```  5087     have uconth: "uniformly_continuous_on ({0..1::real} \<times> (sphere a r)) h"
```
```  5088       by (force intro: compact_Times conth compact_uniformly_continuous)
```
```  5089     let ?g = "\<lambda>x. h (norm (x - a)/r,
```
```  5090                      a + (if x = a then r *\<^sub>R b1 else (r / norm(x - a)) *\<^sub>R (x - a)))"
```
```  5091     let ?g' = "\<lambda>x. h (norm (x - a)/r, a + (r / norm(x - a)) *\<^sub>R (x - a))"
```
```  5092     show ?rhs
```
```  5093     proof (intro exI conjI)
```
```  5094       have "continuous_on (cball a r - {a}) ?g'"
```
```  5095         apply (rule continuous_on_compose2 [OF conth])
```
```  5096          apply (intro continuous_intros)
```
```  5097         using greater apply (auto simp: dist_norm norm_minus_commute)
```
```  5098         done
```
```  5099       then show "continuous_on (cball a r) ?g"
```
```  5100       proof (rule nullhomotopic_from_lemma)
```
```  5101         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
```
```  5102         proof -
```
```  5103           obtain d where "0 < d"
```
```  5104              and d: "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> sphere a r; x' \<in> {0..1} \<times> sphere a r; dist x' x < d\<rbrakk>
```
```  5105                         \<Longrightarrow> dist (h x') (h x) < e"
```
```  5106             using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by auto
```
```  5107           have *: "norm (h (norm (x - a) / r,
```
```  5108                          a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e"
```
```  5109                    if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
```
```  5110           proof -
```
```  5111             have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) =
```
```  5112                   norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
```
```  5113               by (simp add: h)
```
```  5114             also have "\<dots> < e"
```
```  5115               apply (rule d [unfolded dist_norm])
```
```  5116               using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
```
```  5117                 by (auto simp: dist_norm divide_simps)
```
```  5118             finally show ?thesis .
```
```  5119           qed
```
```  5120           show ?thesis
```
```  5121             apply (rule_tac x = "min r (d * r)" in exI)
```
```  5122             using greater \<open>0 < d\<close> by (auto simp: *)
```
```  5123         qed
```
```  5124         show "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> ?g x = ?g' x"
```
```  5125           by auto
```
```  5126       qed
```
```  5127     next
```
```  5128       show "?g ` cball a r \<subseteq> S"
```
```  5129         using greater him \<open>c \<in> S\<close>
```
```  5130         by (force simp: h dist_norm norm_minus_commute)
```
```  5131     next
```
```  5132       show "\<forall>x\<in>sphere a r. ?g x = f x"
```
```  5133         using greater by (auto simp: h dist_norm norm_minus_commute)
```
```  5134     qed
```
```  5135   next
```
```  5136     assume ?rhs
```
```  5137     then obtain g where contg: "continuous_on (cball a r) g"
```
```  5138                     and gim: "g ` cball a r \<subseteq> S"
```
```  5139                     and gf: "\<forall>x \<in> sphere a r. g x = f x"
```
```  5140       by auto
```
```  5141     let ?h = "\<lambda>y. g (a + (fst y) *\<^sub>R (snd y - a))"
```
```  5142     have "continuous_on ({0..1} \<times> sphere a r) ?h"
```
```  5143       apply (rule continuous_on_compose2 [OF contg])
```
```  5144        apply (intro continuous_intros)
```
```  5145       apply (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
```
```  5146       done
```
```  5147     moreover
```
```  5148     have "?h ` ({0..1} \<times> sphere a r) \<subseteq> S"
```
```  5149       by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
```
```  5150     moreover
```
```  5151     have "\<forall>x\<in>sphere a r. ?h (0, x) = g a" "\<forall>x\<in>sphere a r. ?h (1, x) = f x"
```
```  5152       by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
```
```  5153     ultimately
```
```  5154     show ?lhs
```
```  5155       apply (subst homotopic_with_sym)
```
```  5156       apply (rule_tac x="g a" in exI)
```
```  5157       apply (auto simp: homotopic_with)
```
```  5158       done
```
```  5159   qed
```
```  5160   ultimately
```
```  5161   show ?thesis by meson
```
```  5162 qed simp
```
```  5163
```
`  5164 end`