src/HOL/Multivariate_Analysis/Path_Connected.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51478 270b21f3ae0a parent 50935 cfdf19d3ca32 child 51481 ef949192e5d6 permissions -rw-r--r--
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
```     1 (*  Title:      HOL/Multivariate_Analysis/Path_Connected.thy
```
```     2     Author:     Robert Himmelmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Continuous paths and path-connected sets *}
```
```     6
```
```     7 theory Path_Connected
```
```     8 imports Convex_Euclidean_Space
```
```     9 begin
```
```    10
```
```    11 lemma continuous_on_cong: (* MOVE to Topological_Spaces *)
```
```    12   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
```
```    13   unfolding continuous_on_def by (intro ball_cong Lim_cong_within) auto
```
```    14
```
```    15 lemma continuous_on_compose2:
```
```    16   shows "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
```
```    17   using continuous_on_compose[of s f g] by (simp add: comp_def)
```
```    18
```
```    19 subsection {* Paths. *}
```
```    20
```
```    21 definition path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
```
```    22   where "path g \<longleftrightarrow> continuous_on {0 .. 1} g"
```
```    23
```
```    24 definition pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
```
```    25   where "pathstart g = g 0"
```
```    26
```
```    27 definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
```
```    28   where "pathfinish g = g 1"
```
```    29
```
```    30 definition path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
```
```    31   where "path_image g = g ` {0 .. 1}"
```
```    32
```
```    33 definition reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)"
```
```    34   where "reversepath g = (\<lambda>x. g(1 - x))"
```
```    35
```
```    36 definition joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)"
```
```    37     (infixr "+++" 75)
```
```    38   where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
```
```    39
```
```    40 definition simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
```
```    41   where "simple_path g \<longleftrightarrow>
```
```    42     (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
```
```    43
```
```    44 definition injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
```
```    45   where "injective_path g \<longleftrightarrow> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"
```
```    46
```
```    47
```
```    48 subsection {* Some lemmas about these concepts. *}
```
```    49
```
```    50 lemma injective_imp_simple_path: "injective_path g \<Longrightarrow> simple_path g"
```
```    51   unfolding injective_path_def simple_path_def by auto
```
```    52
```
```    53 lemma path_image_nonempty: "path_image g \<noteq> {}"
```
```    54   unfolding path_image_def image_is_empty interval_eq_empty by auto
```
```    55
```
```    56 lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"
```
```    57   unfolding pathstart_def path_image_def by auto
```
```    58
```
```    59 lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"
```
```    60   unfolding pathfinish_def path_image_def by auto
```
```    61
```
```    62 lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
```
```    63   unfolding path_def path_image_def
```
```    64   apply (erule connected_continuous_image)
```
```    65   apply (rule convex_connected, rule convex_real_interval)
```
```    66   done
```
```    67
```
```    68 lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
```
```    69   unfolding path_def path_image_def
```
```    70   by (erule compact_continuous_image, rule compact_interval)
```
```    71
```
```    72 lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"
```
```    73   unfolding reversepath_def by auto
```
```    74
```
```    75 lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"
```
```    76   unfolding pathstart_def reversepath_def pathfinish_def by auto
```
```    77
```
```    78 lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
```
```    79   unfolding pathstart_def reversepath_def pathfinish_def by auto
```
```    80
```
```    81 lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
```
```    82   unfolding pathstart_def joinpaths_def pathfinish_def by auto
```
```    83
```
```    84 lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
```
```    85   unfolding pathstart_def joinpaths_def pathfinish_def by auto
```
```    86
```
```    87 lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g"
```
```    88 proof -
```
```    89   have *: "\<And>g. path_image(reversepath g) \<subseteq> path_image g"
```
```    90     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
```
```    91     apply(rule,rule,erule bexE)
```
```    92     apply(rule_tac x="1 - xa" in bexI)
```
```    93     apply auto
```
```    94     done
```
```    95   show ?thesis
```
```    96     using *[of g] *[of "reversepath g"]
```
```    97     unfolding reversepath_reversepath by auto
```
```    98 qed
```
```    99
```
```   100 lemma path_reversepath[simp]: "path (reversepath g) \<longleftrightarrow> path g"
```
```   101 proof -
```
```   102   have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
```
```   103     unfolding path_def reversepath_def
```
```   104     apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
```
```   105     apply (intro continuous_on_intros)
```
```   106     apply (rule continuous_on_subset[of "{0..1}"], assumption)
```
```   107     apply auto
```
```   108     done
```
```   109   show ?thesis
```
```   110     using *[of "reversepath g"] *[of g]
```
```   111     unfolding reversepath_reversepath
```
```   112     by (rule iffI)
```
```   113 qed
```
```   114
```
```   115 lemmas reversepath_simps =
```
```   116   path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
```
```   117
```
```   118 lemma path_join[simp]:
```
```   119   assumes "pathfinish g1 = pathstart g2"
```
```   120   shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
```
```   121   unfolding path_def pathfinish_def pathstart_def
```
```   122 proof safe
```
```   123   assume cont: "continuous_on {0..1} (g1 +++ g2)"
```
```   124   have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
```
```   125     by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
```
```   126   have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
```
```   127     using assms by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
```
```   128   show "continuous_on {0..1} g1" "continuous_on {0..1} g2"
```
```   129     unfolding g1 g2 by (auto intro!: continuous_on_intros continuous_on_subset[OF cont])
```
```   130 next
```
```   131   assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
```
```   132   have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
```
```   133     by auto
```
```   134   { fix x :: real assume "0 \<le> x" "x \<le> 1" then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
```
```   135       by (intro image_eqI[where x="x/2"]) auto }
```
```   136   note 1 = this
```
```   137   { fix x :: real assume "0 \<le> x" "x \<le> 1" then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
```
```   138       by (intro image_eqI[where x="x/2 + 1/2"]) auto }
```
```   139   note 2 = this
```
```   140   show "continuous_on {0..1} (g1 +++ g2)"
```
```   141     using assms unfolding joinpaths_def 01
```
```   142     by (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_on_intros)
```
```   143        (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
```
```   144 qed
```
```   145
```
```   146 lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)"
```
```   147   unfolding path_image_def joinpaths_def by auto
```
```   148
```
```   149 lemma subset_path_image_join:
```
```   150   assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s"
```
```   151   shows "path_image(g1 +++ g2) \<subseteq> s"
```
```   152   using path_image_join_subset[of g1 g2] and assms by auto
```
```   153
```
```   154 lemma path_image_join:
```
```   155   assumes "pathfinish g1 = pathstart g2"
```
```   156   shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
```
```   157   apply (rule, rule path_image_join_subset, rule)
```
```   158   unfolding Un_iff
```
```   159 proof (erule disjE)
```
```   160   fix x
```
```   161   assume "x \<in> path_image g1"
```
```   162   then obtain y where y: "y\<in>{0..1}" "x = g1 y"
```
```   163     unfolding path_image_def image_iff by auto
```
```   164   then show "x \<in> path_image (g1 +++ g2)"
```
```   165     unfolding joinpaths_def path_image_def image_iff
```
```   166     apply (rule_tac x="(1/2) *\<^sub>R y" in bexI)
```
```   167     apply auto
```
```   168     done
```
```   169 next
```
```   170   fix x
```
```   171   assume "x \<in> path_image g2"
```
```   172   then obtain y where y: "y\<in>{0..1}" "x = g2 y"
```
```   173     unfolding path_image_def image_iff by auto
```
```   174   then show "x \<in> path_image (g1 +++ g2)"
```
```   175     unfolding joinpaths_def path_image_def image_iff
```
```   176     apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI)
```
```   177     using assms(1)[unfolded pathfinish_def pathstart_def]
```
```   178     apply (auto simp add: add_divide_distrib)
```
```   179     done
```
```   180 qed
```
```   181
```
```   182 lemma not_in_path_image_join:
```
```   183   assumes "x \<notin> path_image g1" "x \<notin> path_image g2"
```
```   184   shows "x \<notin> path_image(g1 +++ g2)"
```
```   185   using assms and path_image_join_subset[of g1 g2] by auto
```
```   186
```
```   187 lemma simple_path_reversepath:
```
```   188   assumes "simple_path g"
```
```   189   shows "simple_path (reversepath g)"
```
```   190   using assms
```
```   191   unfolding simple_path_def reversepath_def
```
```   192   apply -
```
```   193   apply (rule ballI)+
```
```   194   apply (erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
```
```   195   apply auto
```
```   196   done
```
```   197
```
```   198 lemma simple_path_join_loop:
```
```   199   assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
```
```   200     "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
```
```   201   shows "simple_path(g1 +++ g2)"
```
```   202   unfolding simple_path_def
```
```   203 proof ((rule ballI)+, rule impI)
```
```   204   let ?g = "g1 +++ g2"
```
```   205   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
```
```   206   fix x y :: real
```
```   207   assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
```
```   208   show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
```
```   209   proof (case_tac "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
```
```   210     assume as: "x \<le> 1 / 2" "y \<le> 1 / 2"
```
```   211     then have "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)"
```
```   212       using xy(3) unfolding joinpaths_def by auto
```
```   213     moreover
```
```   214     have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
```
```   215       by auto
```
```   216     ultimately
```
```   217     show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
```
```   218   next
```
```   219     assume as:"x > 1 / 2" "y > 1 / 2"
```
```   220     then have "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)"
```
```   221       using xy(3) unfolding joinpaths_def by auto
```
```   222     moreover
```
```   223     have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}"
```
```   224       using xy(1,2) as by auto
```
```   225     ultimately
```
```   226     show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
```
```   227   next
```
```   228     assume as:"x \<le> 1 / 2" "y > 1 / 2"
```
```   229     then have "?g x \<in> path_image g1" "?g y \<in> path_image g2"
```
```   230       unfolding path_image_def joinpaths_def
```
```   231       using xy(1,2) by auto
```
```   232     moreover
```
```   233       have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
```
```   234       using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)
```
```   235       by (auto simp add: field_simps)
```
```   236     ultimately
```
```   237     have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
```
```   238     then have "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)
```
```   239       using inj(1)[of "2 *\<^sub>R x" 0] by auto
```
```   240     moreover
```
```   241     have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
```
```   242       unfolding joinpaths_def pathfinish_def using as(2) and xy(2)
```
```   243       using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto
```
```   244     ultimately show ?thesis by auto
```
```   245   next
```
```   246     assume as: "x > 1 / 2" "y \<le> 1 / 2"
```
```   247     then have "?g x \<in> path_image g2" "?g y \<in> path_image g1"
```
```   248       unfolding path_image_def joinpaths_def
```
```   249       using xy(1,2) by auto
```
```   250     moreover
```
```   251       have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
```
```   252       using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)
```
```   253       by (auto simp add: field_simps)
```
```   254     ultimately
```
```   255     have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
```
```   256     then have "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)
```
```   257       using inj(1)[of "2 *\<^sub>R y" 0] by auto
```
```   258     moreover
```
```   259     have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
```
```   260       unfolding joinpaths_def pathfinish_def using as(1) and xy(1)
```
```   261       using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto
```
```   262     ultimately show ?thesis by auto
```
```   263   qed
```
```   264 qed
```
```   265
```
```   266 lemma injective_path_join:
```
```   267   assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
```
```   268     "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
```
```   269   shows "injective_path(g1 +++ g2)"
```
```   270   unfolding injective_path_def
```
```   271 proof (rule, rule, rule)
```
```   272   let ?g = "g1 +++ g2"
```
```   273   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
```
```   274   fix x y
```
```   275   assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
```
```   276   show "x = y"
```
```   277   proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
```
```   278     assume "x \<le> 1 / 2" "y \<le> 1 / 2"
```
```   279     then show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
```
```   280       unfolding joinpaths_def by auto
```
```   281   next
```
```   282     assume "x > 1 / 2" "y > 1 / 2"
```
```   283     then show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
```
```   284       unfolding joinpaths_def by auto
```
```   285   next
```
```   286     assume as: "x \<le> 1 / 2" "y > 1 / 2"
```
```   287     then have "?g x \<in> path_image g1" "?g y \<in> path_image g2"
```
```   288       unfolding path_image_def joinpaths_def
```
```   289       using xy(1,2) by auto
```
```   290     then have "?g x = pathfinish g1" "?g y = pathstart g2"
```
```   291       using assms(4) unfolding assms(3) xy(3) by auto
```
```   292     then show ?thesis
```
```   293       using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)
```
```   294       unfolding pathstart_def pathfinish_def joinpaths_def
```
```   295       by auto
```
```   296   next
```
```   297     assume as:"x > 1 / 2" "y \<le> 1 / 2"
```
```   298     then have "?g x \<in> path_image g2" "?g y \<in> path_image g1"
```
```   299       unfolding path_image_def joinpaths_def
```
```   300       using xy(1,2) by auto
```
```   301     then have "?g x = pathstart g2" "?g y = pathfinish g1"
```
```   302       using assms(4) unfolding assms(3) xy(3) by auto
```
```   303     then show ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)
```
```   304       unfolding pathstart_def pathfinish_def joinpaths_def
```
```   305       by auto
```
```   306   qed
```
```   307 qed
```
```   308
```
```   309 lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
```
```   310
```
```   311
```
```   312 subsection {* Reparametrizing a closed curve to start at some chosen point. *}
```
```   313
```
```   314 definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) =
```
```   315   (\<lambda>x. if (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"
```
```   316
```
```   317 lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
```
```   318   unfolding pathstart_def shiftpath_def by auto
```
```   319
```
```   320 lemma pathfinish_shiftpath:
```
```   321   assumes "0 \<le> a" "pathfinish g = pathstart g"
```
```   322   shows "pathfinish(shiftpath a g) = g a"
```
```   323   using assms unfolding pathstart_def pathfinish_def shiftpath_def
```
```   324   by auto
```
```   325
```
```   326 lemma endpoints_shiftpath:
```
```   327   assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}"
```
```   328   shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
```
```   329   using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
```
```   330
```
```   331 lemma closed_shiftpath:
```
```   332   assumes "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```   333   shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"
```
```   334   using endpoints_shiftpath[OF assms] by auto
```
```   335
```
```   336 lemma path_shiftpath:
```
```   337   assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```   338   shows "path(shiftpath a g)"
```
```   339 proof -
```
```   340   have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto
```
```   341   have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
```
```   342     using assms(2)[unfolded pathfinish_def pathstart_def] by auto
```
```   343   show ?thesis
```
```   344     unfolding path_def shiftpath_def *
```
```   345     apply (rule continuous_on_union)
```
```   346     apply (rule closed_real_atLeastAtMost)+
```
```   347     apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
```
```   348     apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
```
```   349     apply (rule continuous_on_intros)+ prefer 2
```
```   350     apply (rule continuous_on_intros)+
```
```   351     apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
```
```   352     using assms(3) and **
```
```   353     apply (auto, auto simp add: field_simps)
```
```   354     done
```
```   355 qed
```
```   356
```
```   357 lemma shiftpath_shiftpath:
```
```   358   assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}"
```
```   359   shows "shiftpath (1 - a) (shiftpath a g) x = g x"
```
```   360   using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto
```
```   361
```
```   362 lemma path_image_shiftpath:
```
```   363   assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
```
```   364   shows "path_image(shiftpath a g) = path_image g"
```
```   365 proof -
```
```   366   { fix x
```
```   367     assume as:"g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)"
```
```   368     then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
```
```   369     proof (cases "a \<le> x")
```
```   370       case False
```
```   371       then show ?thesis
```
```   372         apply (rule_tac x="1 + x - a" in bexI)
```
```   373         using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
```
```   374         apply (auto simp add: field_simps atomize_not)
```
```   375         done
```
```   376     next
```
```   377       case True
```
```   378       then show ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
```
```   379         by(auto simp add: field_simps)
```
```   380     qed
```
```   381   }
```
```   382   then show ?thesis
```
```   383     using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
```
```   384     by(auto simp add: image_iff)
```
```   385 qed
```
```   386
```
```   387
```
```   388 subsection {* Special case of straight-line paths. *}
```
```   389
```
```   390 definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
```
```   391   where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
```
```   392
```
```   393 lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
```
```   394   unfolding pathstart_def linepath_def by auto
```
```   395
```
```   396 lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"
```
```   397   unfolding pathfinish_def linepath_def by auto
```
```   398
```
```   399 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
```
```   400   unfolding linepath_def by (intro continuous_intros)
```
```   401
```
```   402 lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
```
```   403   using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
```
```   404
```
```   405 lemma path_linepath[intro]: "path(linepath a b)"
```
```   406   unfolding path_def by(rule continuous_on_linepath)
```
```   407
```
```   408 lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
```
```   409   unfolding path_image_def segment linepath_def
```
```   410   apply (rule set_eqI, rule) defer
```
```   411   unfolding mem_Collect_eq image_iff
```
```   412   apply(erule exE)
```
```   413   apply(rule_tac x="u *\<^sub>R 1" in bexI)
```
```   414   apply auto
```
```   415   done
```
```   416
```
```   417 lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
```
```   418   unfolding reversepath_def linepath_def
```
```   419   by auto
```
```   420
```
```   421 lemma injective_path_linepath:
```
```   422   assumes "a \<noteq> b"
```
```   423   shows "injective_path (linepath a b)"
```
```   424 proof -
```
```   425   { fix x y :: "real"
```
```   426     assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
```
```   427     then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps)
```
```   428     with assms have "x = y" by simp }
```
```   429   then show ?thesis
```
```   430     unfolding injective_path_def linepath_def
```
```   431     by (auto simp add: algebra_simps)
```
```   432 qed
```
```   433
```
```   434 lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)"
```
```   435   by(auto intro!: injective_imp_simple_path injective_path_linepath)
```
```   436
```
```   437
```
```   438 subsection {* Bounding a point away from a path. *}
```
```   439
```
```   440 lemma not_on_path_ball:
```
```   441   fixes g :: "real \<Rightarrow> 'a::heine_borel"
```
```   442   assumes "path g" "z \<notin> path_image g"
```
```   443   shows "\<exists>e > 0. ball z e \<inter> (path_image g) = {}"
```
```   444 proof -
```
```   445   obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
```
```   446     using distance_attains_inf[OF _ path_image_nonempty, of g z]
```
```   447     using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
```
```   448   then show ?thesis
```
```   449     apply (rule_tac x="dist z a" in exI)
```
```   450     using assms(2)
```
```   451     apply (auto intro!: dist_pos_lt)
```
```   452     done
```
```   453 qed
```
```   454
```
```   455 lemma not_on_path_cball:
```
```   456   fixes g :: "real \<Rightarrow> 'a::heine_borel"
```
```   457   assumes "path g" "z \<notin> path_image g"
```
```   458   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
```
```   459 proof -
```
```   460   obtain e where "ball z e \<inter> path_image g = {}" "e>0"
```
```   461     using not_on_path_ball[OF assms] by auto
```
```   462   moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
```
```   463   ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto
```
```   464 qed
```
```   465
```
```   466
```
```   467 subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
```
```   468
```
```   469 definition "path_component s x y \<longleftrightarrow>
```
```   470   (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
```
```   471
```
```   472 lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
```
```   473
```
```   474 lemma path_component_mem:
```
```   475   assumes "path_component s x y"
```
```   476   shows "x \<in> s" "y \<in> s"
```
```   477   using assms unfolding path_defs by auto
```
```   478
```
```   479 lemma path_component_refl:
```
```   480   assumes "x \<in> s"
```
```   481   shows "path_component s x x"
```
```   482   unfolding path_defs
```
```   483   apply (rule_tac x="\<lambda>u. x" in exI)
```
```   484   using assms apply (auto intro!:continuous_on_intros) done
```
```   485
```
```   486 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
```
```   487   by (auto intro!: path_component_mem path_component_refl)
```
```   488
```
```   489 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
```
```   490   using assms
```
```   491   unfolding path_component_def
```
```   492   apply (erule exE)
```
```   493   apply (rule_tac x="reversepath g" in exI)
```
```   494   apply auto
```
```   495   done
```
```   496
```
```   497 lemma path_component_trans:
```
```   498   assumes "path_component s x y" "path_component s y z"
```
```   499   shows "path_component s x z"
```
```   500   using assms
```
```   501   unfolding path_component_def
```
```   502   apply -
```
```   503   apply (erule exE)+
```
```   504   apply (rule_tac x="g +++ ga" in exI)
```
```   505   apply (auto simp add: path_image_join)
```
```   506   done
```
```   507
```
```   508 lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow>  path_component s x y \<Longrightarrow> path_component t x y"
```
```   509   unfolding path_component_def by auto
```
```   510
```
```   511
```
```   512 subsection {* Can also consider it as a set, as the name suggests. *}
```
```   513
```
```   514 lemma path_component_set:
```
```   515   "{y. path_component s x y} =
```
```   516     {y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}"
```
```   517   apply (rule set_eqI)
```
```   518   unfolding mem_Collect_eq
```
```   519   unfolding path_component_def
```
```   520   apply auto
```
```   521   done
```
```   522
```
```   523 lemma path_component_subset: "{y. path_component s x y} \<subseteq> s"
```
```   524   apply (rule, rule path_component_mem(2))
```
```   525   apply auto
```
```   526   done
```
```   527
```
```   528 lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s"
```
```   529   apply rule
```
```   530   apply (drule equals0D[of _ x]) defer
```
```   531   apply (rule equals0I)
```
```   532   unfolding mem_Collect_eq
```
```   533   apply (drule path_component_mem(1))
```
```   534   using path_component_refl
```
```   535   apply auto
```
```   536   done
```
```   537
```
```   538
```
```   539 subsection {* Path connectedness of a space. *}
```
```   540
```
```   541 definition "path_connected s \<longleftrightarrow>
```
```   542   (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
```
```   543
```
```   544 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
```
```   545   unfolding path_connected_def path_component_def by auto
```
```   546
```
```   547 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)"
```
```   548   unfolding path_connected_component
```
```   549   apply (rule, rule, rule, rule path_component_subset)
```
```   550   unfolding subset_eq mem_Collect_eq Ball_def
```
```   551   apply auto
```
```   552   done
```
```   553
```
```   554
```
```   555 subsection {* Some useful lemmas about path-connectedness. *}
```
```   556
```
```   557 lemma convex_imp_path_connected:
```
```   558   fixes s :: "'a::real_normed_vector set"
```
```   559   assumes "convex s" shows "path_connected s"
```
```   560   unfolding path_connected_def
```
```   561   apply (rule, rule, rule_tac x = "linepath x y" in exI)
```
```   562   unfolding path_image_linepath
```
```   563   using assms [unfolded convex_contains_segment]
```
```   564   apply auto
```
```   565   done
```
```   566
```
```   567 lemma path_connected_imp_connected:
```
```   568   assumes "path_connected s"
```
```   569   shows "connected s"
```
```   570   unfolding connected_def not_ex
```
```   571   apply (rule, rule, rule ccontr)
```
```   572   unfolding not_not
```
```   573   apply (erule conjE)+
```
```   574 proof -
```
```   575   fix e1 e2
```
```   576   assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
```
```   577   then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
```
```   578   then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
```
```   579     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
```
```   580   have *: "connected {0..1::real}"
```
```   581     by (auto intro!: convex_connected convex_real_interval)
```
```   582   have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}"
```
```   583     using as(3) g(2)[unfolded path_defs] by blast
```
```   584   moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}"
```
```   585     using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto
```
```   586   moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}"
```
```   587     using g(3,4)[unfolded path_defs] using obt
```
```   588     by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
```
```   589   ultimately show False
```
```   590     using *[unfolded connected_local not_ex, rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
```
```   591     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
```
```   592     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)]
```
```   593     by auto
```
```   594 qed
```
```   595
```
```   596 lemma open_path_component:
```
```   597   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
```
```   598   assumes "open s"
```
```   599   shows "open {y. path_component s x y}"
```
```   600   unfolding open_contains_ball
```
```   601 proof
```
```   602   fix y
```
```   603   assume as: "y \<in> {y. path_component s x y}"
```
```   604   then have "y \<in> s"
```
```   605     apply -
```
```   606     apply (rule path_component_mem(2))
```
```   607     unfolding mem_Collect_eq
```
```   608     apply auto
```
```   609     done
```
```   610   then obtain e where e:"e>0" "ball y e \<subseteq> s"
```
```   611     using assms[unfolded open_contains_ball] by auto
```
```   612   show "\<exists>e > 0. ball y e \<subseteq> {y. path_component s x y}"
```
```   613     apply (rule_tac x=e in exI)
```
```   614     apply (rule,rule `e>0`, rule)
```
```   615     unfolding mem_ball mem_Collect_eq
```
```   616   proof -
```
```   617     fix z
```
```   618     assume "dist y z < e"
```
```   619     then show "path_component s x z"
```
```   620       apply (rule_tac path_component_trans[of _ _ y]) defer
```
```   621       apply (rule path_component_of_subset[OF e(2)])
```
```   622       apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
```
```   623       using `e>0` as
```
```   624       apply auto
```
```   625       done
```
```   626   qed
```
```   627 qed
```
```   628
```
```   629 lemma open_non_path_component:
```
```   630   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
```
```   631   assumes "open s"
```
```   632   shows "open(s - {y. path_component s x y})"
```
```   633   unfolding open_contains_ball
```
```   634 proof
```
```   635   fix y
```
```   636   assume as: "y\<in>s - {y. path_component s x y}"
```
```   637   then obtain e where e:"e>0" "ball y e \<subseteq> s"
```
```   638     using assms [unfolded open_contains_ball] by auto
```
```   639   show "\<exists>e>0. ball y e \<subseteq> s - {y. path_component s x y}"
```
```   640     apply (rule_tac x=e in exI)
```
```   641     apply (rule, rule `e>0`, rule, rule) defer
```
```   642   proof (rule ccontr)
```
```   643     fix z
```
```   644     assume "z \<in> ball y e" "\<not> z \<notin> {y. path_component s x y}"
```
```   645     then have "y \<in> {y. path_component s x y}"
```
```   646       unfolding not_not mem_Collect_eq using `e>0`
```
```   647       apply -
```
```   648       apply (rule path_component_trans, assumption)
```
```   649       apply (rule path_component_of_subset[OF e(2)])
```
```   650       apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
```
```   651       apply auto
```
```   652       done
```
```   653     then show False using as by auto
```
```   654   qed (insert e(2), auto)
```
```   655 qed
```
```   656
```
```   657 lemma connected_open_path_connected:
```
```   658   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)
```
```   659   assumes "open s" "connected s"
```
```   660   shows "path_connected s"
```
```   661   unfolding path_connected_component_set
```
```   662 proof (rule, rule, rule path_component_subset, rule)
```
```   663   fix x y
```
```   664   assume "x \<in> s" "y \<in> s"
```
```   665   show "y \<in> {y. path_component s x y}"
```
```   666   proof (rule ccontr)
```
```   667     assume "y \<notin> {y. path_component s x y}"
```
```   668     moreover
```
```   669     have "{y. path_component s x y} \<inter> s \<noteq> {}"
```
```   670       using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto
```
```   671     ultimately
```
```   672     show False
```
```   673       using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
```
```   674       using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"]
```
```   675       by auto
```
```   676   qed
```
```   677 qed
```
```   678
```
```   679 lemma path_connected_continuous_image:
```
```   680   assumes "continuous_on s f" "path_connected s"
```
```   681   shows "path_connected (f ` s)"
```
```   682   unfolding path_connected_def
```
```   683 proof (rule, rule)
```
```   684   fix x' y'
```
```   685   assume "x' \<in> f ` s" "y' \<in> f ` s"
```
```   686   then obtain x y where xy: "x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
```
```   687   guess g using assms(2)[unfolded path_connected_def, rule_format, OF xy(1,2)] ..
```
```   688   then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
```
```   689     unfolding xy
```
```   690     apply (rule_tac x="f \<circ> g" in exI)
```
```   691     unfolding path_defs
```
```   692     using assms(1)
```
```   693     apply (auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"])
```
```   694     done
```
```   695 qed
```
```   696
```
```   697 lemma homeomorphic_path_connectedness:
```
```   698   "s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
```
```   699   unfolding homeomorphic_def homeomorphism_def
```
```   700   apply (erule exE|erule conjE)+
```
```   701   apply rule
```
```   702   apply (drule_tac f=f in path_connected_continuous_image) prefer 3
```
```   703   apply (drule_tac f=g in path_connected_continuous_image)
```
```   704   apply auto
```
```   705   done
```
```   706
```
```   707 lemma path_connected_empty: "path_connected {}"
```
```   708   unfolding path_connected_def by auto
```
```   709
```
```   710 lemma path_connected_singleton: "path_connected {a}"
```
```   711   unfolding path_connected_def pathstart_def pathfinish_def path_image_def
```
```   712   apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv)
```
```   713   apply (simp add: path_def continuous_on_const)
```
```   714   done
```
```   715
```
```   716 lemma path_connected_Un:
```
```   717   assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
```
```   718   shows "path_connected (s \<union> t)"
```
```   719   unfolding path_connected_component
```
```   720 proof (rule, rule)
```
```   721   fix x y
```
```   722   assume as: "x \<in> s \<union> t" "y \<in> s \<union> t"
```
```   723   from assms(3) obtain z where "z \<in> s \<inter> t" by auto
```
```   724   then show "path_component (s \<union> t) x y"
```
```   725     using as and assms(1-2)[unfolded path_connected_component]
```
```   726     apply -
```
```   727     apply (erule_tac[!] UnE)+
```
```   728     apply (rule_tac[2-3] path_component_trans[of _ _ z])
```
```   729     apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2])
```
```   730     done
```
```   731 qed
```
```   732
```
```   733 lemma path_connected_UNION:
```
```   734   assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
```
```   735     and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
```
```   736   shows "path_connected (\<Union>i\<in>A. S i)"
```
```   737   unfolding path_connected_component
```
```   738 proof clarify
```
```   739   fix x i y j
```
```   740   assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
```
```   741   then have "path_component (S i) x z" and "path_component (S j) z y"
```
```   742     using assms by (simp_all add: path_connected_component)
```
```   743   then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y"
```
```   744     using *(1,3) by (auto elim!: path_component_of_subset [rotated])
```
```   745   then show "path_component (\<Union>i\<in>A. S i) x y"
```
```   746     by (rule path_component_trans)
```
```   747 qed
```
```   748
```
```   749
```
```   750 subsection {* sphere is path-connected. *}
```
```   751
```
```   752 lemma path_connected_punctured_universe:
```
```   753   assumes "2 \<le> DIM('a::euclidean_space)"
```
```   754   shows "path_connected((UNIV::'a::euclidean_space set) - {a})"
```
```   755 proof -
```
```   756   let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
```
```   757   let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
```
```   758
```
```   759   have A: "path_connected ?A"
```
```   760     unfolding Collect_bex_eq
```
```   761   proof (rule path_connected_UNION)
```
```   762     fix i :: 'a
```
```   763     assume "i \<in> Basis"
```
```   764     then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" by simp
```
```   765     show "path_connected {x. x \<bullet> i < a \<bullet> i}"
```
```   766       using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
```
```   767       by (simp add: inner_commute)
```
```   768   qed
```
```   769   have B: "path_connected ?B" unfolding Collect_bex_eq
```
```   770   proof (rule path_connected_UNION)
```
```   771     fix i :: 'a
```
```   772     assume "i \<in> Basis"
```
```   773     then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" by simp
```
```   774     show "path_connected {x. a \<bullet> i < x \<bullet> i}"
```
```   775       using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
```
```   776       by (simp add: inner_commute)
```
```   777   qed
```
```   778   obtain S :: "'a set" where "S \<subseteq> Basis" "card S = Suc (Suc 0)"
```
```   779     using ex_card[OF assms] by auto
```
```   780   then obtain b0 b1 :: 'a where "b0 \<in> Basis" "b1 \<in> Basis" "b0 \<noteq> b1"
```
```   781     unfolding card_Suc_eq by auto
```
```   782   then have "a + b0 - b1 \<in> ?A \<inter> ?B" by (auto simp: inner_simps inner_Basis)
```
```   783   then have "?A \<inter> ?B \<noteq> {}" by fast
```
```   784   with A B have "path_connected (?A \<union> ?B)"
```
```   785     by (rule path_connected_Un)
```
```   786   also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
```
```   787     unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
```
```   788   also have "\<dots> = {x. x \<noteq> a}"
```
```   789     unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def)
```
```   790   also have "\<dots> = UNIV - {a}" by auto
```
```   791   finally show ?thesis .
```
```   792 qed
```
```   793
```
```   794 lemma path_connected_sphere:
```
```   795   assumes "2 \<le> DIM('a::euclidean_space)"
```
```   796   shows "path_connected {x::'a::euclidean_space. norm(x - a) = r}"
```
```   797 proof (rule linorder_cases [of r 0])
```
```   798   assume "r < 0"
```
```   799   then have "{x::'a. norm(x - a) = r} = {}" by auto
```
```   800   then show ?thesis using path_connected_empty by simp
```
```   801 next
```
```   802   assume "r = 0"
```
```   803   then show ?thesis using path_connected_singleton by simp
```
```   804 next
```
```   805   assume r: "0 < r"
```
```   806   then have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}"
```
```   807     apply -
```
```   808     apply (rule set_eqI, rule)
```
```   809     unfolding image_iff
```
```   810     apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI)
```
```   811     unfolding mem_Collect_eq norm_scaleR
```
```   812     apply (auto simp add: scaleR_right_diff_distrib)
```
```   813     done
```
```   814   have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})"
```
```   815     apply (rule set_eqI,rule)
```
```   816     unfolding image_iff
```
```   817     apply (rule_tac x=x in bexI)
```
```   818     unfolding mem_Collect_eq
```
```   819     apply (auto split:split_if_asm)
```
```   820     done
```
```   821   have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)"
```
```   822     unfolding field_divide_inverse by (simp add: continuous_on_intros)
```
```   823   then show ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
```
```   824     by (auto intro!: path_connected_continuous_image continuous_on_intros)
```
```   825 qed
```
```   826
```
```   827 lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x - a) = r}"
```
```   828   using path_connected_sphere path_connected_imp_connected by auto
```
```   829
```
```   830 end
```