src/HOL/Analysis/Fashoda_Theorem.thy
author wenzelm
Sun Sep 18 20:33:48 2016 +0200 (2016-09-18)
changeset 63915 bab633745c7f
parent 63627 6ddb43c6b711
child 64267 b9a1486e79be
permissions -rw-r--r--
tuned proofs;
     1 (*  Author:     John Harrison
     2     Author:     Robert Himmelmann, TU Muenchen (translation from HOL light)
     3 *)
     4 
     5 section \<open>Fashoda meet theorem\<close>
     6 
     7 theory Fashoda_Theorem
     8 imports Brouwer_Fixpoint Path_Connected Cartesian_Euclidean_Space
     9 begin
    10 
    11 subsection \<open>Bijections between intervals.\<close>
    12 
    13 definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::euclidean_space"
    14   where "interval_bij =
    15     (\<lambda>(a, b) (u, v) x. (\<Sum>i\<in>Basis. (u\<bullet>i + (x\<bullet>i - a\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (v\<bullet>i - u\<bullet>i)) *\<^sub>R i))"
    16 
    17 lemma interval_bij_affine:
    18   "interval_bij (a,b) (u,v) = (\<lambda>x. (\<Sum>i\<in>Basis. ((v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (x\<bullet>i)) *\<^sub>R i) +
    19     (\<Sum>i\<in>Basis. (u\<bullet>i - (v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (a\<bullet>i)) *\<^sub>R i))"
    20   by (auto simp: setsum.distrib[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff
    21     field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum.cong)
    22 
    23 lemma continuous_interval_bij:
    24   fixes a b :: "'a::euclidean_space"
    25   shows "continuous (at x) (interval_bij (a, b) (u, v))"
    26   by (auto simp add: divide_inverse interval_bij_def intro!: continuous_setsum continuous_intros)
    27 
    28 lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a, b) (u, v))"
    29   apply(rule continuous_at_imp_continuous_on)
    30   apply (rule, rule continuous_interval_bij)
    31   done
    32 
    33 lemma in_interval_interval_bij:
    34   fixes a b u v x :: "'a::euclidean_space"
    35   assumes "x \<in> cbox a b"
    36     and "cbox u v \<noteq> {}"
    37   shows "interval_bij (a, b) (u, v) x \<in> cbox u v"
    38   apply (simp only: interval_bij_def split_conv mem_box inner_setsum_left_Basis cong: ball_cong)
    39   apply safe
    40 proof -
    41   fix i :: 'a
    42   assume i: "i \<in> Basis"
    43   have "cbox a b \<noteq> {}"
    44     using assms by auto
    45   with i have *: "a\<bullet>i \<le> b\<bullet>i" "u\<bullet>i \<le> v\<bullet>i"
    46     using assms(2) by (auto simp add: box_eq_empty)
    47   have x: "a\<bullet>i\<le>x\<bullet>i" "x\<bullet>i\<le>b\<bullet>i"
    48     using assms(1)[unfolded mem_box] using i by auto
    49   have "0 \<le> (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)"
    50     using * x by auto
    51   then show "u \<bullet> i \<le> u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)"
    52     using * by auto
    53   have "((x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (v \<bullet> i - u \<bullet> i) \<le> 1 * (v \<bullet> i - u \<bullet> i)"
    54     apply (rule mult_right_mono)
    55     unfolding divide_le_eq_1
    56     using * x
    57     apply auto
    58     done
    59   then show "u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i) \<le> v \<bullet> i"
    60     using * by auto
    61 qed
    62 
    63 lemma interval_bij_bij:
    64   "\<forall>(i::'a::euclidean_space)\<in>Basis. a\<bullet>i < b\<bullet>i \<and> u\<bullet>i < v\<bullet>i \<Longrightarrow>
    65     interval_bij (a, b) (u, v) (interval_bij (u, v) (a, b) x) = x"
    66   by (auto simp: interval_bij_def euclidean_eq_iff[where 'a='a])
    67 
    68 lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i"
    69   shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
    70   using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
    71 
    72 
    73 subsection \<open>Fashoda meet theorem\<close>
    74 
    75 lemma infnorm_2:
    76   fixes x :: "real^2"
    77   shows "infnorm x = max \<bar>x$1\<bar> \<bar>x$2\<bar>"
    78   unfolding infnorm_cart UNIV_2 by (rule cSup_eq) auto
    79 
    80 lemma infnorm_eq_1_2:
    81   fixes x :: "real^2"
    82   shows "infnorm x = 1 \<longleftrightarrow>
    83     \<bar>x$1\<bar> \<le> 1 \<and> \<bar>x$2\<bar> \<le> 1 \<and> (x$1 = -1 \<or> x$1 = 1 \<or> x$2 = -1 \<or> x$2 = 1)"
    84   unfolding infnorm_2 by auto
    85 
    86 lemma infnorm_eq_1_imp:
    87   fixes x :: "real^2"
    88   assumes "infnorm x = 1"
    89   shows "\<bar>x$1\<bar> \<le> 1" and "\<bar>x$2\<bar> \<le> 1"
    90   using assms unfolding infnorm_eq_1_2 by auto
    91 
    92 lemma fashoda_unit:
    93   fixes f g :: "real \<Rightarrow> real^2"
    94   assumes "f ` {-1 .. 1} \<subseteq> cbox (-1) 1"
    95     and "g ` {-1 .. 1} \<subseteq> cbox (-1) 1"
    96     and "continuous_on {-1 .. 1} f"
    97     and "continuous_on {-1 .. 1} g"
    98     and "f (- 1)$1 = - 1"
    99     and "f 1$1 = 1" "g (- 1) $2 = -1"
   100     and "g 1 $2 = 1"
   101   shows "\<exists>s\<in>{-1 .. 1}. \<exists>t\<in>{-1 .. 1}. f s = g t"
   102 proof (rule ccontr)
   103   assume "\<not> ?thesis"
   104   note as = this[unfolded bex_simps,rule_format]
   105   define sqprojection
   106     where [abs_def]: "sqprojection z = (inverse (infnorm z)) *\<^sub>R z" for z :: "real^2"
   107   define negatex :: "real^2 \<Rightarrow> real^2"
   108     where "negatex x = (vector [-(x$1), x$2])" for x
   109   have lem1: "\<forall>z::real^2. infnorm (negatex z) = infnorm z"
   110     unfolding negatex_def infnorm_2 vector_2 by auto
   111   have lem2: "\<forall>z. z \<noteq> 0 \<longrightarrow> infnorm (sqprojection z) = 1"
   112     unfolding sqprojection_def
   113     unfolding infnorm_mul[unfolded scalar_mult_eq_scaleR]
   114     unfolding abs_inverse real_abs_infnorm
   115     apply (subst infnorm_eq_0[symmetric])
   116     apply auto
   117     done
   118   let ?F = "\<lambda>w::real^2. (f \<circ> (\<lambda>x. x$1)) w - (g \<circ> (\<lambda>x. x$2)) w"
   119   have *: "\<And>i. (\<lambda>x::real^2. x $ i) ` cbox (- 1) 1 = {-1 .. 1}"
   120     apply (rule set_eqI)
   121     unfolding image_iff Bex_def mem_interval_cart interval_cbox_cart
   122     apply rule
   123     defer
   124     apply (rule_tac x="vec x" in exI)
   125     apply auto
   126     done
   127   {
   128     fix x
   129     assume "x \<in> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w) ` (cbox (- 1) (1::real^2))"
   130     then obtain w :: "real^2" where w:
   131         "w \<in> cbox (- 1) 1"
   132         "x = (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w"
   133       unfolding image_iff ..
   134     then have "x \<noteq> 0"
   135       using as[of "w$1" "w$2"]
   136       unfolding mem_interval_cart atLeastAtMost_iff
   137       by auto
   138   } note x0 = this
   139   have 21: "\<And>i::2. i \<noteq> 1 \<Longrightarrow> i = 2"
   140     using UNIV_2 by auto
   141   have 1: "box (- 1) (1::real^2) \<noteq> {}"
   142     unfolding interval_eq_empty_cart by auto
   143   have 2: "continuous_on (cbox (- 1) 1) (negatex \<circ> sqprojection \<circ> ?F)"
   144     apply (intro continuous_intros continuous_on_component)
   145     unfolding *
   146     apply (rule assms)+
   147     apply (subst sqprojection_def)
   148     apply (intro continuous_intros)
   149     apply (simp add: infnorm_eq_0 x0)
   150     apply (rule linear_continuous_on)
   151   proof -
   152     show "bounded_linear negatex"
   153       apply (rule bounded_linearI')
   154       unfolding vec_eq_iff
   155     proof (rule_tac[!] allI)
   156       fix i :: 2
   157       fix x y :: "real^2"
   158       fix c :: real
   159       show "negatex (x + y) $ i =
   160         (negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
   161         apply -
   162         apply (case_tac[!] "i\<noteq>1")
   163         prefer 3
   164         apply (drule_tac[1-2] 21)
   165         unfolding negatex_def
   166         apply (auto simp add:vector_2)
   167         done
   168     qed
   169   qed
   170   have 3: "(negatex \<circ> sqprojection \<circ> ?F) ` cbox (-1) 1 \<subseteq> cbox (-1) 1"
   171     unfolding subset_eq
   172   proof (rule, goal_cases)
   173     case (1 x)
   174     then obtain y :: "real^2" where y:
   175         "y \<in> cbox (- 1) 1"
   176         "x = (negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) y"
   177       unfolding image_iff ..
   178     have "?F y \<noteq> 0"
   179       apply (rule x0)
   180       using y(1)
   181       apply auto
   182       done
   183     then have *: "infnorm (sqprojection (?F y)) = 1"
   184       unfolding y o_def
   185       by - (rule lem2[rule_format])
   186     have "infnorm x = 1"
   187       unfolding *[symmetric] y o_def
   188       by (rule lem1[rule_format])
   189     then show "x \<in> cbox (-1) 1"
   190       unfolding mem_interval_cart interval_cbox_cart infnorm_2
   191       apply -
   192       apply rule
   193     proof -
   194       fix i
   195       assume "max \<bar>x $ 1\<bar> \<bar>x $ 2\<bar> = 1"
   196       then show "(- 1) $ i \<le> x $ i \<and> x $ i \<le> 1 $ i"
   197         apply (cases "i = 1")
   198         defer
   199         apply (drule 21)
   200         apply auto
   201         done
   202     qed
   203   qed
   204   obtain x :: "real^2" where x:
   205       "x \<in> cbox (- 1) 1"
   206       "(negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) x = x"
   207     apply (rule brouwer_weak[of "cbox (- 1) (1::real^2)" "negatex \<circ> sqprojection \<circ> ?F"])
   208     apply (rule compact_cbox convex_box)+
   209     unfolding interior_cbox
   210     apply (rule 1 2 3)+
   211     apply blast
   212     done
   213   have "?F x \<noteq> 0"
   214     apply (rule x0)
   215     using x(1)
   216     apply auto
   217     done
   218   then have *: "infnorm (sqprojection (?F x)) = 1"
   219     unfolding o_def
   220     by (rule lem2[rule_format])
   221   have nx: "infnorm x = 1"
   222     apply (subst x(2)[symmetric])
   223     unfolding *[symmetric] o_def
   224     apply (rule lem1[rule_format])
   225     done
   226   have "\<forall>x i. x \<noteq> 0 \<longrightarrow> (0 < (sqprojection x)$i \<longleftrightarrow> 0 < x$i)"
   227     and "\<forall>x i. x \<noteq> 0 \<longrightarrow> ((sqprojection x)$i < 0 \<longleftrightarrow> x$i < 0)"
   228     apply -
   229     apply (rule_tac[!] allI impI)+
   230   proof -
   231     fix x :: "real^2"
   232     fix i :: 2
   233     assume x: "x \<noteq> 0"
   234     have "inverse (infnorm x) > 0"
   235       using x[unfolded infnorm_pos_lt[symmetric]] by auto
   236     then show "(0 < sqprojection x $ i) = (0 < x $ i)"
   237       and "(sqprojection x $ i < 0) = (x $ i < 0)"
   238       unfolding sqprojection_def vector_component_simps vector_scaleR_component real_scaleR_def
   239       unfolding zero_less_mult_iff mult_less_0_iff
   240       by (auto simp add: field_simps)
   241   qed
   242   note lem3 = this[rule_format]
   243   have x1: "x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}"
   244     using x(1) unfolding mem_interval_cart by auto
   245   then have nz: "f (x $ 1) - g (x $ 2) \<noteq> 0"
   246     unfolding right_minus_eq
   247     apply -
   248     apply (rule as)
   249     apply auto
   250     done
   251   have "x $ 1 = -1 \<or> x $ 1 = 1 \<or> x $ 2 = -1 \<or> x $ 2 = 1"
   252     using nx unfolding infnorm_eq_1_2 by auto
   253   then show False
   254   proof -
   255     fix P Q R S
   256     presume "P \<or> Q \<or> R \<or> S"
   257       and "P \<Longrightarrow> False"
   258       and "Q \<Longrightarrow> False"
   259       and "R \<Longrightarrow> False"
   260       and "S \<Longrightarrow> False"
   261     then show False by auto
   262   next
   263     assume as: "x$1 = 1"
   264     then have *: "f (x $ 1) $ 1 = 1"
   265       using assms(6) by auto
   266     have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
   267       using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
   268       unfolding as negatex_def vector_2
   269       by auto
   270     moreover
   271     from x1 have "g (x $ 2) \<in> cbox (-1) 1"
   272       apply -
   273       apply (rule assms(2)[unfolded subset_eq,rule_format])
   274       apply auto
   275       done
   276     ultimately show False
   277       unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
   278       apply (erule_tac x=1 in allE)
   279       apply auto
   280       done
   281   next
   282     assume as: "x$1 = -1"
   283     then have *: "f (x $ 1) $ 1 = - 1"
   284       using assms(5) by auto
   285     have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
   286       using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
   287       unfolding as negatex_def vector_2
   288       by auto
   289     moreover
   290     from x1 have "g (x $ 2) \<in> cbox (-1) 1"
   291       apply -
   292       apply (rule assms(2)[unfolded subset_eq,rule_format])
   293       apply auto
   294       done
   295     ultimately show False
   296       unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
   297       apply (erule_tac x=1 in allE)
   298       apply auto
   299       done
   300   next
   301     assume as: "x$2 = 1"
   302     then have *: "g (x $ 2) $ 2 = 1"
   303       using assms(8) by auto
   304     have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
   305       using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
   306       unfolding as negatex_def vector_2
   307       by auto
   308     moreover
   309     from x1 have "f (x $ 1) \<in> cbox (-1) 1"
   310       apply -
   311       apply (rule assms(1)[unfolded subset_eq,rule_format])
   312       apply auto
   313       done
   314     ultimately show False
   315       unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
   316       apply (erule_tac x=2 in allE)
   317       apply auto
   318       done
   319   next
   320     assume as: "x$2 = -1"
   321     then have *: "g (x $ 2) $ 2 = - 1"
   322       using assms(7) by auto
   323     have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
   324       using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
   325       unfolding as negatex_def vector_2
   326       by auto
   327     moreover
   328     from x1 have "f (x $ 1) \<in> cbox (-1) 1"
   329       apply -
   330       apply (rule assms(1)[unfolded subset_eq,rule_format])
   331       apply auto
   332       done
   333     ultimately show False
   334       unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
   335       apply (erule_tac x=2 in allE)
   336       apply auto
   337       done
   338   qed auto
   339 qed
   340 
   341 lemma fashoda_unit_path:
   342   fixes f g :: "real \<Rightarrow> real^2"
   343   assumes "path f"
   344     and "path g"
   345     and "path_image f \<subseteq> cbox (-1) 1"
   346     and "path_image g \<subseteq> cbox (-1) 1"
   347     and "(pathstart f)$1 = -1"
   348     and "(pathfinish f)$1 = 1"
   349     and "(pathstart g)$2 = -1"
   350     and "(pathfinish g)$2 = 1"
   351   obtains z where "z \<in> path_image f" and "z \<in> path_image g"
   352 proof -
   353   note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def]
   354   define iscale where [abs_def]: "iscale z = inverse 2 *\<^sub>R (z + 1)" for z :: real
   355   have isc: "iscale ` {- 1..1} \<subseteq> {0..1}"
   356     unfolding iscale_def by auto
   357   have "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. (f \<circ> iscale) s = (g \<circ> iscale) t"
   358   proof (rule fashoda_unit)
   359     show "(f \<circ> iscale) ` {- 1..1} \<subseteq> cbox (- 1) 1" "(g \<circ> iscale) ` {- 1..1} \<subseteq> cbox (- 1) 1"
   360       using isc and assms(3-4) by (auto simp add: image_comp [symmetric])
   361     have *: "continuous_on {- 1..1} iscale"
   362       unfolding iscale_def by (rule continuous_intros)+
   363     show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)"
   364       apply -
   365       apply (rule_tac[!] continuous_on_compose[OF *])
   366       apply (rule_tac[!] continuous_on_subset[OF _ isc])
   367       apply (rule assms)+
   368       done
   369     have *: "(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1"
   370       unfolding vec_eq_iff by auto
   371     show "(f \<circ> iscale) (- 1) $ 1 = - 1"
   372       and "(f \<circ> iscale) 1 $ 1 = 1"
   373       and "(g \<circ> iscale) (- 1) $ 2 = -1"
   374       and "(g \<circ> iscale) 1 $ 2 = 1"
   375       unfolding o_def iscale_def
   376       using assms
   377       by (auto simp add: *)
   378   qed
   379   then obtain s t where st:
   380       "s \<in> {- 1..1}"
   381       "t \<in> {- 1..1}"
   382       "(f \<circ> iscale) s = (g \<circ> iscale) t"
   383     by auto
   384   show thesis
   385     apply (rule_tac z = "f (iscale s)" in that)
   386     using st
   387     unfolding o_def path_image_def image_iff
   388     apply -
   389     apply (rule_tac x="iscale s" in bexI)
   390     prefer 3
   391     apply (rule_tac x="iscale t" in bexI)
   392     using isc[unfolded subset_eq, rule_format]
   393     apply auto
   394     done
   395 qed
   396 
   397 lemma fashoda:
   398   fixes b :: "real^2"
   399   assumes "path f"
   400     and "path g"
   401     and "path_image f \<subseteq> cbox a b"
   402     and "path_image g \<subseteq> cbox a b"
   403     and "(pathstart f)$1 = a$1"
   404     and "(pathfinish f)$1 = b$1"
   405     and "(pathstart g)$2 = a$2"
   406     and "(pathfinish g)$2 = b$2"
   407   obtains z where "z \<in> path_image f" and "z \<in> path_image g"
   408 proof -
   409   fix P Q S
   410   presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" and "Q \<Longrightarrow> thesis" and "S \<Longrightarrow> thesis"
   411   then show thesis
   412     by auto
   413 next
   414   have "cbox a b \<noteq> {}"
   415     using assms(3) using path_image_nonempty[of f] by auto
   416   then have "a \<le> b"
   417     unfolding interval_eq_empty_cart less_eq_vec_def by (auto simp add: not_less)
   418   then show "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)"
   419     unfolding less_eq_vec_def forall_2 by auto
   420 next
   421   assume as: "a$1 = b$1"
   422   have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2"
   423     apply (rule connected_ivt_component_cart)
   424     apply (rule connected_path_image assms)+
   425     apply (rule pathstart_in_path_image)
   426     apply (rule pathfinish_in_path_image)
   427     unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
   428     unfolding pathstart_def
   429     apply (auto simp add: less_eq_vec_def mem_interval_cart)
   430     done
   431   then obtain z :: "real^2" where z: "z \<in> path_image g" "z $ 2 = pathstart f $ 2" ..
   432   have "z \<in> cbox a b"
   433     using z(1) assms(4)
   434     unfolding path_image_def
   435     by blast
   436   then have "z = f 0"
   437     unfolding vec_eq_iff forall_2
   438     unfolding z(2) pathstart_def
   439     using assms(3)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "f 0" 1]
   440     unfolding mem_interval_cart
   441     apply (erule_tac x=1 in allE)
   442     using as
   443     apply auto
   444     done
   445   then show thesis
   446     apply -
   447     apply (rule that[OF _ z(1)])
   448     unfolding path_image_def
   449     apply auto
   450     done
   451 next
   452   assume as: "a$2 = b$2"
   453   have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1"
   454     apply (rule connected_ivt_component_cart)
   455     apply (rule connected_path_image assms)+
   456     apply (rule pathstart_in_path_image)
   457     apply (rule pathfinish_in_path_image)
   458     unfolding assms
   459     using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
   460     unfolding pathstart_def
   461     apply (auto simp add: less_eq_vec_def mem_interval_cart)
   462     done
   463   then obtain z where z: "z \<in> path_image f" "z $ 1 = pathstart g $ 1" ..
   464   have "z \<in> cbox a b"
   465     using z(1) assms(3)
   466     unfolding path_image_def
   467     by blast
   468   then have "z = g 0"
   469     unfolding vec_eq_iff forall_2
   470     unfolding z(2) pathstart_def
   471     using assms(4)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "g 0" 2]
   472     unfolding mem_interval_cart
   473     apply (erule_tac x=2 in allE)
   474     using as
   475     apply auto
   476     done
   477   then show thesis
   478     apply -
   479     apply (rule that[OF z(1)])
   480     unfolding path_image_def
   481     apply auto
   482     done
   483 next
   484   assume as: "a $ 1 < b $ 1 \<and> a $ 2 < b $ 2"
   485   have int_nem: "cbox (-1) (1::real^2) \<noteq> {}"
   486     unfolding interval_eq_empty_cart by auto
   487   obtain z :: "real^2" where z:
   488       "z \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}"
   489       "z \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}"
   490     apply (rule fashoda_unit_path[of "interval_bij (a,b) (- 1,1) \<circ> f" "interval_bij (a,b) (- 1,1) \<circ> g"])
   491     unfolding path_def path_image_def pathstart_def pathfinish_def
   492     apply (rule_tac[1-2] continuous_on_compose)
   493     apply (rule assms[unfolded path_def] continuous_on_interval_bij)+
   494     unfolding subset_eq
   495     apply(rule_tac[1-2] ballI)
   496   proof -
   497     fix x
   498     assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}"
   499     then obtain y where y:
   500         "y \<in> {0..1}"
   501         "x = (interval_bij (a, b) (- 1, 1) \<circ> f) y"
   502       unfolding image_iff ..
   503     show "x \<in> cbox (- 1) 1"
   504       unfolding y o_def
   505       apply (rule in_interval_interval_bij)
   506       using y(1)
   507       using assms(3)[unfolded path_image_def subset_eq] int_nem
   508       apply auto
   509       done
   510   next
   511     fix x
   512     assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}"
   513     then obtain y where y:
   514         "y \<in> {0..1}"
   515         "x = (interval_bij (a, b) (- 1, 1) \<circ> g) y"
   516       unfolding image_iff ..
   517     show "x \<in> cbox (- 1) 1"
   518       unfolding y o_def
   519       apply (rule in_interval_interval_bij)
   520       using y(1)
   521       using assms(4)[unfolded path_image_def subset_eq] int_nem
   522       apply auto
   523       done
   524   next
   525     show "(interval_bij (a, b) (- 1, 1) \<circ> f) 0 $ 1 = -1"
   526       and "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
   527       and "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
   528       and "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1"
   529       using assms as
   530       by (simp_all add: axis_in_Basis cart_eq_inner_axis pathstart_def pathfinish_def interval_bij_def)
   531          (simp_all add: inner_axis)
   532   qed
   533   from z(1) obtain zf where zf:
   534       "zf \<in> {0..1}"
   535       "z = (interval_bij (a, b) (- 1, 1) \<circ> f) zf"
   536     unfolding image_iff ..
   537   from z(2) obtain zg where zg:
   538       "zg \<in> {0..1}"
   539       "z = (interval_bij (a, b) (- 1, 1) \<circ> g) zg"
   540     unfolding image_iff ..
   541   have *: "\<forall>i. (- 1) $ i < (1::real^2) $ i \<and> a $ i < b $ i"
   542     unfolding forall_2
   543     using as
   544     by auto
   545   show thesis
   546     apply (rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
   547     apply (subst zf)
   548     defer
   549     apply (subst zg)
   550     unfolding o_def interval_bij_bij_cart[OF *] path_image_def
   551     using zf(1) zg(1)
   552     apply auto
   553     done
   554 qed
   555 
   556 
   557 subsection \<open>Some slightly ad hoc lemmas I use below\<close>
   558 
   559 lemma segment_vertical:
   560   fixes a :: "real^2"
   561   assumes "a$1 = b$1"
   562   shows "x \<in> closed_segment a b \<longleftrightarrow>
   563     x$1 = a$1 \<and> x$1 = b$1 \<and> (a$2 \<le> x$2 \<and> x$2 \<le> b$2 \<or> b$2 \<le> x$2 \<and> x$2 \<le> a$2)"
   564   (is "_ = ?R")
   565 proof -
   566   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   567   {
   568     presume "?L \<Longrightarrow> ?R" and "?R \<Longrightarrow> ?L"
   569     then show ?thesis
   570       unfolding closed_segment_def mem_Collect_eq
   571       unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps
   572       by blast
   573   }
   574   {
   575     assume ?L
   576     then obtain u where u:
   577         "x $ 1 = (1 - u) * a $ 1 + u * b $ 1"
   578         "x $ 2 = (1 - u) * a $ 2 + u * b $ 2"
   579         "0 \<le> u"
   580         "u \<le> 1"
   581       by blast
   582     { fix b a
   583       assume "b + u * a > a + u * b"
   584       then have "(1 - u) * b > (1 - u) * a"
   585         by (auto simp add:field_simps)
   586       then have "b \<ge> a"
   587         apply (drule_tac mult_left_less_imp_less)
   588         using u
   589         apply auto
   590         done
   591       then have "u * a \<le> u * b"
   592         apply -
   593         apply (rule mult_left_mono[OF _ u(3)])
   594         using u(3-4)
   595         apply (auto simp add: field_simps)
   596         done
   597     } note * = this
   598     {
   599       fix a b
   600       assume "u * b > u * a"
   601       then have "(1 - u) * a \<le> (1 - u) * b"
   602         apply -
   603         apply (rule mult_left_mono)
   604         apply (drule mult_left_less_imp_less)
   605         using u
   606         apply auto
   607         done
   608       then have "a + u * b \<le> b + u * a"
   609         by (auto simp add: field_simps)
   610     } note ** = this
   611     then show ?R
   612       unfolding u assms
   613       using u
   614       by (auto simp add:field_simps not_le intro: * **)
   615   }
   616   {
   617     assume ?R
   618     then show ?L
   619     proof (cases "x$2 = b$2")
   620       case True
   621       then show ?L
   622         apply (rule_tac x="(x$2 - a$2) / (b$2 - a$2)" in exI)
   623         unfolding assms True
   624         using \<open>?R\<close>
   625         apply (auto simp add: field_simps)
   626         done
   627     next
   628       case False
   629       then show ?L
   630         apply (rule_tac x="1 - (x$2 - b$2) / (a$2 - b$2)" in exI)
   631         unfolding assms
   632         using \<open>?R\<close>
   633         apply (auto simp add: field_simps)
   634         done
   635     qed
   636   }
   637 qed
   638 
   639 lemma segment_horizontal:
   640   fixes a :: "real^2"
   641   assumes "a$2 = b$2"
   642   shows "x \<in> closed_segment a b \<longleftrightarrow>
   643     x$2 = a$2 \<and> x$2 = b$2 \<and> (a$1 \<le> x$1 \<and> x$1 \<le> b$1 \<or> b$1 \<le> x$1 \<and> x$1 \<le> a$1)"
   644   (is "_ = ?R")
   645 proof -
   646   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   647   {
   648     presume "?L \<Longrightarrow> ?R" and "?R \<Longrightarrow> ?L"
   649     then show ?thesis
   650       unfolding closed_segment_def mem_Collect_eq
   651       unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps
   652       by blast
   653   }
   654   {
   655     assume ?L
   656     then obtain u where u:
   657         "x $ 1 = (1 - u) * a $ 1 + u * b $ 1"
   658         "x $ 2 = (1 - u) * a $ 2 + u * b $ 2"
   659         "0 \<le> u"
   660         "u \<le> 1"
   661       by blast
   662     {
   663       fix b a
   664       assume "b + u * a > a + u * b"
   665       then have "(1 - u) * b > (1 - u) * a"
   666         by (auto simp add: field_simps)
   667       then have "b \<ge> a"
   668         apply (drule_tac mult_left_less_imp_less)
   669         using u
   670         apply auto
   671         done
   672       then have "u * a \<le> u * b"
   673         apply -
   674         apply (rule mult_left_mono[OF _ u(3)])
   675         using u(3-4)
   676         apply (auto simp add: field_simps)
   677         done
   678     } note * = this
   679     {
   680       fix a b
   681       assume "u * b > u * a"
   682       then have "(1 - u) * a \<le> (1 - u) * b"
   683         apply -
   684         apply (rule mult_left_mono)
   685         apply (drule mult_left_less_imp_less)
   686         using u
   687         apply auto
   688         done
   689       then have "a + u * b \<le> b + u * a"
   690         by (auto simp add: field_simps)
   691     } note ** = this
   692     then show ?R
   693       unfolding u assms
   694       using u
   695       by (auto simp add: field_simps not_le intro: * **)
   696   }
   697   {
   698     assume ?R
   699     then show ?L
   700     proof (cases "x$1 = b$1")
   701       case True
   702       then show ?L
   703         apply (rule_tac x="(x$1 - a$1) / (b$1 - a$1)" in exI)
   704         unfolding assms True
   705         using \<open>?R\<close>
   706         apply (auto simp add: field_simps)
   707         done
   708     next
   709       case False
   710       then show ?L
   711         apply (rule_tac x="1 - (x$1 - b$1) / (a$1 - b$1)" in exI)
   712         unfolding assms
   713         using \<open>?R\<close>
   714         apply (auto simp add: field_simps)
   715         done
   716     qed
   717   }
   718 qed
   719 
   720 
   721 subsection \<open>Useful Fashoda corollary pointed out to me by Tom Hales\<close>
   722 
   723 lemma fashoda_interlace:
   724   fixes a :: "real^2"
   725   assumes "path f"
   726     and "path g"
   727     and "path_image f \<subseteq> cbox a b"
   728     and "path_image g \<subseteq> cbox a b"
   729     and "(pathstart f)$2 = a$2"
   730     and "(pathfinish f)$2 = a$2"
   731     and "(pathstart g)$2 = a$2"
   732     and "(pathfinish g)$2 = a$2"
   733     and "(pathstart f)$1 < (pathstart g)$1"
   734     and "(pathstart g)$1 < (pathfinish f)$1"
   735     and "(pathfinish f)$1 < (pathfinish g)$1"
   736   obtains z where "z \<in> path_image f" and "z \<in> path_image g"
   737 proof -
   738   have "cbox a b \<noteq> {}"
   739     using path_image_nonempty[of f] using assms(3) by auto
   740   note ab=this[unfolded interval_eq_empty_cart not_ex forall_2 not_less]
   741   have "pathstart f \<in> cbox a b"
   742     and "pathfinish f \<in> cbox a b"
   743     and "pathstart g \<in> cbox a b"
   744     and "pathfinish g \<in> cbox a b"
   745     using pathstart_in_path_image pathfinish_in_path_image
   746     using assms(3-4)
   747     by auto
   748   note startfin = this[unfolded mem_interval_cart forall_2]
   749   let ?P1 = "linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2]) +++
   750      linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f) +++ f +++
   751      linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2]) +++
   752      linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2])"
   753   let ?P2 = "linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g) +++ g +++
   754      linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1]) +++
   755      linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1]) +++
   756      linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3])"
   757   let ?a = "vector[a$1 - 2, a$2 - 3]"
   758   let ?b = "vector[b$1 + 2, b$2 + 3]"
   759   have P1P2: "path_image ?P1 = path_image (linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2])) \<union>
   760       path_image (linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f)) \<union> path_image f \<union>
   761       path_image (linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2])) \<union>
   762       path_image (linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2]))"
   763     "path_image ?P2 = path_image(linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g)) \<union> path_image g \<union>
   764       path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union>
   765       path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
   766       path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
   767       by(auto simp add: path_image_join path_linepath)
   768   have abab: "cbox a b \<subseteq> cbox ?a ?b"
   769     unfolding interval_cbox_cart[symmetric]
   770     by (auto simp add:less_eq_vec_def forall_2 vector_2)
   771   obtain z where
   772     "z \<in> path_image
   773           (linepath (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) +++
   774            linepath (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f) +++
   775            f +++
   776            linepath (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) +++
   777            linepath (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]))"
   778     "z \<in> path_image
   779           (linepath (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g) +++
   780            g +++
   781            linepath (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1]) +++
   782            linepath (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1]) +++
   783            linepath (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]))"
   784     apply (rule fashoda[of ?P1 ?P2 ?a ?b])
   785     unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2
   786   proof -
   787     show "path ?P1" and "path ?P2"
   788       using assms by auto
   789     have "path_image ?P1 \<subseteq> cbox ?a ?b"
   790       unfolding P1P2 path_image_linepath
   791       apply (rule Un_least)+
   792       defer 3
   793       apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format])
   794       unfolding mem_interval_cart forall_2 vector_2
   795       using ab startfin abab assms(3)
   796       using assms(9-)
   797       unfolding assms
   798       apply (auto simp add: field_simps box_def)
   799       done
   800     then show "path_image ?P1 \<subseteq> cbox ?a ?b" .
   801     have "path_image ?P2 \<subseteq> cbox ?a ?b"
   802       unfolding P1P2 path_image_linepath
   803       apply (rule Un_least)+
   804       defer 2
   805       apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format])
   806       unfolding mem_interval_cart forall_2 vector_2
   807       using ab startfin abab assms(4)
   808       using assms(9-)
   809       unfolding assms
   810       apply (auto simp add: field_simps box_def)
   811       done
   812     then show "path_image ?P2 \<subseteq> cbox ?a ?b" .
   813     show "a $ 1 - 2 = a $ 1 - 2"
   814       and "b $ 1 + 2 = b $ 1 + 2"
   815       and "pathstart g $ 2 - 3 = a $ 2 - 3"
   816       and "b $ 2 + 3 = b $ 2 + 3"
   817       by (auto simp add: assms)
   818   qed
   819   note z=this[unfolded P1P2 path_image_linepath]
   820   show thesis
   821     apply (rule that[of z])
   822   proof -
   823     have "(z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<or>
   824       z \<in> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) \<or>
   825       z \<in> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<or>
   826       z \<in> closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]) \<Longrightarrow>
   827     (((z \<in> closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g)) \<or>
   828       z \<in> closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1])) \<or>
   829       z \<in> closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1])) \<or>
   830       z \<in> closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]) \<Longrightarrow> False"
   831     proof (simp only: segment_vertical segment_horizontal vector_2, goal_cases)
   832       case prems: 1
   833       have "pathfinish f \<in> cbox a b"
   834         using assms(3) pathfinish_in_path_image[of f] by auto
   835       then have "1 + b $ 1 \<le> pathfinish f $ 1 \<Longrightarrow> False"
   836         unfolding mem_interval_cart forall_2 by auto
   837       then have "z$1 \<noteq> pathfinish f$1"
   838         using prems(2)
   839         using assms ab
   840         by (auto simp add: field_simps)
   841       moreover have "pathstart f \<in> cbox a b"
   842         using assms(3) pathstart_in_path_image[of f]
   843         by auto
   844       then have "1 + b $ 1 \<le> pathstart f $ 1 \<Longrightarrow> False"
   845         unfolding mem_interval_cart forall_2
   846         by auto
   847       then have "z$1 \<noteq> pathstart f$1"
   848         using prems(2) using assms ab
   849         by (auto simp add: field_simps)
   850       ultimately have *: "z$2 = a$2 - 2"
   851         using prems(1)
   852         by auto
   853       have "z$1 \<noteq> pathfinish g$1"
   854         using prems(2)
   855         using assms ab
   856         by (auto simp add: field_simps *)
   857       moreover have "pathstart g \<in> cbox a b"
   858         using assms(4) pathstart_in_path_image[of g]
   859         by auto
   860       note this[unfolded mem_interval_cart forall_2]
   861       then have "z$1 \<noteq> pathstart g$1"
   862         using prems(1)
   863         using assms ab
   864         by (auto simp add: field_simps *)
   865       ultimately have "a $ 2 - 1 \<le> z $ 2 \<and> z $ 2 \<le> b $ 2 + 3 \<or> b $ 2 + 3 \<le> z $ 2 \<and> z $ 2 \<le> a $ 2 - 1"
   866         using prems(2)
   867         unfolding * assms
   868         by (auto simp add: field_simps)
   869       then show False
   870         unfolding * using ab by auto
   871     qed
   872     then have "z \<in> path_image f \<or> z \<in> path_image g"
   873       using z unfolding Un_iff by blast
   874     then have z': "z \<in> cbox a b"
   875       using assms(3-4)
   876       by auto
   877     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow>
   878       z = pathstart f \<or> z = pathfinish f"
   879       unfolding vec_eq_iff forall_2 assms
   880       by auto
   881     with z' show "z \<in> path_image f"
   882       using z(1)
   883       unfolding Un_iff mem_interval_cart forall_2
   884       apply -
   885       apply (simp only: segment_vertical segment_horizontal vector_2)
   886       unfolding assms
   887       apply auto
   888       done
   889     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow>
   890       z = pathstart g \<or> z = pathfinish g"
   891       unfolding vec_eq_iff forall_2 assms
   892       by auto
   893     with z' show "z \<in> path_image g"
   894       using z(2)
   895       unfolding Un_iff mem_interval_cart forall_2
   896       apply (simp only: segment_vertical segment_horizontal vector_2)
   897       unfolding assms
   898       apply auto
   899       done
   900   qed
   901 qed
   902 
   903 (** The Following still needs to be translated. Maybe I will do that later.
   904 
   905 (* ------------------------------------------------------------------------- *)
   906 (* Complement in dimension N >= 2 of set homeomorphic to any interval in     *)
   907 (* any dimension is (path-)connected. This naively generalizes the argument  *)
   908 (* in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer        *)
   909 (* fixed point theorem", American Mathematical Monthly 1984.                 *)
   910 (* ------------------------------------------------------------------------- *)
   911 
   912 let RETRACTION_INJECTIVE_IMAGE_INTERVAL = prove
   913  (`!p:real^M->real^N a b.
   914         ~(interval[a,b] = {}) /\
   915         p continuous_on interval[a,b] /\
   916         (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ p x = p y ==> x = y)
   917         ==> ?f. f continuous_on (:real^N) /\
   918                 IMAGE f (:real^N) SUBSET (IMAGE p (interval[a,b])) /\
   919                 (!x. x IN (IMAGE p (interval[a,b])) ==> f x = x)`,
   920   REPEAT STRIP_TAC THEN
   921   FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
   922   DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN
   923   SUBGOAL_THEN `(q:real^N->real^M) continuous_on
   924                 (IMAGE p (interval[a:real^M,b]))`
   925   ASSUME_TAC THENL
   926    [MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL];
   927     ALL_TAC] THEN
   928   MP_TAC(ISPECL [`q:real^N->real^M`;
   929                  `IMAGE (p:real^M->real^N)
   930                  (interval[a,b])`;
   931                  `a:real^M`; `b:real^M`]
   932         TIETZE_CLOSED_INTERVAL) THEN
   933   ASM_SIMP_TAC[COMPACT_INTERVAL; COMPACT_CONTINUOUS_IMAGE;
   934                COMPACT_IMP_CLOSED] THEN
   935   ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
   936   DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^M` STRIP_ASSUME_TAC) THEN
   937   EXISTS_TAC `(p:real^M->real^N) o (r:real^N->real^M)` THEN
   938   REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM; IN_UNIV] THEN
   939   CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
   940   MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
   941   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ]
   942         CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);;
   943 
   944 let UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
   945  (`!s:real^N->bool a b:real^M.
   946         s homeomorphic (interval[a,b])
   947         ==> !x. ~(x IN s) ==> ~bounded(path_component((:real^N) DIFF s) x)`,
   948   REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
   949   REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
   950   MAP_EVERY X_GEN_TAC [`p':real^N->real^M`; `p:real^M->real^N`] THEN
   951   DISCH_TAC THEN
   952   SUBGOAL_THEN
   953    `!x y. x IN interval[a,b] /\ y IN interval[a,b] /\
   954           (p:real^M->real^N) x = p y ==> x = y`
   955   ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
   956   FIRST_X_ASSUM(MP_TAC o funpow 4 CONJUNCT2) THEN
   957   DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) ASSUME_TAC) THEN
   958   ASM_CASES_TAC `interval[a:real^M,b] = {}` THEN
   959   ASM_REWRITE_TAC[IMAGE_CLAUSES; DIFF_EMPTY; PATH_COMPONENT_UNIV;
   960                   NOT_BOUNDED_UNIV] THEN
   961   ABBREV_TAC `s = (:real^N) DIFF (IMAGE p (interval[a:real^M,b]))` THEN
   962   X_GEN_TAC `c:real^N` THEN REPEAT STRIP_TAC THEN
   963   SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
   964   SUBGOAL_THEN `bounded((path_component s c) UNION
   965                         (IMAGE (p:real^M->real^N) (interval[a,b])))`
   966   MP_TAC THENL
   967    [ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED; COMPACT_IMP_BOUNDED;
   968                  COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
   969     ALL_TAC] THEN
   970   DISCH_THEN(MP_TAC o SPEC `c:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN
   971   REWRITE_TAC[UNION_SUBSET] THEN
   972   DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
   973   MP_TAC(ISPECL [`p:real^M->real^N`; `a:real^M`; `b:real^M`]
   974     RETRACTION_INJECTIVE_IMAGE_INTERVAL) THEN
   975   ASM_REWRITE_TAC[SUBSET; IN_UNIV] THEN
   976   DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` MP_TAC) THEN
   977   DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
   978    (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
   979   REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_TAC THEN
   980   ABBREV_TAC `q = \z:real^N. if z IN path_component s c then r(z) else z` THEN
   981   SUBGOAL_THEN
   982     `(q:real^N->real^N) continuous_on
   983      (closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)))`
   984   MP_TAC THENL
   985    [EXPAND_TAC "q" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN
   986     REWRITE_TAC[CLOSED_CLOSURE; CONTINUOUS_ON_ID; GSYM OPEN_CLOSED] THEN
   987     REPEAT CONJ_TAC THENL
   988      [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
   989       ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
   990                    COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
   991       ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV];
   992       ALL_TAC] THEN
   993     X_GEN_TAC `z:real^N` THEN
   994     REWRITE_TAC[SET_RULE `~(z IN (s DIFF t) /\ z IN t)`] THEN
   995     STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
   996     MP_TAC(ISPECL
   997      [`path_component s (z:real^N)`; `path_component s (c:real^N)`]
   998      OPEN_INTER_CLOSURE_EQ_EMPTY) THEN
   999     ASM_REWRITE_TAC[GSYM DISJOINT; PATH_COMPONENT_DISJOINT] THEN ANTS_TAC THENL
  1000      [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
  1001       ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
  1002                    COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
  1003       REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
  1004       DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[] THEN
  1005       GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN] THEN
  1006       REWRITE_TAC[PATH_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]];
  1007     ALL_TAC] THEN
  1008   SUBGOAL_THEN
  1009    `closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)) =
  1010     (:real^N)`
  1011   SUBST1_TAC THENL
  1012    [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> t UNION (UNIV DIFF s) = UNIV`) THEN
  1013     REWRITE_TAC[CLOSURE_SUBSET];
  1014     DISCH_TAC] THEN
  1015   MP_TAC(ISPECL
  1016    [`(\x. &2 % c - x) o
  1017      (\x. c + B / norm(x - c) % (x - c)) o (q:real^N->real^N)`;
  1018     `cball(c:real^N,B)`]
  1019     BROUWER) THEN
  1020   REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; COMPACT_CBALL; CONVEX_CBALL] THEN
  1021   ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_LT_IMP_LE; REAL_NOT_LT] THEN
  1022   SUBGOAL_THEN `!x. ~((q:real^N->real^N) x = c)` ASSUME_TAC THENL
  1023    [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN
  1024     REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN COND_CASES_TAC THEN
  1025     ASM SET_TAC[PATH_COMPONENT_REFL_EQ];
  1026     ALL_TAC] THEN
  1027   REPEAT CONJ_TAC THENL
  1028    [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
  1029     SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
  1030     MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
  1031      [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN
  1032     MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
  1033     MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
  1034     SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
  1035     REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN
  1036     MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN
  1037     MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
  1038     ASM_REWRITE_TAC[FORALL_IN_IMAGE; NORM_EQ_0; VECTOR_SUB_EQ] THEN
  1039     SUBGOAL_THEN
  1040      `(\x:real^N. lift(norm(x - c))) = (lift o norm) o (\x. x - c)`
  1041     SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
  1042     MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
  1043     ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST;
  1044                  CONTINUOUS_ON_LIFT_NORM];
  1045     REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CBALL; o_THM; dist] THEN
  1046     X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
  1047     REWRITE_TAC[VECTOR_ARITH `c - (&2 % c - (c + x)) = x`] THEN
  1048     REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
  1049     ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
  1050     ASM_REAL_ARITH_TAC;
  1051     REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(c /\ b) <=> c ==> ~b`] THEN
  1052     REWRITE_TAC[IN_CBALL; o_THM; dist] THEN
  1053     X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
  1054     REWRITE_TAC[VECTOR_ARITH `&2 % c - (c + x') = x <=> --x' = x - c`] THEN
  1055     ASM_CASES_TAC `(x:real^N) IN path_component s c` THENL
  1056      [MATCH_MP_TAC(NORM_ARITH `norm(y) < B /\ norm(x) = B ==> ~(--x = y)`) THEN
  1057       REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
  1058       ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
  1059       ASM_SIMP_TAC[REAL_ARITH `&0 < B ==> abs B = B`] THEN
  1060       UNDISCH_TAC `path_component s c SUBSET ball(c:real^N,B)` THEN
  1061       REWRITE_TAC[SUBSET; IN_BALL] THEN ASM_MESON_TAC[dist; NORM_SUB];
  1062       EXPAND_TAC "q" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN
  1063       REWRITE_TAC[VECTOR_ARITH `--(c % x) = x <=> (&1 + c) % x = vec 0`] THEN
  1064       ASM_REWRITE_TAC[DE_MORGAN_THM; VECTOR_SUB_EQ; VECTOR_MUL_EQ_0] THEN
  1065       SUBGOAL_THEN `~(x:real^N = c)` ASSUME_TAC THENL
  1066        [ASM_MESON_TAC[PATH_COMPONENT_REFL; IN]; ALL_TAC] THEN
  1067       ASM_REWRITE_TAC[] THEN
  1068       MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`) THEN
  1069       ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]]);;
  1070 
  1071 let PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
  1072  (`!s:real^N->bool a b:real^M.
  1073         2 <= dimindex(:N) /\ s homeomorphic interval[a,b]
  1074         ==> path_connected((:real^N) DIFF s)`,
  1075   REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
  1076   FIRST_ASSUM(MP_TAC o MATCH_MP
  1077     UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
  1078   ASM_REWRITE_TAC[SET_RULE `~(x IN s) <=> x IN (UNIV DIFF s)`] THEN
  1079   ABBREV_TAC `t = (:real^N) DIFF s` THEN
  1080   DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
  1081   STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN
  1082   REWRITE_TAC[COMPACT_INTERVAL] THEN
  1083   DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
  1084   REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM] THEN
  1085   X_GEN_TAC `B:real` THEN STRIP_TAC THEN
  1086   SUBGOAL_THEN `(?u:real^N. u IN path_component t x /\ B < norm(u)) /\
  1087                 (?v:real^N. v IN path_component t y /\ B < norm(v))`
  1088   STRIP_ASSUME_TAC THENL
  1089    [ASM_MESON_TAC[BOUNDED_POS; REAL_NOT_LE]; ALL_TAC] THEN
  1090   MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `u:real^N` THEN
  1091   CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
  1092   MATCH_MP_TAC PATH_COMPONENT_SYM THEN
  1093   MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `v:real^N` THEN
  1094   CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
  1095   MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN
  1096   EXISTS_TAC `(:real^N) DIFF cball(vec 0,B)` THEN CONJ_TAC THENL
  1097    [EXPAND_TAC "t" THEN MATCH_MP_TAC(SET_RULE
  1098      `s SUBSET t ==> (u DIFF t) SUBSET (u DIFF s)`) THEN
  1099     ASM_REWRITE_TAC[SUBSET; IN_CBALL_0];
  1100     MP_TAC(ISPEC `cball(vec 0:real^N,B)`
  1101        PATH_CONNECTED_COMPLEMENT_BOUNDED_CONVEX) THEN
  1102     ASM_REWRITE_TAC[BOUNDED_CBALL; CONVEX_CBALL] THEN
  1103     REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
  1104     DISCH_THEN MATCH_MP_TAC THEN
  1105     ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_CBALL_0; REAL_NOT_LE]]);;
  1106 
  1107 (* ------------------------------------------------------------------------- *)
  1108 (* In particular, apply all these to the special case of an arc.             *)
  1109 (* ------------------------------------------------------------------------- *)
  1110 
  1111 let RETRACTION_ARC = prove
  1112  (`!p. arc p
  1113        ==> ?f. f continuous_on (:real^N) /\
  1114                IMAGE f (:real^N) SUBSET path_image p /\
  1115                (!x. x IN path_image p ==> f x = x)`,
  1116   REWRITE_TAC[arc; path; path_image] THEN REPEAT STRIP_TAC THEN
  1117   MATCH_MP_TAC RETRACTION_INJECTIVE_IMAGE_INTERVAL THEN
  1118   ASM_REWRITE_TAC[INTERVAL_EQ_EMPTY_CART_1; DROP_VEC; REAL_NOT_LT; REAL_POS]);;
  1119 
  1120 let PATH_CONNECTED_ARC_COMPLEMENT = prove
  1121  (`!p. 2 <= dimindex(:N) /\ arc p
  1122        ==> path_connected((:real^N) DIFF path_image p)`,
  1123   REWRITE_TAC[arc; path] THEN REPEAT STRIP_TAC THEN SIMP_TAC[path_image] THEN
  1124   MP_TAC(ISPECL [`path_image p:real^N->bool`; `vec 0:real^1`; `vec 1:real^1`]
  1125     PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
  1126   ASM_REWRITE_TAC[path_image] THEN DISCH_THEN MATCH_MP_TAC THEN
  1127   ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
  1128   MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
  1129   EXISTS_TAC `p:real^1->real^N` THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]);;
  1130 
  1131 let CONNECTED_ARC_COMPLEMENT = prove
  1132  (`!p. 2 <= dimindex(:N) /\ arc p
  1133        ==> connected((:real^N) DIFF path_image p)`,
  1134   SIMP_TAC[PATH_CONNECTED_ARC_COMPLEMENT; PATH_CONNECTED_IMP_CONNECTED]);; *)
  1135 
  1136 end