src/HOL/Multivariate_Analysis/Fashoda.thy

modernize lemmas about 'continuous' and 'continuous_on';
improve automation of continuity proofs;

(* Author:                     John Harrison
   Translation from HOL light: Robert Himmelmann, TU Muenchen *)

header {* Fashoda meet theorem. *}

theory Fashoda
imports Brouwer_Fixpoint Path_Connected Cartesian_Euclidean_Space
begin

subsection {*Fashoda meet theorem. *}

lemma infnorm_2: "infnorm (x::real^2) = max (abs(x$1)) (abs(x$2))"
  unfolding infnorm_cart UNIV_2 apply(rule Sup_eq) by auto

lemma infnorm_eq_1_2: "infnorm (x::real^2) = 1 \<longleftrightarrow>
        (abs(x$1) \<le> 1 \<and> abs(x$2) \<le> 1 \<and> (x$1 = -1 \<or> x$1 = 1 \<or> x$2 = -1 \<or> x$2 = 1))"
  unfolding infnorm_2 by auto

lemma infnorm_eq_1_imp: assumes "infnorm (x::real^2) = 1" shows "abs(x$1) \<le> 1" "abs(x$2) \<le> 1"
  using assms unfolding infnorm_eq_1_2 by auto

lemma fashoda_unit: fixes f g::"real \<Rightarrow> real^2"
  assumes "f ` {- 1..1} \<subseteq> {- 1..1}" "g ` {- 1..1} \<subseteq> {- 1..1}"
  "continuous_on {- 1..1} f"  "continuous_on {- 1..1} g"
  "f (- 1)$1 = - 1" "f 1$1 = 1" "g (- 1) $2 = -1" "g 1 $2 = 1"
  shows "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. f s = g t" proof(rule ccontr)
  case goal1 note as = this[unfolded bex_simps,rule_format]
  def sqprojection \<equiv> "\<lambda>z::real^2. (inverse (infnorm z)) *\<^sub>R z" 
  def negatex \<equiv> "\<lambda>x::real^2. (vector [-(x$1), x$2])::real^2" 
  have lem1:"\<forall>z::real^2. infnorm(negatex z) = infnorm z"
    unfolding negatex_def infnorm_2 vector_2 by auto
  have lem2:"\<forall>z. z\<noteq>0 \<longrightarrow> infnorm(sqprojection z) = 1" unfolding sqprojection_def
    unfolding infnorm_mul[unfolded smult_conv_scaleR] unfolding abs_inverse real_abs_infnorm
    apply(subst infnorm_eq_0[THEN sym]) by auto
  let ?F = "(\<lambda>w::real^2. (f \<circ> (\<lambda>x. x$1)) w - (g \<circ> (\<lambda>x. x$2)) w)"
  have *:"\<And>i. (\<lambda>x::real^2. x $ i) ` {- 1..1} = {- 1..1::real}"
    apply(rule set_eqI) unfolding image_iff Bex_def mem_interval_cart apply rule defer 
    apply(rule_tac x="vec x" in exI) by auto
  { fix x assume "x \<in> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w) ` {- 1..1::real^2}"
    then guess w unfolding image_iff .. note w = this
    hence "x \<noteq> 0" using as[of "w$1" "w$2"] unfolding mem_interval_cart by auto} note x0=this
  have 21:"\<And>i::2. i\<noteq>1 \<Longrightarrow> i=2" using UNIV_2 by auto
  have 1:"{- 1<..<1::real^2} \<noteq> {}" unfolding interval_eq_empty_cart by auto
  have 2:"continuous_on {- 1..1} (negatex \<circ> sqprojection \<circ> ?F)"
    apply(intro continuous_on_intros continuous_on_component)
    unfolding * apply(rule assms)+
    apply(subst sqprojection_def)
    apply(intro continuous_on_intros)
    apply(simp add: infnorm_eq_0 x0)
    apply(rule linear_continuous_on)
  proof-
    show "bounded_linear negatex" apply(rule bounded_linearI') unfolding vec_eq_iff proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
      show "negatex (x + y) $ i = (negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
        apply-apply(case_tac[!] "i\<noteq>1") prefer 3 apply(drule_tac[1-2] 21) 
        unfolding negatex_def by(auto simp add:vector_2 ) qed
  qed
  have 3:"(negatex \<circ> sqprojection \<circ> ?F) ` {- 1..1} \<subseteq> {- 1..1}" unfolding subset_eq apply rule proof-
    case goal1 then guess y unfolding image_iff .. note y=this have "?F y \<noteq> 0" apply(rule x0) using y(1) by auto
    hence *:"infnorm (sqprojection (?F y)) = 1" unfolding y o_def apply- by(rule lem2[rule_format])
    have "infnorm x = 1" unfolding *[THEN sym] y o_def by(rule lem1[rule_format])
    thus "x\<in>{- 1..1}" unfolding mem_interval_cart infnorm_2 apply- apply rule
    proof-case goal1 thus ?case apply(cases "i=1") defer apply(drule 21) by auto qed qed
  guess x apply(rule brouwer_weak[of "{- 1..1::real^2}" "negatex \<circ> sqprojection \<circ> ?F"])
    apply(rule compact_interval convex_interval)+ unfolding interior_closed_interval
    apply(rule 1 2 3)+ . note x=this
  have "?F x \<noteq> 0" apply(rule x0) using x(1) by auto
  hence *:"infnorm (sqprojection (?F x)) = 1" unfolding o_def by(rule lem2[rule_format])
  have nx:"infnorm x = 1" apply(subst x(2)[THEN sym]) unfolding *[THEN sym] o_def by(rule lem1[rule_format])
  have "\<forall>x i. x \<noteq> 0 \<longrightarrow> (0 < (sqprojection x)$i \<longleftrightarrow> 0 < x$i)"    "\<forall>x i. x \<noteq> 0 \<longrightarrow> ((sqprojection x)$i < 0 \<longleftrightarrow> x$i < 0)"
    apply- apply(rule_tac[!] allI impI)+ proof- fix x::"real^2" and i::2 assume x:"x\<noteq>0"
    have "inverse (infnorm x) > 0" using x[unfolded infnorm_pos_lt[THEN sym]] by auto
    thus "(0 < sqprojection x $ i) = (0 < x $ i)"   "(sqprojection x $ i < 0) = (x $ i < 0)"
      unfolding sqprojection_def vector_component_simps vector_scaleR_component real_scaleR_def
      unfolding zero_less_mult_iff mult_less_0_iff by(auto simp add:field_simps) qed
  note lem3 = this[rule_format]
  have x1:"x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}" using x(1) unfolding mem_interval_cart by auto
  hence nz:"f (x $ 1) - g (x $ 2) \<noteq> 0" unfolding right_minus_eq apply-apply(rule as) by auto
  have "x $ 1 = -1 \<or> x $ 1 = 1 \<or> x $ 2 = -1 \<or> x $ 2 = 1" using nx unfolding infnorm_eq_1_2 by auto 
  thus False proof- fix P Q R S 
    presume "P \<or> Q \<or> R \<or> S" "P\<Longrightarrow>False" "Q\<Longrightarrow>False" "R\<Longrightarrow>False" "S\<Longrightarrow>False" thus False by auto
  next assume as:"x$1 = 1"
    hence *:"f (x $ 1) $ 1 = 1" using assms(6) by auto
    have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
      unfolding as negatex_def vector_2 by auto moreover
    from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
    ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
      apply(erule_tac x=1 in allE) by auto 
  next assume as:"x$1 = -1"
    hence *:"f (x $ 1) $ 1 = - 1" using assms(5) by auto
    have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
      unfolding as negatex_def vector_2 by auto moreover
    from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
    ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
      apply(erule_tac x=1 in allE) by auto
  next assume as:"x$2 = 1"
    hence *:"g (x $ 2) $ 2 = 1" using assms(8) by auto
    have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
      unfolding as negatex_def vector_2 by auto moreover
    from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
    ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
     apply(erule_tac x=2 in allE) by auto
 next assume as:"x$2 = -1"
    hence *:"g (x $ 2) $ 2 = - 1" using assms(7) by auto
    have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
      unfolding as negatex_def vector_2 by auto moreover
    from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
    ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
      apply(erule_tac x=2 in allE) by auto qed(auto) qed

lemma fashoda_unit_path: fixes f ::"real \<Rightarrow> real^2" and g ::"real \<Rightarrow> real^2"
  assumes "path f" "path g" "path_image f \<subseteq> {- 1..1}" "path_image g \<subseteq> {- 1..1}"
  "(pathstart f)$1 = -1" "(pathfinish f)$1 = 1"  "(pathstart g)$2 = -1" "(pathfinish g)$2 = 1"
  obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
  note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def]
  def iscale \<equiv> "\<lambda>z::real. inverse 2 *\<^sub>R (z + 1)"
  have isc:"iscale ` {- 1..1} \<subseteq> {0..1}" unfolding iscale_def by(auto)
  have "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. (f \<circ> iscale) s = (g \<circ> iscale) t" proof(rule fashoda_unit) 
    show "(f \<circ> iscale) ` {- 1..1} \<subseteq> {- 1..1}" "(g \<circ> iscale) ` {- 1..1} \<subseteq> {- 1..1}"
      using isc and assms(3-4) unfolding image_compose by auto
    have *:"continuous_on {- 1..1} iscale" unfolding iscale_def by(rule continuous_on_intros)+
    show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)"
      apply-apply(rule_tac[!] continuous_on_compose[OF *]) apply(rule_tac[!] continuous_on_subset[OF _ isc])
      by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding vec_eq_iff by auto
    show "(f \<circ> iscale) (- 1) $ 1 = - 1" "(f \<circ> iscale) 1 $ 1 = 1" "(g \<circ> iscale) (- 1) $ 2 = -1" "(g \<circ> iscale) 1 $ 2 = 1"
      unfolding o_def iscale_def using assms by(auto simp add:*) qed
  then guess s .. from this(2) guess t .. note st=this
  show thesis apply(rule_tac z="f (iscale s)" in that)
    using st `s\<in>{- 1..1}` unfolding o_def path_image_def image_iff apply-
    apply(rule_tac x="iscale s" in bexI) prefer 3 apply(rule_tac x="iscale t" in bexI)
    using isc[unfolded subset_eq, rule_format] by auto qed

(* move *)
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
  unfolding interval_bij_cart split_conv vec_eq_iff vec_lambda_beta
  apply(rule,insert assms,erule_tac x=i in allE) by auto

lemma fashoda: fixes b::"real^2"
  assumes "path f" "path g" "path_image f \<subseteq> {a..b}" "path_image g \<subseteq> {a..b}"
  "(pathstart f)$1 = a$1" "(pathfinish f)$1 = b$1"
  "(pathstart g)$2 = a$2" "(pathfinish g)$2 = b$2"
  obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
  fix P Q S presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" "Q \<Longrightarrow> thesis" "S \<Longrightarrow> thesis" thus thesis by auto
next have "{a..b} \<noteq> {}" using assms(3) using path_image_nonempty by auto
  hence "a \<le> b" unfolding interval_eq_empty_cart less_eq_vec_def by(auto simp add: not_less)
  thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding less_eq_vec_def forall_2 by auto
next assume as:"a$1 = b$1" have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2" apply(rule connected_ivt_component_cart)
    apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
    unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
    unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
  have "z \<in> {a..b}" using z(1) assms(4) unfolding path_image_def by blast 
  hence "z = f 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
    using assms(3)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "f 0" 1]
    unfolding mem_interval_cart apply(erule_tac x=1 in allE) using as by auto
  thus thesis apply-apply(rule that[OF _ z(1)]) unfolding path_image_def by auto
next assume as:"a$2 = b$2" have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1" apply(rule connected_ivt_component_cart)
    apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
    unfolding assms using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
    unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
  have "z \<in> {a..b}" using z(1) assms(3) unfolding path_image_def by blast 
  hence "z = g 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
    using assms(4)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "g 0" 2]
    unfolding mem_interval_cart apply(erule_tac x=2 in allE) using as by auto
  thus thesis apply-apply(rule that[OF z(1)]) unfolding path_image_def by auto
next assume as:"a $ 1 < b $ 1 \<and> a $ 2 < b $ 2"
  have int_nem:"{- 1..1::real^2} \<noteq> {}" unfolding interval_eq_empty_cart by auto
  guess z apply(rule fashoda_unit_path[of "interval_bij (a,b) (- 1,1) \<circ> f" "interval_bij (a,b) (- 1,1) \<circ> g"]) 
    unfolding path_def path_image_def pathstart_def pathfinish_def
    apply(rule_tac[1-2] continuous_on_compose) apply(rule assms[unfolded path_def] continuous_on_interval_bij)+
    unfolding subset_eq apply(rule_tac[1-2] ballI)
  proof- fix x assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}"
    then guess y unfolding image_iff .. note y=this
    show "x \<in> {- 1..1}" unfolding y o_def apply(rule in_interval_interval_bij)
      using y(1) using assms(3)[unfolded path_image_def subset_eq] int_nem by auto
  next fix x assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}"
    then guess y unfolding image_iff .. note y=this
    show "x \<in> {- 1..1}" unfolding y o_def apply(rule in_interval_interval_bij)
      using y(1) using assms(4)[unfolded path_image_def subset_eq] int_nem by auto
  next show "(interval_bij (a, b) (- 1, 1) \<circ> f) 0 $ 1 = -1"
      "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
      "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
      "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1"
      unfolding interval_bij_cart vector_component_simps o_def split_conv
      unfolding assms[unfolded pathstart_def pathfinish_def] using as by auto qed note z=this
  from z(1) guess zf unfolding image_iff .. note zf=this
  from z(2) guess zg unfolding image_iff .. note zg=this
  have *:"\<forall>i. (- 1) $ i < (1::real^2) $ i \<and> a $ i < b $ i" unfolding forall_2 using as by auto
  show thesis apply(rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
    apply(subst zf) defer apply(subst zg) unfolding o_def interval_bij_bij_cart[OF *] path_image_def
    using zf(1) zg(1) by auto qed

subsection {*Some slightly ad hoc lemmas I use below*}

lemma segment_vertical: fixes a::"real^2" assumes "a$1 = b$1"
  shows "x \<in> closed_segment a b \<longleftrightarrow> (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))" (is "_ = ?R")
proof- 
  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"
  { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
  { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
    { fix b a assume "b + u * a > a + u * b"
      hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
      hence "b \<ge> a" apply(drule_tac mult_less_imp_less_left) using u by auto
      hence "u * a \<le> u * b" apply-apply(rule mult_left_mono[OF _ u(3)]) 
        using u(3-4) by(auto simp add:field_simps) } note * = this
    { fix a b assume "u * b > u * a" hence "(1 - u) * a \<le> (1 - u) * b" apply-apply(rule mult_left_mono)
        apply(drule mult_less_imp_less_left) using u by auto
      hence "a + u * b \<le> b + u * a" by(auto simp add:field_simps) } note ** = this
    thus ?R unfolding u assms using u by(auto simp add:field_simps not_le intro:* **) }
  { assume ?R thus ?L proof(cases "x$2 = b$2")
      case True thus ?L apply(rule_tac x="(x$2 - a$2) / (b$2 - a$2)" in exI) unfolding assms True
        using `?R` by(auto simp add:field_simps)
    next case False thus ?L apply(rule_tac x="1 - (x$2 - b$2) / (a$2 - b$2)" in exI) unfolding assms using `?R`
        by(auto simp add:field_simps)
    qed } qed

lemma segment_horizontal: fixes a::"real^2" assumes "a$2 = b$2"
  shows "x \<in> closed_segment a b \<longleftrightarrow> (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))" (is "_ = ?R")
proof- 
  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"
  { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
  { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
    { fix b a assume "b + u * a > a + u * b"
      hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
      hence "b \<ge> a" apply(drule_tac mult_less_imp_less_left) using u by auto
      hence "u * a \<le> u * b" apply-apply(rule mult_left_mono[OF _ u(3)]) 
        using u(3-4) by(auto simp add:field_simps) } note * = this
    { fix a b assume "u * b > u * a" hence "(1 - u) * a \<le> (1 - u) * b" apply-apply(rule mult_left_mono)
        apply(drule mult_less_imp_less_left) using u by auto
      hence "a + u * b \<le> b + u * a" by(auto simp add:field_simps) } note ** = this
    thus ?R unfolding u assms using u by(auto simp add:field_simps not_le intro:* **) }
  { assume ?R thus ?L proof(cases "x$1 = b$1")
      case True thus ?L apply(rule_tac x="(x$1 - a$1) / (b$1 - a$1)" in exI) unfolding assms True
        using `?R` by(auto simp add:field_simps)
    next case False thus ?L apply(rule_tac x="1 - (x$1 - b$1) / (a$1 - b$1)" in exI) unfolding assms using `?R`
        by(auto simp add:field_simps)
    qed } qed

subsection {*useful Fashoda corollary pointed out to me by Tom Hales. *}

lemma fashoda_interlace: fixes a::"real^2"
  assumes "path f" "path g"
  "path_image f \<subseteq> {a..b}" "path_image g \<subseteq> {a..b}"
  "(pathstart f)$2 = a$2" "(pathfinish f)$2 = a$2"
  "(pathstart g)$2 = a$2" "(pathfinish g)$2 = a$2"
  "(pathstart f)$1 < (pathstart g)$1" "(pathstart g)$1 < (pathfinish f)$1"
  "(pathfinish f)$1 < (pathfinish g)$1"
  obtains z where "z \<in> path_image f" "z \<in> path_image g"
proof-
  have "{a..b} \<noteq> {}" using path_image_nonempty using assms(3) by auto
  note ab=this[unfolded interval_eq_empty_cart not_ex forall_2 not_less]
  have "pathstart f \<in> {a..b}" "pathfinish f \<in> {a..b}" "pathstart g \<in> {a..b}" "pathfinish g \<in> {a..b}"
    using pathstart_in_path_image pathfinish_in_path_image using assms(3-4) by auto
  note startfin = this[unfolded mem_interval_cart forall_2]
  let ?P1 = "linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2]) +++
     linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f) +++ f +++
     linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2]) +++
     linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2])" 
  let ?P2 = "linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g) +++ g +++
     linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1]) +++
     linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1]) +++
     linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3])"
  let ?a = "vector[a$1 - 2, a$2 - 3]"
  let ?b = "vector[b$1 + 2, b$2 + 3]"
  have P1P2:"path_image ?P1 = path_image (linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2])) \<union>
      path_image (linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f)) \<union> path_image f \<union>
      path_image (linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2])) \<union>
      path_image (linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2]))"
    "path_image ?P2 = path_image(linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g)) \<union> path_image g \<union>
      path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union>
      path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
      path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
      by(auto simp add: path_image_join path_linepath)
  have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:less_eq_vec_def forall_2 vector_2)
  guess z apply(rule fashoda[of ?P1 ?P2 ?a ?b])
    unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 proof-
    show "path ?P1" "path ?P2" using assms by auto
    have "path_image ?P1 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 3
      apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
      unfolding mem_interval_cart forall_2 vector_2 using ab startfin abab assms(3)
      using assms(9-) unfolding assms by(auto simp add:field_simps)
    thus "path_image ?P1  \<subseteq> {?a .. ?b}" .
    have "path_image ?P2 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 2
      apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
      unfolding mem_interval_cart forall_2 vector_2 using ab startfin abab assms(4)
      using assms(9-) unfolding assms  by(auto simp add:field_simps)
    thus "path_image ?P2  \<subseteq> {?a .. ?b}" . 
    show "a $ 1 - 2 = a $ 1 - 2"  "b $ 1 + 2 = b $ 1 + 2" "pathstart g $ 2 - 3 = a $ 2 - 3"  "b $ 2 + 3 = b $ 2 + 3"
      by(auto simp add: assms)
  qed note z=this[unfolded P1P2 path_image_linepath]
  show thesis apply(rule that[of z]) proof-
    have "(z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<or>
     z \<in> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) \<or>
   z \<in> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<or>
  z \<in> closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]) \<Longrightarrow>
  (((z \<in> closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g)) \<or>
    z \<in> closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1])) \<or>
   z \<in> closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1])) \<or>
  z \<in> closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]) \<Longrightarrow> False"
      apply(simp only: segment_vertical segment_horizontal vector_2) proof- case goal1 note as=this
      have "pathfinish f \<in> {a..b}" using assms(3) pathfinish_in_path_image[of f] by auto 
      hence "1 + b $ 1 \<le> pathfinish f $ 1 \<Longrightarrow> False" unfolding mem_interval_cart forall_2 by auto
      hence "z$1 \<noteq> pathfinish f$1" using as(2) using assms ab by(auto simp add:field_simps)
      moreover have "pathstart f \<in> {a..b}" using assms(3) pathstart_in_path_image[of f] by auto 
      hence "1 + b $ 1 \<le> pathstart f $ 1 \<Longrightarrow> False" unfolding mem_interval_cart forall_2 by auto
      hence "z$1 \<noteq> pathstart f$1" using as(2) using assms ab by(auto simp add:field_simps)
      ultimately have *:"z$2 = a$2 - 2" using goal1(1) by auto
      have "z$1 \<noteq> pathfinish g$1" using as(2) using assms ab by(auto simp add:field_simps *)
      moreover have "pathstart g \<in> {a..b}" using assms(4) pathstart_in_path_image[of g] by auto 
      note this[unfolded mem_interval_cart forall_2]
      hence "z$1 \<noteq> pathstart g$1" using as(1) using assms ab by(auto simp add:field_simps *)
      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"
        using as(2) unfolding * assms by(auto simp add:field_simps)
      thus False unfolding * using ab by auto
    qed hence "z \<in> path_image f \<or> z \<in> path_image g" using z unfolding Un_iff by blast
    hence z':"z\<in>{a..b}" using assms(3-4) by auto
    have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow> (z = pathstart f \<or> z = pathfinish f)"
      unfolding vec_eq_iff forall_2 assms by auto
    with z' show "z\<in>path_image f" using z(1) unfolding Un_iff mem_interval_cart forall_2 apply-
      apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
    have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow> (z = pathstart g \<or> z = pathfinish g)"
      unfolding vec_eq_iff forall_2 assms by auto
    with z' show "z\<in>path_image g" using z(2) unfolding Un_iff mem_interval_cart forall_2 apply-
      apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
  qed qed

(** The Following still needs to be translated. Maybe I will do that later.

(* ------------------------------------------------------------------------- *)
(* Complement in dimension N >= 2 of set homeomorphic to any interval in     *)
(* any dimension is (path-)connected. This naively generalizes the argument  *)
(* in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer        *)
(* fixed point theorem", American Mathematical Monthly 1984.                 *)
(* ------------------------------------------------------------------------- *)

let RETRACTION_INJECTIVE_IMAGE_INTERVAL = prove
 (`!p:real^M->real^N a b.
        ~(interval[a,b] = {}) /\
        p continuous_on interval[a,b] /\
        (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ p x = p y ==> x = y)
        ==> ?f. f continuous_on (:real^N) /\
                IMAGE f (:real^N) SUBSET (IMAGE p (interval[a,b])) /\
                (!x. x IN (IMAGE p (interval[a,b])) ==> f x = x)`,
  REPEAT STRIP_TAC THEN
  FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
  DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN
  SUBGOAL_THEN `(q:real^N->real^M) continuous_on
                (IMAGE p (interval[a:real^M,b]))`
  ASSUME_TAC THENL
   [MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL];
    ALL_TAC] THEN
  MP_TAC(ISPECL [`q:real^N->real^M`;
                 `IMAGE (p:real^M->real^N)
                 (interval[a,b])`;
                 `a:real^M`; `b:real^M`]
        TIETZE_CLOSED_INTERVAL) THEN
  ASM_SIMP_TAC[COMPACT_INTERVAL; COMPACT_CONTINUOUS_IMAGE;
               COMPACT_IMP_CLOSED] THEN
  ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
  DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^M` STRIP_ASSUME_TAC) THEN
  EXISTS_TAC `(p:real^M->real^N) o (r:real^N->real^M)` THEN
  REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM; IN_UNIV] THEN
  CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
  MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
  FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ]
        CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);;

let UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
 (`!s:real^N->bool a b:real^M.
        s homeomorphic (interval[a,b])
        ==> !x. ~(x IN s) ==> ~bounded(path_component((:real^N) DIFF s) x)`,
  REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
  REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
  MAP_EVERY X_GEN_TAC [`p':real^N->real^M`; `p:real^M->real^N`] THEN
  DISCH_TAC THEN
  SUBGOAL_THEN
   `!x y. x IN interval[a,b] /\ y IN interval[a,b] /\
          (p:real^M->real^N) x = p y ==> x = y`
  ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
  FIRST_X_ASSUM(MP_TAC o funpow 4 CONJUNCT2) THEN
  DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) ASSUME_TAC) THEN
  ASM_CASES_TAC `interval[a:real^M,b] = {}` THEN
  ASM_REWRITE_TAC[IMAGE_CLAUSES; DIFF_EMPTY; PATH_COMPONENT_UNIV;
                  NOT_BOUNDED_UNIV] THEN
  ABBREV_TAC `s = (:real^N) DIFF (IMAGE p (interval[a:real^M,b]))` THEN
  X_GEN_TAC `c:real^N` THEN REPEAT STRIP_TAC THEN
  SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
  SUBGOAL_THEN `bounded((path_component s c) UNION
                        (IMAGE (p:real^M->real^N) (interval[a,b])))`
  MP_TAC THENL
   [ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED; COMPACT_IMP_BOUNDED;
                 COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
    ALL_TAC] THEN
  DISCH_THEN(MP_TAC o SPEC `c:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN
  REWRITE_TAC[UNION_SUBSET] THEN
  DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
  MP_TAC(ISPECL [`p:real^M->real^N`; `a:real^M`; `b:real^M`]
    RETRACTION_INJECTIVE_IMAGE_INTERVAL) THEN
  ASM_REWRITE_TAC[SUBSET; IN_UNIV] THEN
  DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` MP_TAC) THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
   (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
  REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_TAC THEN
  ABBREV_TAC `q = \z:real^N. if z IN path_component s c then r(z) else z` THEN
  SUBGOAL_THEN
    `(q:real^N->real^N) continuous_on
     (closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)))`
  MP_TAC THENL
   [EXPAND_TAC "q" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN
    REWRITE_TAC[CLOSED_CLOSURE; CONTINUOUS_ON_ID; GSYM OPEN_CLOSED] THEN
    REPEAT CONJ_TAC THENL
     [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
      ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
                   COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
      ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV];
      ALL_TAC] THEN
    X_GEN_TAC `z:real^N` THEN
    REWRITE_TAC[SET_RULE `~(z IN (s DIFF t) /\ z IN t)`] THEN
    STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
    MP_TAC(ISPECL
     [`path_component s (z:real^N)`; `path_component s (c:real^N)`]
     OPEN_INTER_CLOSURE_EQ_EMPTY) THEN
    ASM_REWRITE_TAC[GSYM DISJOINT; PATH_COMPONENT_DISJOINT] THEN ANTS_TAC THENL
     [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
      ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
                   COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
      REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
      DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[] THEN
      GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN] THEN
      REWRITE_TAC[PATH_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]];
    ALL_TAC] THEN
  SUBGOAL_THEN
   `closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)) =
    (:real^N)`
  SUBST1_TAC THENL
   [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> t UNION (UNIV DIFF s) = UNIV`) THEN
    REWRITE_TAC[CLOSURE_SUBSET];
    DISCH_TAC] THEN
  MP_TAC(ISPECL
   [`(\x. &2 % c - x) o
     (\x. c + B / norm(x - c) % (x - c)) o (q:real^N->real^N)`;
    `cball(c:real^N,B)`]
    BROUWER) THEN
  REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; COMPACT_CBALL; CONVEX_CBALL] THEN
  ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_LT_IMP_LE; REAL_NOT_LT] THEN
  SUBGOAL_THEN `!x. ~((q:real^N->real^N) x = c)` ASSUME_TAC THENL
   [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN
    REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN COND_CASES_TAC THEN
    ASM SET_TAC[PATH_COMPONENT_REFL_EQ];
    ALL_TAC] THEN
  REPEAT CONJ_TAC THENL
   [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
    SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
    MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
     [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN
    MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
    MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
    SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
    REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN
    MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN
    MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
    ASM_REWRITE_TAC[FORALL_IN_IMAGE; NORM_EQ_0; VECTOR_SUB_EQ] THEN
    SUBGOAL_THEN
     `(\x:real^N. lift(norm(x - c))) = (lift o norm) o (\x. x - c)`
    SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
    MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
    ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST;
                 CONTINUOUS_ON_LIFT_NORM];
    REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CBALL; o_THM; dist] THEN
    X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
    REWRITE_TAC[VECTOR_ARITH `c - (&2 % c - (c + x)) = x`] THEN
    REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
    ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
    ASM_REAL_ARITH_TAC;
    REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(c /\ b) <=> c ==> ~b`] THEN
    REWRITE_TAC[IN_CBALL; o_THM; dist] THEN
    X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
    REWRITE_TAC[VECTOR_ARITH `&2 % c - (c + x') = x <=> --x' = x - c`] THEN
    ASM_CASES_TAC `(x:real^N) IN path_component s c` THENL
     [MATCH_MP_TAC(NORM_ARITH `norm(y) < B /\ norm(x) = B ==> ~(--x = y)`) THEN
      REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
      ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
      ASM_SIMP_TAC[REAL_ARITH `&0 < B ==> abs B = B`] THEN
      UNDISCH_TAC `path_component s c SUBSET ball(c:real^N,B)` THEN
      REWRITE_TAC[SUBSET; IN_BALL] THEN ASM_MESON_TAC[dist; NORM_SUB];
      EXPAND_TAC "q" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN
      REWRITE_TAC[VECTOR_ARITH `--(c % x) = x <=> (&1 + c) % x = vec 0`] THEN
      ASM_REWRITE_TAC[DE_MORGAN_THM; VECTOR_SUB_EQ; VECTOR_MUL_EQ_0] THEN
      SUBGOAL_THEN `~(x:real^N = c)` ASSUME_TAC THENL
       [ASM_MESON_TAC[PATH_COMPONENT_REFL; IN]; ALL_TAC] THEN
      ASM_REWRITE_TAC[] THEN
      MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`) THEN
      ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]]);;

let PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
 (`!s:real^N->bool a b:real^M.
        2 <= dimindex(:N) /\ s homeomorphic interval[a,b]
        ==> path_connected((:real^N) DIFF s)`,
  REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
  FIRST_ASSUM(MP_TAC o MATCH_MP
    UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
  ASM_REWRITE_TAC[SET_RULE `~(x IN s) <=> x IN (UNIV DIFF s)`] THEN
  ABBREV_TAC `t = (:real^N) DIFF s` THEN
  DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
  STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN
  REWRITE_TAC[COMPACT_INTERVAL] THEN
  DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
  REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM] THEN
  X_GEN_TAC `B:real` THEN STRIP_TAC THEN
  SUBGOAL_THEN `(?u:real^N. u IN path_component t x /\ B < norm(u)) /\
                (?v:real^N. v IN path_component t y /\ B < norm(v))`
  STRIP_ASSUME_TAC THENL
   [ASM_MESON_TAC[BOUNDED_POS; REAL_NOT_LE]; ALL_TAC] THEN
  MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `u:real^N` THEN
  CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
  MATCH_MP_TAC PATH_COMPONENT_SYM THEN
  MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `v:real^N` THEN
  CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
  MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN
  EXISTS_TAC `(:real^N) DIFF cball(vec 0,B)` THEN CONJ_TAC THENL
   [EXPAND_TAC "t" THEN MATCH_MP_TAC(SET_RULE
     `s SUBSET t ==> (u DIFF t) SUBSET (u DIFF s)`) THEN
    ASM_REWRITE_TAC[SUBSET; IN_CBALL_0];
    MP_TAC(ISPEC `cball(vec 0:real^N,B)`
       PATH_CONNECTED_COMPLEMENT_BOUNDED_CONVEX) THEN
    ASM_REWRITE_TAC[BOUNDED_CBALL; CONVEX_CBALL] THEN
    REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
    DISCH_THEN MATCH_MP_TAC THEN
    ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_CBALL_0; REAL_NOT_LE]]);;

(* ------------------------------------------------------------------------- *)
(* In particular, apply all these to the special case of an arc.             *)
(* ------------------------------------------------------------------------- *)

let RETRACTION_ARC = prove
 (`!p. arc p
       ==> ?f. f continuous_on (:real^N) /\
               IMAGE f (:real^N) SUBSET path_image p /\
               (!x. x IN path_image p ==> f x = x)`,
  REWRITE_TAC[arc; path; path_image] THEN REPEAT STRIP_TAC THEN
  MATCH_MP_TAC RETRACTION_INJECTIVE_IMAGE_INTERVAL THEN
  ASM_REWRITE_TAC[INTERVAL_EQ_EMPTY_CART_1; DROP_VEC; REAL_NOT_LT; REAL_POS]);;

let PATH_CONNECTED_ARC_COMPLEMENT = prove
 (`!p. 2 <= dimindex(:N) /\ arc p
       ==> path_connected((:real^N) DIFF path_image p)`,
  REWRITE_TAC[arc; path] THEN REPEAT STRIP_TAC THEN SIMP_TAC[path_image] THEN
  MP_TAC(ISPECL [`path_image p:real^N->bool`; `vec 0:real^1`; `vec 1:real^1`]
    PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
  ASM_REWRITE_TAC[path_image] THEN DISCH_THEN MATCH_MP_TAC THEN
  ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
  MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
  EXISTS_TAC `p:real^1->real^N` THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]);;

let CONNECTED_ARC_COMPLEMENT = prove
 (`!p. 2 <= dimindex(:N) /\ arc p
       ==> connected((:real^N) DIFF path_image p)`,
  SIMP_TAC[PATH_CONNECTED_ARC_COMPLEMENT; PATH_CONNECTED_IMP_CONNECTED]);; *)

end