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products/Sources/formale Sprachen/Isabelle/HOL/Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 124 kB

Quelle Homeomorphism.thy Sprache: Isabelle

(*  Title:      HOL/Analysis/Homeomorphism.thy
    Author: LC Paulson (ported from HOL Light)
*)

section \<open>Homeomorphism Theorems\<close>

theory Homeomorphism
imports Homotopy
begin

lemma homeomorphic_spheres':
  fixes a ::"'a::euclidean_space" and b ::"'b::euclidean_space"
  assumes "0 < \" and dimeq: "DIM('a) = DIM('b)"
  shows "(sphere a \) homeomorphic (sphere b \)"
proof -
  obtain f :: "'a\'b" and g where "linear f" "linear g"
     and fg: "\x. norm(f x) = norm x" "\y. norm(g y) = norm y" "\x. g(f x) = x" "\y. f(g y) = y"
    by (blast intro: isomorphisms_UNIV_UNIV [OF dimeq])
  then have "continuous_on UNIV f" "continuous_on UNIV g"
    using linear_continuous_on linear_linear by blast+
  then show ?thesis
    unfolding homeomorphic_minimal
    apply(rule_tac x="\x. b + f(x - a)" in exI)
    apply(rule_tac x="\x. a + g(x - b)" in exI)
    using assms
    apply (force intro: continuous_intros
                  continuous_on_compose2 [of _ f] continuous_on_compose2 [of _ g] simp: dist_commute dist_norm fg)
    done
qed

lemma homeomorphic_spheres_gen:
    fixes a :: "'a::euclidean_space" and b :: "'b::euclidean_space"
  assumes "0 < r" "0 < s" "DIM('a::euclidean_space) = DIM('b::euclidean_space)"
  shows "(sphere a r homeomorphic sphere b s)"
  using assms homeomorphic_trans [OF homeomorphic_spheres homeomorphic_spheres'] by auto

subsection \<open>Homeomorphism of all convex compact sets with nonempty interior\<close>

proposition
  fixes S :: "'a::euclidean_space set"
  assumes "compact S" and 0: "0 \ rel_interior S"
      and star: "\x. x \ S \ open_segment 0 x \ rel_interior S"
    shows starlike_compact_projective1_0:
            "S - rel_interior S homeomorphic sphere 0 1 \ affine hull S"
            (is "?SMINUS homeomorphic ?SPHER")
      and starlike_compact_projective2_0:
            "S homeomorphic cball 0 1 \ affine hull S"
            (is "S homeomorphic ?CBALL")
proof -
  have starI: "(u *\<^sub>R x) \ rel_interior S" if "x \ S" "0 \ u" "u < 1" for x u
  proof (cases "x=0 \ u=0")
    case True with 0 show ?thesis by force
  next
    case False with that show ?thesis
      by (auto simp: in_segment intro: star [THEN subsetD])
  qed
  have "0 \ S" using assms rel_interior_subset by auto
  define proj where "proj \ \x::'a. x /\<^sub>R norm x"
  have eqI: "x = y" if "proj x = proj y" "norm x = norm y" for x y
    using that  by (force simp: proj_def)
  then have iff_eq: "\x y. (proj x = proj y \ norm x = norm y) \ x = y"
    by blast
  have projI: "x \ affine hull S \ proj x \ affine hull S" for x
    by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_mul proj_def)
  have nproj1 [simp]: "x \ 0 \ norm(proj x) = 1" for x
    by (simp add: proj_def)
  have proj0_iff [simp]: "proj x = 0 \ x = 0" for x
    by (simp add: proj_def)
  have cont_proj: "continuous_on (UNIV - {0}) proj"
    unfolding proj_def by (rule continuous_intros | force)+
  have proj_spherI: "\x. \x \ affine hull S; x \ 0\ \ proj x \ ?SPHER"
    by (simp add: projI)
  have "bounded S" "closed S"
    using \<open>compact S\<close> compact_eq_bounded_closed by blast+
  have inj_on_proj: "inj_on proj (S - rel_interior S)"
  proof
    fix x y
    assume x: "x \ S - rel_interior S"
       and y: "y \ S - rel_interior S" and eq: "proj x = proj y"
    then have xynot: "x \ 0" "y \ 0" "x \ S" "y \ S" "x \ rel_interior S" "y \ rel_interior S"
      using 0 by auto
    consider "norm x = norm y" | "norm x < norm y" | "norm x > norm y" by linarith
    then show "x = y"
    proof cases
      assume "norm x = norm y"
        with iff_eq eq show "x = y" by blast
    next
      assume *: "norm x < norm y"
      have "x /\<^sub>R norm x = norm x *\<^sub>R (x /\<^sub>R norm x) /\<^sub>R norm (norm x *\<^sub>R (x /\<^sub>R norm x))"
        by force
      then have "proj ((norm x / norm y) *\<^sub>R y) = proj x"
        by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
      then have [simp]: "(norm x / norm y) *\<^sub>R y = x"
        by (rule eqI) (simp add: \<open>y \<noteq> 0\<close>)
      have no: "0 \ norm x / norm y" "norm x / norm y < 1"
        using * by (auto simp: field_split_simps)
      then show "x = y"
        using starI [OF \<open>y \<in> S\<close> no] xynot by auto
    next
      assume *: "norm x > norm y"
      have "y /\<^sub>R norm y = norm y *\<^sub>R (y /\<^sub>R norm y) /\<^sub>R norm (norm y *\<^sub>R (y /\<^sub>R norm y))"
        by force
      then have "proj ((norm y / norm x) *\<^sub>R x) = proj y"
        by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
      then have [simp]: "(norm y / norm x) *\<^sub>R x = y"
        by (rule eqI) (simp add: \<open>x \<noteq> 0\<close>)
      have no: "0 \ norm y / norm x" "norm y / norm x < 1"
        using * by (auto simp: field_split_simps)
      then show "x = y"
        using starI [OF \<open>x \<in> S\<close> no] xynot by auto
    qed
  qed
  have "\surf. homeomorphism (S - rel_interior S) ?SPHER proj surf"
  proof (rule homeomorphism_compact)
    show "compact (S - rel_interior S)"
       using \<open>compact S\<close> compact_rel_boundary by blast
    show "continuous_on (S - rel_interior S) proj"
      using 0 by (blast intro: continuous_on_subset [OF cont_proj])
    show "proj ` (S - rel_interior S) = ?SPHER"
    proof
      show "proj ` (S - rel_interior S) \ ?SPHER"
        using 0 by (force simp: hull_inc projI intro: nproj1)
      show "?SPHER \ proj ` (S - rel_interior S)"
      proof (clarsimp simp: proj_def)
        fix x
        assume "x \ affine hull S" and nox: "norm x = 1"
        then have "x \ 0" by auto
        obtain d where "0 < d" and dx: "(d *\<^sub>R x) \ rel_frontier S"
          and ri: "\e. \0 \ e; e < d\ \ (e *\<^sub>R x) \ rel_interior S"
          using ray_to_rel_frontier [OF \<open>bounded S\<close> 0] \<open>x \<in> affine hull S\<close> \<open>x \<noteq> 0\<close> by auto
        show "x \ (\x. x /\<^sub>R norm x) ` (S - rel_interior S)"
        proof
          show "x = d *\<^sub>R x /\<^sub>R norm (d *\<^sub>R x)"
            using \<open>0 < d\<close> by (auto simp: nox)
          show "d *\<^sub>R x \ S - rel_interior S"
            using dx \<open>closed S\<close> by (auto simp: rel_frontier_def)
        qed
      qed
    qed
  qed (rule inj_on_proj)
  then obtain surf where surf: "homeomorphism (S - rel_interior S) ?SPHER proj surf"
    by blast
  then have cont_surf: "continuous_on (proj ` (S - rel_interior S)) surf"
    by (auto simp: homeomorphism_def)
  have surf_nz: "\x. x \ ?SPHER \ surf x \ 0"
    by (metis "0" DiffE homeomorphism_def imageI surf)
  have cont_nosp: "continuous_on (?SPHER) (\x. norm x *\<^sub>R ((surf o proj) x))"
  proof (intro continuous_intros)
    show "continuous_on (sphere 0 1 \ affine hull S) proj"
      by (rule continuous_on_subset [OF cont_proj], force)
    show "continuous_on (proj ` (sphere 0 1 \ affine hull S)) surf"
      by (intro continuous_on_subset [OF cont_surf]) (force simp: homeomorphism_image1 [OF surf] dest: proj_spherI)
  qed
  have surfpS: "\x. \norm x = 1; x \ affine hull S\ \ surf (proj x) \ S"
    by (metis (full_types) DiffE \<open>0 \<in> S\<close> homeomorphism_def image_eqI norm_zero proj_spherI real_vector.scale_zero_left scaleR_one surf)
  have *: "\y. norm y = 1 \ y \ affine hull S \ x = surf (proj y)"
          if "x \ S" "x \ rel_interior S" for x
  proof -
    have "proj x \ ?SPHER"
      by (metis (full_types) "0" hull_inc proj_spherI that)
    moreover have "surf (proj x) = x"
      by (metis Diff_iff homeomorphism_def surf that)
    ultimately show ?thesis
      by (metis \<open>\<And>x. x \<in> ?SPHER \<Longrightarrow> surf x \<noteq> 0\<close> hull_inc inverse_1 local.proj_def norm_sgn projI scaleR_one sgn_div_norm that(1))
  qed
  have surfp_notin: "\x. \norm x = 1; x \ affine hull S\ \ surf (proj x) \ rel_interior S"
    by (metis (full_types) DiffE one_neq_zero homeomorphism_def image_eqI norm_zero proj_spherI surf)
  have no_sp_im: "(\x. norm x *\<^sub>R surf (proj x)) ` (?SPHER) = S - rel_interior S"
    by (auto simp: surfpS image_def Bex_def surfp_notin *)
  have inj_spher: "inj_on (\x. norm x *\<^sub>R surf (proj x)) ?SPHER"
  proof
    fix x y
    assume xy: "x \ ?SPHER" "y \ ?SPHER"
       and eq: " norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
    then have "norm x = 1" "norm y = 1" "x \ affine hull S" "y \ affine hull S"
      using 0 by auto
    with eq show "x = y"
      by (simp add: proj_def) (metis surf xy homeomorphism_def)
  qed
  have co01: "compact ?SPHER"
    by (simp add: compact_Int_closed)
  show "?SMINUS homeomorphic ?SPHER"
    using homeomorphic_def surf by blast
  have proj_scaleR: "\a x. 0 < a \ proj (a *\<^sub>R x) = proj x"
    by (simp add: proj_def)
  have cont_sp0: "continuous_on (affine hull S - {0}) (surf o proj)"
  proof (rule continuous_on_compose [OF continuous_on_subset [OF cont_proj]])
    show "continuous_on (proj ` (affine hull S - {0})) surf"
      using homeomorphism_image1 proj_spherI surf by (intro continuous_on_subset [OF cont_surf]) fastforce
  qed auto
  obtain B where "B>0" and B: "\x. x \ S \ norm x \ B"
    by (metis compact_imp_bounded \<open>compact S\<close> bounded_pos_less less_eq_real_def)
  have cont_nosp: "continuous (at x within ?CBALL) (\x. norm x *\<^sub>R surf (proj x))"
         if "norm x \ 1" "x \ affine hull S" for x
  proof (cases "x=0")
    case True
    have "(norm \ 0) (at 0 within cball 0 1 \ affine hull S)"
      by (simp add: tendsto_norm_zero eventually_at)
    with True show ?thesis
      apply (simp add: continuous_within)
      apply (rule lim_null_scaleR_bounded [where B=B])
      using B \<open>0 < B\<close> local.proj_def projI surfpS by (auto simp: eventually_at)
  next
    case False
    then have "\\<^sub>F x in at x. (x \ affine hull S - {0}) = (x \ affine hull S)"
      by (force simp: False eventually_at)
    moreover
    have "continuous (at x within affine hull S - {0}) (\x. surf (proj x))"
      using cont_sp0 False that by (auto simp add: continuous_on_eq_continuous_within)
    ultimately have *: "continuous (at x within affine hull S) (\x. surf (proj x))"
      by (simp add: continuous_within Lim_transform_within_set continuous_on_eq_continuous_within)
    show ?thesis
      by (intro continuous_within_subset [where S = "affine hull S", OF _ Int_lower2] continuous_intros *)
  qed
  have cont_nosp2: "continuous_on ?CBALL (\x. norm x *\<^sub>R ((surf o proj) x))"
    by (simp add: continuous_on_eq_continuous_within cont_nosp)
  have "norm y *\<^sub>R surf (proj y) \ S" if "y \ cball 0 1" and yaff: "y \ affine hull S" for y
  proof (cases "y=0")
    case True then show ?thesis
      by (simp add: \<open>0 \<in> S\<close>)
  next
    case False
    then have "norm y *\<^sub>R surf (proj y) = norm y *\<^sub>R surf (proj (y /\<^sub>R norm y))"
      by (simp add: proj_def)
    have "norm y \ 1" using that by simp
    have "surf (proj (y /\<^sub>R norm y)) \ S"
      using False local.proj_def nproj1 projI surfpS yaff by blast
    then have "surf (proj y) \ S"
      by (simp add: False proj_def)
    then show "norm y *\<^sub>R surf (proj y) \ S"
      by (metis dual_order.antisym le_less_linear norm_ge_zero rel_interior_subset scaleR_one
                starI subset_eq \<open>norm y \<le> 1\<close>)
  qed
  moreover have "x \ (\x. norm x *\<^sub>R surf (proj x)) ` (?CBALL)" if "x \ S" for x
  proof (cases "x=0")
    case True with that hull_inc  show ?thesis by fastforce
  next
    case False
    then have psp: "proj (surf (proj x)) = proj x"
      by (metis homeomorphism_def hull_inc proj_spherI surf that)
    have nxx: "norm x *\<^sub>R proj x = x"
      by (simp add: False local.proj_def)
    have affineI: "(1 / norm (surf (proj x))) *\<^sub>R x \ affine hull S"
      by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_clauses(4) that)
    have sproj_nz: "surf (proj x) \ 0"
      by (metis False proj0_iff psp)
    then have "proj x = proj (proj x)"
      by (metis False nxx proj_scaleR zero_less_norm_iff)
    moreover have scaleproj: "\a r. r *\<^sub>R proj a = (r / norm a) *\<^sub>R a"
      by (simp add: divide_inverse local.proj_def)
    ultimately have "(norm (surf (proj x)) / norm x) *\<^sub>R x \ rel_interior S"
      by (metis (no_types) sproj_nz divide_self_if hull_inc norm_eq_zero nproj1 projI psp scaleR_one surfp_notin that)
    then have "(norm (surf (proj x)) / norm x) \ 1"
      using starI [OF that] by (meson starI [OF that] le_less_linear norm_ge_zero zero_le_divide_iff)
    then have nole: "norm x \ norm (surf (proj x))"
      by (simp add: le_divide_eq_1)
    let ?inx = "x /\<^sub>R norm (surf (proj x))"
    show ?thesis
    proof
      show "x = norm ?inx *\<^sub>R surf (proj ?inx)"
        by (simp add: field_simps) (metis inverse_eq_divide nxx positive_imp_inverse_positive proj_scaleR psp scaleproj sproj_nz zero_less_norm_iff)
      qed (auto simp: field_split_simps nole affineI)
  qed
  ultimately have im_cball: "(\x. norm x *\<^sub>R surf (proj x)) ` ?CBALL = S"
    by blast
  have inj_cball: "inj_on (\x. norm x *\<^sub>R surf (proj x)) ?CBALL"
  proof
    fix x y
    assume "x \ ?CBALL" "y \ ?CBALL"
       and eq: "norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
    then have x: "x \ affine hull S" and y: "y \ affine hull S"
      using 0 by auto
    show "x = y"
    proof (cases "x=0 \ y=0")
      case True then show "x = y" using eq proj_spherI surf_nz x y by force
    next
      case False
      with x y have speq: "surf (proj x) = surf (proj y)"
        by (metis eq homeomorphism_apply2 proj_scaleR proj_spherI surf zero_less_norm_iff)
      then have "norm x = norm y"
        by (metis \<open>x \<in> affine hull S\<close> \<open>y \<in> affine hull S\<close> eq proj_spherI real_vector.scale_cancel_right surf_nz)
      moreover have "proj x = proj y"
        by (metis (no_types) False speq homeomorphism_apply2 proj_spherI surf x y)
      ultimately show "x = y"
        using eq eqI by blast
    qed
  qed
  have co01: "compact ?CBALL"
    by (simp add: compact_Int_closed)
  show "S homeomorphic ?CBALL"
    using homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball] homeomorphic_sym by blast
qed

corollary
  fixes S :: "'a::euclidean_space set"
  assumes "compact S" and a: "a \ rel_interior S"
      and star: "\x. x \ S \ open_segment a x \ rel_interior S"
    shows starlike_compact_projective1:
            "S - rel_interior S homeomorphic sphere a 1 \ affine hull S"
      and starlike_compact_projective2:
            "S homeomorphic cball a 1 \ affine hull S"
proof -
  have 1: "compact ((+) (-a) ` S)" by (meson assms compact_translation)
  have 2: "0 \ rel_interior ((+) (-a) ` S)"
    using a rel_interior_translation [of "- a" S] by (simp cong: image_cong_simp)
  have 3: "open_segment 0 x \ rel_interior ((+) (-a) ` S)" if "x \ ((+) (-a) ` S)" for x
  proof -
    have "x+a \ S" using that by auto
    then have "open_segment a (x+a) \ rel_interior S" by (metis star)
    then show ?thesis using open_segment_translation [of a 0 x]
      using rel_interior_translation [of "- a" S] by (fastforce simp add: ac_simps image_iff cong: image_cong_simp)
  qed
  have "S - rel_interior S homeomorphic ((+) (-a) ` S) - rel_interior ((+) (-a) ` S)"
    by (metis rel_interior_translation translation_diff homeomorphic_translation)
  also have "... homeomorphic sphere 0 1 \ affine hull ((+) (-a) ` S)"
    by (rule starlike_compact_projective1_0 [OF 1 2 3])
  also have "... = (+) (-a) ` (sphere a 1 \ affine hull S)"
    by (metis affine_hull_translation left_minus sphere_translation translation_Int)
  also have "... homeomorphic sphere a 1 \ affine hull S"
    using homeomorphic_translation homeomorphic_sym by blast
  finally show "S - rel_interior S homeomorphic sphere a 1 \ affine hull S" .

  have "S homeomorphic ((+) (-a) ` S)"
    by (metis homeomorphic_translation)
  also have "... homeomorphic cball 0 1 \ affine hull ((+) (-a) ` S)"
    by (rule starlike_compact_projective2_0 [OF 1 2 3])
  also have "... = (+) (-a) ` (cball a 1 \ affine hull S)"
    by (metis affine_hull_translation left_minus cball_translation translation_Int)
  also have "... homeomorphic cball a 1 \ affine hull S"
    using homeomorphic_translation homeomorphic_sym by blast
  finally show "S homeomorphic cball a 1 \ affine hull S" .
qed

corollary starlike_compact_projective_special:
  assumes "compact S"
    and cb01: "cball (0::'a::euclidean_space) 1 \ S"
    and scale: "\x u. \x \ S; 0 \ u; u < 1\ \ u *\<^sub>R x \ S - frontier S"
  shows "S homeomorphic (cball (0::'a::euclidean_space) 1)"
proof -
  have "ball 0 1 \ interior S"
    using cb01 interior_cball interior_mono by blast
  then have 0: "0 \ rel_interior S"
    by (meson centre_in_ball subsetD interior_subset_rel_interior le_numeral_extra(2) not_le)
  have [simp]: "affine hull S = UNIV"
    using \<open>ball 0 1 \<subseteq> interior S\<close> by (auto intro!: affine_hull_nonempty_interior)
  have star: "open_segment 0 x \ rel_interior S" if "x \ S" for x
  proof
    fix p assume "p \ open_segment 0 x"
    then obtain u where "x \ 0" and u: "0 \ u" "u < 1" and p: "u *\<^sub>R x = p"
      by (auto simp: in_segment)
    then show "p \ rel_interior S"
      using scale [OF that u] closure_subset frontier_def interior_subset_rel_interior by fastforce
  qed
  show ?thesis
    using starlike_compact_projective2_0 [OF \<open>compact S\<close> 0 star] by simp
qed

lemma homeomorphic_convex_lemma:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "convex S" "compact S" "convex T" "compact T"
      and affeq: "aff_dim S = aff_dim T"
    shows "(S - rel_interior S) homeomorphic (T - rel_interior T) \
           S homeomorphic T"
proof (cases "rel_interior S = {} \ rel_interior T = {}")
  case True
    then show ?thesis
      by (metis Diff_empty affeq \<open>convex S\<close> \<open>convex T\<close> aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty)
next
  case False
  then obtain a b where a: "a \ rel_interior S" and b: "b \ rel_interior T" by auto
  have starS: "\x. x \ S \ open_segment a x \ rel_interior S"
    using rel_interior_closure_convex_segment
          a \<open>convex S\<close> closure_subset subsetCE by blast
  have starT: "\x. x \ T \ open_segment b x \ rel_interior T"
    using rel_interior_closure_convex_segment
          b \<open>convex T\<close> closure_subset subsetCE by blast
  let ?aS = "(+) (-a) ` S" and ?bT = "(+) (-b) ` T"
  have 0: "0 \ affine hull ?aS" "0 \ affine hull ?bT"
    by (metis a b subsetD hull_inc image_eqI left_minus rel_interior_subset)+
  have subs: "subspace (span ?aS)" "subspace (span ?bT)"
    by (rule subspace_span)+
  moreover
  have "dim (span ((+) (- a) ` S)) = dim (span ((+) (- b) ` T))"
    by (metis 0 aff_dim_translation_eq aff_dim_zero affeq dim_span nat_int)
  ultimately obtain f g where "linear f" "linear g"
                and fim: "f ` span ?aS = span ?bT"
                and gim: "g ` span ?bT = span ?aS"
                and fno: "\x. x \ span ?aS \ norm(f x) = norm x"
                and gno: "\x. x \ span ?bT \ norm(g x) = norm x"
                and gf: "\x. x \ span ?aS \ g(f x) = x"
                and fg: "\x. x \ span ?bT \ f(g x) = x"
    by (rule isometries_subspaces) blast
  have [simp]: "continuous_on A f" for A
    using \<open>linear f\<close> linear_conv_bounded_linear linear_continuous_on by blast
  have [simp]: "continuous_on B g" for B
    using \<open>linear g\<close> linear_conv_bounded_linear linear_continuous_on by blast
  have eqspanS: "affine hull ?aS = span ?aS"
    by (metis a affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
  have eqspanT: "affine hull ?bT = span ?bT"
    by (metis b affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
  have "S homeomorphic cball a 1 \ affine hull S"
    by (rule starlike_compact_projective2 [OF \<open>compact S\<close> a starS])
  also have "... homeomorphic (+) (-a) ` (cball a 1 \ affine hull S)"
    by (metis homeomorphic_translation)
  also have "... = cball 0 1 \ (+) (-a) ` (affine hull S)"
    by (auto simp: dist_norm)
  also have "... = cball 0 1 \ span ?aS"
    using eqspanS affine_hull_translation by blast
  also have "... homeomorphic cball 0 1 \ span ?bT"
  proof (rule homeomorphicI)
    show fim1: "f ` (cball 0 1 \ span ?aS) = cball 0 1 \ span ?bT"
    proof
      show "f ` (cball 0 1 \ span ?aS) \ cball 0 1 \ span ?bT"
        using fim fno by auto
      show "cball 0 1 \ span ?bT \ f ` (cball 0 1 \ span ?aS)"
        by clarify (metis IntI fg gim gno image_eqI mem_cball_0)
    qed
    show "g ` (cball 0 1 \ span ?bT) = cball 0 1 \ span ?aS"
    proof
      show "g ` (cball 0 1 \ span ?bT) \ cball 0 1 \ span ?aS"
        using gim gno by auto
      show "cball 0 1 \ span ?aS \ g ` (cball 0 1 \ span ?bT)"
        by clarify (metis IntI fim1 gf image_eqI)
    qed
  qed (auto simp: fg gf)
  also have "... = cball 0 1 \ (+) (-b) ` (affine hull T)"
    using eqspanT affine_hull_translation by blast
  also have "... = (+) (-b) ` (cball b 1 \ affine hull T)"
    by (auto simp: dist_norm)
  also have "... homeomorphic (cball b 1 \ affine hull T)"
    by (metis homeomorphic_translation homeomorphic_sym)
  also have "... homeomorphic T"
    by (metis starlike_compact_projective2 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
  finally have 1: "S homeomorphic T" .

  have "S - rel_interior S homeomorphic sphere a 1 \ affine hull S"
    by (rule starlike_compact_projective1 [OF \<open>compact S\<close> a starS])
  also have "... homeomorphic (+) (-a) ` (sphere a 1 \ affine hull S)"
    by (metis homeomorphic_translation)
  also have "... = sphere 0 1 \ (+) (-a) ` (affine hull S)"
    by (auto simp: dist_norm)
  also have "... = sphere 0 1 \ span ?aS"
    using eqspanS affine_hull_translation by blast
  also have "... homeomorphic sphere 0 1 \ span ?bT"
  proof (rule homeomorphicI)
    show fim1: "f ` (sphere 0 1 \ span ?aS) = sphere 0 1 \ span ?bT"
    proof
      show "f ` (sphere 0 1 \ span ?aS) \ sphere 0 1 \ span ?bT"
        using fim fno by auto
      show "sphere 0 1 \ span ?bT \ f ` (sphere 0 1 \ span ?aS)"
        by clarify (metis IntI fg gim gno image_eqI mem_sphere_0)
    qed
    show "g ` (sphere 0 1 \ span ?bT) = sphere 0 1 \ span ?aS"
    proof
      show "g ` (sphere 0 1 \ span ?bT) \ sphere 0 1 \ span ?aS"
        using gim gno by auto
      show "sphere 0 1 \ span ?aS \ g ` (sphere 0 1 \ span ?bT)"
        by clarify (metis IntI fim1 gf image_eqI)
    qed
  qed (auto simp: fg gf)
  also have "... = sphere 0 1 \ (+) (-b) ` (affine hull T)"
    using eqspanT affine_hull_translation by blast
  also have "... = (+) (-b) ` (sphere b 1 \ affine hull T)"
    by (auto simp: dist_norm)
  also have "... homeomorphic (sphere b 1 \ affine hull T)"
    by (metis homeomorphic_translation homeomorphic_sym)
  also have "... homeomorphic T - rel_interior T"
    by (metis starlike_compact_projective1 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
  finally have 2: "S - rel_interior S homeomorphic T - rel_interior T" .
  show ?thesis
    using 1 2 by blast
qed

lemma homeomorphic_convex_compact_sets:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "convex S" "compact S" "convex T" "compact T"
      and affeq: "aff_dim S = aff_dim T"
    shows "S homeomorphic T"
using homeomorphic_convex_lemma [OF assms] assms
by (auto simp: rel_frontier_def)

lemma homeomorphic_rel_frontiers_convex_bounded_sets:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "convex S" "bounded S" "convex T" "bounded T"
      and affeq: "aff_dim S = aff_dim T"
    shows  "rel_frontier S homeomorphic rel_frontier T"
using assms homeomorphic_convex_lemma [of "closure S" "closure T"]
by (simp add: rel_frontier_def convex_rel_interior_closure)

subsection\<open>Homeomorphisms between punctured spheres and affine sets\<close>
text\<open>Including the famous stereoscopic projection of the 3-D sphere to the complex plane\<close>

text\<open>The special case with centre 0 and radius 1\<close>
lemma homeomorphic_punctured_affine_sphere_affine_01:
  assumes "b \ sphere 0 1" "affine T" "0 \ T" "b \ T" "affine p"
      and affT: "aff_dim T = aff_dim p + 1"
    shows "(sphere 0 1 \ T) - {b} homeomorphic p"
proof -
  have [simp]: "norm b = 1" "b\b = 1"
    using assms by (auto simp: norm_eq_1)
  have [simp]: "T \ {v. b\v = 0} \ {}"
    using \<open>0 \<in> T\<close> by auto
  have [simp]: "\ T \ {v. b\v = 0}"
    using \<open>norm b = 1\<close> \<open>b \<in> T\<close> by auto
  define f where "f \ \x. 2 *\<^sub>R b + (2 / (1 - b\x)) *\<^sub>R (x - b)"
  define g where "g \ \y. b + (4 / (norm y ^ 2 + 4)) *\<^sub>R (y - 2 *\<^sub>R b)"
  have fg[simp]: "\x. \x \ T; b\x = 0\ \ f (g x) = x"
    unfolding f_def g_def by (simp add: algebra_simps field_split_simps add_nonneg_eq_0_iff)
  have no: "(norm (f x))\<^sup>2 = 4 * (1 + b \ x) / (1 - b \ x)"
    if "norm x = 1" and "b \ x \ 1" for x
    using that sum_sqs_eq [of 1 "b \ x"]
    apply (simp flip: dot_square_norm add: norm_eq_1 nonzero_eq_divide_eq)
    apply (simp add: f_def vector_add_divide_simps inner_simps)
    apply (auto simp add: field_split_simps inner_commute)
    done
  have [simp]: "\u::real. 8 + u * (u * 8) = u * 16 \ u=1"
    by algebra
  have gf[simp]: "\x. \norm x = 1; b \ x \ 1\ \ g (f x) = x"
    unfolding g_def no by (auto simp: f_def field_split_simps)
  have g1: "norm (g x) = 1" if "x \ T" and "b \ x = 0" for x
    using that
    apply (simp only: g_def)
    apply (rule power2_eq_imp_eq)
    apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps)
    apply (simp add: algebra_simps inner_commute)
    done
  have ne1: "b \ g x \ 1" if "x \ T" and "b \ x = 0" for x
    using that unfolding g_def
    apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff)
    apply (auto simp: algebra_simps)
    done
  have "subspace T"
    by (simp add: assms subspace_affine)
  have gT: "\x. \x \ T; b \ x = 0\ \ g x \ T"
    unfolding g_def
    by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
  have "f ` {x. norm x = 1 \ b\x \ 1} \ {x. b\x = 0}"
    unfolding f_def using \<open>norm b = 1\<close> norm_eq_1
    by (force simp: field_simps inner_add_right inner_diff_right)
  moreover have "f ` T \ T"
    unfolding f_def using assms \<open>subspace T\<close>
    by (auto simp add: inner_add_right inner_diff_right mem_affine_3_minus subspace_mul)
  moreover have "{x. b\x = 0} \ T \ f ` ({x. norm x = 1 \ b\x \ 1} \ T)"
    by clarify (metis (mono_tags) IntI ne1 fg gT g1 imageI mem_Collect_eq)
  ultimately have imf: "f ` ({x. norm x = 1 \ b\x \ 1} \ T) = {x. b\x = 0} \ T"
    by blast
  have no4: "\y. b\y = 0 \ norm ((y\y + 4) *\<^sub>R b + 4 *\<^sub>R (y - 2 *\<^sub>R b)) = y\y + 4"
    apply (rule power2_eq_imp_eq)
    apply (simp_all flip: dot_square_norm)
    apply (auto simp: power2_eq_square algebra_simps inner_commute)
    done
  have [simp]: "\x. \norm x = 1; b \ x \ 1\ \ b \ f x = 0"
    by (simp add: f_def algebra_simps field_split_simps)
  have [simp]: "\x. \x \ T; norm x = 1; b \ x \ 1\ \ f x \ T"
    unfolding f_def
    by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
  have "g ` {x. b\x = 0} \ {x. norm x = 1 \ b\x \ 1}"
    unfolding g_def
    apply (clarsimp simp: no4 vector_add_divide_simps divide_simps add_nonneg_eq_0_iff dot_square_norm [symmetric])
    apply (auto simp: algebra_simps)
    done
  moreover have "g ` T \ T"
    unfolding g_def
    by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
  moreover have "{x. norm x = 1 \ b\x \ 1} \ T \ g ` ({x. b\x = 0} \ T)"
    by clarify (metis (mono_tags, lifting) IntI gf image_iff imf mem_Collect_eq)
  ultimately have img: "g ` ({x. b\x = 0} \ T) = {x. norm x = 1 \ b\x \ 1} \ T"
    by blast
  have aff: "affine ({x. b\x = 0} \ T)"
    by (blast intro: affine_hyperplane assms)
  have contf: "continuous_on ({x. norm x = 1 \ b\x \ 1} \ T) f"
    unfolding f_def by (rule continuous_intros | force)+
  have contg: "continuous_on ({x. b\x = 0} \ T) g"
    unfolding g_def by (rule continuous_intros | force simp: add_nonneg_eq_0_iff)+
  have "(sphere 0 1 \ T) - {b} = {x. norm x = 1 \ (b\x \ 1)} \ T"
    using  \<open>norm b = 1\<close> by (auto simp: norm_eq_1) (metis vector_eq  \<open>b\<bullet>b = 1\<close>)
  also have "... homeomorphic {x. b\x = 0} \ T"
    by (rule homeomorphicI [OF imf img contf contg]) auto
  also have "... homeomorphic p"
  proof (rule homeomorphic_affine_sets [OF aff \<open>affine p\<close>])
    show "aff_dim ({x. b \ x = 0} \ T) = aff_dim p"
      by (simp add: Int_commute aff_dim_affine_Int_hyperplane [OF \<open>affine T\<close>] affT)
  qed
  finally show ?thesis .
qed

theorem homeomorphic_punctured_affine_sphere_affine:
  fixes a :: "'a :: euclidean_space"
  assumes "0 < r" "b \ sphere a r" "affine T" "a \ T" "b \ T" "affine p"
      and aff: "aff_dim T = aff_dim p + 1"
    shows "(sphere a r \ T) - {b} homeomorphic p"
proof -
  have "a \ b" using assms by auto
  then have inj: "inj (\x::'a. x /\<^sub>R norm (a - b))"
    by (simp add: inj_on_def)
  have "((sphere a r \ T) - {b}) homeomorphic
        (+) (-a) ` ((sphere a r \<inter> T) - {b})"
    by (rule homeomorphic_translation)
  also have "... homeomorphic (*\<^sub>R) (inverse r) ` (+) (- a) ` (sphere a r \ T - {b})"
    by (metis \<open>0 < r\<close> homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl)
  also have "... = sphere 0 1 \ ((*\<^sub>R) (inverse r) ` (+) (- a) ` T) - {(b - a) /\<^sub>R r}"
    using assms by (auto simp: dist_norm norm_minus_commute divide_simps)
  also have "... homeomorphic p"
    using assms affine_translation [symmetric, of "- a"] aff_dim_translation_eq [of "- a"]
    by (intro homeomorphic_punctured_affine_sphere_affine_01) (auto simp: dist_norm norm_minus_commute affine_scaling inj)
  finally show ?thesis .
qed

corollary homeomorphic_punctured_sphere_affine:
  fixes a :: "'a :: euclidean_space"
  assumes "0 < r" and b: "b \ sphere a r"
      and "affine T" and affS: "aff_dim T + 1 = DIM('a)"
    shows "(sphere a r - {b}) homeomorphic T"
  using homeomorphic_punctured_affine_sphere_affine [of r b a UNIV T] assms by auto

corollary homeomorphic_punctured_sphere_hyperplane:
  fixes a :: "'a :: euclidean_space"
  assumes "0 < r" and b: "b \ sphere a r"
    and "c \ 0"
  shows "(sphere a r - {b}) homeomorphic {x::'a. c \ x = d}"
  using assms
  by (intro homeomorphic_punctured_sphere_affine) (auto simp: affine_hyperplane of_nat_diff)

proposition homeomorphic_punctured_sphere_affine_gen:
  fixes a :: "'a :: euclidean_space"
  assumes "convex S" "bounded S" and a: "a \ rel_frontier S"
      and "affine T" and affS: "aff_dim S = aff_dim T + 1"
    shows "rel_frontier S - {a} homeomorphic T"
proof -
  obtain U :: "'a set" where "affine U" "convex U" and affdS: "aff_dim U = aff_dim S"
    using choose_affine_subset [OF affine_UNIV aff_dim_geq]
    by (meson aff_dim_affine_hull affine_affine_hull affine_imp_convex)
  have "S \ {}" using assms by auto
  then obtain z where "z \ U"
    by (metis aff_dim_negative_iff equals0I affdS)
  then have bne: "ball z 1 \ U \ {}" by force
  then have [simp]: "aff_dim(ball z 1 \ U) = aff_dim U"
    using aff_dim_convex_Int_open [OF \<open>convex U\<close> open_ball]
    by (fastforce simp add: Int_commute)
  have "rel_frontier S homeomorphic rel_frontier (ball z 1 \ U)"
    by (rule homeomorphic_rel_frontiers_convex_bounded_sets)
       (auto simp: \<open>affine U\<close> affine_imp_convex convex_Int affdS assms)
  also have "... = sphere z 1 \ U"
    using convex_affine_rel_frontier_Int [of "ball z 1" U]
    by (simp add: \<open>affine U\<close> bne)
  finally have "rel_frontier S homeomorphic sphere z 1 \ U" .
  then obtain h k where him: "h ` rel_frontier S = sphere z 1 \ U"
                    and kim: "k ` (sphere z 1 \ U) = rel_frontier S"
                    and hcon: "continuous_on (rel_frontier S) h"
                    and kcon: "continuous_on (sphere z 1 \ U) k"
                    and kh:  "\x. x \ rel_frontier S \ k(h(x)) = x"
                    and hk:  "\y. y \ sphere z 1 \ U \ h(k(y)) = y"
    unfolding homeomorphic_def homeomorphism_def by auto
  have "rel_frontier S - {a} homeomorphic (sphere z 1 \ U) - {h a}"
  proof (rule homeomorphicI)
    show h: "h ` (rel_frontier S - {a}) = sphere z 1 \ U - {h a}"
      using him a kh by auto metis
    show "k ` (sphere z 1 \ U - {h a}) = rel_frontier S - {a}"
      by (force simp: h [symmetric] image_comp o_def kh)
  qed (auto intro: continuous_on_subset hcon kcon simp: kh hk)
  also have "... homeomorphic T"
    by (rule homeomorphic_punctured_affine_sphere_affine)
       (use a him in \<open>auto simp: affS affdS \<open>affine T\<close> \<open>affine U\<close> \<open>z \<in> U\<close>\<close>)
  finally show ?thesis .
qed

text\<open> When dealing with AR, ANR and ANR later, it's useful to know that every set
  is homeomorphic to a closed subset of a convex set, and
  if the set is locally compact we can take the convex set to be the universe.\<close>

proposition homeomorphic_closedin_convex:
  fixes S :: "'m::euclidean_space set"
  assumes "aff_dim S < DIM('n)"
  obtains U and T :: "'n::euclidean_space set"
     where "convex U" "U \ {}" "closedin (top_of_set U) T"
           "S homeomorphic T"
proof (cases "S = {}")
  case True then show ?thesis
    by (rule_tac U=UNIV and T="{}" in that) auto
next
  case False
  then obtain a where "a \ S" by auto
  obtain i::'n where i: "i \ Basis" "i \ 0"
    using SOME_Basis Basis_zero by force
  have "0 \ affine hull ((+) (- a) ` S)"
    by (simp add: \<open>a \<in> S\<close> hull_inc)
  then have "dim ((+) (- a) ` S) = aff_dim ((+) (- a) ` S)"
    by (simp add: aff_dim_zero)
  also have "... < DIM('n)"
    by (simp add: aff_dim_translation_eq_subtract assms cong: image_cong_simp)
  finally have dd: "dim ((+) (- a) ` S) < DIM('n)"
    by linarith
  have span: "span {x. i \ x = 0} = {x. i \ x = 0}"
    using span_eq_iff [symmetric, of "{x. i \ x = 0}"] subspace_hyperplane [of i] by simp
  have "dim ((+) (- a) ` S) \ dim {x. i \ x = 0}"
    using dd by (simp add: dim_hyperplane [OF \<open>i \<noteq> 0\<close>])
  then obtain T where "subspace T" and Tsub: "T \ {x. i \ x = 0}"
    and dimT: "dim T = dim ((+) (- a) ` S)"
    by (rule choose_subspace_of_subspace) (simp add: span)
  have "subspace (span ((+) (- a) ` S))"
    using subspace_span by blast
  then obtain h k where "linear h" "linear k"
               and heq: "h ` span ((+) (- a) ` S) = T"
               and keq:"k ` T = span ((+) (- a) ` S)"
               and hinv [simp]:  "\x. x \ span ((+) (- a) ` S) \ k(h x) = x"
               and kinv [simp]:  "\x. x \ T \ h(k x) = x"
    by (auto simp: dimT intro: isometries_subspaces [OF _ \<open>subspace T\<close>] dimT)
  have hcont: "continuous_on A h" and kcont: "continuous_on B k" for A B
    using \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_conv_bounded_linear by blast+
  have ihhhh[simp]: "\x. x \ S \ i \ h (x - a) = 0"
    using Tsub [THEN subsetD] heq span_superset by fastforce
  have "sphere 0 1 - {i} homeomorphic {x. i \ x = 0}"
  proof (rule homeomorphic_punctured_sphere_affine)
    show "affine {x. i \ x = 0}"
      by (auto simp: affine_hyperplane)
    show "aff_dim {x. i \ x = 0} + 1 = int DIM('n)"
      using i by force
  qed (use i in auto)
  then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i \ x = 0} f g"
    by (force simp: homeomorphic_def)
  show ?thesis
  proof
    have "h ` (+) (- a) ` S \ T"
      using heq span_superset span_linear_image by blast
    then have "g ` h ` (+) (- a) ` S \ g ` {x. i \ x = 0}"
      using Tsub by (simp add: image_mono)
    also have "... \ sphere 0 1 - {i}"
      by (simp add: fg [unfolded homeomorphism_def])
    finally have gh_sub_sph: "(g \ h) ` (+) (- a) ` S \ sphere 0 1 - {i}"
      by (metis image_comp)
    then have gh_sub_cb: "(g \ h) ` (+) (- a) ` S \ cball 0 1"
      by (metis Diff_subset order_trans sphere_cball)
    have [simp]: "\u. u \ S \ norm (g (h (u - a))) = 1"
      using gh_sub_sph [THEN subsetD] by (auto simp: o_def)
    show "convex (ball 0 1 \ (g \ h) ` (+) (- a) ` S)"
      by (meson ball_subset_cball convex_intermediate_ball gh_sub_cb sup.bounded_iff sup.cobounded1)
    show "closedin (top_of_set (ball 0 1 \ (g \ h) ` (+) (- a) ` S)) ((g \ h) ` (+) (- a) ` S)"
      unfolding closedin_closed
      by (rule_tac x="sphere 0 1" in exI) auto
    have ghcont: "continuous_on ((\x. x - a) ` S) (\x. g (h x))"
      by (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force)
    have kfcont: "continuous_on ((\x. g (h (x - a))) ` S) (\x. k (f x))"
    proof (rule continuous_on_compose2 [OF kcont])
      show "continuous_on ((\x. g (h (x - a))) ` S) f"
        using homeomorphism_cont1 [OF fg] gh_sub_sph by (fastforce intro: continuous_on_subset)
    qed auto
    have "S homeomorphic (+) (- a) ` S"
      by (fact homeomorphic_translation)
    also have "\ homeomorphic (g \ h) ` (+) (- a) ` S"
      apply (simp add: homeomorphic_def homeomorphism_def cong: image_cong_simp)
      apply (rule_tac x="g \ h" in exI)
      apply (rule_tac x="k \ f" in exI)
      apply (auto simp: ghcont kfcont span_base homeomorphism_apply2 [OF fg] image_comp cong: image_cong_simp)
      done
    finally show "S homeomorphic (g \ h) ` (+) (- a) ` S" .
  qed auto
qed

subsection\<open>Locally compact sets in an open set\<close>

text\<open> Locally compact sets are closed in an open set and are homeomorphic
  to an absolutely closed set if we have one more dimension to play with.\<close>

lemma locally_compact_open_Int_closure:
  fixes S :: "'a :: metric_space set"
  assumes "locally compact S"
  obtains T where "open T" "S = T \ closure S"
proof -
  have "\x\S. \T v u. u = S \ T \ x \ u \ u \ v \ v \ S \ open T \ compact v"
    by (metis assms locally_compact openin_open)
  then obtain t v where
        tv: "\x. x \ S
             \<Longrightarrow> v x \<subseteq> S \<and> open (t x) \<and> compact (v x) \<and> (\<exists>u. x \<in> u \<and> u \<subseteq> v x \<and> u = S \<inter> t x)"
    by metis
  then have o: "open (\(t ` S))"
    by blast
  have "S = \ (v ` S)"
    using tv by blast
  also have "... = \(t ` S) \ closure S"
  proof
    show "\(v ` S) \ \(t ` S) \ closure S"
      by clarify (meson IntD2 IntI UN_I closure_subset subsetD tv)
    have "t x \ closure S \ v x" if "x \ S" for x
    proof -
      have "t x \ closure S \ closure (t x \ S)"
        by (simp add: open_Int_closure_subset that tv)
      also have "... \ v x"
        by (metis Int_commute closure_minimal compact_imp_closed that tv)
      finally show ?thesis .
    qed
    then show "\(t ` S) \ closure S \ \(v ` S)"
      by blast
  qed
  finally have e: "S = \(t ` S) \ closure S" .
  show ?thesis
    by (rule that [OF o e])
qed

lemma locally_compact_closedin_open:
    fixes S :: "'a :: metric_space set"
    assumes "locally compact S"
    obtains T where "open T" "closedin (top_of_set T) S"
  by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int)

lemma locally_compact_homeomorphism_projection_closed:
  assumes "locally compact S"
  obtains T and f :: "'a \ 'a :: euclidean_space \ 'b :: euclidean_space"
  where "closed T" "homeomorphism S T f fst"
proof (cases "closed S")
  case True
  show ?thesis
  proof
    show "homeomorphism S (S \ {0}) (\x. (x, 0)) fst"
      by (auto simp: homeomorphism_def continuous_intros)
  qed (use True closed_Times in auto)
next
  case False
    obtain U where "open U" and US: "U \ closure S = S"
      by (metis locally_compact_open_Int_closure [OF assms])
    with False have Ucomp: "-U \ {}"
      using closure_eq by auto
    have [simp]: "closure (- U) = -U"
      by (simp add: \<open>open U\<close> closed_Compl)
    define f :: "'a \ 'a \ 'b" where "f \ \x. (x, One /\<^sub>R setdist {x} (- U))"
    have "continuous_on U (\x. (x, One /\<^sub>R setdist {x} (- U)))"
    proof (intro continuous_intros continuous_on_setdist)
      show "\x\U. setdist {x} (- U) \ 0"
        by (simp add: Ucomp setdist_eq_0_sing_1)
    qed
    then have homU: "homeomorphism U (f`U) f fst"
      by (auto simp: f_def homeomorphism_def image_iff continuous_intros)
    have cloS: "closedin (top_of_set U) S"
      by (metis US closed_closure closedin_closed_Int)
    have cont: "isCont ((\x. setdist {x} (- U)) o fst) z" for z :: "'a \ 'b"
      by (rule continuous_at_compose continuous_intros continuous_at_setdist)+
    have setdist1D: "setdist {a} (- U) *\<^sub>R b = One \ setdist {a} (- U) \ 0" for a::'a and b::'b
      by force
    have *: "r *\<^sub>R b = One \ b = (1 / r) *\<^sub>R One" for r and b::'b
      by (metis One_non_0 nonzero_divide_eq_eq real_vector.scale_eq_0_iff real_vector.scale_scale scaleR_one)
    have "\a b::'b. setdist {a} (- U) *\<^sub>R b = One \ (a,b) \ (\x. (x, (1 / setdist {x} (- U)) *\<^sub>R One)) ` U"
      by (metis (mono_tags, lifting) "*" ComplI image_eqI setdist1D setdist_sing_in_set)
    then have "f ` U = (\z. (setdist {fst z} (- U) *\<^sub>R snd z)) -` {One}"
      by (auto simp: f_def setdist_eq_0_sing_1 field_simps Ucomp)
    then have clfU: "closed (f ` U)"
      by (force intro: continuous_intros cont [unfolded o_def] continuous_closed_vimage)
    have "closed (f ` S)"
      by (metis closedin_closed_trans [OF _ clfU] homeomorphism_imp_closed_map [OF homU cloS])
    then show ?thesis
      by (metis US homU homeomorphism_of_subsets inf_sup_ord(1) that)
qed

lemma locally_compact_closed_Int_open:
  fixes S :: "'a :: euclidean_space set"
  shows "locally compact S \ (\U V. closed U \ open V \ S = U \ V)" (is "?lhs = ?rhs")
proof
  show "?lhs \ ?rhs"
    by (metis closed_closure inf_commute locally_compact_open_Int_closure)
  show "?rhs \ ?lhs"
    by (meson closed_imp_locally_compact locally_compact_Int open_imp_locally_compact)
qed

lemma lowerdim_embeddings:
  assumes  "DIM('a) < DIM('b)"
  obtains f :: "'a::euclidean_space*real \ 'b::euclidean_space"
      and g :: "'b \ 'a*real"
      and j :: 'b
  where "linear f" "linear g" "\z. g (f z) = z" "j \ Basis" "\x. f(x,0) \ j = 0"
proof -
  let ?B = "Basis :: ('a*real) set"
  have b01: "(0,1) \ ?B"
    by (simp add: Basis_prod_def)
  have "DIM('a * real) \ DIM('b)"
    by (simp add: Suc_leI assms)
  then obtain basf :: "'a*real \ 'b" where sbf: "basf ` ?B \ Basis" and injbf: "inj_on basf Basis"
    by (metis finite_Basis card_le_inj)
  define basg:: "'b \ 'a * real" where
    "basg \ \i. if i \ basf ` Basis then inv_into Basis basf i else (0,1)"
  have bgf[simp]: "basg (basf i) = i" if "i \ Basis" for i
    using inv_into_f_f injbf that by (force simp: basg_def)
  have sbg: "basg ` Basis \ ?B"
    by (force simp: basg_def injbf b01)
  define f :: "'a*real \ 'b" where "f \ \u. \j\Basis. (u \ basg j) *\<^sub>R j"
  define g :: "'b \ 'a*real" where "g \ \z. (\i\Basis. (z \ basf i) *\<^sub>R i)"
  show ?thesis
  proof
    show "linear f"
      unfolding f_def
      by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
    show "linear g"
      unfolding g_def
      by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
    have *: "(\a \ Basis. a \ basf b * (x \ basg a)) = x \ b" if "b \ Basis" for x b
      using sbf that by auto
    show gf: "g (f x) = x" for x
    proof (rule euclidean_eqI)
      show "\b. b \ Basis \ g (f x) \ b = x \ b"
        using f_def g_def sbf by auto
    qed
    show "basf(0,1) \ Basis"
      using b01 sbf by auto
    then show "f(x,0) \ basf(0,1) = 0" for x
      unfolding f_def inner_sum_left
      using b01 inner_not_same_Basis
      by (fastforce intro: comm_monoid_add_class.sum.neutral)
  qed
qed

proposition locally_compact_homeomorphic_closed:
  fixes S :: "'a::euclidean_space set"
  assumes "locally compact S" and dimlt: "DIM('a) < DIM('b)"
  obtains T :: "'b::euclidean_space set" where "closed T" "S homeomorphic T"
proof -
  obtain U:: "('a*real)set" and h
    where "closed U" and homU: "homeomorphism S U h fst"
    using locally_compact_homeomorphism_projection_closed assms by metis
  obtain f :: "'a*real \ 'b" and g :: "'b \ 'a*real"
    where "linear f" "linear g" and gf [simp]: "\z. g (f z) = z"
    using lowerdim_embeddings [OF dimlt] by metis
  then have "inj f"
    by (metis injI)
  have gfU: "g ` f ` U = U"
    by (simp add: image_comp o_def)
  have "S homeomorphic U"
    using homU homeomorphic_def by blast
  also have "... homeomorphic f ` U"
  proof (rule homeomorphicI [OF refl gfU])
    show "continuous_on U f"
      by (meson \<open>inj f\<close> \<open>linear f\<close> homeomorphism_cont2 linear_homeomorphism_image)
    show "continuous_on (f ` U) g"
      using \<open>linear g\<close> linear_continuous_on linear_conv_bounded_linear by blast
  qed (auto simp: o_def)
  finally show ?thesis
    using \<open>closed U\<close> \<open>inj f\<close> \<open>linear f\<close> closed_injective_linear_image that by blast
qed

lemma homeomorphic_convex_compact_lemma:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S"
    and "compact S"
    and "cball 0 1 \ S"
  shows "S homeomorphic (cball (0::'a) 1)"
proof (rule starlike_compact_projective_special[OF assms(2-3)])
  fix x u
  assume "x \ S" and "0 \ u" and "u < (1::real)"
  have "open (ball (u *\<^sub>R x) (1 - u))"
    by (rule open_ball)
  moreover have "u *\<^sub>R x \ ball (u *\<^sub>R x) (1 - u)"
    unfolding centre_in_ball using \<open>u < 1\<close> by simp
  moreover have "ball (u *\<^sub>R x) (1 - u) \ S"
  proof
    fix y
    assume "y \ ball (u *\<^sub>R x) (1 - u)"
    then have "dist (u *\<^sub>R x) y < 1 - u"
      unfolding mem_ball .
    with \<open>u < 1\<close> have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
      by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
    with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \ S" ..
    with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \ S"
      using \<open>x \<in> S\<close> \<open>0 \<le> u\<close> \<open>u < 1\<close> [THEN less_imp_le] by (rule convexD_alt)
    then show "y \ S" using \u < 1\
      by simp
  qed
  ultimately have "u *\<^sub>R x \ interior S" ..
  then show "u *\<^sub>R x \ S - frontier S"
    using frontier_def and interior_subset by auto
qed

proposition homeomorphic_convex_compact_cball:
  fixes e :: real
    and S :: "'a::euclidean_space set"
  assumes S: "convex S" "compact S" "interior S \ {}" and "e > 0"
  shows "S homeomorphic (cball (b::'a) e)"
proof (rule homeomorphic_trans[OF _ homeomorphic_balls(2)])
  obtain a where "a \ interior S"
    using assms by auto
  then show "S homeomorphic cball (0::'a) 1"
    by (metis (no_types) aff_dim_cball S compact_cball convex_cball
        homeomorphic_convex_lemma interior_rel_interior_gen zero_less_one)
qed (use \<open>e>0\<close> in auto)

corollary homeomorphic_convex_compact:
  fixes S :: "'a::euclidean_space set"
    and T :: "'a set"
  assumes "convex S" "compact S" "interior S \ {}"
    and "convex T" "compact T" "interior T \ {}"
  shows "S homeomorphic T"
  using assms
  by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)

lemma homeomorphic_closed_intervals:
  fixes a :: "'a::euclidean_space" and b and c :: "'a::euclidean_space" and d
  assumes "box a b \ {}" and "box c d \ {}"
    shows "(cbox a b) homeomorphic (cbox c d)"
  by (simp add: assms homeomorphic_convex_compact)

lemma homeomorphic_closed_intervals_real:
  fixes a::real and b and c::real and d
  assumes "a and "c
  shows "{a..b} homeomorphic {c..d}"
  using assms by (auto intro: homeomorphic_convex_compact)

subsection\<open>Covering spaces and lifting results for them\<close>

definition\<^marker>\<open>tag important\<close> covering_space
           :: "'a::topological_space set \ ('a \ 'b) \ 'b::topological_space set \ bool"
  where
  "covering_space c p S \
       continuous_on c p \<and> p ` c = S \<and>
       (\<forall>x \<in> S. \<exists>T. x \<in> T \<and> openin (top_of_set S) T \<and>
                    (\<exists>v. \<Union>v = c \<inter> p -` T \<and>
                        (\<forall>u \<in> v. openin (top_of_set c) u) \<and>
                        pairwise disjnt v \<and>
                        (\<forall>u \<in> v. \<exists>q. homeomorphism u T p q)))"

lemma covering_spaceI [intro?]:
  assumes "continuous_on c p" "p ` c = S"
          "\x. x \ S \ \T. x \ T \ openin (top_of_set S) T \
                             (\<exists>v. \<Union>v = c \<inter> p -` T \<and> (\<forall>u \<in> v. openin (top_of_set c) u) \<and>
                             pairwise disjnt v \<and> (\<forall>u \<in> v. \<exists>q. homeomorphism u T p q))"
  shows "covering_space c p S"
  using assms unfolding covering_space_def by auto

lemma covering_space_imp_continuous: "covering_space c p S \ continuous_on c p"
  by (simp add: covering_space_def)

lemma covering_space_imp_surjective: "covering_space c p S \ p ` c = S"
  by (simp add: covering_space_def)

lemma homeomorphism_imp_covering_space: "homeomorphism S T f g \ covering_space S f T"
  apply (clarsimp simp add: homeomorphism_def covering_space_def)
  apply (rule_tac x=T in exI, simp)
  apply (rule_tac x="{S}" in exI, auto)
  done

lemma covering_space_local_homeomorphism:
  assumes "covering_space c p S" "x \ c"
  obtains T u q where "x \ T" "openin (top_of_set c) T"
                      "p x \ u" "openin (top_of_set S) u"
                      "homeomorphism T u p q"
  using assms
  by (clarsimp simp add: covering_space_def) (metis IntI UnionE vimage_eq)

lemma covering_space_local_homeomorphism_alt:
  assumes p: "covering_space c p S" and "y \ S"
  obtains x T U q where "p x = y"
                        "x \ T" "openin (top_of_set c) T"
                        "y \ U" "openin (top_of_set S) U"
                          "homeomorphism T U p q"
proof -
  obtain x where "p x = y" "x \ c"
    using assms covering_space_imp_surjective by blast
  show ?thesis
    using that \<open>p x = y\<close> by (auto intro: covering_space_local_homeomorphism [OF p \<open>x \<in> c\<close>])
qed

proposition covering_space_open_map:
  fixes S :: "'a :: metric_space set" and T :: "'b :: metric_space set"
  assumes p: "covering_space c p S" and T: "openin (top_of_set c) T"
    shows "openin (top_of_set S) (p ` T)"
proof -
  have pce: "p ` c = S"
   and covs:
       "\x. x \ S \
            \<exists>X VS. x \<in> X \<and> openin (top_of_set S) X \<and>
                  \<Union>VS = c \<inter> p -` X \<and>
                  (\<forall>u \<in> VS. openin (top_of_set c) u) \<and>
                  pairwise disjnt VS \<and>
                  (\<forall>u \<in> VS. \<exists>q. homeomorphism u X p q)"
    using p by (auto simp: covering_space_def)
  have "T \ c" by (metis openin_euclidean_subtopology_iff T)
  have "\X. openin (top_of_set S) X \ y \ X \ X \ p ` T"
          if "y \ p ` T" for y
  proof -
    have "y \ S" using \T \ c\ pce that by blast
    obtain U VS where "y \ U" and U: "openin (top_of_set S) U"
                  and VS: "\VS = c \ p -` U"
                  and openVS: "\V \ VS. openin (top_of_set c) V"
                  and homVS: "\V. V \ VS \ \q. homeomorphism V U p q"
      using covs [OF \<open>y \<in> S\<close>] by auto
    obtain x where "x \ c" "p x \ U" "x \ T" "p x = y"
      using T [unfolded openin_euclidean_subtopology_iff] \<open>y \<in> U\<close> \<open>y \<in> p ` T\<close> by blast
    with VS obtain V where "x \ V" "V \ VS" by auto
    then obtain q where q: "homeomorphism V U p q" using homVS by blast
    then have ptV: "p ` (T \ V) = U \ q -` (T \ V)"
      using VS \<open>V \<in> VS\<close> by (auto simp: homeomorphism_def)
    have ocv: "openin (top_of_set c) V"
      by (simp add: \<open>V \<in> VS\<close> openVS)
    have "openin (top_of_set (q ` U)) (T \ V)"
      using q unfolding homeomorphism_def
      by (metis T inf.absorb_iff2 ocv openin_imp_subset openin_subtopology_Int subtopology_subtopology)
    then have "openin (top_of_set U) (U \ q -` (T \ V))"
      using continuous_on_open homeomorphism_def q by blast
    then have os: "openin (top_of_set S) (U \ q -` (T \ V))"
      using openin_trans [of U] by (simp add: Collect_conj_eq U)
    show ?thesis
    proof (intro exI conjI)
      show "openin (top_of_set S) (p ` (T \ V))"
        by (simp only: ptV os)
    qed (use \<open>p x = y\<close> \<open>x \<in> V\<close> \<open>x \<in> T\<close> in auto)
  qed
  with openin_subopen show ?thesis by blast
qed

lemma covering_space_lift_unique_gen:
  fixes f :: "'a::topological_space \ 'b::topological_space"
  fixes g1 :: "'a \ 'c::real_normed_vector"
  assumes cov: "covering_space c p S"
      and eq: "g1 a = g2 a"
      and f: "continuous_on T f"  "f \ T \ S"
      and g1: "continuous_on T g1"  "g1 \ T \ c"
      and fg1: "\x. x \ T \ f x = p(g1 x)"
      and g2: "continuous_on T g2"  "g2 \ T \ c"
      and fg2: "\x. x \ T \ f x = p(g2 x)"
      and u_compt: "U \ components T" and "a \ U" "x \ U"
    shows "g1 x = g2 x"
proof -
  have "U \ T" by (rule in_components_subset [OF u_compt])
  define G12 where "G12 \ {x \ U. g1 x - g2 x = 0}"
  have "connected U" by (rule in_components_connected [OF u_compt])
  have contu: "continuous_on U g1" "continuous_on U g2"
       using \<open>U \<subseteq> T\<close> continuous_on_subset g1 g2 by blast+
  have o12: "openin (top_of_set U) G12"
  unfolding G12_def
  proof (subst openin_subopen, clarify)
    fix z
    assume z: "z \ U" "g1 z - g2 z = 0"
    obtain v w q where "g1 z \ v" and ocv: "openin (top_of_set c) v"
      and "p (g1 z) \ w" and osw: "openin (top_of_set S) w"
      and hom: "homeomorphism v w p q"
    proof (rule covering_space_local_homeomorphism [OF cov])
      show "g1 z \ c"
        using \<open>U \<subseteq> T\<close> \<open>z \<in> U\<close> g1(2) by blast
    qed auto
    have "g2 z \ v" using \g1 z \ v\ z by auto
    have gg: "U \ g -` v = U \ g -` (v \ g ` U)" for g
      by auto
    have "openin (top_of_set (g1 ` U)) (v \ g1 ` U)"
      using ocv \<open>U \<subseteq> T\<close> g1 by (fastforce simp add: openin_open)
    then have 1: "openin (top_of_set U) (U \ g1 -` v)"
      unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
    have "openin (top_of_set (g2 ` U)) (v \ g2 ` U)"
      using ocv \<open>U \<subseteq> T\<close> g2 by (fastforce simp add: openin_open)
    then have 2: "openin (top_of_set U) (U \ g2 -` v)"
      unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
    let ?T = "(U \ g1 -` v) \ (U \ g2 -` v)"
    show "\T. openin (top_of_set U) T \ z \ T \ T \ {z \ U. g1 z - g2 z = 0}"
    proof (intro exI conjI)
      show "openin (top_of_set U) ?T"
        using "1" "2" by blast
      show "z \ ?T"
        using z by (simp add: \<open>g1 z \<in> v\<close> \<open>g2 z \<in> v\<close>)
      show "?T \ {z \ U. g1 z - g2 z = 0}"
        using hom
        by (clarsimp simp: homeomorphism_def) (metis \<open>U \<subseteq> T\<close> fg1 fg2 subsetD)
    qed
  qed
  have c12: "closedin (top_of_set U) G12"
    unfolding G12_def
    by (intro continuous_intros continuous_closedin_preimage_constant contu)
  have "G12 = {} \ G12 = U"
    by (intro connected_clopen [THEN iffD1, rule_format] \<open>connected U\<close> conjI o12 c12)
  with eq \<open>a \<in> U\<close> have "\<And>x. x \<in> U \<Longrightarrow> g1 x - g2 x = 0" by (auto simp: G12_def)
  then show ?thesis
    using \<open>x \<in> U\<close> by force
qed

proposition covering_space_lift_unique:
  fixes f :: "'a::topological_space \ 'b::topological_space"
  fixes g1 :: "'a \ 'c::real_normed_vector"
  assumes "covering_space c p S"
          "g1 a = g2 a"
          "continuous_on T f"  "f \ T \ S"
          "continuous_on T g1"  "g1 \ T \ c" "\x. x \ T \ f x = p(g1 x)"
          "continuous_on T g2"  "g2 \ T \ c" "\x. x \ T \ f x = p(g2 x)"
          "connected T"  "a \ T" "x \ T"
   shows "g1 x = g2 x"
  using covering_space_lift_unique_gen [of c p S] in_components_self assms ex_in_conv
  by blast

lemma covering_space_locally:
  fixes p :: "'a::real_normed_vector \ 'b::real_normed_vector"
  assumes loc: "locally \ C" and cov: "covering_space C p S"
      and pim: "\T. \T \ C; \ T\ \ \(p ` T)"
    shows "locally \ S"
proof -
  have "locally \ (p ` C)"
  proof (rule locally_open_map_image [OF loc])
    show "continuous_on C p"
      using cov covering_space_imp_continuous by blast
    show "\T. openin (top_of_set C) T \ openin (top_of_set (p ` C)) (p ` T)"
      using cov covering_space_imp_surjective covering_space_open_map by blast
  qed (simp add: pim)
  then show ?thesis
    using covering_space_imp_surjective [OF cov] by metis
qed

proposition covering_space_locally_eq:
  fixes p :: "'a::real_normed_vector \ 'b::real_normed_vector"
  assumes cov: "covering_space C p S"
      and pim: "\T. \T \ C; \ T\ \ \(p ` T)"
      and qim: "\q U. \U \ S; continuous_on U q; \ U\ \ \(q ` U)"
    shows "locally \ S \ locally \ C"
         (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  show ?rhs
  proof (rule locallyI)
    fix V x
    assume V: "openin (top_of_set C) V" and "x \ V"
    have "p x \ p ` C"
      by (metis IntE V \<open>x \<in> V\<close> imageI openin_open)
    then obtain T \<V> where "p x \<in> T"
                      and opeT: "openin (top_of_set S) T"
                      and veq: "\\ = C \ p -` T"
                      and ope: "\U\\. openin (top_of_set C) U"
                      and hom: "\U\\. \q. homeomorphism U T p q"
      using cov unfolding covering_space_def by (blast intro: that)
    have "x \ \\"
      using V veq \<open>p x \<in> T\<close> \<open>x \<in> V\<close> openin_imp_subset by fastforce
    then obtain U where "x \ U" "U \ \"
      by blast
    then obtain q where opeU: "openin (top_of_set C) U" and q: "homeomorphism U T p q"
      using ope hom by blast
    with V have "openin (top_of_set C) (U \ V)"
      by blast
    then have UV: "openin (top_of_set S) (p ` (U \ V))"
      using cov covering_space_open_map by blast
    obtain W W' where opeW: "openin (top_of_set S) W" and "\ W'" "p x \ W" "W \ W'" and W'sub: "W' \ p ` (U \ V)"
      using locallyE [OF L UV] \<open>x \<in> U\<close> \<open>x \<in> V\<close> by blast
    then have "W \ T"
      by (metis Int_lower1 q homeomorphism_image1 image_Int_subset order_trans)
    show "\U Z. openin (top_of_set C) U \
                 \<phi> Z \<and> x \<in> U \<and> U \<subseteq> Z \<and> Z \<subseteq> V"
    proof (intro exI conjI)
      have "openin (top_of_set T) W"
        by (meson opeW opeT openin_imp_subset openin_subset_trans \<open>W \<subseteq> T\<close>)
      then have "openin (top_of_set U) (q ` W)"
        by (meson homeomorphism_imp_open_map homeomorphism_symD q)
      then show "openin (top_of_set C) (q ` W)"
        using opeU openin_trans by blast
      show "\ (q ` W')"
        by (metis (mono_tags, lifting) Int_subset_iff UV W'sub \\ W'\ continuous_on_subset dual_order.trans homeomorphism_def image_Int_subset openin_imp_subset q qim)
      show "x \ q ` W"
        by (metis \<open>p x \<in> W\<close> \<open>x \<in> U\<close> homeomorphism_def imageI q)
      show "q ` W \ q ` W'"
        using \<open>W \<subseteq> W'\<close> by blast
      have "W' \ p ` V"
        using W'sub by blast
      then show "q ` W' \ V"
        using W'sub homeomorphism_apply1 [OF q] by auto
      qed
  qed
next
  assume ?rhs
  then show ?lhs
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