author paulson Mon, 03 Oct 2016 13:01:01 +0100 changeset 64006 0de4736dad8b parent 63999 5649a993666d child 64007 041cda506fb2
new theorems including the theory FurtherTopology
 src/HOL/Analysis/Analysis.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Brouwer_Fixpoint.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Convex_Euclidean_Space.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/FurtherTopology.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Path_Connected.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Analysis/Analysis.thy	Sun Oct 02 21:05:14 2016 +0200
+++ b/src/HOL/Analysis/Analysis.thy	Mon Oct 03 13:01:01 2016 +0100
@@ -10,6 +10,7 @@
Bounded_Continuous_Function
Weierstrass_Theorems
Polytope
+  FurtherTopology
Poly_Roots
Conformal_Mappings
Generalised_Binomial_Theorem```
```--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Sun Oct 02 21:05:14 2016 +0200
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Oct 03 13:01:01 2016 +0100
@@ -1975,7 +1975,7 @@

text \<open>So we get the no-retraction theorem.\<close>

-lemma no_retraction_cball:
+theorem no_retraction_cball:
fixes a :: "'a::euclidean_space"
assumes "e > 0"
shows "\<not> (frontier (cball a e) retract_of (cball a e))"
@@ -2001,6 +2001,26 @@
using x assms by auto
qed

+corollary contractible_sphere:
+  fixes a :: "'a::euclidean_space"
+  shows "contractible(sphere a r) \<longleftrightarrow> r \<le> 0"
+proof (cases "0 < r")
+  case True
+  then show ?thesis
+    unfolding contractible_def nullhomotopic_from_sphere_extension
+    using no_retraction_cball [OF True, of a]
+    by (auto simp: retract_of_def retraction_def)
+next
+  case False
+  then show ?thesis
+    unfolding contractible_def nullhomotopic_from_sphere_extension
+    apply (rule_tac x=id in exI)
+    apply (auto simp: continuous_on_def)
+    apply (meson dist_not_less_zero le_less less_le_trans)
+    done
+qed
+
subsection\<open>Retractions\<close>

lemma retraction:```
```--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Sun Oct 02 21:05:14 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Oct 03 13:01:01 2016 +0100
@@ -7495,6 +7495,11 @@
by (auto simp: closed_segment_commute)
qed

+lemma open_segment_eq_real_ivl:
+  fixes a b::real
+  shows "open_segment a b = (if a \<le> b then {a<..<b} else {b<..<a})"
+by (auto simp: closed_segment_eq_real_ivl open_segment_def split: if_split_asm)
+
lemma closed_segment_real_eq:
fixes u::real shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
@@ -11353,6 +11358,81 @@
by (metis connected_segment convex_contains_segment ends_in_segment imageI
is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])

+lemma continuous_injective_image_segment_1:
+  fixes f :: "'a::euclidean_space \<Rightarrow> real"
+  assumes contf: "continuous_on (closed_segment a b) f"
+      and injf: "inj_on f (closed_segment a b)"
+  shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
+proof
+  show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
+    by (metis subset_continuous_image_segment_1 contf)
+  show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
+  proof (cases "a = b")
+    case True
+    then show ?thesis by auto
+  next
+    case False
+    then have fnot: "f a \<noteq> f b"
+      using inj_onD injf by fastforce
+    moreover
+    have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
+    proof (clarsimp simp add: open_segment_def)
+      assume fa: "f a \<in> closed_segment (f c) (f b)"
+      moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
+        by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
+      ultimately have "f a \<in> f ` closed_segment c b"
+        by blast
+      then have a: "a \<in> closed_segment c b"
+        by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+      have cb: "closed_segment c b \<subseteq> closed_segment a b"
+        by (simp add: closed_segment_subset that)
+      show "f a = f c"
+      proof (rule between_antisym)
+        show "between (f c, f b) (f a)"
+          by (simp add: between_mem_segment fa)
+        show "between (f a, f b) (f c)"
+          by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
+      qed
+    qed
+    moreover
+    have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
+    proof (clarsimp simp add: open_segment_def fnot eq_commute)
+      assume fb: "f b \<in> closed_segment (f a) (f c)"
+      moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
+        by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
+      ultimately have "f b \<in> f ` closed_segment a c"
+        by blast
+      then have b: "b \<in> closed_segment a c"
+        by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+      have ca: "closed_segment a c \<subseteq> closed_segment a b"
+        by (simp add: closed_segment_subset that)
+      show "f b = f c"
+      proof (rule between_antisym)
+        show "between (f c, f a) (f b)"
+          by (simp add: between_commute between_mem_segment fb)
+        show "between (f b, f a) (f c)"
+          by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
+      qed
+    qed
+    ultimately show ?thesis
+      by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
+  qed
+qed
+
+lemma continuous_injective_image_open_segment_1:
+  fixes f :: "'a::euclidean_space \<Rightarrow> real"
+  assumes contf: "continuous_on (closed_segment a b) f"
+      and injf: "inj_on f (closed_segment a b)"
+    shows "f ` (open_segment a b) = open_segment (f a) (f b)"
+proof -
+  have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
+    by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
+  also have "... = open_segment (f a) (f b)"
+    using continuous_injective_image_segment_1 [OF assms]
+    by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
+  finally show ?thesis .
+qed
+
lemma collinear_imp_coplanar:
"collinear s ==> coplanar s"
by (metis collinear_affine_hull coplanar_def insert_absorb2)```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/FurtherTopology.thy	Mon Oct 03 13:01:01 2016 +0100
@@ -0,0 +1,1891 @@
+section \<open>Extending Continous Maps, etc..\<close>
+
+text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
+
+theory "FurtherTopology"
+  imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope
+
+begin
+
+subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
+
+lemma spheremap_lemma1:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
+      and "S \<subseteq> T"
+      and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
+    shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
+proof
+  assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
+  have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
+    using subspace_mul \<open>subspace S\<close> by blast
+  have subS01: "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0}) \<subseteq> sphere 0 1 \<inter> S"
+    using \<open>subspace S\<close> subspace_mul by fastforce
+  then have diff_f': "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0})"
+    by (rule differentiable_on_subset [OF diff_f])
+  define g where "g \<equiv> \<lambda>x. norm x *\<^sub>R f(inverse(norm x) *\<^sub>R x)"
+  have gdiff: "g differentiable_on S - {0}"
+    unfolding g_def
+    by (rule diff_f' derivative_intros differentiable_on_compose [where f=f] | force)+
+  have geq: "g ` (S - {0}) = T - {0}"
+  proof
+    have "g ` (S - {0}) \<subseteq> T"
+      apply (auto simp: g_def subspace_mul [OF \<open>subspace T\<close>])
+      apply (metis (mono_tags, lifting) DiffI subS01 subspace_mul [OF \<open>subspace T\<close>] fim image_subset_iff inf_le2 singletonD)
+      done
+    moreover have "g ` (S - {0}) \<subseteq> UNIV - {0}"
+    proof (clarsimp simp: g_def)
+      fix y
+      assume "y \<in> S" and f0: "f (y /\<^sub>R norm y) = 0"
+      then have "y \<noteq> 0 \<Longrightarrow> y /\<^sub>R norm y \<in> sphere 0 1 \<inter> S"
+        by (auto simp: subspace_mul [OF \<open>subspace S\<close>])
+      then show "y = 0"
+        by (metis fim f0 Int_iff image_iff mem_sphere_0 norm_eq_zero zero_neq_one)
+    qed
+    ultimately show "g ` (S - {0}) \<subseteq> T - {0}"
+      by auto
+  next
+    have *: "sphere 0 1 \<inter> T \<subseteq> f ` (sphere 0 1 \<inter> S)"
+      using fim by (simp add: image_subset_iff)
+    have "x \<in> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+          if "x \<in> T" "x \<noteq> 0" for x
+    proof -
+      have "x /\<^sub>R norm x \<in> T"
+        using \<open>subspace T\<close> subspace_mul that by blast
+      then show ?thesis
+        using * [THEN subsetD, of "x /\<^sub>R norm x"] that apply clarsimp
+        apply (rule_tac x="norm x *\<^sub>R xa" in image_eqI, simp)
+        apply (metis norm_eq_zero right_inverse scaleR_one scaleR_scaleR)
+        using \<open>subspace S\<close> subspace_mul apply force
+        done
+    qed
+    then have "T - {0} \<subseteq> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+      by force
+    then show "T - {0} \<subseteq> g ` (S - {0})"
+  qed
+  define T' where "T' \<equiv> {y. \<forall>x \<in> T. orthogonal x y}"
+  have "subspace T'"
+    by (simp add: subspace_orthogonal_to_vectors T'_def)
+  have dim_eq: "dim T' + dim T = DIM('a)"
+    using dim_subspace_orthogonal_to_vectors [of T UNIV] \<open>subspace T\<close>
+    by (simp add: dim_UNIV T'_def)
+  have "\<exists>v1 v2. v1 \<in> span T \<and> (\<forall>w \<in> span T. orthogonal v2 w) \<and> x = v1 + v2" for x
+    by (force intro: orthogonal_subspace_decomp_exists [of T x])
+  then obtain p1 p2 where p1span: "p1 x \<in> span T"
+                      and "\<And>w. w \<in> span T \<Longrightarrow> orthogonal (p2 x) w"
+                      and eq: "p1 x + p2 x = x" for x
+    by metis
+  then have p1: "\<And>z. p1 z \<in> T" and ortho: "\<And>w. w \<in> T \<Longrightarrow> orthogonal (p2 x) w" for x
+    using span_eq \<open>subspace T\<close> by blast+
+  then have p2: "\<And>z. p2 z \<in> T'"
+    by (simp add: T'_def orthogonal_commute)
+  have p12_eq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p1(x + y) = x \<and> p2(x + y) = y"
+  proof (rule orthogonal_subspace_decomp_unique [OF eq p1span, where T=T'])
+    show "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p2 (x + y) \<in> span T'"
+      using span_eq p2 \<open>subspace T'\<close> by blast
+    show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+      using T'_def by blast
+  qed (auto simp: span_superset)
+  then have "\<And>c x. p1 (c *\<^sub>R x) = c *\<^sub>R p1 x \<and> p2 (c *\<^sub>R x) = c *\<^sub>R p2 x"
+    by (metis eq \<open>subspace T\<close> \<open>subspace T'\<close> p1 p2 scaleR_add_right subspace_mul)
+  moreover have lin_add: "\<And>x y. p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
+  proof (rule orthogonal_subspace_decomp_unique [OF _ p1span, where T=T'])
+    show "\<And>x y. p1 (x + y) + p2 (x + y) = p1 x + p1 y + (p2 x + p2 y)"
+    show  "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+      using T'_def by blast
+  qed (auto simp: p1span p2 span_superset subspace_add)
+  ultimately have "linear p1" "linear p2"
+    by unfold_locales auto
+  have "(\<lambda>z. g (p1 z)) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+    apply (rule differentiable_on_compose [where f=g])
+    apply (rule linear_imp_differentiable_on [OF \<open>linear p1\<close>])
+    apply (rule differentiable_on_subset [OF gdiff])
+    using p12_eq \<open>S \<subseteq> T\<close> apply auto
+    done
+  then have diff: "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+    by (intro derivative_intros linear_imp_differentiable_on [OF \<open>linear p2\<close>])
+  have "dim {x + y |x y. x \<in> S - {0} \<and> y \<in> T'} \<le> dim {x + y |x y. x \<in> S  \<and> y \<in> T'}"
+    by (blast intro: dim_subset)
+  also have "... = dim S + dim T' - dim (S \<inter> T')"
+    using dim_sums_Int [OF \<open>subspace S\<close> \<open>subspace T'\<close>]
+  also have "... < DIM('a)"
+    using dimST dim_eq by auto
+  finally have neg: "negligible {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+    by (rule negligible_lowdim)
+  have "negligible ((\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'})"
+    by (rule negligible_differentiable_image_negligible [OF order_refl neg diff])
+  then have "negligible {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+  proof (rule negligible_subset)
+    have "\<lbrakk>t' \<in> T'; s \<in> S; s \<noteq> 0\<rbrakk>
+          \<Longrightarrow> g s + t' \<in> (\<lambda>x. g (p1 x) + p2 x) `
+                         {x + t' |x t'. x \<in> S \<and> x \<noteq> 0 \<and> t' \<in> T'}" for t' s
+      apply (rule_tac x="s + t'" in image_eqI)
+      using \<open>S \<subseteq> T\<close> p12_eq by auto
+    then show "{x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}
+          \<subseteq> (\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+      by auto
+  qed
+  moreover have "- T' \<subseteq> {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+  proof clarsimp
+    fix z assume "z \<notin> T'"
+    show "\<exists>x y. z = x + y \<and> x \<in> g ` (S - {0}) \<and> y \<in> T'"
+      apply (rule_tac x="p1 z" in exI)
+      apply (rule_tac x="p2 z" in exI)
+      apply (simp add: p1 eq p2 geq)
+      by (metis \<open>z \<notin> T'\<close> add.left_neutral eq p2)
+  qed
+  ultimately have "negligible (-T')"
+    using negligible_subset by blast
+  moreover have "negligible T'"
+    using negligible_lowdim
+  ultimately have  "negligible (-T' \<union> T')"
+    by (metis negligible_Un_eq)
+  then show False
+    using negligible_Un_eq non_negligible_UNIV by simp
+qed
+
+
+lemma spheremap_lemma2:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes ST: "subspace S" "subspace T" "dim S < dim T"
+      and "S \<subseteq> T"
+      and contf: "continuous_on (sphere 0 1 \<inter> S) f"
+      and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
+    shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
+proof -
+  have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
+    using fim by (simp add: image_subset_iff)
+  have "compact (sphere 0 1 \<inter> S)"
+    by (simp add: \<open>subspace S\<close> closed_subspace compact_Int_closed)
+  then obtain g where pfg: "polynomial_function g" and gim: "g ` (sphere 0 1 \<inter> S) \<subseteq> T"
+                and g12: "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> norm(f x - g x) < 1/2"
+    apply (rule Stone_Weierstrass_polynomial_function_subspace [OF _ contf _ \<open>subspace T\<close>, of "1/2"])
+    using fim apply auto
+    done
+  have gnz: "g x \<noteq> 0" if "x \<in> sphere 0 1 \<inter> S" for x
+  proof -
+    have "norm (f x) = 1"
+      using fim that by (simp add: image_subset_iff)
+    then show ?thesis
+      using g12 [OF that] by auto
+  qed
+  have diffg: "g differentiable_on sphere 0 1 \<inter> S"
+    by (metis pfg differentiable_on_polynomial_function)
+  define h where "h \<equiv> \<lambda>x. inverse(norm(g x)) *\<^sub>R g x"
+  have h: "x \<in> sphere 0 1 \<inter> S \<Longrightarrow> h x \<in> sphere 0 1 \<inter> T" for x
+    unfolding h_def
+    using gnz [of x]
+    by (auto simp: subspace_mul [OF \<open>subspace T\<close>] subsetD [OF gim])
+  have diffh: "h differentiable_on sphere 0 1 \<inter> S"
+    unfolding h_def
+    apply (intro derivative_intros diffg differentiable_on_compose [OF diffg])
+    using gnz apply auto
+    done
+  have homfg: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) f g"
+  proof (rule homotopic_with_linear [OF contf])
+    show "continuous_on (sphere 0 1 \<inter> S) g"
+      using pfg by (simp add: differentiable_imp_continuous_on diffg)
+  next
+    have non0fg: "0 \<notin> closed_segment (f x) (g x)" if "norm x = 1" "x \<in> S" for x
+    proof -
+      have "f x \<in> sphere 0 1"
+        using fim that by (simp add: image_subset_iff)
+      moreover have "norm(f x - g x) < 1/2"
+        apply (rule g12)
+        using that by force
+      ultimately show ?thesis
+        by (auto simp: norm_minus_commute dest: segment_bound)
+    qed
+    show "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> T - {0}"
+      apply (simp add: subset_Diff_insert non0fg)
+      apply (rule hull_minimal)
+       using fim image_eqI gim apply force
+      apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+      done
+  qed
+  obtain d where d: "d \<in> (sphere 0 1 \<inter> T) - h ` (sphere 0 1 \<inter> S)"
+    using h spheremap_lemma1 [OF ST \<open>S \<subseteq> T\<close> diffh] by force
+  then have non0hd: "0 \<notin> closed_segment (h x) (- d)" if "norm x = 1" "x \<in> S" for x
+    using midpoint_between [of 0 "h x" "-d"] that h [of x]
+    by (auto simp: between_mem_segment midpoint_def)
+  have conth: "continuous_on (sphere 0 1 \<inter> S) h"
+    using differentiable_imp_continuous_on diffh by blast
+  have hom_hd: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) h (\<lambda>x. -d)"
+    apply (rule homotopic_with_linear [OF conth continuous_on_const])
+    apply (simp add: subset_Diff_insert non0hd)
+    apply (rule hull_minimal)
+     using h d apply (force simp: subspace_neg [OF \<open>subspace T\<close>])
+    apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+    done
+  have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
+    by (intro continuous_intros) auto
+  have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
+    by (fastforce simp: assms(2) subspace_mul)
+  obtain c where homhc: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
+    apply (rule_tac c="-d" in that)
+    apply (rule homotopic_with_eq)
+       apply (rule homotopic_compose_continuous_left [OF hom_hd conT0 sub0T])
+    using d apply (auto simp: h_def)
+    done
+  show ?thesis
+    apply (rule_tac x=c in exI)
+    apply (rule homotopic_with_trans [OF _ homhc])
+    apply (rule homotopic_with_eq)
+       apply (rule homotopic_compose_continuous_left [OF homfg conT0 sub0T])
+      apply (auto simp: h_def)
+    done
+qed
+
+
+lemma spheremap_lemma3:
+  assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
+  obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
+                  "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
+proof (cases "S = {}")
+  case True
+  with \<open>subspace U\<close> subspace_0 show ?thesis
+    by (rule_tac T = "{0}" in that) auto
+next
+  case False
+  then obtain a where "a \<in> S"
+    by auto
+  then have affS: "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
+    by (metis hull_inc aff_dim_eq_dim)
+  with affSU have "dim ((\<lambda>x. -a+x) ` S) \<le> dim U"
+    by linarith
+  with choose_subspace_of_subspace
+  obtain T where "subspace T" "T \<subseteq> span U" and dimT: "dim T = dim ((\<lambda>x. -a+x) ` S)" .
+  show ?thesis
+  proof (rule that [OF \<open>subspace T\<close>])
+    show "T \<subseteq> U"
+      using span_eq \<open>subspace U\<close> \<open>T \<subseteq> span U\<close> by blast
+    show "aff_dim T = aff_dim S"
+      using dimT \<open>subspace T\<close> affS aff_dim_subspace by fastforce
+    show "rel_frontier S homeomorphic sphere 0 1 \<inter> T"
+    proof -
+      have "aff_dim (ball 0 1 \<inter> T) = aff_dim (T)"
+        by (metis IntI interior_ball \<open>subspace T\<close> aff_dim_convex_Int_nonempty_interior centre_in_ball empty_iff inf_commute subspace_0 subspace_imp_convex zero_less_one)
+      then have affS_eq: "aff_dim S = aff_dim (ball 0 1 \<inter> T)"
+        using \<open>aff_dim T = aff_dim S\<close> by simp
+      have "rel_frontier S homeomorphic rel_frontier(ball 0 1 \<inter> T)"
+        apply (rule homeomorphic_rel_frontiers_convex_bounded_sets [OF \<open>convex S\<close> \<open>bounded S\<close>])
+          apply (simp add: \<open>subspace T\<close> convex_Int subspace_imp_convex)
+        apply (rule affS_eq)
+        done
+      also have "... = frontier (ball 0 1) \<inter> T"
+        apply (rule convex_affine_rel_frontier_Int [OF convex_ball])
+         apply (simp add: \<open>subspace T\<close> subspace_imp_affine)
+        using \<open>subspace T\<close> subspace_0 by force
+      also have "... = sphere 0 1 \<inter> T"
+        by auto
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+
+proposition inessential_spheremap_lowdim_gen:
+  fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes "convex S" "bounded S" "convex T" "bounded T"
+      and affST: "aff_dim S < aff_dim T"
+      and contf: "continuous_on (rel_frontier S) f"
+      and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
+  obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+proof (cases "S = {}")
+  case True
+  then show ?thesis
+next
+  case False
+  then show ?thesis
+  proof (cases "T = {}")
+    case True
+    then show ?thesis
+      using fim that by auto
+  next
+    case False
+    obtain T':: "'a set"
+      where "subspace T'" and affT': "aff_dim T' = aff_dim T"
+        and homT: "rel_frontier T homeomorphic sphere 0 1 \<inter> T'"
+      apply (rule spheremap_lemma3 [OF \<open>bounded T\<close> \<open>convex T\<close> subspace_UNIV, where 'b='a])
+       apply (simp add: dim_UNIV aff_dim_le_DIM)
+      using \<open>T \<noteq> {}\<close> by blast
+    with homeomorphic_imp_homotopy_eqv
+    have relT: "sphere 0 1 \<inter> T'  homotopy_eqv rel_frontier T"
+      using homotopy_eqv_sym by blast
+    have "aff_dim S \<le> int (dim T')"
+      using affT' \<open>subspace T'\<close> affST aff_dim_subspace by force
+    with spheremap_lemma3 [OF \<open>bounded S\<close> \<open>convex S\<close> \<open>subspace T'\<close>] \<open>S \<noteq> {}\<close>
+    obtain S':: "'a set" where "subspace S'" "S' \<subseteq> T'"
+       and affS': "aff_dim S' = aff_dim S"
+       and homT: "rel_frontier S homeomorphic sphere 0 1 \<inter> S'"
+        by metis
+    with homeomorphic_imp_homotopy_eqv
+    have relS: "sphere 0 1 \<inter> S'  homotopy_eqv rel_frontier S"
+      using homotopy_eqv_sym by blast
+    have dimST': "dim S' < dim T'"
+      by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> affS' affST affT' less_irrefl not_le subspace_dim_equal)
+    have "\<exists>c. homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+      apply (rule homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim])
+      apply (rule homotopy_eqv_cohomotopic_triviality_null[OF relS, THEN iffD1, rule_format])
+       apply (metis dimST' \<open>subspace S'\<close>  \<open>subspace T'\<close>  \<open>S' \<subseteq> T'\<close> spheremap_lemma2, blast)
+      done
+    with that show ?thesis by blast
+  qed
+qed
+
+lemma inessential_spheremap_lowdim:
+  fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  assumes
+   "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
+   obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
+proof (cases "s \<le> 0")
+  case True then show ?thesis
+    by (meson nullhomotopic_into_contractible f contractible_sphere that)
+next
+  case False
+  show ?thesis
+  proof (cases "r \<le> 0")
+    case True then show ?thesis
+      by (meson f nullhomotopic_from_contractible contractible_sphere that)
+  next
+    case False
+    with \<open>~ s \<le> 0\<close> have "r > 0" "s > 0" by auto
+    show ?thesis
+      apply (rule inessential_spheremap_lowdim_gen [of "cball a r" "cball b s" f])
+      using  \<open>0 < r\<close> \<open>0 < s\<close> assms(1)
+             apply (simp_all add: f aff_dim_cball)
+      using that by blast
+  qed
+qed
+
+
+
+subsection\<open> Some technical lemmas about extending maps from cell complexes.\<close>
+
+lemma extending_maps_Union_aux:
+  assumes fin: "finite \<F>"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+      and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+   shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+using assms
+proof (induction \<F>)
+  case empty show ?case by simp
+next
+  case (insert S \<F>)
+  then obtain f where contf: "continuous_on (S) f" and fim: "f ` S \<subseteq> T" and feq: "\<forall>x \<in> S \<inter> K. f x = h x"
+    by (meson insertI1)
+  obtain g where contg: "continuous_on (\<Union>\<F>) g" and gim: "g ` \<Union>\<F> \<subseteq> T" and geq: "\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x"
+    using insert by auto
+  have fg: "f x = g x" if "x \<in> T" "T \<in> \<F>" "x \<in> S" for x T
+  proof -
+    have "T \<inter> S \<subseteq> K \<or> S = T"
+      using that by (metis (no_types) insert.prems(2) insertCI)
+    then show ?thesis
+      using UnionI feq geq \<open>S \<notin> \<F>\<close> subsetD that by fastforce
+  qed
+  show ?case
+    apply (rule_tac x="\<lambda>x. if x \<in> S then f x else g x" in exI, simp)
+    apply (intro conjI continuous_on_cases)
+    apply (simp_all add: insert closed_Union contf contg)
+    using fim gim feq geq
+    apply (force simp: insert closed_Union contf contg inf_commute intro: fg)+
+    done
+qed
+
+lemma extending_maps_Union:
+  assumes fin: "finite \<F>"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+      and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~ X \<subseteq> Y; ~ Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+    shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+apply (simp add: Union_maximal_sets [OF fin, symmetric])
+apply (rule extending_maps_Union_aux)
+apply (simp_all add: Union_maximal_sets [OF fin] assms)
+by (metis K psubsetI)
+
+
+lemma extend_map_lemma:
+  assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
+      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+      and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
+      and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
+      and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+  obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof (cases "\<F> - \<G> = {}")
+  case True
+  then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
+  then show ?thesis
+    apply (rule_tac g=f in that)
+      using contf continuous_on_subset apply blast
+     using fim apply blast
+    by simp
+next
+  case False
+  then have "0 \<le> aff_dim T"
+    by (metis aff aff_dim_empty aff_dim_geq aff_dim_negative_iff all_not_in_conv not_less)
+  then obtain i::nat where i: "int i = aff_dim T"
+    by (metis nonneg_eq_int)
+  have Union_empty_eq: "\<Union>{D. D = {} \<and> P D} = {}" for P :: "'a set \<Rightarrow> bool"
+    by auto
+  have extendf: "\<exists>g. continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) g \<and>
+                     g ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) \<subseteq> rel_frontier T \<and>
+                     (\<forall>x \<in> \<Union>\<G>. g x = f x)"
+       if "i \<le> aff_dim T" for i::nat
+  using that
+  proof (induction i)
+    case 0 then show ?case
+      apply (rule_tac x=f in exI)
+      apply (intro conjI)
+      using contf continuous_on_subset apply blast
+      using fim apply blast
+      by simp
+  next
+    case (Suc p)
+    with \<open>bounded T\<close> have "rel_frontier T \<noteq> {}"
+      by (auto simp: rel_frontier_eq_empty affine_bounded_eq_lowdim [of T])
+    then obtain t where t: "t \<in> rel_frontier T" by auto
+    have ple: "int p \<le> aff_dim T" using Suc.prems by force
+    obtain h where conth: "continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})) h"
+               and him: "h ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}))
+                         \<subseteq> rel_frontier T"
+               and heq: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+      using Suc.IH [OF ple] by auto
+    let ?Faces = "{D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D \<le> p}"
+    have extendh: "\<exists>g. continuous_on D g \<and>
+                       g ` D \<subseteq> rel_frontier T \<and>
+                       (\<forall>x \<in> D \<inter> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+      if D: "D \<in> \<G> \<union> ?Faces" for D
+    proof (cases "D \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})")
+      case True
+      then show ?thesis
+        apply (rule_tac x=h in exI)
+        apply (intro conjI)
+        apply (blast intro: continuous_on_subset [OF conth])
+        using him apply blast
+        by simp
+    next
+      case False
+      note notDsub = False
+      show ?thesis
+      proof (cases "\<exists>a. D = {a}")
+        case True
+        then obtain a where "D = {a}" by auto
+        with notDsub t show ?thesis
+          by (rule_tac x="\<lambda>x. t" in exI) simp
+      next
+        case False
+        have "D \<noteq> {}" using notDsub by auto
+        have Dnotin: "D \<notin> \<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+          using notDsub by auto
+        then have "D \<notin> \<G>" by simp
+        have "D \<in> ?Faces - {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+          using Dnotin that by auto
+        then obtain C where "C \<in> \<F>" "D face_of C" and affD: "aff_dim D = int p"
+          by auto
+        then have "bounded D"
+          using face_of_polytope_polytope poly polytope_imp_bounded by blast
+        then have [simp]: "\<not> affine D"
+          using affine_bounded_eq_trivial False \<open>D \<noteq> {}\<close> \<open>bounded D\<close> by blast
+        have "{F. F facet_of D} \<subseteq> {E. E face_of C \<and> aff_dim E < int p}"
+          apply clarify
+          apply (metis \<open>D face_of C\<close> affD eq_iff face_of_trans facet_of_def zle_diff1_eq)
+          done
+        moreover have "polyhedron D"
+          using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
+        ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
+          by (simp add: rel_frontier_of_polyhedron Union_mono)
+        then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
+          using \<open>C \<in> \<F>\<close> him by blast
+        have "convex D"
+          by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
+        have affD_lessT: "aff_dim D < aff_dim T"
+          using Suc.prems affD by linarith
+        have contDh: "continuous_on (rel_frontier D) h"
+          using \<open>C \<in> \<F>\<close> relf_sub by (blast intro: continuous_on_subset [OF conth])
+        then have *: "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
+                      (\<exists>g. continuous_on UNIV g \<and>  range g \<subseteq> rel_frontier T \<and>
+                           (\<forall>x\<in>rel_frontier D. g x = h x))"
+          apply (rule nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
+          apply (simp_all add: assms rel_frontier_eq_empty him_relf)
+          done
+        have "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D)
+              (rel_frontier T) h (\<lambda>x. c))"
+          by (metis inessential_spheremap_lowdim_gen
+                 [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
+        then obtain g where contg: "continuous_on UNIV g"
+                        and gim: "range g \<subseteq> rel_frontier T"
+                        and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
+          by (metis *)
+        have "D \<inter> E \<subseteq> rel_frontier D"
+             if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
+        proof (rule face_of_subset_rel_frontier)
+          show "D \<inter> E face_of D"
+            using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
+            apply auto
+            apply (meson face_of_Int_subface \<open>\<G> \<subseteq> \<F>\<close> face_of_refl_eq poly polytope_imp_convex subsetD)
+            using face_of_Int_subface apply blast
+            done
+          show "D \<inter> E \<noteq> D"
+            using that notDsub by auto
+        qed
+        then show ?thesis
+          apply (rule_tac x=g in exI)
+          apply (intro conjI ballI)
+            using continuous_on_subset contg apply blast
+           using gim apply blast
+          using gh by fastforce
+      qed
+    qed
+    have intle: "i < 1 + int j \<longleftrightarrow> i \<le> int j" for i j
+      by auto
+    have "finite \<G>"
+      using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> rev_finite_subset by blast
+    then have fin: "finite (\<G> \<union> ?Faces)"
+      apply simp
+      apply (rule_tac B = "\<Union>{{D. D face_of C}| C. C \<in> \<F>}" in finite_subset)
+       by (auto simp: \<open>finite \<F>\<close> finite_polytope_faces poly)
+    have clo: "closed S" if "S \<in> \<G> \<union> ?Faces" for S
+      using that \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly polytope_imp_closed by blast
+    have K: "X \<inter> Y \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int p})"
+                if "X \<in> \<G> \<union> ?Faces" "Y \<in> \<G> \<union> ?Faces" "~ Y \<subseteq> X" for X Y
+    proof -
+      have ff: "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+        if XY: "X face_of D" "Y face_of E" and DE: "D \<in> \<F>" "E \<in> \<F>" for D E
+        apply (rule face_of_Int_subface [OF _ _ XY])
+        apply (auto simp: face DE)
+        done
+      show ?thesis
+        using that
+        apply auto
+        apply (drule_tac x="X \<inter> Y" in spec, safe)
+        using ff face_of_imp_convex [of X] face_of_imp_convex [of Y]
+        apply (fastforce dest: face_of_aff_dim_lt)
+        by (meson face_of_trans ff)
+    qed
+    obtain g where "continuous_on (\<Union>(\<G> \<union> ?Faces)) g"
+                   "g ` \<Union>(\<G> \<union> ?Faces) \<subseteq> rel_frontier T"
+                   "(\<forall>x \<in> \<Union>(\<G> \<union> ?Faces) \<inter>
+                          \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+      apply (rule exE [OF extending_maps_Union [OF fin extendh clo K]], blast+)
+      done
+    then show ?case
+      apply (simp add: intle local.heq [symmetric], blast)
+      done
+  qed
+  have eq: "\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i}) = \<Union>\<F>"
+  proof
+    show "\<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> \<Union>\<F>"
+      apply (rule Union_subsetI)
+      using \<open>\<G> \<subseteq> \<F>\<close> face_of_imp_subset  apply force
+      done
+    show "\<Union>\<F> \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < i})"
+      apply (rule Union_mono)
+      using face  apply (fastforce simp: aff i)
+      done
+  qed
+  have "int i \<le> aff_dim T" by (simp add: i)
+  then show ?thesis
+    using extendf [of i] unfolding eq by (metis that)
+qed
+
+lemma extend_map_lemma_cofinite0:
+  assumes "finite \<F>"
+      and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+    shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and>
+                 continuous_on (\<Union>\<F> - C) g \<and> g ` (\<Union>\<F> - C) \<subseteq> T
+                  \<and> (\<forall>x \<in> (\<Union>\<F> - C) \<inter> K. g x = h x)"
+  using assms
+proof induction
+  case empty then show ?case
+    by force
+next
+  case (insert X \<F>)
+  then have "closed X" and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+        and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+        and pwX: "\<And>Y. Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
+        and pwF: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) \<F>"
+  obtain C g where C: "finite C" "disjnt C U" "card C \<le> card \<F>"
+               and contg: "continuous_on (\<Union>\<F> - C) g"
+               and gim: "g ` (\<Union>\<F> - C) \<subseteq> T"
+               and gh:  "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+    using insert.IH [OF pwF \<F> clo] by auto
+  obtain a f where "a \<notin> U"
+               and contf: "continuous_on (X - {a}) f"
+               and fim: "f ` (X - {a}) \<subseteq> T"
+               and fh: "(\<forall>x \<in> X \<inter> K. f x = h x)"
+    using insert.prems by (meson insertI1)
+  show ?case
+  proof (intro exI conjI)
+    show "finite (insert a C)"
+    show "disjnt (insert a C) U"
+      using C \<open>a \<notin> U\<close> by simp
+    show "card (insert a C) \<le> card (insert X \<F>)"
+      by (simp add: C card_insert_if insert.hyps le_SucI)
+    have "closed (\<Union>\<F>)"
+      using clo insert.hyps by blast
+    have "continuous_on (X - insert a C \<union> (\<Union>\<F> - insert a C)) (\<lambda>x. if x \<in> X then f x else g x)"
+       apply (rule continuous_on_cases_local)
+        using \<open>closed X\<close> apply blast
+        using \<open>closed (\<Union>\<F>)\<close> apply blast
+        using contf apply (force simp: elim: continuous_on_subset)
+        using contg apply (force simp: elim: continuous_on_subset)
+        using fh gh insert.hyps pwX by fastforce
+    then show "continuous_on (\<Union>insert X \<F> - insert a C) (\<lambda>a. if a \<in> X then f a else g a)"
+      by (blast intro: continuous_on_subset)
+    show "\<forall>x\<in>(\<Union>insert X \<F> - insert a C) \<inter> K. (if x \<in> X then f x else g x) = h x"
+      using gh by (auto simp: fh)
+    show "(\<lambda>a. if a \<in> X then f a else g a) ` (\<Union>insert X \<F> - insert a C) \<subseteq> T"
+      using fim gim by auto force
+  qed
+qed
+
+
+lemma extend_map_lemma_cofinite1:
+assumes "finite \<F>"
+    and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
+    and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+    and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~(X \<subseteq> Y); ~(Y \<subseteq> X)\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+  obtains C g where "finite C" "disjnt C U" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+                    "g ` (\<Union>\<F> - C) \<subseteq> T"
+                    "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+proof -
+  let ?\<F> = "{X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
+  have [simp]: "\<Union>?\<F> = \<Union>\<F>"
+    by (simp add: Union_maximal_sets assms)
+  have fin: "finite ?\<F>"
+    by (force intro: finite_subset [OF _ \<open>finite \<F>\<close>])
+  have pw: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) ?\<F>"
+    by (simp add: pairwise_def) (metis K psubsetI)
+  have "card {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y} \<le> card \<F>"
+    by (simp add: \<open>finite \<F>\<close> card_mono)
+  moreover
+  obtain C g where "finite C \<and> disjnt C U \<and> card C \<le> card ?\<F> \<and>
+                 continuous_on (\<Union>?\<F> - C) g \<and> g ` (\<Union>?\<F> - C) \<subseteq> T
+                  \<and> (\<forall>x \<in> (\<Union>?\<F> - C) \<inter> K. g x = h x)"
+    apply (rule exE [OF extend_map_lemma_cofinite0 [OF fin pw, of U T h]])
+      apply (fastforce intro!:  clo \<F>)+
+    done
+  ultimately show ?thesis
+    by (rule_tac C=C and g=g in that) auto
+qed
+
+
+lemma extend_map_lemma_cofinite:
+  assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
+      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+      and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+      and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+  obtains C g where
+     "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+     "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof -
+  define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
+  have "finite \<G>"
+    using assms finite_subset by blast
+  moreover have "finite (\<Union>{{D. D face_of C} |C. C \<in> \<F>})"
+    apply (rule finite_Union)
+     apply (simp add: \<open>finite \<F>\<close>)
+    using finite_polytope_faces poly by auto
+  ultimately have "finite \<H>"
+    apply (rule finite_subset [of _ "\<Union> {{D. D face_of C} | C. C \<in> \<F>}"], auto)
+    done
+  have *: "\<And>X Y. \<lbrakk>X \<in> \<H>; Y \<in> \<H>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+    unfolding \<H>_def
+    apply (elim UnE bexE CollectE DiffE)
+    using subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] apply (simp_all add: face)
+      apply (meson subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] face face_of_Int_subface face_of_imp_subset face_of_refl poly polytope_imp_convex)+
+    done
+  obtain h where conth: "continuous_on (\<Union>\<H>) h" and him: "h ` (\<Union>\<H>) \<subseteq> rel_frontier T"
+             and hf: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+    using \<open>finite \<H>\<close>
+    unfolding \<H>_def
+    apply (rule extend_map_lemma [OF _ Un_upper1 T _ _ _ contf fim])
+    using \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly apply fastforce
+    using * apply (auto simp: \<H>_def)
+    done
+  have "bounded (\<Union>\<G>)"
+    using \<open>finite \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> poly polytope_imp_bounded by blast
+  then have "\<Union>\<G> \<noteq> UNIV"
+    by auto
+  then obtain a where a: "a \<notin> \<Union>\<G>"
+    by blast
+  have \<F>: "\<exists>a g. a \<notin> \<Union>\<G> \<and> continuous_on (D - {a}) g \<and>
+                  g ` (D - {a}) \<subseteq> rel_frontier T \<and> (\<forall>x \<in> D \<inter> \<Union>\<H>. g x = h x)"
+       if "D \<in> \<F>" for D
+  proof (cases "D \<subseteq> \<Union>\<H>")
+    case True
+    then show ?thesis
+      apply (rule_tac x=a in exI)
+      apply (rule_tac x=h in exI)
+      using him apply (blast intro!: \<open>a \<notin> \<Union>\<G>\<close> continuous_on_subset [OF conth]) +
+      done
+  next
+    case False
+    note D_not_subset = False
+    show ?thesis
+    proof (cases "D \<in> \<G>")
+      case True
+      with D_not_subset show ?thesis
+        by (auto simp: \<H>_def)
+    next
+      case False
+      then have affD: "aff_dim D \<le> aff_dim T"
+        by (simp add: \<open>D \<in> \<F>\<close> aff)
+      show ?thesis
+      proof (cases "rel_interior D = {}")
+        case True
+        with \<open>D \<in> \<F>\<close> poly a show ?thesis
+          by (force simp: rel_interior_eq_empty polytope_imp_convex)
+      next
+        case False
+        then obtain b where brelD: "b \<in> rel_interior D"
+          by blast
+        have "polyhedron D"
+          by (simp add: poly polytope_imp_polyhedron that)
+        have "rel_frontier D retract_of affine hull D - {b}"
+          by (simp add: rel_frontier_retract_of_punctured_affine_hull poly polytope_imp_bounded polytope_imp_convex that brelD)
+        then obtain r where relfD: "rel_frontier D \<subseteq> affine hull D - {b}"
+                        and contr: "continuous_on (affine hull D - {b}) r"
+                        and rim: "r ` (affine hull D - {b}) \<subseteq> rel_frontier D"
+                        and rid: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> r x = x"
+          by (auto simp: retract_of_def retraction_def)
+        show ?thesis
+        proof (intro exI conjI ballI)
+          show "b \<notin> \<Union>\<G>"
+          proof clarify
+            fix E
+            assume "b \<in> E" "E \<in> \<G>"
+            then have "E \<inter> D face_of E \<and> E \<inter> D face_of D"
+              using \<open>\<G> \<subseteq> \<F>\<close> face that by auto
+            with face_of_subset_rel_frontier \<open>E \<in> \<G>\<close> \<open>b \<in> E\<close> brelD rel_interior_subset [of D]
+                 D_not_subset rel_frontier_def \<H>_def
+            show False
+              by blast
+          qed
+          have "r ` (D - {b}) \<subseteq> r ` (affine hull D - {b})"
+            by (simp add: Diff_mono hull_subset image_mono)
+          also have "... \<subseteq> rel_frontier D"
+            by (rule rim)
+          also have "... \<subseteq> \<Union>{E. E face_of D \<and> aff_dim E < aff_dim T}"
+            using affD
+            by (force simp: rel_frontier_of_polyhedron [OF \<open>polyhedron D\<close>] facet_of_def)
+          also have "... \<subseteq> \<Union>(\<H>)"
+            using D_not_subset \<H>_def that by fastforce
+          finally have rsub: "r ` (D - {b}) \<subseteq> \<Union>(\<H>)" .
+          show "continuous_on (D - {b}) (h \<circ> r)"
+            apply (intro conjI \<open>b \<notin> \<Union>\<G>\<close> continuous_on_compose)
+               apply (rule continuous_on_subset [OF contr])
+            apply (simp add: Diff_mono hull_subset)
+            apply (rule continuous_on_subset [OF conth rsub])
+            done
+          show "(h \<circ> r) ` (D - {b}) \<subseteq> rel_frontier T"
+            using brelD him rsub by fastforce
+          show "(h \<circ> r) x = h x" if x: "x \<in> D \<inter> \<Union>\<H>" for x
+          proof -
+            consider A where "x \<in> D" "A \<in> \<G>" "x \<in> A"
+                 | A B where "x \<in> D" "A face_of B" "B \<in> \<F>" "B \<notin> \<G>" "aff_dim A < aff_dim T" "x \<in> A"
+              using x by (auto simp: \<H>_def)
+            then have xrel: "x \<in> rel_frontier D"
+            proof cases
+              case 1 show ?thesis
+              proof (rule face_of_subset_rel_frontier [THEN subsetD])
+                show "D \<inter> A face_of D"
+                  using \<open>A \<in> \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> face \<open>D \<in> \<F>\<close> by blast
+                show "D \<inter> A \<noteq> D"
+                  using \<open>A \<in> \<G>\<close> D_not_subset \<H>_def by blast
+              qed (auto simp: 1)
+            next
+              case 2 show ?thesis
+              proof (rule face_of_subset_rel_frontier [THEN subsetD])
+                show "D \<inter> A face_of D"
+                  apply (rule face_of_Int_subface [of D B _ A, THEN conjunct1])
+                     apply (simp_all add: 2 \<open>D \<in> \<F>\<close> face)
+                   apply (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex face_of_refl)
+                  done
+                show "D \<inter> A \<noteq> D"
+                  using "2" D_not_subset \<H>_def by blast
+              qed (auto simp: 2)
+            qed
+            show ?thesis
+              by (simp add: rid xrel)
+          qed
+        qed
+      qed
+    qed
+  qed
+  have clo: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+    by (simp add: poly polytope_imp_closed)
+  obtain C g where "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+                   "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+               and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> \<Union>\<H> \<Longrightarrow> g x = h x"
+  proof (rule extend_map_lemma_cofinite1 [OF \<open>finite \<F>\<close> \<F> clo])
+    show "X \<inter> Y \<subseteq> \<Union>\<H>" if XY: "X \<in> \<F>" "Y \<in> \<F>" and "\<not> X \<subseteq> Y" "\<not> Y \<subseteq> X" for X Y
+    proof (cases "X \<in> \<G>")
+      case True
+      then show ?thesis
+        by (auto simp: \<H>_def)
+    next
+      case False
+      have "X \<inter> Y \<noteq> X"
+        using \<open>\<not> X \<subseteq> Y\<close> by blast
+      with XY
+      show ?thesis
+        by (clarsimp simp: \<H>_def)
+           (metis Diff_iff Int_iff aff antisym_conv face face_of_aff_dim_lt face_of_refl
+                  not_le poly polytope_imp_convex)
+    qed
+  qed (blast)+
+  with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis
+    apply (rule_tac C=C and g=g in that)
+     apply (auto simp: disjnt_def hf [symmetric] \<H>_def intro!: gh)
+    done
+qed
+
+text\<open>The next two proofs are similar\<close>
+theorem extend_map_cell_complex_to_sphere:
+  assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+      and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
+      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+  obtains g where "continuous_on (\<Union>\<F>) g"
+     "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+    using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+  have "compact S"
+    by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+  then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+    using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+  obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+             and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+             and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+             and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+             and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+  proof (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly _ face])
+    show "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+  qed auto
+  obtain h where conth: "continuous_on (\<Union>\<G>) h" and him: "h ` \<Union>\<G> \<subseteq> rel_frontier T" and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+  proof (rule extend_map_lemma [of \<G> "\<G> \<inter> Pow V" T g])
+    show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+      by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+  qed (use \<open>finite \<G>\<close> T polyG affG faceG gim in fastforce)+
+  show ?thesis
+  proof
+    show "continuous_on (\<Union>\<F>) h"
+      using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+    show "h ` \<Union>\<F> \<subseteq> rel_frontier T"
+      using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+    show "h x = f x" if "x \<in> S" for x
+    proof -
+      have "x \<in> \<Union>\<G>"
+        using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> that by auto
+      then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+      then have "diameter X < d" "bounded X"
+        by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+      then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+        by fastforce
+      have "h x = g x"
+        apply (rule hg)
+        using \<open>X \<in> \<G>\<close> \<open>X \<subseteq> V\<close> \<open>x \<in> X\<close> by blast
+      also have "... = f x"
+        by (simp add: gf that)
+      finally show "h x = f x" .
+    qed
+  qed
+qed
+
+
+theorem extend_map_cell_complex_to_sphere_cofinite:
+  assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+      and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+  obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
+     "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+    using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+  have "compact S"
+    by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+  then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+    using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+  obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+             and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+             and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+             and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+             and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+    by (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly aff face]) auto
+  obtain C h where "finite C" and dis: "disjnt C (\<Union>(\<G> \<inter> Pow V))"
+               and card: "card C \<le> card \<G>" and conth: "continuous_on (\<Union>\<G> - C) h"
+               and him: "h ` (\<Union>\<G> - C) \<subseteq> rel_frontier T"
+               and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+  proof (rule extend_map_lemma_cofinite [of \<G> "\<G> \<inter> Pow V" T g])
+    show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+      by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+    show "g ` \<Union>(\<G> \<inter> Pow V) \<subseteq> rel_frontier T"
+      using gim by force
+  qed (auto intro: \<open>finite \<G>\<close> T polyG affG dest: faceG)
+  have Ssub: "S \<subseteq> \<Union>(\<G> \<inter> Pow V)"
+  proof
+    fix x
+    assume "x \<in> S"
+    then have "x \<in> \<Union>\<G>"
+      using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> by auto
+    then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+    then have "diameter X < d" "bounded X"
+      by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+    then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+      by fastforce
+    then show "x \<in> \<Union>(\<G> \<inter> Pow V)"
+      using \<open>X \<in> \<G>\<close> \<open>x \<in> X\<close> by blast
+  qed
+  show ?thesis
+  proof
+    show "continuous_on (\<Union>\<F>-C) h"
+      using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+    show "h ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+      using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+    show "h x = f x" if "x \<in> S" for x
+    proof -
+      have "h x = g x"
+        apply (rule hg)
+        using Ssub that by blast
+      also have "... = f x"
+        by (simp add: gf that)
+      finally show "h x = f x" .
+    qed
+    show "disjnt C S"
+      using dis Ssub  by (meson disjnt_iff subset_eq)
+  qed (intro \<open>finite C\<close>)
+qed
+
+
+
+subsection\<open> Special cases and corollaries involving spheres.\<close>
+
+lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
+  by (auto simp: disjnt_def)
+
+proposition extend_map_affine_to_sphere_cofinite_simple:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "compact S" "convex U" "bounded U"
+      and aff: "aff_dim T \<le> aff_dim U"
+      and "S \<subseteq> T" and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> rel_frontier U"
+ obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+                   "g ` (T - K) \<subseteq> rel_frontier U"
+                   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+  have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
+              g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
+       if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U"  for T
+  proof (cases "S = {}")
+    case True
+    show ?thesis
+    proof (cases "rel_frontier U = {}")
+      case True
+      with \<open>bounded U\<close> have "aff_dim U \<le> 0"
+        using affine_bounded_eq_lowdim rel_frontier_eq_empty by auto
+      with aff have "aff_dim T \<le> 0" by auto
+      then obtain a where "T \<subseteq> {a}"
+        using \<open>affine T\<close> affine_bounded_eq_lowdim affine_bounded_eq_trivial by auto
+      then show ?thesis
+        using \<open>S = {}\<close> fim
+        by (metis Diff_cancel contf disjnt_empty2 finite.emptyI finite_insert finite_subset)
+    next
+      case False
+      then obtain a where "a \<in> rel_frontier U"
+        by auto
+      then show ?thesis
+        using continuous_on_const [of _ a] \<open>S = {}\<close> by force
+    qed
+  next
+    case False
+    have "bounded S"
+      by (simp add: \<open>compact S\<close> compact_imp_bounded)
+    then obtain b where b: "S \<subseteq> cbox (-b) b"
+      using bounded_subset_cbox_symmetric by blast
+    define bbox where "bbox \<equiv> cbox (-(b+One)) (b+One)"
+    have "cbox (-b) b \<subseteq> bbox"
+      by (auto simp: bbox_def algebra_simps intro!: subset_box_imp)
+    with b \<open>S \<subseteq> T\<close> have "S \<subseteq> bbox \<inter> T"
+      by auto
+    then have Ssub: "S \<subseteq> \<Union>{bbox \<inter> T}"
+      by auto
+    then have "aff_dim (bbox \<inter> T) \<le> aff_dim U"
+      by (metis aff aff_dim_subset inf_commute inf_le1 order_trans)
+    obtain K g where K: "finite K" "disjnt K S"
+                 and contg: "continuous_on (\<Union>{bbox \<inter> T} - K) g"
+                 and gim: "g ` (\<Union>{bbox \<inter> T} - K) \<subseteq> rel_frontier U"
+                 and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+    proof (rule extend_map_cell_complex_to_sphere_cofinite
+              [OF _ Ssub _ \<open>convex U\<close> \<open>bounded U\<close> _ _ _ contf fim])
+      show "closed S"
+        using \<open>compact S\<close> compact_eq_bounded_closed by auto
+      show poly: "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> polytope X"
+        by (simp add: polytope_Int_polyhedron bbox_def polytope_interval affine_imp_polyhedron \<open>affine T\<close>)
+      show "\<And>X Y. \<lbrakk>X \<in> {bbox \<inter> T}; Y \<in> {bbox \<inter> T}\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+        by (simp add:poly face_of_refl polytope_imp_convex)
+      show "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> aff_dim X \<le> aff_dim U"
+        by (simp add: \<open>aff_dim (bbox \<inter> T) \<le> aff_dim U\<close>)
+    qed auto
+    define fro where "fro \<equiv> \<lambda>d. frontier(cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+    obtain d where d12: "1/2 \<le> d" "d \<le> 1" and dd: "disjnt K (fro d)"
+    proof (rule disjoint_family_elem_disjnt [OF _ \<open>finite K\<close>])
+      show "infinite {1/2..1::real}"
+      have dis1: "disjnt (fro x) (fro y)" if "x<y" for x y
+        by (auto simp: algebra_simps that subset_box_imp disjnt_Diff1 frontier_def fro_def)
+      then show "disjoint_family_on fro {1/2..1}"
+        by (auto simp: disjoint_family_on_def disjnt_def neq_iff)
+    qed auto
+    define c where "c \<equiv> b + d *\<^sub>R One"
+    have cbsub: "cbox (-b) b \<subseteq> box (-c) c"  "cbox (-b) b \<subseteq> cbox (-c) c"  "cbox (-c) c \<subseteq> bbox"
+      using d12 by (auto simp: algebra_simps subset_box_imp c_def bbox_def)
+    have clo_cbT: "closed (cbox (- c) c \<inter> T)"
+      by (simp add: affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+    have cpT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+      using \<open>S \<noteq> {}\<close> b cbsub(2) \<open>S \<subseteq> T\<close> by fastforce
+    have "closest_point (cbox (- c) c \<inter> T) x \<notin> K" if "x \<in> T" "x \<notin> K" for x
+    proof (cases "x \<in> cbox (-c) c")
+      case True with that show ?thesis
+    next
+      case False
+      have int_ne: "interior (cbox (-c) c) \<inter> T \<noteq> {}"
+        using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b \<open>cbox (- b) b \<subseteq> box (- c) c\<close> by force
+      have "convex T"
+        by (meson \<open>affine T\<close> affine_imp_convex)
+      then have "x \<in> affine hull (cbox (- c) c \<inter> T)"
+          by (metis Int_commute Int_iff \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> cbsub(1) \<open>x \<in> T\<close> affine_hull_convex_Int_nonempty_interior all_not_in_conv b hull_inc inf.orderE interior_cbox)
+      then have "x \<in> affine hull (cbox (- c) c \<inter> T) - rel_interior (cbox (- c) c \<inter> T)"
+        by (meson DiffI False Int_iff rel_interior_subset subsetCE)
+      then have "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+        by (rule closest_point_in_rel_frontier [OF clo_cbT cpT_ne])
+      moreover have "(rel_frontier (cbox (- c) c \<inter> T)) \<subseteq> fro d"
+        apply (subst convex_affine_rel_frontier_Int [OF _  \<open>affine T\<close> int_ne])
+         apply (auto simp: fro_def c_def)
+        done
+      ultimately show ?thesis
+        using dd  by (force simp: disjnt_def)
+    qed
+    then have cpt_subset: "closest_point (cbox (- c) c \<inter> T) ` (T - K) \<subseteq> \<Union>{bbox \<inter> T} - K"
+      using closest_point_in_set [OF clo_cbT cpT_ne] cbsub(3) by force
+    show ?thesis
+    proof (intro conjI ballI exI)
+      have "continuous_on (T - K) (closest_point (cbox (- c) c \<inter> T))"
+        apply (rule continuous_on_closest_point)
+        using \<open>S \<noteq> {}\<close> cbsub(2) b that
+        by (auto simp: affine_imp_convex convex_Int affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+      then show "continuous_on (T - K) (g \<circ> closest_point (cbox (- c) c \<inter> T))"
+        by (metis continuous_on_compose continuous_on_subset [OF contg cpt_subset])
+      have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> g ` (\<Union>{bbox \<inter> T} - K)"
+        by (metis image_comp image_mono cpt_subset)
+      also have "... \<subseteq> rel_frontier U"
+        by (rule gim)
+      finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
+      show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
+      proof -
+        have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
+          unfolding o_def
+          by (metis IntI \<open>S \<subseteq> T\<close> b cbsub(2) closest_point_self subset_eq that)
+        also have "... = f x"
+          by (simp add: that gf)
+        finally show ?thesis .
+      qed
+    qed (auto simp: K)
+  qed
+  then obtain K g where "finite K" "disjnt K S"
+               and contg: "continuous_on (affine hull T - K) g"
+               and gim:  "g ` (affine hull T - K) \<subseteq> rel_frontier U"
+               and gf:   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+    by (metis aff affine_affine_hull aff_dim_affine_hull
+              order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
+  then obtain K g where "finite K" "disjnt K S"
+               and contg: "continuous_on (T - K) g"
+               and gim:  "g ` (T - K) \<subseteq> rel_frontier U"
+               and gf:   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+    by (rule_tac K=K and g=g in that) (auto simp: hull_inc elim: continuous_on_subset)
+  then show ?thesis
+    by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
+qed
+
+subsection\<open>Extending maps to spheres\<close>
+
+(*Up to extend_map_affine_to_sphere_cofinite_gen*)
+
+lemma closedin_closed_subset:
+ "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+             \<Longrightarrow> closedin (subtopology euclidean T) S"
+  by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
+
+lemma extend_map_affine_to_sphere1:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
+  assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
+      and fim: "f ` (U - K) \<subseteq> T"
+      and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+      and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
+  obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "K = {}")
+  case True
+  then show ?thesis
+    by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
+next
+  case False
+  have "S \<subseteq> U"
+    using clo closedin_limpt by blast
+  then have "(U - S) \<inter> K \<noteq> {}"
+    by (metis Diff_triv False Int_Diff K disjnt_def inf.absorb_iff2 inf_commute)
+  then have "\<Union>(components (U - S)) \<inter> K \<noteq> {}"
+    using Union_components by simp
+  then obtain C0 where C0: "C0 \<in> components (U - S)" "C0 \<inter> K \<noteq> {}"
+    by blast
+  have "convex U"
+    by (simp add: affine_imp_convex \<open>affine U\<close>)
+  then have "locally connected U"
+    by (rule convex_imp_locally_connected)
+  have "\<exists>a g. a \<in> C \<and> a \<in> L \<and> continuous_on (S \<union> (C - {a})) g \<and>
+              g ` (S \<union> (C - {a})) \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x)"
+       if C: "C \<in> components (U - S)" and CK: "C \<inter> K \<noteq> {}" for C
+  proof -
+    have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
+      by (simp_all add: in_components_subset comps that)
+    then obtain a where a: "a \<in> C" "a \<in> L" by auto
+    have opeUC: "openin (subtopology euclidean U) C"
+    proof (rule openin_trans)
+      show "openin (subtopology euclidean (U-S)) C"
+        by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
+      show "openin (subtopology euclidean U) (U - S)"
+        by (simp add: clo openin_diff)
+    qed
+    then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
+      using openin_contains_cball by (metis \<open>a \<in> C\<close>)
+    then have "ball a d \<inter> U \<subseteq> C"
+      by auto
+    obtain h k where homhk: "homeomorphism (S \<union> C) (S \<union> C) h k"
+                 and subC: "{x. (~ (h x = x \<and> k x = x))} \<subseteq> C"
+                 and bou: "bounded {x. (~ (h x = x \<and> k x = x))}"
+                 and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
+    proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
+      show "openin (subtopology euclidean C) (ball a d \<inter> U)"
+        by (metis Topology_Euclidean_Space.open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
+      show "openin (subtopology euclidean (affine hull C)) C"
+        by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
+      show "ball a d \<inter> U \<noteq> {}"
+        using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+      show "finite (C \<inter> K)"
+        by (simp add: \<open>finite K\<close>)
+      show "S \<union> C \<subseteq> affine hull C"
+        by (metis \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> opeUC affine_hull_eq affine_hull_openin all_not_in_conv assms(2) sup.bounded_iff)
+      show "connected C"
+        by (metis C in_components_connected)
+    qed auto
+    have a_BU: "a \<in> ball a d \<inter> U"
+      using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+    have "rel_frontier (cball a d \<inter> U) retract_of (affine hull (cball a d \<inter> U) - {a})"
+      apply (rule rel_frontier_retract_of_punctured_affine_hull)
+        apply (auto simp: \<open>convex U\<close> convex_Int)
+      by (metis \<open>affine U\<close> convex_cball empty_iff interior_cball a_BU rel_interior_convex_Int_affine)
+    moreover have "rel_frontier (cball a d \<inter> U) = frontier (cball a d) \<inter> U"
+      apply (rule convex_affine_rel_frontier_Int)
+      using a_BU by (force simp: \<open>affine U\<close>)+
+    moreover have "affine hull (cball a d \<inter> U) = U"
+      by (metis \<open>convex U\<close> a_BU affine_hull_convex_Int_nonempty_interior affine_hull_eq \<open>affine U\<close> equals0D inf.commute interior_cball)
+    ultimately have "frontier (cball a d) \<inter> U retract_of (U - {a})"
+      by metis
+    then obtain r where contr: "continuous_on (U - {a}) r"
+                    and rim: "r ` (U - {a}) \<subseteq> sphere a d"  "r ` (U - {a}) \<subseteq> U"
+                    and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
+      using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
+    define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
+    have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
+      using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
+    have Uaeq: "U - {a} = (cball a d - {a}) \<inter> U \<union> (U - ball a d)"
+      using \<open>0 < d\<close> by auto
+    have jim: "j ` (S \<union> (C - {a})) \<subseteq> (S \<union> C) - ball a d"
+    proof clarify
+      fix y  assume "y \<in> S \<union> (C - {a})"
+      then have "y \<in> U - {a}"
+        using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+      then have "r y \<in> sphere a d"
+        using rim by auto
+      then show "j y \<in> S \<union> C - ball a d"
+        using \<open>r y \<in> sphere a d\<close> \<open>y \<in> U - {a}\<close> \<open>y \<in> S \<union> (C - {a})\<close> d rim by fastforce
+    qed
+    have contj: "continuous_on (U - {a}) j"
+      unfolding j_def Uaeq
+    proof (intro continuous_on_cases_local continuous_on_id, simp_all add: req closedin_closed Uaeq [symmetric])
+      show "\<exists>T. closed T \<and> (cball a d - {a}) \<inter> U = (U - {a}) \<inter> T"
+          apply (rule_tac x="(cball a d) \<inter> U" in exI)
+        using affine_closed \<open>affine U\<close> by blast
+      show "\<exists>T. closed T \<and> U - ball a d = (U - {a}) \<inter> T"
+         apply (rule_tac x="U - ball a d" in exI)
+        using \<open>0 < d\<close>  by (force simp: affine_closed \<open>affine U\<close> closed_Diff)
+      show "continuous_on ((cball a d - {a}) \<inter> U) r"
+        by (force intro: continuous_on_subset [OF contr])
+    qed
+    have fT: "x \<in> U - K \<Longrightarrow> f x \<in> T" for x
+      using fim by blast
+    show ?thesis
+    proof (intro conjI exI)
+      show "continuous_on (S \<union> (C - {a})) (f \<circ> k \<circ> j)"
+      proof (intro continuous_on_compose)
+        show "continuous_on (S \<union> (C - {a})) j"
+          apply (rule continuous_on_subset [OF contj])
+          using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+        show "continuous_on (j ` (S \<union> (C - {a}))) k"
+          apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
+          using jim \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def by fastforce
+        show "continuous_on (k ` j ` (S \<union> (C - {a}))) f"
+        proof (clarify intro!: continuous_on_subset [OF contf])
+          fix y  assume "y \<in> S \<union> (C - {a})"
+          have ky: "k y \<in> S \<union> C"
+            using homeomorphism_image2 [OF homhk] \<open>y \<in> S \<union> (C - {a})\<close> by blast
+          have jy: "j y \<in> S \<union> C - ball a d"
+            using Un_iff \<open>y \<in> S \<union> (C - {a})\<close> jim by auto
+          show "k (j y) \<in> U - K"
+            apply safe
+            using \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close>  homeomorphism_image2 [OF homhk] jy apply blast
+            by (metis DiffD1 DiffD2 Int_iff Un_iff \<open>disjnt K S\<close> disjnt_def empty_iff hin homeomorphism_apply2 homeomorphism_image2 homhk imageI jy)
+        qed
+      qed
+      have ST: "\<And>x. x \<in> S \<Longrightarrow> (f \<circ> k \<circ> j) x \<in> T"
+        apply (metis DiffI \<open>S \<subseteq> U\<close> \<open>disjnt K S\<close> subsetD disjnt_iff fim image_subset_iff)
+        done
+      moreover have "(f \<circ> k \<circ> j) x \<in> T" if "x \<in> C" "x \<noteq> a" "x \<notin> S" for x
+      proof -
+        have rx: "r x \<in> sphere a d"
+          using \<open>C \<subseteq> U\<close> rim that by fastforce
+        have jj: "j x \<in> S \<union> C - ball a d"
+          using jim that by blast
+        have "k (j x) = j x \<longrightarrow> k (j x) \<in> C \<or> j x \<in> C"
+          by (metis Diff_iff Int_iff Un_iff \<open>S \<subseteq> U\<close> subsetD d j_def jj rx sphere_cball that(1))
+        then have "k (j x) \<in> C"
+          using homeomorphism_apply2 [OF homhk, of "j x"]   \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> a rx
+          by (metis (mono_tags, lifting) Diff_iff subsetD jj mem_Collect_eq subC)
+        with jj \<open>C \<subseteq> U\<close> show ?thesis
+          apply safe
+          using ST j_def apply fastforce
+          apply (auto simp: not_less intro!: fT)
+          by (metis DiffD1 DiffD2 Int_iff hin homeomorphism_apply2 [OF homhk] jj)
+      qed
+      ultimately show "(f \<circ> k \<circ> j) ` (S \<union> (C - {a})) \<subseteq> T"
+        by force
+      show "\<forall>x\<in>S. (f \<circ> k \<circ> j) x = f x" using kj by simp
+    qed (auto simp: a)
+  qed
+  then obtain a h where
+    ah: "\<And>C. \<lbrakk>C \<in> components (U - S); C \<inter> K \<noteq> {}\<rbrakk>
+           \<Longrightarrow> a C \<in> C \<and> a C \<in> L \<and> continuous_on (S \<union> (C - {a C})) (h C) \<and>
+               h C ` (S \<union> (C - {a C})) \<subseteq> T \<and> (\<forall>x \<in> S. h C x = f x)"
+    using that by metis
+  define F where "F \<equiv> {C \<in> components (U - S). C \<inter> K \<noteq> {}}"
+  define G where "G \<equiv> {C \<in> components (U - S). C \<inter> K = {}}"
+  define UF where "UF \<equiv> (\<Union>C\<in>F. C - {a C})"
+  have "C0 \<in> F"
+    by (auto simp: F_def C0)
+  have "finite F"
+  proof (subst finite_image_iff [of "\<lambda>C. C \<inter> K" F, symmetric])
+    show "inj_on (\<lambda>C. C \<inter> K) F"
+      unfolding F_def inj_on_def
+      using components_nonoverlap by blast
+    show "finite ((\<lambda>C. C \<inter> K) ` F)"
+      unfolding F_def
+      by (rule finite_subset [of _ "Pow K"]) (auto simp: \<open>finite K\<close>)
+  qed
+  obtain g where contg: "continuous_on (S \<union> UF) g"
+             and gh: "\<And>x i. \<lbrakk>i \<in> F; x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))\<rbrakk>
+                            \<Longrightarrow> g x = h i x"
+  proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>, of "S \<union> UF" "\<lambda>C. S \<union> (C - {a C})" h])
+    show "S \<union> UF \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
+      using \<open>C0 \<in> F\<close> by (force simp: UF_def)
+    show "closedin (subtopology euclidean (S \<union> UF)) (S \<union> (C - {a C}))"
+         if "C \<in> F" for C
+    proof (rule closedin_closed_subset [of U "S \<union> C"])
+      show "closedin (subtopology euclidean U) (S \<union> C)"
+        apply (rule closedin_Un_complement_component [OF \<open>locally connected U\<close> clo])
+        using F_def that by blast
+    next
+      have "x = a C'" if "C' \<in> F"  "x \<in> C'" "x \<notin> U" for x C'
+      proof -
+        have "\<forall>A. x \<in> \<Union>A \<or> C' \<notin> A"
+          using \<open>x \<in> C'\<close> by blast
+        with that show "x = a C'"
+          by (metis (lifting) DiffD1 F_def Union_components mem_Collect_eq)
+      qed
+      then show "S \<union> UF \<subseteq> U"
+        using \<open>S \<subseteq> U\<close> by (force simp: UF_def)
+    next
+      show "S \<union> (C - {a C}) = (S \<union> C) \<inter> (S \<union> UF)"
+        using F_def UF_def components_nonoverlap that by auto
+    qed
+  next
+    show "continuous_on (S \<union> (C' - {a C'})) (h C')" if "C' \<in> F" for C'
+      using ah F_def that by blast
+    show "\<And>i j x. \<lbrakk>i \<in> F; j \<in> F;
+                   x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
+                  \<Longrightarrow> h i x = h j x"
+      using components_eq by (fastforce simp: components_eq F_def ah)
+  qed blast
+  have SU': "S \<union> \<Union>G \<union> (S \<union> UF) \<subseteq> U"
+    using \<open>S \<subseteq> U\<close> in_components_subset by (auto simp: F_def G_def UF_def)
+  have clo1: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
+  proof (rule closedin_closed_subset [OF _ SU'])
+    have *: "\<And>C. C \<in> F \<Longrightarrow> openin (subtopology euclidean U) C"
+      unfolding F_def
+      by clarify (metis (no_types, lifting) \<open>locally connected U\<close> clo closedin_def locally_diff_closed openin_components_locally_connected openin_trans topspace_euclidean_subtopology)
+    show "closedin (subtopology euclidean U) (U - UF)"
+      unfolding UF_def
+      by (force intro: openin_delete *)
+    show "S \<union> \<Union>G = (U - UF) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+      using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+        apply (metis Diff_iff UnionI Union_components)
+       apply (metis DiffD1 UnionI Union_components)
+      by (metis (no_types, lifting) IntI components_nonoverlap empty_iff)
+  qed
+  have clo2: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
+  proof (rule closedin_closed_subset [OF _ SU'])
+    show "closedin (subtopology euclidean U) (\<Union>C\<in>F. S \<union> C)"
+      apply (rule closedin_Union)
+       apply (simp add: \<open>finite F\<close>)
+      using F_def \<open>locally connected U\<close> clo closedin_Un_complement_component by blast
+    show "S \<union> UF = (\<Union>C\<in>F. S \<union> C) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+      using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+      using C0 apply blast
+      by (metis components_nonoverlap disjnt_def disjnt_iff)
+  qed
+  have SUG: "S \<union> \<Union>G \<subseteq> U - K"
+    using \<open>S \<subseteq> U\<close> K apply (auto simp: G_def disjnt_iff)
+    by (meson Diff_iff subsetD in_components_subset)
+  then have contf': "continuous_on (S \<union> \<Union>G) f"
+    by (rule continuous_on_subset [OF contf])
+  have contg': "continuous_on (S \<union> UF) g"
+    apply (rule continuous_on_subset [OF contg])
+    using \<open>S \<subseteq> U\<close> by (auto simp: F_def G_def)
+  have  "\<And>x. \<lbrakk>S \<subseteq> U; x \<in> S\<rbrakk> \<Longrightarrow> f x = g x"
+    by (subst gh) (auto simp: ah C0 intro: \<open>C0 \<in> F\<close>)
+  then have f_eq_g: "\<And>x. x \<in> S \<union> UF \<and> x \<in> S \<union> \<Union>G \<Longrightarrow> f x = g x"
+    using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def dest: in_components_subset)
+    using components_eq by blast
+  have cont: "continuous_on (S \<union> \<Union>G \<union> (S \<union> UF)) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+    by (blast intro: continuous_on_cases_local [OF clo1 clo2 contf' contg' f_eq_g, of "\<lambda>x. x \<in> S \<union> \<Union>G"])
+  show ?thesis
+  proof
+    have UF: "\<Union>F - L \<subseteq> UF"
+      unfolding F_def UF_def using ah by blast
+    have "U - S - L = \<Union>(components (U - S)) - L"
+      by simp
+    also have "... = \<Union>F \<union> \<Union>G - L"
+      unfolding F_def G_def by blast
+    also have "... \<subseteq> UF \<union> \<Union>G"
+      using UF by blast
+    finally have "U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)"
+      by blast
+    then show "continuous_on (U - L) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+      by (rule continuous_on_subset [OF cont])
+    have "((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq>  ((U - L) \<inter> (-S \<inter> UF))"
+      using \<open>U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)\<close> by auto
+    moreover have "g ` ((U - L) \<inter> (-S \<inter> UF)) \<subseteq> T"
+    proof -
+      have "g x \<in> T" if "x \<in> U" "x \<notin> L" "x \<notin> S" "C \<in> F" "x \<in> C" "x \<noteq> a C" for x C
+      proof (subst gh)
+        show "x \<in> (S \<union> UF) \<inter> (S \<union> (C - {a C}))"
+          using that by (auto simp: UF_def)
+        show "h C x \<in> T"
+          using ah that by (fastforce simp add: F_def)
+      qed (rule that)
+      then show ?thesis
+        by (force simp: UF_def)
+    qed
+    ultimately have "g ` ((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> T"
+      using image_mono order_trans by blast
+    moreover have "f ` ((U - L) \<inter> (S \<union> \<Union>G)) \<subseteq> T"
+      using fim SUG by blast
+    ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) ` (U - L) \<subseteq> T"
+       by force
+    show "\<And>x. x \<in> S \<Longrightarrow> (if x \<in> S \<union> \<Union>G then f x else g x) = f x"
+      by (simp add: F_def G_def)
+  qed
+qed
+
+
+lemma extend_map_affine_to_sphere2:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+      and affTU: "aff_dim T \<le> aff_dim U"
+      and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> rel_frontier U"
+      and ovlap: "\<And>C. C \<in> components(T - S) \<Longrightarrow> C \<inter> L \<noteq> {}"
+    obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
+                      "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
+                      "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+  obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
+               and contg: "continuous_on (T - K) g"
+               and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+               and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+     using assms extend_map_affine_to_sphere_cofinite_simple by metis
+  have "(\<exists>y C. C \<in> components (T - S) \<and> x \<in> C \<and> y \<in> C \<and> y \<in> L)" if "x \<in> K" for x
+  proof -
+    have "x \<in> T-S"
+      using \<open>K \<subseteq> T\<close> \<open>disjnt K S\<close> disjnt_def that by fastforce
+    then obtain C where "C \<in> components(T - S)" "x \<in> C"
+      by (metis UnionE Union_components)
+    with ovlap [of C] show ?thesis
+      by blast
+  qed
+  then obtain \<xi> where \<xi>: "\<And>x. x \<in> K \<Longrightarrow> \<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
+    by metis
+  obtain h where conth: "continuous_on (T - \<xi> ` K) h"
+             and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
+             and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
+  proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
+    show cloTS: "closedin (subtopology euclidean T) S"
+      by (simp add: \<open>compact S\<close> \<open>S \<subseteq> T\<close> closed_subset compact_imp_closed)
+    show "\<And>C. \<lbrakk>C \<in> components (T - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> \<xi> ` K \<noteq> {}"
+      using \<xi> components_eq by blast
+  qed (use K in auto)
+  show ?thesis
+  proof
+    show *: "\<xi> ` K \<subseteq> L"
+      using \<xi> by blast
+    show "finite (\<xi> ` K)"
+    show "\<xi> ` K \<subseteq> T"
+      by clarify (meson \<xi> Diff_iff contra_subsetD in_components_subset)
+    show "continuous_on (T - \<xi> ` K) h"
+      by (rule conth)
+    show "disjnt (\<xi> ` K) S"
+      using K
+      apply (auto simp: disjnt_def)
+      by (metis \<xi> DiffD2 UnionI Union_components)
+  qed (simp_all add: him hg gf)
+qed
+
+
+proposition extend_map_affine_to_sphere_cofinite_gen:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+      and aff: "aff_dim T \<le> aff_dim U"
+      and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> rel_frontier U"
+      and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+                   "g ` (T - K) \<subseteq> rel_frontier U"
+                   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "S = {}")
+  case True
+  show ?thesis
+  proof (cases "rel_frontier U = {}")
+    case True
+    with aff have "aff_dim T \<le> 0"
+      using affine_bounded_eq_lowdim \<open>bounded U\<close> order_trans by auto
+    with aff_dim_geq [of T] consider "aff_dim T = -1" |  "aff_dim T = 0"
+      by linarith
+    then show ?thesis
+    proof cases
+      assume "aff_dim T = -1"
+      then have "T = {}"
+      then show ?thesis
+        by (rule_tac K="{}" in that) auto
+    next
+      assume "aff_dim T = 0"
+      then obtain a where "T = {a}"
+        using aff_dim_eq_0 by blast
+      then have "a \<in> L"
+        using dis [of "{a}"] \<open>S = {}\<close> by (auto simp: in_components_self)
+      with \<open>S = {}\<close> \<open>T = {a}\<close> show ?thesis
+        by (rule_tac K="{a}" and g=f in that) auto
+    qed
+  next
+    case False
+    then obtain y where "y \<in> rel_frontier U"
+      by auto
+    with \<open>S = {}\<close> show ?thesis
+      by (rule_tac K="{}" and g="\<lambda>x. y" in that)  (auto simp: continuous_on_const)
+  qed
+next
+  case False
+  have "bounded S"
+    by (simp add: assms compact_imp_bounded)
+  then obtain b where b: "S \<subseteq> cbox (-b) b"
+    using bounded_subset_cbox_symmetric by blast
+  define LU where "LU \<equiv> L \<union> (\<Union> {C \<in> components (T - S). ~bounded C} - cbox (-(b+One)) (b+One))"
+  obtain K g where "finite K" "K \<subseteq> LU" "K \<subseteq> T" "disjnt K S"
+               and contg: "continuous_on (T - K) g"
+               and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+               and gf:  "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+  proof (rule extend_map_affine_to_sphere2 [OF SUT aff contf fim])
+    show "C \<inter> LU \<noteq> {}" if "C \<in> components (T - S)" for C
+    proof (cases "bounded C")
+      case True
+      with dis that show ?thesis
+        unfolding LU_def by fastforce
+    next
+      case False
+      then have "\<not> bounded (\<Union>{C \<in> components (T - S). \<not> bounded C})"
+        by (metis (no_types, lifting) Sup_upper bounded_subset mem_Collect_eq that)
+      then show ?thesis
+        apply (clarsimp simp: LU_def Int_Un_distrib Diff_Int_distrib Int_UN_distrib)
+        by (metis (no_types, lifting) False Sup_upper bounded_cbox bounded_subset inf.orderE mem_Collect_eq that)
+    qed
+  qed blast
+  have *: False if "x \<in> cbox (- b - m *\<^sub>R One) (b + m *\<^sub>R One)"
+                   "x \<notin> box (- b - n *\<^sub>R One) (b + n *\<^sub>R One)"
+                   "0 \<le> m" "m < n" "n \<le> 1" for m n x
+    using that by (auto simp: mem_box algebra_simps)
+  have "disjoint_family_on (\<lambda>d. frontier (cbox (- b - d *\<^sub>R One) (b + d *\<^sub>R One))) {1 / 2..1}"
+    by (auto simp: disjoint_family_on_def neq_iff frontier_def dest: *)
+  then obtain d where d12: "1/2 \<le> d" "d \<le> 1"
+                  and ddis: "disjnt K (frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One)))"
+    using disjoint_family_elem_disjnt [of "{1/2..1::real}" K "\<lambda>d. frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"]
+    by (auto simp: \<open>finite K\<close>)
+  define c where "c \<equiv> b + d *\<^sub>R One"
+  have cbsub: "cbox (-b) b \<subseteq> box (-c) c"
+              "cbox (-b) b \<subseteq> cbox (-c) c"
+              "cbox (-c) c \<subseteq> cbox (-(b+One)) (b+One)"
+    using d12 by (simp_all add: subset_box c_def inner_diff_left inner_left_distrib)
+  have clo_cT: "closed (cbox (- c) c \<inter> T)"
+    using affine_closed \<open>affine T\<close> by blast
+  have cT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+    using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub by fastforce
+  have S_sub_cc: "S \<subseteq> cbox (- c) c"
+    using \<open>cbox (- b) b \<subseteq> cbox (- c) c\<close> b by auto
+  show ?thesis
+  proof
+    show "finite (K \<inter> cbox (-(b+One)) (b+One))"
+      using \<open>finite K\<close> by blast
+    show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> L"
+      using \<open>K \<subseteq> LU\<close> by (auto simp: LU_def)
+    show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> T"
+      using \<open>K \<subseteq> T\<close> by auto
+    show "disjnt (K \<inter> cbox (- (b + One)) (b + One)) S"
+      using \<open>disjnt K S\<close>  by (simp add: disjnt_def disjoint_eq_subset_Compl inf.coboundedI1)
+    have cloTK: "closest_point (cbox (- c) c \<inter> T) x \<in> T - K"
+                if "x \<in> T" and Knot: "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+    proof (cases "x \<in> cbox (- c) c")
+      case True
+      with \<open>x \<in> T\<close> show ?thesis
+        using cbsub(3) Knot  by (force simp: closest_point_self)
+    next
+      case False
+      have clo_in_rf: "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+      proof (intro closest_point_in_rel_frontier [OF clo_cT cT_ne] DiffI notI)
+        have "T \<inter> interior (cbox (- c) c) \<noteq> {}"
+          using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+        then show "x \<in> affine hull (cbox (- c) c \<inter> T)"
+          by (simp add: Int_commute affine_hull_affine_Int_nonempty_interior \<open>affine T\<close> hull_inc that(1))
+      next
+        show "False" if "x \<in> rel_interior (cbox (- c) c \<inter> T)"
+        proof -
+          have "interior (cbox (- c) c) \<inter> T \<noteq> {}"
+            using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+          then have "affine hull (T \<inter> cbox (- c) c) = T"
+            using affine_hull_convex_Int_nonempty_interior [of T "cbox (- c) c"]
+            by (simp add: affine_imp_convex \<open>affine T\<close> inf_commute)
+          then show ?thesis
+            by (meson subsetD le_inf_iff rel_interior_subset that False)
+        qed
+      qed
+      have "closest_point (cbox (- c) c \<inter> T) x \<notin> K"
+      proof
+        assume inK: "closest_point (cbox (- c) c \<inter> T) x \<in> K"
+        have "\<And>x. x \<in> K \<Longrightarrow> x \<notin> frontier (cbox (- (b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+          by (metis ddis disjnt_iff)
+        then show False
+          by (metis DiffI Int_iff \<open>affine T\<close> cT_ne c_def clo_cT clo_in_rf closest_point_in_set
+                    convex_affine_rel_frontier_Int convex_box(1) empty_iff frontier_cbox inK interior_cbox)
+      qed
+      then show ?thesis
+        using cT_ne clo_cT closest_point_in_set by blast
+    qed
+    show "continuous_on (T - K \<inter> cbox (- (b + One)) (b + One)) (g \<circ> closest_point (cbox (-c) c \<inter> T))"
+      apply (intro continuous_on_compose continuous_on_closest_point continuous_on_subset [OF contg])
+         apply (simp_all add: clo_cT affine_imp_convex \<open>affine T\<close> convex_Int cT_ne)
+      using cloTK by blast
+    have "g (closest_point (cbox (- c) c \<inter> T) x) \<in> rel_frontier U"
+         if "x \<in> T" "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+      apply (rule gim [THEN subsetD])
+      using that cloTK by blast
+    then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K \<inter> cbox (- (b + One)) (b + One))
+               \<subseteq> rel_frontier U"
+      by force
+    show "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x"
+      by simp (metis (mono_tags, lifting) IntI \<open>S \<subseteq> T\<close> cT_ne clo_cT closest_point_refl gf subsetD S_sub_cc)
+  qed
+qed
+
+
+corollary extend_map_affine_to_sphere_cofinite:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes SUT: "compact S" "affine T" "S \<subseteq> T"
+      and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
+      and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> sphere a r"
+      and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+  obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+                    "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "r = 0")
+  case True
+  with fim show ?thesis
+    by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
+next
+  case False
+  with assms have "0 < r" by auto
+  then have "aff_dim T \<le> aff_dim (cball a r)"
+    by (simp add: aff aff_dim_cball)
+  then show ?thesis
+    apply (rule extend_map_affine_to_sphere_cofinite_gen
+            [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf])
+    using fim apply (auto simp: assms False that dest: dis)
+    done
+qed
+
+corollary extend_map_UNIV_to_sphere_cofinite:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+      and SUT: "compact S"
+      and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> sphere a r"
+      and dis: "\<And>C. \<lbrakk>C \<in> components(- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+  obtains K g where "finite K" "K \<subseteq> L" "disjnt K S" "continuous_on (- K) g"
+                    "g ` (- K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_affine_to_sphere_cofinite
+        [OF \<open>compact S\<close> affine_UNIV subset_UNIV _ \<open>0 \<le> r\<close> contf fim dis])
+ apply (auto simp: assms that Compl_eq_Diff_UNIV [symmetric])
+done
+
+corollary extend_map_UNIV_to_sphere_no_bounded_component:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+      and SUT: "compact S"
+      and contf: "continuous_on S f"
+      and fim: "f ` S \<subseteq> sphere a r"
+      and dis: "\<And>C. C \<in> components(- S) \<Longrightarrow> \<not> bounded C"
+  obtains g where "continuous_on UNIV g" "g ` UNIV \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_UNIV_to_sphere_cofinite [OF aff \<open>0 \<le> r\<close> \<open>compact S\<close> contf fim, of "{}"])
+   apply (auto simp: that dest: dis)
+done
+
+theorem Borsuk_separation_theorem_gen:
+  fixes S :: "'a::euclidean_space set"
+  assumes "compact S"
+    shows "(\<forall>c \<in> components(- S). ~bounded c) \<longleftrightarrow>
+           (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+                \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+       (is "?lhs = ?rhs")
+proof
+  assume L [rule_format]: ?lhs
+  show ?rhs
+  proof clarify
+    fix f :: "'a \<Rightarrow> 'a"
+    assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> sphere 0 1"
+    obtain g where contg: "continuous_on UNIV g" and gim: "range g \<subseteq> sphere 0 1"
+               and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+      by (rule extend_map_UNIV_to_sphere_no_bounded_component [OF _ _ \<open>compact S\<close> contf fim L]) auto
+    then show "\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)"
+      using nullhomotopic_from_contractible [OF contg gim]
+      by (metis assms compact_imp_closed contf empty_iff fim homotopic_with_equal nullhomotopic_into_sphere_extension)
+  qed
+next
+  assume R [rule_format]: ?rhs
+  show ?lhs
+    unfolding components_def
+  proof clarify
+    fix a
+    assume "a \<notin> S" and a: "bounded (connected_component_set (- S) a)"
+    have cont: "continuous_on S (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))"
+      apply (intro continuous_intros)
+      using \<open>a \<notin> S\<close> by auto
+    have im: "(\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) ` S \<subseteq> sphere 0 1"
+      by clarsimp (metis \<open>a \<notin> S\<close> eq_iff_diff_eq_0 left_inverse norm_eq_zero)
+    show False
+      using R cont im Borsuk_map_essential_bounded_component [OF \<open>compact S\<close> \<open>a \<notin> S\<close>] a by blast
+  qed
+qed
+
+
+corollary Borsuk_separation_theorem:
+  fixes S :: "'a::euclidean_space set"
+  assumes "compact S" and 2: "2 \<le> DIM('a)"
+    shows "connected(- S) \<longleftrightarrow>
+           (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+                \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+       (is "?lhs = ?rhs")
+proof
+  assume L: ?lhs
+  show ?rhs
+  proof (cases "S = {}")
+    case True
+    then show ?thesis by auto
+  next
+    case False
+    then have "(\<forall>c\<in>components (- S). \<not> bounded c)"
+      by (metis L assms(1) bounded_empty cobounded_imp_unbounded compact_imp_bounded in_components_maximal order_refl)
+    then show ?thesis
+      by (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>])
+  qed
+next
+  assume R: ?rhs
+  then show ?lhs
+    apply (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>, symmetric])
+    apply (auto simp: components_def connected_iff_eq_connected_component_set)
+    using connected_component_in apply fastforce
+    using cobounded_unique_unbounded_component [OF _ 2, of "-S"] \<open>compact S\<close> compact_eq_bounded_closed by fastforce
+qed
+
+
+lemma homotopy_eqv_separation:
+  fixes S :: "'a::euclidean_space set" and T :: "'a set"
+  assumes "S homotopy_eqv T" and "compact S" and "compact T"
+  shows "connected(- S) \<longleftrightarrow> connected(- T)"
+proof -
+  consider "DIM('a) = 1" | "2 \<le> DIM('a)"
+    by (metis DIM_ge_Suc0 One_nat_def Suc_1 dual_order.antisym not_less_eq_eq)
+  then show ?thesis
+  proof cases
+    case 1
+    then show ?thesis
+      using bounded_connected_Compl_1 compact_imp_bounded homotopy_eqv_empty1 homotopy_eqv_empty2 assms by metis
+  next
+    case 2
+    with assms show ?thesis
+      by (simp add: Borsuk_separation_theorem homotopy_eqv_cohomotopic_triviality_null)
+  qed
+qed
+
+lemma Jordan_Brouwer_separation:
+  fixes S :: "'a::euclidean_space set" and a::'a
+  assumes hom: "S homeomorphic sphere a r" and "0 < r"
+    shows "\<not> connected(- S)"
+proof -
+  have "- sphere a r \<inter> ball a r \<noteq> {}"
+    using \<open>0 < r\<close> by (simp add: Int_absorb1 subset_eq)
+  moreover
+  have eq: "- sphere a r - ball a r = - cball a r"
+    by auto
+  have "- cball a r \<noteq> {}"
+  proof -
+    have "frontier (cball a r) \<noteq> {}"
+      using \<open>0 < r\<close> by auto
+    then show ?thesis
+      by (metis frontier_complement frontier_empty)
+  qed
+  with eq have "- sphere a r - ball a r \<noteq> {}"
+    by auto
+  moreover
+  have "connected (- S) = connected (- sphere a r)"
+  proof (rule homotopy_eqv_separation)
+    show "S homotopy_eqv sphere a r"
+      using hom homeomorphic_imp_homotopy_eqv by blast
+    show "compact (sphere a r)"
+      by simp
+    then show " compact S"
+      using hom homeomorphic_compactness by blast
+  qed
+  ultimately show ?thesis
+    using connected_Int_frontier [of "- sphere a r" "ball a r"] by (auto simp: \<open>0 < r\<close>)
+qed
+
+
+lemma Jordan_Brouwer_frontier:
+  fixes S :: "'a::euclidean_space set" and a::'a
+  assumes S: "S homeomorphic sphere a r" and T: "T \<in> components(- S)" and 2: "2 \<le> DIM('a)"
+    shows "frontier T = S"
+proof (cases r rule: linorder_cases)
+  assume "r < 0"
+  with S T show ?thesis by auto
+next
+  assume "r = 0"
+  with S T card_eq_SucD obtain b where "S = {b}"
+    by (auto simp: homeomorphic_finite [of "{a}" S])
+  have "components (- {b}) = { -{b}}"
+    using T \<open>S = {b}\<close> by (auto simp: components_eq_sing_iff connected_punctured_universe 2)
+  with T show ?thesis
+    by (metis \<open>S = {b}\<close> cball_trivial frontier_cball frontier_complement singletonD sphere_trivial)
+next
+  assume "r > 0"
+  have "compact S"
+    using homeomorphic_compactness compact_sphere S by blast
+  show ?thesis
+  proof (rule frontier_minimal_separating_closed)
+    show "closed S"
+      using \<open>compact S\<close> compact_eq_bounded_closed by blast
+    show "\<not> connected (- S)"
+      using Jordan_Brouwer_separation S \<open>0 < r\<close> by blast
+    obtain f g where hom: "homeomorphism S (sphere a r) f g"
+      using S by (auto simp: homeomorphic_def)
+    show "connected (- T)" if "closed T" "T \<subset> S" for T
+    proof -
+      have "f ` T \<subseteq> sphere a r"
+        using \<open>T \<subset> S\<close> hom homeomorphism_image1 by blast
+      moreover have "f ` T \<noteq> sphere a r"
+        using \<open>T \<subset> S\<close> hom
+        by (metis homeomorphism_image2 homeomorphism_of_subsets order_refl psubsetE)
+      ultimately have "f ` T \<subset> sphere a r" by blast
+      then have "connected (- f ` T)"
+        by (rule psubset_sphere_Compl_connected [OF _ \<open>0 < r\<close> 2])
+      moreover have "compact T"
+        using \<open>compact S\<close> bounded_subset compact_eq_bounded_closed that by blast
+      moreover then have "compact (f ` T)"
+        by (meson compact_continuous_image continuous_on_subset hom homeomorphism_def psubsetE \<open>T \<subset> S\<close>)
+      moreover have "T homotopy_eqv f ` T"
+        by (meson \<open>f ` T \<subseteq> sphere a r\<close> dual_order.strict_implies_order hom homeomorphic_def homeomorphic_imp_homotopy_eqv homeomorphism_of_subsets \<open>T \<subset> S\<close>)
+      ultimately show ?thesis
+        using homotopy_eqv_separation [of T "f`T"] by blast
+    qed
+  qed (rule T)
+qed
+
+lemma Jordan_Brouwer_nonseparation:
+  fixes S :: "'a::euclidean_space set" and a::'a
+  assumes S: "S homeomorphic sphere a r" and "T \<subset> S" and 2: "2 \<le> DIM('a)"
+    shows "connected(- T)"
+proof -
+  have *: "connected(C \<union> (S - T))" if "C \<in> components(- S)" for C
+  proof (rule connected_intermediate_closure)
+    show "connected C"
+      using in_components_connected that by auto
+    have "S = frontier C"
+      using "2" Jordan_Brouwer_frontier S that by blast
+    with closure_subset show "C \<union> (S - T) \<subseteq> closure C"
+      by (auto simp: frontier_def)
+  qed auto
+  have "components(- S) \<noteq> {}"
+    by (metis S bounded_empty cobounded_imp_unbounded compact_eq_bounded_closed compact_sphere
+              components_eq_empty homeomorphic_compactness)
+  then have "- T = (\<Union>C \<in> components(- S). C \<union> (S - T))"
+    using Union_components [of "-S"] \<open>T \<subset> S\<close> by auto
+  then show ?thesis
+    apply (rule ssubst)
+    apply (rule connected_Union)
+    using \<open>T \<subset> S\<close> apply (auto simp: *)
+    done
+qed
+
+end```
```--- a/src/HOL/Analysis/Path_Connected.thy	Sun Oct 02 21:05:14 2016 +0200
+++ b/src/HOL/Analysis/Path_Connected.thy	Mon Oct 03 13:01:01 2016 +0100
@@ -1883,6 +1883,10 @@
finally show ?thesis .
qed

+corollary connected_punctured_universe:
+  "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})"
+  by (simp add: path_connected_punctured_universe path_connected_imp_connected)
+
lemma path_connected_sphere:
assumes "2 \<le> DIM('a::euclidean_space)"
shows "path_connected {x::'a. norm (x - a) = r}"
@@ -2104,6 +2108,32 @@
thus ?case by (metis Diff_insert)
qed

+lemma psubset_sphere_Compl_connected:
+  fixes S :: "'a::euclidean_space set"
+  assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)"
+  shows "connected(- S)"
+proof -
+  have "S \<subseteq> sphere a r"
+    using S by blast
+  obtain b where "dist a b = r" and "b \<notin> S"
+    using S mem_sphere by blast
+  have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}"
+    by (auto simp: )
+  have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}"
+    using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast
+  moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}"
+    apply (rule connected_intermediate_closure [of "ball a r"])
+    using assms by auto
+  moreover
+  have "connected {x. r \<le> dist a x \<and> x \<notin> S}"
+    apply (rule connected_intermediate_closure [of "- cball a r"])
+    using assms apply (auto intro: connected_complement_bounded_convex)
+    apply (metis ComplI interior_cball interior_closure mem_ball not_less)
+    done
+  ultimately show ?thesis
+    by (simp add: CS connected_Un)
+qed
+
subsection\<open>Relations between components and path components\<close>

lemma open_connected_component:
@@ -2505,9 +2535,9 @@
{ fix y
assume y1: "y \<in> closure (connected_component_set S x)"
and y2: "y \<notin> interior (connected_component_set S x)"
-    have 1: "y \<in> closure S"
+    have "y \<in> closure S"
using y1 closure_mono connected_component_subset by blast
-    have "z \<in> interior (connected_component_set S x)"
+    moreover have "z \<in> interior (connected_component_set S x)"
if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z
proof -
have "ball y e \<subseteq> connected_component_set S y"
@@ -2516,12 +2546,12 @@
done
then show ?thesis
using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
-        by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in subsetD
-                       dist_commute mem_Collect_eq mem_ball mem_interior \<open>0 < e\<close> y2)
+        by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2)
qed
-    then have 2: "y \<notin> interior S"
+    then have "y \<notin> interior S"
using y2 by (force simp: open_contains_ball_eq [OF open_interior])
-    note 1 2
+    ultimately have "y \<in> frontier S"
+      by (auto simp: frontier_def)
}
then show ?thesis by (auto simp: frontier_def)
qed
@@ -2565,6 +2595,49 @@
by (rule order_trans [OF frontier_Union_subset_closure])
(auto simp: closure_subset_eq)

+lemma frontier_of_components_subset:
+  fixes S :: "'a::real_normed_vector set"
+  shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S"
+  by (metis Path_Connected.frontier_of_connected_component_subset components_iff)
+
+lemma frontier_of_components_closed_complement:
+  fixes S :: "'a::real_normed_vector set"
+  shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S"
+  using frontier_complement frontier_of_components_subset frontier_subset_eq by blast
+
+lemma frontier_minimal_separating_closed:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "closed S"
+      and nconn: "~ connected(- S)"
+      and C: "C \<in> components (- S)"
+      and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)"
+    shows "frontier C = S"
+proof (rule ccontr)
+  assume "frontier C \<noteq> S"
+  then have "frontier C \<subset> S"
+    using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast
+  then have "connected(- (frontier C))"
+  have "\<not> connected(- (frontier C))"
+    unfolding connected_def not_not
+  proof (intro exI conjI)
+    show "open C"
+      using C \<open>closed S\<close> open_components by blast
+    show "open (- closure C)"
+      by blast
+    show "C \<inter> - closure C \<inter> - frontier C = {}"
+      using closure_subset by blast
+    show "C \<inter> - frontier C \<noteq> {}"
+      using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce
+    show "- frontier C \<subseteq> C \<union> - closure C"
+      by (simp add: \<open>open C\<close> closed_Compl frontier_closures)
+    then show "- closure C \<inter> - frontier C \<noteq> {}"
+      by (metis (no_types, lifting) C Compl_subset_Compl_iff \<open>frontier C \<subset> S\<close> compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb)
+  qed
+  then show False
+    using \<open>connected (- frontier C)\<close> by blast
+qed
+
lemma connected_component_UNIV [simp]:
fixes x :: "'a::real_normed_vector"
shows "connected_component_set UNIV x = UNIV"
@@ -6140,6 +6213,51 @@
apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)

+lemma homotopy_eqv_homotopic_triviality_null_imp:
+  fixes S :: "'a::real_normed_vector set"
+    and T :: "'b::real_normed_vector set"
+    and U :: "'c::real_normed_vector set"
+  assumes "S homotopy_eqv T"
+      and f: "continuous_on U f" "f ` U \<subseteq> T"
+      and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+                      \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
+    shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
+proof -
+  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+               and k: "continuous_on T k" "k ` T \<subseteq> S"
+               and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+                        "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+    using assms by (auto simp: homotopy_eqv_def)
+  obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
+    apply (rule exE [OF homSU [of "k \<circ> f"]])
+    apply (intro continuous_on_compose h)
+    using k f  apply (force elim!: continuous_on_subset)+
+    done
+  then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+    apply (rule homotopic_with_compose_continuous_left [where Y=S])
+    using h by auto
+  moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
+    apply (rule homotopic_with_compose_continuous_right [where X=T])
+      apply (simp add: hom homotopic_with_symD)
+     using f apply auto
+    done
+  ultimately show ?thesis
+    using homotopic_with_trans by (fastforce simp add: o_def)
+qed
+
+lemma homotopy_eqv_homotopic_triviality_null:
+  fixes S :: "'a::real_normed_vector set"
+    and T :: "'b::real_normed_vector set"
+    and U :: "'c::real_normed_vector set"
+  assumes "S homotopy_eqv T"
+    shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
+                  \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
+           (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
+                  \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
+
lemma homotopy_eqv_contractible_sets:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"```