--- a/src/HOL/Analysis/FurtherTopology.thy Tue Oct 18 16:05:24 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3098 +0,0 @@
-section \<open>Extending Continous Maps, Invariance of Domain, 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 Complex_Transcendental
-
-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})"
- by (simp add: g_def)
- 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)"
- by (simp add: add.assoc add.left_commute eq)
- 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>]
- by (simp add: algebra_simps)
- 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
- by (metis add.commute assms(3) diff_add_inverse2 diff_self_eq_0 dim_eq le_add1 le_antisym linordered_semidom_class.add_diff_inverse not_less0)
- 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 (simp add: segment_convex_hull)
- 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 (simp add: segment_convex_hull)
- 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 (simp add: bounded_Int)
- 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
- by (simp add: that)
-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>"
- by (simp add: Union_mono)
- 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 (simp add: Union_empty_eq)
- 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>"
- by (simp_all add: pairwise_insert)
- 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)"
- by (simp add: 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)
- apply (simp_all add: closedin_closed)
- 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 (simp add: \<H>_def)
- 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"
- by (simp add: aff)
- 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}"
- by (simp add: infinite_Icc)
- 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
- by (simp add: closest_point_self)
- 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"
- apply (simp add: j_def)
- 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 (simp add: kj)
- 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)"
- by (simp add: 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"
- apply (simp add: rel_frontier_eq_empty)
- 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 = {}"
- by (simp add: aff_dim_empty)
- 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
-
-subsection\<open> Invariance of domain and corollaries\<close>
-
-lemma invariance_of_domain_ball:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on (cball a r) f" and "0 < r"
- and inj: "inj_on f (cball a r)"
- shows "open(f ` ball a r)"
-proof (cases "DIM('a) = 1")
- case True
- obtain h::"'a\<Rightarrow>real" and k
- where "linear h" "linear k" "h ` UNIV = UNIV" "k ` UNIV = UNIV"
- "\<And>x. norm(h x) = norm x" "\<And>x. norm(k x) = norm x"
- "\<And>x. k(h x) = x" "\<And>x. h(k x) = x"
- apply (rule isomorphisms_UNIV_UNIV [where 'M='a and 'N=real])
- using True
- apply force
- by (metis UNIV_I UNIV_eq_I imageI)
- have cont: "continuous_on S h" "continuous_on T k" for S T
- by (simp_all add: \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_linear)
- have "continuous_on (h ` cball a r) (h \<circ> f \<circ> k)"
- apply (intro continuous_on_compose cont continuous_on_subset [OF contf])
- apply (auto simp: \<open>\<And>x. k (h x) = x\<close>)
- done
- moreover have "is_interval (h ` cball a r)"
- by (simp add: is_interval_connected_1 \<open>linear h\<close> linear_continuous_on linear_linear connected_continuous_image)
- moreover have "inj_on (h \<circ> f \<circ> k) (h ` cball a r)"
- using inj by (simp add: inj_on_def) (metis \<open>\<And>x. k (h x) = x\<close>)
- ultimately have *: "\<And>T. \<lbrakk>open T; T \<subseteq> h ` cball a r\<rbrakk> \<Longrightarrow> open ((h \<circ> f \<circ> k) ` T)"
- using injective_eq_1d_open_map_UNIV by blast
- have "open ((h \<circ> f \<circ> k) ` (h ` ball a r))"
- by (rule *) (auto simp: \<open>linear h\<close> \<open>range h = UNIV\<close> open_surjective_linear_image)
- then have "open ((h \<circ> f) ` ball a r)"
- by (simp add: image_comp \<open>\<And>x. k (h x) = x\<close> cong: image_cong)
- then show ?thesis
- apply (simp add: image_comp [symmetric])
- apply (metis open_bijective_linear_image_eq \<open>linear h\<close> \<open>\<And>x. k (h x) = x\<close> \<open>range h = UNIV\<close> bijI inj_on_def)
- done
-next
- case False
- then have 2: "DIM('a) \<ge> 2"
- by (metis DIM_ge_Suc0 One_nat_def Suc_1 antisym not_less_eq_eq)
- have fimsub: "f ` ball a r \<subseteq> - f ` sphere a r"
- using inj by clarsimp (metis inj_onD less_eq_real_def mem_cball order_less_irrefl)
- have hom: "f ` sphere a r homeomorphic sphere a r"
- by (meson compact_sphere contf continuous_on_subset homeomorphic_compact homeomorphic_sym inj inj_on_subset sphere_cball)
- then have nconn: "\<not> connected (- f ` sphere a r)"
- by (rule Jordan_Brouwer_separation) (auto simp: \<open>0 < r\<close>)
- obtain C where C: "C \<in> components (- f ` sphere a r)" and "bounded C"
- apply (rule cobounded_has_bounded_component [OF _ nconn])
- apply (simp_all add: 2)
- by (meson compact_imp_bounded compact_continuous_image_eq compact_sphere contf inj sphere_cball)
- moreover have "f ` (ball a r) = C"
- proof
- have "C \<noteq> {}"
- by (rule in_components_nonempty [OF C])
- show "C \<subseteq> f ` ball a r"
- proof (rule ccontr)
- assume nonsub: "\<not> C \<subseteq> f ` ball a r"
- have "- f ` cball a r \<subseteq> C"
- proof (rule components_maximal [OF C])
- have "f ` cball a r homeomorphic cball a r"
- using compact_cball contf homeomorphic_compact homeomorphic_sym inj by blast
- then show "connected (- f ` cball a r)"
- by (auto intro: connected_complement_homeomorphic_convex_compact 2)
- show "- f ` cball a r \<subseteq> - f ` sphere a r"
- by auto
- then show "C \<inter> - f ` cball a r \<noteq> {}"
- using \<open>C \<noteq> {}\<close> in_components_subset [OF C] nonsub
- using image_iff by fastforce
- qed
- then have "bounded (- f ` cball a r)"
- using bounded_subset \<open>bounded C\<close> by auto
- then have "\<not> bounded (f ` cball a r)"
- using cobounded_imp_unbounded by blast
- then show "False"
- using compact_continuous_image [OF contf] compact_cball compact_imp_bounded by blast
- qed
- with \<open>C \<noteq> {}\<close> have "C \<inter> f ` ball a r \<noteq> {}"
- by (simp add: inf.absorb_iff1)
- then show "f ` ball a r \<subseteq> C"
- by (metis components_maximal [OF C _ fimsub] connected_continuous_image ball_subset_cball connected_ball contf continuous_on_subset)
- qed
- moreover have "open (- f ` sphere a r)"
- using hom compact_eq_bounded_closed compact_sphere homeomorphic_compactness by blast
- ultimately show ?thesis
- using open_components by blast
-qed
-
-
-text\<open>Proved by L. E. J. Brouwer (1912)\<close>
-theorem invariance_of_domain:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes "continuous_on S f" "open S" "inj_on f S"
- shows "open(f ` S)"
- unfolding open_subopen [of "f`S"]
-proof clarify
- fix a
- assume "a \<in> S"
- obtain \<delta> where "\<delta> > 0" and \<delta>: "cball a \<delta> \<subseteq> S"
- using \<open>open S\<close> \<open>a \<in> S\<close> open_contains_cball_eq by blast
- show "\<exists>T. open T \<and> f a \<in> T \<and> T \<subseteq> f ` S"
- proof (intro exI conjI)
- show "open (f ` (ball a \<delta>))"
- by (meson \<delta> \<open>0 < \<delta>\<close> assms continuous_on_subset inj_on_subset invariance_of_domain_ball)
- show "f a \<in> f ` ball a \<delta>"
- by (simp add: \<open>0 < \<delta>\<close>)
- show "f ` ball a \<delta> \<subseteq> f ` S"
- using \<delta> ball_subset_cball by blast
- qed
-qed
-
-lemma inv_of_domain_ss0:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
- and "subspace S" and dimS: "dim S = DIM('b::euclidean_space)"
- and ope: "openin (subtopology euclidean S) U"
- shows "openin (subtopology euclidean S) (f ` U)"
-proof -
- have "U \<subseteq> S"
- using ope openin_imp_subset by blast
- have "(UNIV::'b set) homeomorphic S"
- by (simp add: \<open>subspace S\<close> dimS dim_UNIV homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism (UNIV::'b set) S h k"
- using homeomorphic_def by blast
- have homkh: "homeomorphism S (k ` S) k h"
- using homhk homeomorphism_image2 homeomorphism_sym by fastforce
- have "open ((k \<circ> f \<circ> h) ` k ` U)"
- proof (rule invariance_of_domain)
- show "continuous_on (k ` U) (k \<circ> f \<circ> h)"
- proof (intro continuous_intros)
- show "continuous_on (k ` U) h"
- by (meson continuous_on_subset [OF homeomorphism_cont1 [OF homhk]] top_greatest)
- show "continuous_on (h ` k ` U) f"
- apply (rule continuous_on_subset [OF contf], clarify)
- apply (metis homhk homeomorphism_def ope openin_imp_subset rev_subsetD)
- done
- show "continuous_on (f ` h ` k ` U) k"
- apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
- using fim homhk homeomorphism_apply2 ope openin_subset by fastforce
- qed
- have ope_iff: "\<And>T. open T \<longleftrightarrow> openin (subtopology euclidean (k ` S)) T"
- using homhk homeomorphism_image2 open_openin by fastforce
- show "open (k ` U)"
- by (simp add: ope_iff homeomorphism_imp_open_map [OF homkh ope])
- show "inj_on (k \<circ> f \<circ> h) (k ` U)"
- apply (clarsimp simp: inj_on_def)
- by (metis subsetD fim homeomorphism_apply2 [OF homhk] image_subset_iff inj_on_eq_iff injf \<open>U \<subseteq> S\<close>)
- qed
- moreover
- have eq: "f ` U = h ` (k \<circ> f \<circ> h \<circ> k) ` U"
- apply (auto simp: image_comp [symmetric])
- apply (metis homkh \<open>U \<subseteq> S\<close> fim homeomorphism_image2 homeomorphism_of_subsets homhk imageI subset_UNIV)
- by (metis \<open>U \<subseteq> S\<close> subsetD fim homeomorphism_def homhk image_eqI)
- ultimately show ?thesis
- by (metis (no_types, hide_lams) homeomorphism_imp_open_map homhk image_comp open_openin subtopology_UNIV)
-qed
-
-lemma inv_of_domain_ss1:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
- and "subspace S"
- and ope: "openin (subtopology euclidean S) U"
- shows "openin (subtopology euclidean S) (f ` U)"
-proof -
- define S' where "S' \<equiv> {y. \<forall>x \<in> S. orthogonal x y}"
- have "subspace S'"
- by (simp add: S'_def subspace_orthogonal_to_vectors)
- define g where "g \<equiv> \<lambda>z::'a*'a. ((f \<circ> fst)z, snd z)"
- have "openin (subtopology euclidean (S \<times> S')) (g ` (U \<times> S'))"
- proof (rule inv_of_domain_ss0)
- show "continuous_on (U \<times> S') g"
- apply (simp add: g_def)
- apply (intro continuous_intros continuous_on_compose2 [OF contf continuous_on_fst], auto)
- done
- show "g ` (U \<times> S') \<subseteq> S \<times> S'"
- using fim by (auto simp: g_def)
- show "inj_on g (U \<times> S')"
- using injf by (auto simp: g_def inj_on_def)
- show "subspace (S \<times> S')"
- by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> subspace_Times)
- show "openin (subtopology euclidean (S \<times> S')) (U \<times> S')"
- by (simp add: openin_Times [OF ope])
- have "dim (S \<times> S') = dim S + dim S'"
- by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> dim_Times)
- also have "... = DIM('a)"
- using dim_subspace_orthogonal_to_vectors [OF \<open>subspace S\<close> subspace_UNIV]
- by (simp add: add.commute S'_def)
- finally show "dim (S \<times> S') = DIM('a)" .
- qed
- moreover have "g ` (U \<times> S') = f ` U \<times> S'"
- by (auto simp: g_def image_iff)
- moreover have "0 \<in> S'"
- using \<open>subspace S'\<close> subspace_affine by blast
- ultimately show ?thesis
- by (auto simp: openin_Times_eq)
-qed
-
-
-corollary invariance_of_domain_subspaces:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S"
- shows "openin (subtopology euclidean V) (f ` S)"
-proof -
- obtain V' where "subspace V'" "V' \<subseteq> U" "dim V' = dim V"
- using choose_subspace_of_subspace [OF VU]
- by (metis span_eq \<open>subspace U\<close>)
- then have "V homeomorphic V'"
- by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism V V' h k"
- using homeomorphic_def by blast
- have eq: "f ` S = k ` (h \<circ> f) ` S"
- proof -
- have "k ` h ` f ` S = f ` S"
- by (meson fim homeomorphism_def homeomorphism_of_subsets homhk subset_refl)
- then show ?thesis
- by (simp add: image_comp)
- qed
- show ?thesis
- unfolding eq
- proof (rule homeomorphism_imp_open_map)
- show homkh: "homeomorphism V' V k h"
- by (simp add: homeomorphism_symD homhk)
- have hfV': "(h \<circ> f) ` S \<subseteq> V'"
- using fim homeomorphism_image1 homhk by fastforce
- moreover have "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
- proof (rule inv_of_domain_ss1)
- show "continuous_on S (h \<circ> f)"
- by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
- show "inj_on (h \<circ> f) S"
- apply (clarsimp simp: inj_on_def)
- by (metis fim homeomorphism_apply2 [OF homkh] image_subset_iff inj_onD injf)
- show "(h \<circ> f) ` S \<subseteq> U"
- using \<open>V' \<subseteq> U\<close> hfV' by auto
- qed (auto simp: assms)
- ultimately show "openin (subtopology euclidean V') ((h \<circ> f) ` S)"
- using openin_subset_trans \<open>V' \<subseteq> U\<close> by force
- qed
-qed
-
-corollary invariance_of_dimension_subspaces:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and "subspace U" "subspace V"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "dim U \<le> dim V"
-proof -
- have "False" if "dim V < dim U"
- proof -
- obtain T where "subspace T" "T \<subseteq> U" "dim T = dim V"
- using choose_subspace_of_subspace [of "dim V" U]
- by (metis span_eq \<open>subspace U\<close> \<open>dim V < dim U\<close> linear not_le)
- then have "V homeomorphic T"
- by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism V T h k"
- using homeomorphic_def by blast
- have "continuous_on S (h \<circ> f)"
- by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
- moreover have "(h \<circ> f) ` S \<subseteq> U"
- using \<open>T \<subseteq> U\<close> fim homeomorphism_image1 homhk by fastforce
- moreover have "inj_on (h \<circ> f) S"
- apply (clarsimp simp: inj_on_def)
- by (metis fim homeomorphism_apply1 homhk image_subset_iff inj_onD injf)
- ultimately have ope_hf: "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
- using invariance_of_domain_subspaces [OF ope \<open>subspace U\<close> \<open>subspace U\<close>] by auto
- have "(h \<circ> f) ` S \<subseteq> T"
- using fim homeomorphism_image1 homhk by fastforce
- then show ?thesis
- by (metis dim_openin \<open>dim T = dim V\<close> ope_hf \<open>subspace U\<close> \<open>S \<noteq> {}\<close> dim_subset image_is_empty not_le that)
- qed
- then show ?thesis
- using not_less by blast
-qed
-
-corollary invariance_of_domain_affine_sets:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S"
- shows "openin (subtopology euclidean V) (f ` S)"
-proof (cases "S = {}")
- case True
- then show ?thesis by auto
-next
- case False
- obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
- using False fim ope openin_contains_cball by fastforce
- have "openin (subtopology euclidean (op + (- b) ` V)) ((op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S)"
- proof (rule invariance_of_domain_subspaces)
- show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
- by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
- show "subspace (op + (- a) ` U)"
- by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
- show "subspace (op + (- b) ` V)"
- by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
- show "dim (op + (- b) ` V) \<le> dim (op + (- a) ` U)"
- by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
- show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
- by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
- show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
- using fim by auto
- show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
- by (auto simp: inj_on_def) (meson inj_onD injf)
- qed
- then show ?thesis
- by (metis (no_types, lifting) homeomorphism_imp_open_map homeomorphism_translation image_comp translation_galois)
-qed
-
-corollary invariance_of_dimension_affine_sets:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and aff: "affine U" "affine V"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "aff_dim U \<le> aff_dim V"
-proof -
- obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
- using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
- have "dim (op + (- a) ` U) \<le> dim (op + (- b) ` V)"
- proof (rule invariance_of_dimension_subspaces)
- show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
- by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
- show "subspace (op + (- a) ` U)"
- by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
- show "subspace (op + (- b) ` V)"
- by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
- show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
- by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
- show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
- using fim by auto
- show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
- by (auto simp: inj_on_def) (meson inj_onD injf)
- qed (use \<open>S \<noteq> {}\<close> in auto)
- then show ?thesis
- by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
-qed
-
-corollary invariance_of_dimension:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes contf: "continuous_on S f" and "open S"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "DIM('a) \<le> DIM('b)"
- using invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
- by auto
-
-
-corollary continuous_injective_image_subspace_dim_le:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "subspace S" "subspace T"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
- and injf: "inj_on f S"
- shows "dim S \<le> dim T"
- apply (rule invariance_of_dimension_subspaces [of S S _ f])
- using assms by (auto simp: subspace_affine)
-
-lemma invariance_of_dimension_convex_domain:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "convex S"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> affine hull T"
- and injf: "inj_on f S"
- shows "aff_dim S \<le> aff_dim T"
-proof (cases "S = {}")
- case True
- then show ?thesis by (simp add: aff_dim_geq)
-next
- case False
- have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
- proof (rule invariance_of_dimension_affine_sets)
- show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
- by (simp add: openin_rel_interior)
- show "continuous_on (rel_interior S) f"
- using contf continuous_on_subset rel_interior_subset by blast
- show "f ` rel_interior S \<subseteq> affine hull T"
- using fim rel_interior_subset by blast
- show "inj_on f (rel_interior S)"
- using inj_on_subset injf rel_interior_subset by blast
- show "rel_interior S \<noteq> {}"
- by (simp add: False \<open>convex S\<close> rel_interior_eq_empty)
- qed auto
- then show ?thesis
- by simp
-qed
-
-
-lemma homeomorphic_convex_sets_le:
- assumes "convex S" "S homeomorphic T"
- shows "aff_dim S \<le> aff_dim T"
-proof -
- obtain h k where homhk: "homeomorphism S T h k"
- using homeomorphic_def assms by blast
- show ?thesis
- proof (rule invariance_of_dimension_convex_domain [OF \<open>convex S\<close>])
- show "continuous_on S h"
- using homeomorphism_def homhk by blast
- show "h ` S \<subseteq> affine hull T"
- by (metis homeomorphism_def homhk hull_subset)
- show "inj_on h S"
- by (meson homeomorphism_apply1 homhk inj_on_inverseI)
- qed
-qed
-
-lemma homeomorphic_convex_sets:
- assumes "convex S" "convex T" "S homeomorphic T"
- shows "aff_dim S = aff_dim T"
- by (meson assms dual_order.antisym homeomorphic_convex_sets_le homeomorphic_sym)
-
-lemma homeomorphic_convex_compact_sets_eq:
- assumes "convex S" "compact S" "convex T" "compact T"
- shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
- by (meson assms homeomorphic_convex_compact_sets homeomorphic_convex_sets)
-
-lemma invariance_of_domain_gen:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
- shows "open(f ` S)"
- using invariance_of_domain_subspaces [of UNIV S UNIV f] assms by auto
-
-lemma injective_into_1d_imp_open_map_UNIV:
- fixes f :: "'a::euclidean_space \<Rightarrow> real"
- assumes "open T" "continuous_on S f" "inj_on f S" "T \<subseteq> S"
- shows "open (f ` T)"
- apply (rule invariance_of_domain_gen [OF \<open>open T\<close>])
- using assms apply (auto simp: elim: continuous_on_subset subset_inj_on)
- done
-
-lemma continuous_on_inverse_open:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
- shows "continuous_on (f ` S) g"
-proof (clarsimp simp add: continuous_openin_preimage_eq)
- fix T :: "'a set"
- assume "open T"
- have eq: "{x. x \<in> f ` S \<and> g x \<in> T} = f ` (S \<inter> T)"
- by (auto simp: gf)
- show "openin (subtopology euclidean (f ` S)) {x \<in> f ` S. g x \<in> T}"
- apply (subst eq)
- apply (rule open_openin_trans)
- apply (rule invariance_of_domain_gen)
- using assms
- apply auto
- using inj_on_inverseI apply auto[1]
- by (metis \<open>open T\<close> continuous_on_subset inj_onI inj_on_subset invariance_of_domain_gen openin_open openin_open_eq)
-qed
-
-lemma invariance_of_domain_homeomorphism:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
- obtains g where "homeomorphism S (f ` S) f g"
-proof
- show "homeomorphism S (f ` S) f (inv_into S f)"
- by (simp add: assms continuous_on_inverse_open homeomorphism_def)
-qed
-
-corollary invariance_of_domain_homeomorphic:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
- shows "S homeomorphic (f ` S)"
- using invariance_of_domain_homeomorphism [OF assms]
- by (meson homeomorphic_def)
-
-lemma continuous_image_subset_interior:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
- shows "f ` (interior S) \<subseteq> interior(f ` S)"
- apply (rule interior_maximal)
- apply (simp add: image_mono interior_subset)
- apply (rule invariance_of_domain_gen)
- using assms
- apply (auto simp: subset_inj_on interior_subset continuous_on_subset)
- done
-
-lemma homeomorphic_interiors_same_dimension:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" and dimeq: "DIM('a) = DIM('b)"
- shows "(interior S) homeomorphic (interior T)"
- using assms [unfolded homeomorphic_minimal]
- unfolding homeomorphic_def
-proof (clarify elim!: ex_forward)
- fix f g
- assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
- by (auto simp: inj_on_def intro: rev_image_eqI) metis+
- have fim: "f ` interior S \<subseteq> interior T"
- using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
- have gim: "g ` interior T \<subseteq> interior S"
- using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
- show "homeomorphism (interior S) (interior T) f g"
- unfolding homeomorphism_def
- proof (intro conjI ballI)
- show "\<And>x. x \<in> interior S \<Longrightarrow> g (f x) = x"
- by (meson \<open>\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x\<close> subsetD interior_subset)
- have "interior T \<subseteq> f ` interior S"
- proof
- fix x assume "x \<in> interior T"
- then have "g x \<in> interior S"
- using gim by blast
- then show "x \<in> f ` interior S"
- by (metis T \<open>x \<in> interior T\<close> image_iff interior_subset subsetCE)
- qed
- then show "f ` interior S = interior T"
- using fim by blast
- show "continuous_on (interior S) f"
- by (metis interior_subset continuous_on_subset contf)
- show "\<And>y. y \<in> interior T \<Longrightarrow> f (g y) = y"
- by (meson T subsetD interior_subset)
- have "interior S \<subseteq> g ` interior T"
- proof
- fix x assume "x \<in> interior S"
- then have "f x \<in> interior T"
- using fim by blast
- then show "x \<in> g ` interior T"
- by (metis S \<open>x \<in> interior S\<close> image_iff interior_subset subsetCE)
- qed
- then show "g ` interior T = interior S"
- using gim by blast
- show "continuous_on (interior T) g"
- by (metis interior_subset continuous_on_subset contg)
- qed
-qed
-
-lemma homeomorphic_open_imp_same_dimension:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "open S" "S \<noteq> {}" "open T" "T \<noteq> {}"
- shows "DIM('a) = DIM('b)"
- using assms
- apply (simp add: homeomorphic_minimal)
- apply (rule order_antisym; metis inj_onI invariance_of_dimension)
- done
-
-lemma homeomorphic_interiors:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "interior S = {} \<longleftrightarrow> interior T = {}"
- shows "(interior S) homeomorphic (interior T)"
-proof (cases "interior T = {}")
- case True
- with assms show ?thesis by auto
-next
- case False
- then have "DIM('a) = DIM('b)"
- using assms
- apply (simp add: homeomorphic_minimal)
- apply (rule order_antisym; metis continuous_on_subset inj_onI inj_on_subset interior_subset invariance_of_dimension open_interior)
- done
- then show ?thesis
- by (rule homeomorphic_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
-qed
-
-lemma homeomorphic_frontiers_same_dimension:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "closed S" "closed T" and dimeq: "DIM('a) = DIM('b)"
- shows "(frontier S) homeomorphic (frontier T)"
- using assms [unfolded homeomorphic_minimal]
- unfolding homeomorphic_def
-proof (clarify elim!: ex_forward)
- fix f g
- assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
- by (auto simp: inj_on_def intro: rev_image_eqI) metis+
- have "g ` interior T \<subseteq> interior S"
- using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
- then have fim: "f ` frontier S \<subseteq> frontier T"
- apply (simp add: frontier_def)
- using continuous_image_subset_interior assms(2) assms(3) S by auto
- have "f ` interior S \<subseteq> interior T"
- using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
- then have gim: "g ` frontier T \<subseteq> frontier S"
- apply (simp add: frontier_def)
- using continuous_image_subset_interior T assms(2) assms(3) by auto
- show "homeomorphism (frontier S) (frontier T) f g"
- unfolding homeomorphism_def
- proof (intro conjI ballI)
- show gf: "\<And>x. x \<in> frontier S \<Longrightarrow> g (f x) = x"
- by (simp add: S assms(2) frontier_def)
- show fg: "\<And>y. y \<in> frontier T \<Longrightarrow> f (g y) = y"
- by (simp add: T assms(3) frontier_def)
- have "frontier T \<subseteq> f ` frontier S"
- proof
- fix x assume "x \<in> frontier T"
- then have "g x \<in> frontier S"
- using gim by blast
- then show "x \<in> f ` frontier S"
- by (metis fg \<open>x \<in> frontier T\<close> imageI)
- qed
- then show "f ` frontier S = frontier T"
- using fim by blast
- show "continuous_on (frontier S) f"
- by (metis Diff_subset assms(2) closure_eq contf continuous_on_subset frontier_def)
- have "frontier S \<subseteq> g ` frontier T"
- proof
- fix x assume "x \<in> frontier S"
- then have "f x \<in> frontier T"
- using fim by blast
- then show "x \<in> g ` frontier T"
- by (metis gf \<open>x \<in> frontier S\<close> imageI)
- qed
- then show "g ` frontier T = frontier S"
- using gim by blast
- show "continuous_on (frontier T) g"
- by (metis Diff_subset assms(3) closure_closed contg continuous_on_subset frontier_def)
- qed
-qed
-
-lemma homeomorphic_frontiers:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "closed S" "closed T"
- "interior S = {} \<longleftrightarrow> interior T = {}"
- shows "(frontier S) homeomorphic (frontier T)"
-proof (cases "interior T = {}")
- case True
- then show ?thesis
- by (metis Diff_empty assms closure_eq frontier_def)
-next
- case False
- show ?thesis
- apply (rule homeomorphic_frontiers_same_dimension)
- apply (simp_all add: assms)
- using False assms homeomorphic_interiors homeomorphic_open_imp_same_dimension by blast
-qed
-
-lemma continuous_image_subset_rel_interior:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
- and TS: "aff_dim T \<le> aff_dim S"
- shows "f ` (rel_interior S) \<subseteq> rel_interior(f ` S)"
-proof (rule rel_interior_maximal)
- show "f ` rel_interior S \<subseteq> f ` S"
- by(simp add: image_mono rel_interior_subset)
- show "openin (subtopology euclidean (affine hull f ` S)) (f ` rel_interior S)"
- proof (rule invariance_of_domain_affine_sets)
- show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
- by (simp add: openin_rel_interior)
- show "aff_dim (affine hull f ` S) \<le> aff_dim (affine hull S)"
- by (metis aff_dim_affine_hull aff_dim_subset fim TS order_trans)
- show "f ` rel_interior S \<subseteq> affine hull f ` S"
- by (meson \<open>f ` rel_interior S \<subseteq> f ` S\<close> hull_subset order_trans)
- show "continuous_on (rel_interior S) f"
- using contf continuous_on_subset rel_interior_subset by blast
- show "inj_on f (rel_interior S)"
- using inj_on_subset injf rel_interior_subset by blast
- qed auto
-qed
-
-lemma homeomorphic_rel_interiors_same_dimension:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
- shows "(rel_interior S) homeomorphic (rel_interior T)"
- using assms [unfolded homeomorphic_minimal]
- unfolding homeomorphic_def
-proof (clarify elim!: ex_forward)
- fix f g
- assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
- by (auto simp: inj_on_def intro: rev_image_eqI) metis+
- have fim: "f ` rel_interior S \<subseteq> rel_interior T"
- by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
- have gim: "g ` rel_interior T \<subseteq> rel_interior S"
- by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
- show "homeomorphism (rel_interior S) (rel_interior T) f g"
- unfolding homeomorphism_def
- proof (intro conjI ballI)
- show gf: "\<And>x. x \<in> rel_interior S \<Longrightarrow> g (f x) = x"
- using S rel_interior_subset by blast
- show fg: "\<And>y. y \<in> rel_interior T \<Longrightarrow> f (g y) = y"
- using T mem_rel_interior_ball by blast
- have "rel_interior T \<subseteq> f ` rel_interior S"
- proof
- fix x assume "x \<in> rel_interior T"
- then have "g x \<in> rel_interior S"
- using gim by blast
- then show "x \<in> f ` rel_interior S"
- by (metis fg \<open>x \<in> rel_interior T\<close> imageI)
- qed
- moreover have "f ` rel_interior S \<subseteq> rel_interior T"
- by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
- ultimately show "f ` rel_interior S = rel_interior T"
- by blast
- show "continuous_on (rel_interior S) f"
- using contf continuous_on_subset rel_interior_subset by blast
- have "rel_interior S \<subseteq> g ` rel_interior T"
- proof
- fix x assume "x \<in> rel_interior S"
- then have "f x \<in> rel_interior T"
- using fim by blast
- then show "x \<in> g ` rel_interior T"
- by (metis gf \<open>x \<in> rel_interior S\<close> imageI)
- qed
- then show "g ` rel_interior T = rel_interior S"
- using gim by blast
- show "continuous_on (rel_interior T) g"
- using contg continuous_on_subset rel_interior_subset by blast
- qed
-qed
-
-lemma homeomorphic_rel_interiors:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
- shows "(rel_interior S) homeomorphic (rel_interior T)"
-proof (cases "rel_interior T = {}")
- case True
- with assms show ?thesis by auto
-next
- case False
- obtain f g
- where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- using assms [unfolded homeomorphic_minimal] by auto
- have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
- apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
- apply (simp_all add: openin_rel_interior False assms)
- using contf continuous_on_subset rel_interior_subset apply blast
- apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
- apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
- done
- moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
- apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
- apply (simp_all add: openin_rel_interior False assms)
- using contg continuous_on_subset rel_interior_subset apply blast
- apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
- apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
- done
- ultimately have "aff_dim S = aff_dim T" by force
- then show ?thesis
- by (rule homeomorphic_rel_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
-qed
-
-
-lemma homeomorphic_rel_boundaries_same_dimension:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
- shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
- using assms [unfolded homeomorphic_minimal]
- unfolding homeomorphic_def
-proof (clarify elim!: ex_forward)
- fix f g
- assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
- by (auto simp: inj_on_def intro: rev_image_eqI) metis+
- have fim: "f ` rel_interior S \<subseteq> rel_interior T"
- by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
- have gim: "g ` rel_interior T \<subseteq> rel_interior S"
- by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
- show "homeomorphism (S - rel_interior S) (T - rel_interior T) f g"
- unfolding homeomorphism_def
- proof (intro conjI ballI)
- show gf: "\<And>x. x \<in> S - rel_interior S \<Longrightarrow> g (f x) = x"
- using S rel_interior_subset by blast
- show fg: "\<And>y. y \<in> T - rel_interior T \<Longrightarrow> f (g y) = y"
- using T mem_rel_interior_ball by blast
- show "f ` (S - rel_interior S) = T - rel_interior T"
- using S fST fim gim by auto
- show "continuous_on (S - rel_interior S) f"
- using contf continuous_on_subset rel_interior_subset by blast
- show "g ` (T - rel_interior T) = S - rel_interior S"
- using T gTS gim fim by auto
- show "continuous_on (T - rel_interior T) g"
- using contg continuous_on_subset rel_interior_subset by blast
- qed
-qed
-
-lemma homeomorphic_rel_boundaries:
- fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
- assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
- shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
-proof (cases "rel_interior T = {}")
- case True
- with assms show ?thesis by auto
-next
- case False
- obtain f g
- where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- using assms [unfolded homeomorphic_minimal] by auto
- have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
- apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
- apply (simp_all add: openin_rel_interior False assms)
- using contf continuous_on_subset rel_interior_subset apply blast
- apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
- apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
- done
- moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
- apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
- apply (simp_all add: openin_rel_interior False assms)
- using contg continuous_on_subset rel_interior_subset apply blast
- apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
- apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
- done
- ultimately have "aff_dim S = aff_dim T" by force
- then show ?thesis
- by (rule homeomorphic_rel_boundaries_same_dimension [OF \<open>S homeomorphic T\<close>])
-qed
-
-proposition uniformly_continuous_homeomorphism_UNIV_trivial:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
- assumes contf: "uniformly_continuous_on S f" and hom: "homeomorphism S UNIV f g"
- shows "S = UNIV"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (metis UNIV_I hom empty_iff homeomorphism_def image_eqI)
-next
- case False
- have "inj g"
- by (metis UNIV_I hom homeomorphism_apply2 injI)
- then have "open (g ` UNIV)"
- by (blast intro: invariance_of_domain hom homeomorphism_cont2)
- then have "open S"
- using hom homeomorphism_image2 by blast
- moreover have "complete S"
- unfolding complete_def
- proof clarify
- fix \<sigma>
- assume \<sigma>: "\<forall>n. \<sigma> n \<in> S" and "Cauchy \<sigma>"
- have "Cauchy (f o \<sigma>)"
- using uniformly_continuous_imp_Cauchy_continuous \<open>Cauchy \<sigma>\<close> \<sigma> contf by blast
- then obtain l where "(f \<circ> \<sigma>) \<longlonglongrightarrow> l"
- by (auto simp: convergent_eq_Cauchy [symmetric])
- show "\<exists>l\<in>S. \<sigma> \<longlonglongrightarrow> l"
- proof
- show "g l \<in> S"
- using hom homeomorphism_image2 by blast
- have "(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l"
- by (meson UNIV_I \<open>(f \<circ> \<sigma>) \<longlonglongrightarrow> l\<close> continuous_on_sequentially hom homeomorphism_cont2)
- then show "\<sigma> \<longlonglongrightarrow> g l"
- proof -
- have "\<forall>n. \<sigma> n = (g \<circ> (f \<circ> \<sigma>)) n"
- by (metis (no_types) \<sigma> comp_eq_dest_lhs hom homeomorphism_apply1)
- then show ?thesis
- by (metis (no_types) LIMSEQ_iff \<open>(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l\<close>)
- qed
- qed
- qed
- then have "closed S"
- by (simp add: complete_eq_closed)
- ultimately show ?thesis
- using clopen [of S] False by simp
-qed
-
-subsection\<open>The power, squaring and exponential functions as covering maps\<close>
-
-proposition covering_space_power_punctured_plane:
- assumes "0 < n"
- shows "covering_space (- {0}) (\<lambda>z::complex. z^n) (- {0})"
-proof -
- consider "n = 1" | "2 \<le> n" using assms by linarith
- then obtain e where "0 < e"
- and e: "\<And>w z. cmod(w - z) < e * cmod z \<Longrightarrow> (w^n = z^n \<longleftrightarrow> w = z)"
- proof cases
- assume "n = 1" then show ?thesis
- by (rule_tac e=1 in that) auto
- next
- assume "2 \<le> n"
- have eq_if_pow_eq:
- "w = z" if lt: "cmod (w - z) < 2 * sin (pi / real n) * cmod z"
- and eq: "w^n = z^n" for w z
- proof (cases "z = 0")
- case True with eq assms show ?thesis by (auto simp: power_0_left)
- next
- case False
- then have "z \<noteq> 0" by auto
- have "(w/z)^n = 1"
- by (metis False divide_self_if eq power_divide power_one)
- then obtain j where j: "w / z = exp (2 * of_real pi * \<i> * j / n)" and "j < n"
- using Suc_leI assms \<open>2 \<le> n\<close> complex_roots_unity [THEN eqset_imp_iff, of n "w/z"]
- by force
- have "cmod (w/z - 1) < 2 * sin (pi / real n)"
- using lt assms \<open>z \<noteq> 0\<close> by (simp add: divide_simps norm_divide)
- then have "cmod (exp (\<i> * of_real (2 * pi * j / n)) - 1) < 2 * sin (pi / real n)"
- by (simp add: j field_simps)
- then have "2 * \<bar>sin((2 * pi * j / n) / 2)\<bar> < 2 * sin (pi / real n)"
- by (simp only: dist_exp_ii_1)
- then have sin_less: "sin((pi * j / n)) < sin (pi / real n)"
- by (simp add: field_simps)
- then have "w / z = 1"
- proof (cases "j = 0")
- case True then show ?thesis by (auto simp: j)
- next
- case False
- then have "sin (pi / real n) \<le> sin((pi * j / n))"
- proof (cases "j / n \<le> 1/2")
- case True
- show ?thesis
- apply (rule sin_monotone_2pi_le)
- using \<open>j \<noteq> 0 \<close> \<open>j < n\<close> True
- apply (auto simp: field_simps intro: order_trans [of _ 0])
- done
- next
- case False
- then have seq: "sin(pi * j / n) = sin(pi * (n - j) / n)"
- using \<open>j < n\<close> by (simp add: algebra_simps diff_divide_distrib of_nat_diff)
- show ?thesis
- apply (simp only: seq)
- apply (rule sin_monotone_2pi_le)
- using \<open>j < n\<close> False
- apply (auto simp: field_simps intro: order_trans [of _ 0])
- done
- qed
- with sin_less show ?thesis by force
- qed
- then show ?thesis by simp
- qed
- show ?thesis
- apply (rule_tac e = "2 * sin(pi / n)" in that)
- apply (force simp: \<open>2 \<le> n\<close> sin_pi_divide_n_gt_0)
- apply (meson eq_if_pow_eq)
- done
- qed
- have zn1: "continuous_on (- {0}) (\<lambda>z::complex. z^n)"
- by (rule continuous_intros)+
- have zn2: "(\<lambda>z::complex. z^n) ` (- {0}) = - {0}"
- using assms by (auto simp: image_def elim: exists_complex_root_nonzero [where n = n])
- have zn3: "\<exists>T. z^n \<in> T \<and> open T \<and> 0 \<notin> T \<and>
- (\<exists>v. \<Union>v = {x. x \<noteq> 0 \<and> x^n \<in> T} \<and>
- (\<forall>u\<in>v. open u \<and> 0 \<notin> u) \<and>
- pairwise disjnt v \<and>
- (\<forall>u\<in>v. Ex (homeomorphism u T (\<lambda>z. z^n))))"
- if "z \<noteq> 0" for z::complex
- proof -
- def d \<equiv> "min (1/2) (e/4) * norm z"
- have "0 < d"
- by (simp add: d_def \<open>0 < e\<close> \<open>z \<noteq> 0\<close>)
- have iff_x_eq_y: "x^n = y^n \<longleftrightarrow> x = y"
- if eq: "w^n = z^n" and x: "x \<in> ball w d" and y: "y \<in> ball w d" for w x y
- proof -
- have [simp]: "norm z = norm w" using that
- by (simp add: assms power_eq_imp_eq_norm)
- show ?thesis
- proof (cases "w = 0")
- case True with \<open>z \<noteq> 0\<close> assms eq
- show ?thesis by (auto simp: power_0_left)
- next
- case False
- have "cmod (x - y) < 2*d"
- using x y
- by (simp add: dist_norm [symmetric]) (metis dist_commute mult_2 dist_triangle_less_add)
- also have "... \<le> 2 * e / 4 * norm w"
- using \<open>e > 0\<close> by (simp add: d_def min_mult_distrib_right)
- also have "... = e * (cmod w / 2)"
- by simp
- also have "... \<le> e * cmod y"
- apply (rule mult_left_mono)
- using \<open>e > 0\<close> y
- apply (simp_all add: dist_norm d_def min_mult_distrib_right del: divide_const_simps)
- apply (metis dist_0_norm dist_complex_def dist_triangle_half_l linorder_not_less order_less_irrefl)
- done
- finally have "cmod (x - y) < e * cmod y" .
- then show ?thesis by (rule e)
- qed
- qed
- then have inj: "inj_on (\<lambda>w. w^n) (ball z d)"
- by (simp add: inj_on_def)
- have cont: "continuous_on (ball z d) (\<lambda>w. w ^ n)"
- by (intro continuous_intros)
- have noncon: "\<not> (\<lambda>w::complex. w^n) constant_on UNIV"
- by (metis UNIV_I assms constant_on_def power_one zero_neq_one zero_power)
- have open_imball: "open ((\<lambda>w. w^n) ` ball z d)"
- by (rule invariance_of_domain [OF cont open_ball inj])
- have im_eq: "(\<lambda>w. w^n) ` ball z' d = (\<lambda>w. w^n) ` ball z d"
- if z': "z'^n = z^n" for z'
- proof -
- have nz': "norm z' = norm z" using that assms power_eq_imp_eq_norm by blast
- have "(w \<in> (\<lambda>w. w^n) ` ball z' d) = (w \<in> (\<lambda>w. w^n) ` ball z d)" for w
- proof (cases "w=0")
- case True with assms show ?thesis
- by (simp add: image_def ball_def nz')
- next
- case False
- have "z' \<noteq> 0" using \<open>z \<noteq> 0\<close> nz' by force
- have [simp]: "(z*x / z')^n = x^n" if "x \<noteq> 0" for x
- using z' that by (simp add: field_simps \<open>z \<noteq> 0\<close>)
- have [simp]: "cmod (z - z * x / z') = cmod (z' - x)" if "x \<noteq> 0" for x
- proof -
- have "cmod (z - z * x / z') = cmod z * cmod (1 - x / z')"
- by (metis (no_types) ab_semigroup_mult_class.mult_ac(1) complex_divide_def mult.right_neutral norm_mult right_diff_distrib')
- also have "... = cmod z' * cmod (1 - x / z')"
- by (simp add: nz')
- also have "... = cmod (z' - x)"
- by (simp add: \<open>z' \<noteq> 0\<close> diff_divide_eq_iff norm_divide)
- finally show ?thesis .
- qed
- have [simp]: "(z'*x / z)^n = x^n" if "x \<noteq> 0" for x
- using z' that by (simp add: field_simps \<open>z \<noteq> 0\<close>)
- have [simp]: "cmod (z' - z' * x / z) = cmod (z - x)" if "x \<noteq> 0" for x
- proof -
- have "cmod (z * (1 - x * inverse z)) = cmod (z - x)"
- by (metis \<open>z \<noteq> 0\<close> diff_divide_distrib divide_complex_def divide_self_if nonzero_eq_divide_eq semiring_normalization_rules(7))
- then show ?thesis
- by (metis (no_types) mult.assoc complex_divide_def mult.right_neutral norm_mult nz' right_diff_distrib')
- qed
- show ?thesis
- unfolding image_def ball_def
- apply safe
- apply simp_all
- apply (rule_tac x="z/z' * x" in exI)
- using assms False apply (simp add: dist_norm)
- apply (rule_tac x="z'/z * x" in exI)
- using assms False apply (simp add: dist_norm)
- done
- qed
- then show ?thesis by blast
- qed
- have ex_ball: "\<exists>B. (\<exists>z'. B = ball z' d \<and> z'^n = z^n) \<and> x \<in> B"
- if "x \<noteq> 0" and eq: "x^n = w^n" and dzw: "dist z w < d" for x w
- proof -
- have "w \<noteq> 0" by (metis assms power_eq_0_iff that(1) that(2))
- have [simp]: "cmod x = cmod w"
- using assms power_eq_imp_eq_norm eq by blast
- have [simp]: "cmod (x * z / w - x) = cmod (z - w)"
- proof -
- have "cmod (x * z / w - x) = cmod x * cmod (z / w - 1)"
- by (metis (no_types) mult.right_neutral norm_mult right_diff_distrib' times_divide_eq_right)
- also have "... = cmod w * cmod (z / w - 1)"
- by simp
- also have "... = cmod (z - w)"
- by (simp add: \<open>w \<noteq> 0\<close> divide_diff_eq_iff nonzero_norm_divide)
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule_tac x="ball (z / w * x) d" in exI)
- using \<open>d > 0\<close> that
- apply (simp add: ball_eq_ball_iff)
- apply (simp add: \<open>z \<noteq> 0\<close> \<open>w \<noteq> 0\<close> field_simps)
- apply (simp add: dist_norm)
- done
- qed
- have ball1: "\<Union>{ball z' d |z'. z'^n = z^n} = {x. x \<noteq> 0 \<and> x^n \<in> (\<lambda>w. w^n) ` ball z d}"
- apply (rule equalityI)
- prefer 2 apply (force simp: ex_ball, clarsimp)
- apply (subst im_eq [symmetric], assumption)
- using assms
- apply (force simp: dist_norm d_def min_mult_distrib_right dest: power_eq_imp_eq_norm)
- done
- have ball2: "pairwise disjnt {ball z' d |z'. z'^n = z^n}"
- proof (clarsimp simp add: pairwise_def disjnt_iff)
- fix \<xi> \<zeta> x
- assume "\<xi>^n = z^n" "\<zeta>^n = z^n" "ball \<xi> d \<noteq> ball \<zeta> d"
- and "dist \<xi> x < d" "dist \<zeta> x < d"
- then have "dist \<xi> \<zeta> < d+d"
- using dist_triangle_less_add by blast
- then have "cmod (\<xi> - \<zeta>) < 2*d"
- by (simp add: dist_norm)
- also have "... \<le> e * cmod z"
- using mult_right_mono \<open>0 < e\<close> that by (auto simp: d_def)
- finally have "cmod (\<xi> - \<zeta>) < e * cmod z" .
- with e have "\<xi> = \<zeta>"
- by (metis \<open>\<xi>^n = z^n\<close> \<open>\<zeta>^n = z^n\<close> assms power_eq_imp_eq_norm)
- then show "False"
- using \<open>ball \<xi> d \<noteq> ball \<zeta> d\<close> by blast
- qed
- have ball3: "Ex (homeomorphism (ball z' d) ((\<lambda>w. w^n) ` ball z d) (\<lambda>z. z^n))"
- if zeq: "z'^n = z^n" for z'
- proof -
- have inj: "inj_on (\<lambda>z. z ^ n) (ball z' d)"
- by (meson iff_x_eq_y inj_onI zeq)
- show ?thesis
- apply (rule invariance_of_domain_homeomorphism [of "ball z' d" "\<lambda>z. z^n"])
- apply (rule open_ball continuous_intros order_refl inj)+
- apply (force simp: im_eq [OF zeq])
- done
- qed
- show ?thesis
- apply (rule_tac x = "(\<lambda>w. w^n) ` (ball z d)" in exI)
- apply (intro conjI open_imball)
- using \<open>d > 0\<close> apply simp
- using \<open>z \<noteq> 0\<close> assms apply (force simp: d_def)
- apply (rule_tac x="{ ball z' d |z'. z'^n = z^n}" in exI)
- apply (intro conjI ball1 ball2)
- apply (force simp: assms d_def power_eq_imp_eq_norm that, clarify)
- by (metis ball3)
- qed
- show ?thesis
- using assms
- apply (simp add: covering_space_def zn1 zn2)
- apply (subst zn2 [symmetric])
- apply (simp add: openin_open_eq open_Compl)
- apply (blast intro: zn3)
- done
-qed
-
-corollary covering_space_square_punctured_plane:
- "covering_space (- {0}) (\<lambda>z::complex. z^2) (- {0})"
- by (simp add: covering_space_power_punctured_plane)
-
-
-
-proposition covering_space_exp_punctured_plane:
- "covering_space UNIV (\<lambda>z::complex. exp z) (- {0})"
-proof (simp add: covering_space_def, intro conjI ballI)
- show "continuous_on UNIV (\<lambda>z::complex. exp z)"
- by (rule continuous_on_exp [OF continuous_on_id])
- show "range exp = - {0::complex}"
- by auto (metis exp_Ln range_eqI)
- show "\<exists>T. z \<in> T \<and> openin (subtopology euclidean (- {0})) T \<and>
- (\<exists>v. \<Union>v = {z. exp z \<in> T} \<and> (\<forall>u\<in>v. open u) \<and> disjoint v \<and>
- (\<forall>u\<in>v. \<exists>q. homeomorphism u T exp q))"
- if "z \<in> - {0::complex}" for z
- proof -
- have "z \<noteq> 0"
- using that by auto
- have inj_exp: "inj_on exp (ball (Ln z) 1)"
- apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
- using pi_ge_two by (simp add: ball_subset_ball_iff)
- define \<V> where "\<V> \<equiv> range (\<lambda>n. (\<lambda>x. x + of_real (2 * of_int n * pi) * ii) ` (ball(Ln z) 1))"
- show ?thesis
- proof (intro exI conjI)
- show "z \<in> exp ` (ball(Ln z) 1)"
- by (metis \<open>z \<noteq> 0\<close> centre_in_ball exp_Ln rev_image_eqI zero_less_one)
- have "open (- {0::complex})"
- by blast
- moreover have "inj_on exp (ball (Ln z) 1)"
- apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
- using pi_ge_two by (simp add: ball_subset_ball_iff)
- ultimately show "openin (subtopology euclidean (- {0})) (exp ` ball (Ln z) 1)"
- by (auto simp: openin_open_eq invariance_of_domain continuous_on_exp [OF continuous_on_id])
- show "\<Union>\<V> = {w. exp w \<in> exp ` ball (Ln z) 1}"
- by (auto simp: \<V>_def Complex_Transcendental.exp_eq image_iff)
- show "\<forall>V\<in>\<V>. open V"
- by (auto simp: \<V>_def inj_on_def continuous_intros invariance_of_domain)
- have xy: "2 \<le> cmod (2 * of_int x * of_real pi * \<i> - 2 * of_int y * of_real pi * \<i>)"
- if "x < y" for x y
- proof -
- have "1 \<le> abs (x - y)"
- using that by linarith
- then have "1 \<le> cmod (of_int x - of_int y) * 1"
- by (metis mult.right_neutral norm_of_int of_int_1_le_iff of_int_abs of_int_diff)
- also have "... \<le> cmod (of_int x - of_int y) * of_real pi"
- apply (rule mult_left_mono)
- using pi_ge_two by auto
- also have "... \<le> cmod ((of_int x - of_int y) * of_real pi * \<i>)"
- by (simp add: norm_mult)
- also have "... \<le> cmod (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>)"
- by (simp add: algebra_simps)
- finally have "1 \<le> cmod (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>)" .
- then have "2 * 1 \<le> cmod (2 * (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>))"
- by (metis mult_le_cancel_left_pos norm_mult_numeral1 zero_less_numeral)
- then show ?thesis
- by (simp add: algebra_simps)
- qed
- show "disjoint \<V>"
- apply (clarsimp simp add: \<V>_def pairwise_def disjnt_def add.commute [of _ "x*y" for x y]
- image_add_ball ball_eq_ball_iff)
- apply (rule disjoint_ballI)
- apply (auto simp: dist_norm neq_iff)
- by (metis norm_minus_commute xy)+
- show "\<forall>u\<in>\<V>. \<exists>q. homeomorphism u (exp ` ball (Ln z) 1) exp q"
- proof
- fix u
- assume "u \<in> \<V>"
- then obtain n where n: "u = (\<lambda>x. x + of_real (2 * of_int n * pi) * ii) ` (ball(Ln z) 1)"
- by (auto simp: \<V>_def)
- have "compact (cball (Ln z) 1)"
- by simp
- moreover have "continuous_on (cball (Ln z) 1) exp"
- by (rule continuous_on_exp [OF continuous_on_id])
- moreover have "inj_on exp (cball (Ln z) 1)"
- apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
- using pi_ge_two by (simp add: cball_subset_ball_iff)
- ultimately obtain \<gamma> where hom: "homeomorphism (cball (Ln z) 1) (exp ` cball (Ln z) 1) exp \<gamma>"
- using homeomorphism_compact by blast
- have eq1: "exp ` u = exp ` ball (Ln z) 1"
- unfolding n
- apply (auto simp: algebra_simps)
- apply (rename_tac w)
- apply (rule_tac x = "w + \<i> * (of_int n * (of_real pi * 2))" in image_eqI)
- apply (auto simp: image_iff)
- done
- have \<gamma>exp: "\<gamma> (exp x) + 2 * of_int n * of_real pi * \<i> = x" if "x \<in> u" for x
- proof -
- have "exp x = exp (x - 2 * of_int n * of_real pi * \<i>)"
- by (simp add: exp_eq)
- then have "\<gamma> (exp x) = \<gamma> (exp (x - 2 * of_int n * of_real pi * \<i>))"
- by simp
- also have "... = x - 2 * of_int n * of_real pi * \<i>"
- apply (rule homeomorphism_apply1 [OF hom])
- using \<open>x \<in> u\<close> by (auto simp: n)
- finally show ?thesis
- by simp
- qed
- have exp2n: "exp (\<gamma> (exp x) + 2 * of_int n * complex_of_real pi * \<i>) = exp x"
- if "dist (Ln z) x < 1" for x
- using that by (auto simp: exp_eq homeomorphism_apply1 [OF hom])
- have cont: "continuous_on (exp ` ball (Ln z) 1) (\<lambda>x. \<gamma> x + 2 * of_int n * complex_of_real pi * \<i>)"
- apply (intro continuous_intros)
- apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF hom]])
- apply (force simp:)
- done
- show "\<exists>q. homeomorphism u (exp ` ball (Ln z) 1) exp q"
- apply (rule_tac x="(\<lambda>x. x + of_real(2 * n * pi) * ii) \<circ> \<gamma>" in exI)
- unfolding homeomorphism_def
- apply (intro conjI ballI eq1 continuous_on_exp [OF continuous_on_id])
- apply (auto simp: \<gamma>exp exp2n cont n)
- apply (simp add: homeomorphism_apply1 [OF hom])
- apply (simp add: image_comp [symmetric])
- using hom homeomorphism_apply1 apply (force simp: image_iff)
- done
- qed
- qed
- qed
-qed
-
-end