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(* Title: HOL/Analysis/Homeomorphism.thy
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Author: LC Paulson (ported from HOL Light)
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*)
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section \<open>Homeomorphism Theorems\<close>
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theory Homeomorphism
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imports Path_Connected
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begin
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subsection \<open>Homeomorphism of all convex compact sets with nonempty interior\<close>
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proposition ray_to_rel_frontier:
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fixes a :: "'a::real_inner"
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assumes "bounded S"
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and a: "a \<in> rel_interior S"
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and aff: "(a + l) \<in> affine hull S"
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and "l \<noteq> 0"
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obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
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"\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
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proof -
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have aaff: "a \<in> affine hull S"
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by (meson a hull_subset rel_interior_subset rev_subsetD)
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let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
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obtain B where "B > 0" and B: "S \<subseteq> ball a B"
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using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
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have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
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by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
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with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
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using rel_interior_subset subsetCE by blast
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with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
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using divide_pos_pos zero_less_norm_iff by fastforce
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have bdd: "bdd_below ?D"
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by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
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have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
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\<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
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using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
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define d where "d = Inf ?D"
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obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
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proof -
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obtain e where "e>0"
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and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
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using relin_Ex a by blast
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show thesis
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proof (rule_tac \<epsilon> = "e / norm l" in that)
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show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
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next
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show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
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proof (rule e)
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show "a + \<eta> *\<^sub>R l \<in> affine hull S"
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by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
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show "dist (a + \<eta> *\<^sub>R l) a < e"
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using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
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qed
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qed
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qed
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have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
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unfolding d_def using cInf_lower [OF _ bdd]
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by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
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have "\<epsilon> \<le> d"
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unfolding d_def
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apply (rule cInf_greatest [OF nonMT])
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using \<epsilon> dual_order.strict_implies_order le_less_linear by blast
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with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
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have "a + d *\<^sub>R l \<notin> rel_interior S"
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proof
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assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
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obtain e where "e > 0"
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and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
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using relin_Ex adl by blast
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have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
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proof (rule cInf_greatest [OF nonMT], clarsimp)
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fix x::real
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assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
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show "d + e / norm l \<le> x"
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proof (cases "x < d")
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case True with inint nonrel \<open>0 < x\<close>
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show ?thesis by auto
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next
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case False
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then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
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by (simp add: field_simps \<open>l \<noteq> 0\<close>)
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have ain: "a + x *\<^sub>R l \<in> affine hull S"
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by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
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show ?thesis
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using e [OF ain] nonrel dle by force
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qed
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qed
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then show False
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using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] divide_simps)
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qed
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moreover have "a + d *\<^sub>R l \<in> closure S"
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proof (clarsimp simp: closure_approachable)
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fix \<eta>::real assume "0 < \<eta>"
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have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
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apply (rule subsetD [OF rel_interior_subset inint])
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using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
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have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
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by (metis min_def mult_left_mono norm_ge_zero order_refl)
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also have "... < \<eta>"
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using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: divide_simps)
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finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
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show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
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apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
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using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
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done
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qed
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ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
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by (simp add: rel_frontier_def)
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show ?thesis
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by (rule that [OF \<open>0 < d\<close> infront inint])
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qed
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corollary ray_to_frontier:
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fixes a :: "'a::euclidean_space"
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assumes "bounded S"
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and a: "a \<in> interior S"
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and "l \<noteq> 0"
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obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
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"\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
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proof -
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have "interior S = rel_interior S"
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using a rel_interior_nonempty_interior by auto
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then have "a \<in> rel_interior S"
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using a by simp
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then show ?thesis
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apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
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using a affine_hull_nonempty_interior apply blast
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by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
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qed
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proposition
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fixes S :: "'a::euclidean_space set"
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assumes "compact S" and 0: "0 \<in> rel_interior S"
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and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S"
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shows starlike_compact_projective1_0:
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"S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S"
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(is "?SMINUS homeomorphic ?SPHER")
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and starlike_compact_projective2_0:
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"S homeomorphic cball 0 1 \<inter> affine hull S"
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(is "S homeomorphic ?CBALL")
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proof -
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have starI: "(u *\<^sub>R x) \<in> rel_interior S" if "x \<in> S" "0 \<le> u" "u < 1" for x u
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proof (cases "x=0 \<or> u=0")
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case True with 0 show ?thesis by force
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next
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case False with that show ?thesis
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by (auto simp: in_segment intro: star [THEN subsetD])
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qed
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have "0 \<in> S" using assms rel_interior_subset by auto
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define proj where "proj \<equiv> \<lambda>x::'a. x /\<^sub>R norm x"
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have eqI: "x = y" if "proj x = proj y" "norm x = norm y" for x y
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using that by (force simp: proj_def)
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then have iff_eq: "\<And>x y. (proj x = proj y \<and> norm x = norm y) \<longleftrightarrow> x = y"
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by blast
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have projI: "x \<in> affine hull S \<Longrightarrow> proj x \<in> affine hull S" for x
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by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_mul proj_def)
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have nproj1 [simp]: "x \<noteq> 0 \<Longrightarrow> norm(proj x) = 1" for x
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by (simp add: proj_def)
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have proj0_iff [simp]: "proj x = 0 \<longleftrightarrow> x = 0" for x
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by (simp add: proj_def)
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have cont_proj: "continuous_on (UNIV - {0}) proj"
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unfolding proj_def by (rule continuous_intros | force)+
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have proj_spherI: "\<And>x. \<lbrakk>x \<in> affine hull S; x \<noteq> 0\<rbrakk> \<Longrightarrow> proj x \<in> ?SPHER"
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by (simp add: projI)
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have "bounded S" "closed S"
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using \<open>compact S\<close> compact_eq_bounded_closed by blast+
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have inj_on_proj: "inj_on proj (S - rel_interior S)"
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proof
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fix x y
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assume x: "x \<in> S - rel_interior S"
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and y: "y \<in> S - rel_interior S" and eq: "proj x = proj y"
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then have xynot: "x \<noteq> 0" "y \<noteq> 0" "x \<in> S" "y \<in> S" "x \<notin> rel_interior S" "y \<notin> rel_interior S"
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using 0 by auto
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consider "norm x = norm y" | "norm x < norm y" | "norm x > norm y" by linarith
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then show "x = y"
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proof cases
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assume "norm x = norm y"
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with iff_eq eq show "x = y" by blast
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next
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assume *: "norm x < norm y"
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have "x /\<^sub>R norm x = norm x *\<^sub>R (x /\<^sub>R norm x) /\<^sub>R norm (norm x *\<^sub>R (x /\<^sub>R norm x))"
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by force
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then have "proj ((norm x / norm y) *\<^sub>R y) = proj x"
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by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
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then have [simp]: "(norm x / norm y) *\<^sub>R y = x"
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by (rule eqI) (simp add: \<open>y \<noteq> 0\<close>)
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have no: "0 \<le> norm x / norm y" "norm x / norm y < 1"
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using * by (auto simp: divide_simps)
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then show "x = y"
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using starI [OF \<open>y \<in> S\<close> no] xynot by auto
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next
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assume *: "norm x > norm y"
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have "y /\<^sub>R norm y = norm y *\<^sub>R (y /\<^sub>R norm y) /\<^sub>R norm (norm y *\<^sub>R (y /\<^sub>R norm y))"
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by force
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then have "proj ((norm y / norm x) *\<^sub>R x) = proj y"
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by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
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then have [simp]: "(norm y / norm x) *\<^sub>R x = y"
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by (rule eqI) (simp add: \<open>x \<noteq> 0\<close>)
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have no: "0 \<le> norm y / norm x" "norm y / norm x < 1"
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using * by (auto simp: divide_simps)
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then show "x = y"
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using starI [OF \<open>x \<in> S\<close> no] xynot by auto
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qed
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qed
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have "\<exists>surf. homeomorphism (S - rel_interior S) ?SPHER proj surf"
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proof (rule homeomorphism_compact)
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show "compact (S - rel_interior S)"
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using \<open>compact S\<close> compact_rel_boundary by blast
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show "continuous_on (S - rel_interior S) proj"
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using 0 by (blast intro: continuous_on_subset [OF cont_proj])
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show "proj ` (S - rel_interior S) = ?SPHER"
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proof
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show "proj ` (S - rel_interior S) \<subseteq> ?SPHER"
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using 0 by (force simp: hull_inc projI intro: nproj1)
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show "?SPHER \<subseteq> proj ` (S - rel_interior S)"
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proof (clarsimp simp: proj_def)
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fix x
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assume "x \<in> affine hull S" and nox: "norm x = 1"
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then have "x \<noteq> 0" by auto
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obtain d where "0 < d" and dx: "(d *\<^sub>R x) \<in> rel_frontier S"
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and ri: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (e *\<^sub>R x) \<in> rel_interior S"
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using ray_to_rel_frontier [OF \<open>bounded S\<close> 0] \<open>x \<in> affine hull S\<close> \<open>x \<noteq> 0\<close> by auto
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show "x \<in> (\<lambda>x. x /\<^sub>R norm x) ` (S - rel_interior S)"
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apply (rule_tac x="d *\<^sub>R x" in image_eqI)
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using \<open>0 < d\<close>
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using dx \<open>closed S\<close> apply (auto simp: rel_frontier_def divide_simps nox)
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done
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qed
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qed
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qed (rule inj_on_proj)
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then obtain surf where surf: "homeomorphism (S - rel_interior S) ?SPHER proj surf"
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by blast
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then have cont_surf: "continuous_on (proj ` (S - rel_interior S)) surf"
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by (auto simp: homeomorphism_def)
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have surf_nz: "\<And>x. x \<in> ?SPHER \<Longrightarrow> surf x \<noteq> 0"
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by (metis "0" DiffE homeomorphism_def imageI surf)
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have cont_nosp: "continuous_on (?SPHER) (\<lambda>x. norm x *\<^sub>R ((surf o proj) x))"
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apply (rule continuous_intros)+
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apply (rule continuous_on_subset [OF cont_proj], force)
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apply (rule continuous_on_subset [OF cont_surf])
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apply (force simp: homeomorphism_image1 [OF surf] dest: proj_spherI)
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done
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have surfpS: "\<And>x. \<lbrakk>norm x = 1; x \<in> affine hull S\<rbrakk> \<Longrightarrow> surf (proj x) \<in> S"
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by (metis (full_types) DiffE \<open>0 \<in> S\<close> homeomorphism_def image_eqI norm_zero proj_spherI real_vector.scale_zero_left scaleR_one surf)
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have *: "\<exists>y. norm y = 1 \<and> y \<in> affine hull S \<and> x = surf (proj y)"
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if "x \<in> S" "x \<notin> rel_interior S" for x
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proof -
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have "proj x \<in> ?SPHER"
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by (metis (full_types) "0" hull_inc proj_spherI that)
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moreover have "surf (proj x) = x"
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by (metis Diff_iff homeomorphism_def surf that)
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ultimately show ?thesis
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by (metis \<open>\<And>x. x \<in> ?SPHER \<Longrightarrow> surf x \<noteq> 0\<close> hull_inc inverse_1 local.proj_def norm_sgn projI scaleR_one sgn_div_norm that(1))
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qed
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have surfp_notin: "\<And>x. \<lbrakk>norm x = 1; x \<in> affine hull S\<rbrakk> \<Longrightarrow> surf (proj x) \<notin> rel_interior S"
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by (metis (full_types) DiffE one_neq_zero homeomorphism_def image_eqI norm_zero proj_spherI surf)
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have no_sp_im: "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` (?SPHER) = S - rel_interior S"
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by (auto simp: surfpS image_def Bex_def surfp_notin *)
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have inj_spher: "inj_on (\<lambda>x. norm x *\<^sub>R surf (proj x)) ?SPHER"
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proof
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fix x y
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assume xy: "x \<in> ?SPHER" "y \<in> ?SPHER"
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and eq: " norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
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then have "norm x = 1" "norm y = 1" "x \<in> affine hull S" "y \<in> affine hull S"
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using 0 by auto
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with eq show "x = y"
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by (simp add: proj_def) (metis surf xy homeomorphism_def)
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qed
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have co01: "compact ?SPHER"
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by (simp add: closed_affine_hull compact_Int_closed)
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show "?SMINUS homeomorphic ?SPHER"
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apply (subst homeomorphic_sym)
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274 |
apply (rule homeomorphic_compact [OF co01 cont_nosp [unfolded o_def] no_sp_im inj_spher])
|
|
275 |
done
|
|
276 |
have proj_scaleR: "\<And>a x. 0 < a \<Longrightarrow> proj (a *\<^sub>R x) = proj x"
|
|
277 |
by (simp add: proj_def)
|
|
278 |
have cont_sp0: "continuous_on (affine hull S - {0}) (surf o proj)"
|
|
279 |
apply (rule continuous_on_compose [OF continuous_on_subset [OF cont_proj]], force)
|
|
280 |
apply (rule continuous_on_subset [OF cont_surf])
|
|
281 |
using homeomorphism_image1 proj_spherI surf by fastforce
|
|
282 |
obtain B where "B>0" and B: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
|
|
283 |
by (metis compact_imp_bounded \<open>compact S\<close> bounded_pos_less less_eq_real_def)
|
|
284 |
have cont_nosp: "continuous (at x within ?CBALL) (\<lambda>x. norm x *\<^sub>R surf (proj x))"
|
|
285 |
if "norm x \<le> 1" "x \<in> affine hull S" for x
|
|
286 |
proof (cases "x=0")
|
|
287 |
case True
|
|
288 |
show ?thesis using True
|
|
289 |
apply (simp add: continuous_within)
|
|
290 |
apply (rule lim_null_scaleR_bounded [where B=B])
|
|
291 |
apply (simp_all add: tendsto_norm_zero eventually_at)
|
|
292 |
apply (rule_tac x=B in exI)
|
|
293 |
using B surfpS proj_def projI apply (auto simp: \<open>B > 0\<close>)
|
|
294 |
done
|
|
295 |
next
|
|
296 |
case False
|
|
297 |
then have "\<forall>\<^sub>F x in at x. (x \<in> affine hull S - {0}) = (x \<in> affine hull S)"
|
|
298 |
apply (simp add: eventually_at)
|
|
299 |
apply (rule_tac x="norm x" in exI)
|
|
300 |
apply (auto simp: False)
|
|
301 |
done
|
|
302 |
with cont_sp0 have *: "continuous (at x within affine hull S) (\<lambda>x. surf (proj x))"
|
|
303 |
apply (simp add: continuous_on_eq_continuous_within)
|
|
304 |
apply (drule_tac x=x in bspec, force simp: False that)
|
|
305 |
apply (simp add: continuous_within Lim_transform_within_set)
|
|
306 |
done
|
|
307 |
show ?thesis
|
|
308 |
apply (rule continuous_within_subset [where s = "affine hull S", OF _ Int_lower2])
|
|
309 |
apply (rule continuous_intros *)+
|
|
310 |
done
|
|
311 |
qed
|
|
312 |
have cont_nosp2: "continuous_on ?CBALL (\<lambda>x. norm x *\<^sub>R ((surf o proj) x))"
|
|
313 |
by (simp add: continuous_on_eq_continuous_within cont_nosp)
|
|
314 |
have "norm y *\<^sub>R surf (proj y) \<in> S" if "y \<in> cball 0 1" and yaff: "y \<in> affine hull S" for y
|
|
315 |
proof (cases "y=0")
|
|
316 |
case True then show ?thesis
|
|
317 |
by (simp add: \<open>0 \<in> S\<close>)
|
|
318 |
next
|
|
319 |
case False
|
|
320 |
then have "norm y *\<^sub>R surf (proj y) = norm y *\<^sub>R surf (proj (y /\<^sub>R norm y))"
|
|
321 |
by (simp add: proj_def)
|
|
322 |
have "norm y \<le> 1" using that by simp
|
|
323 |
have "surf (proj (y /\<^sub>R norm y)) \<in> S"
|
|
324 |
apply (rule surfpS)
|
|
325 |
using proj_def projI yaff
|
|
326 |
by (auto simp: False)
|
|
327 |
then have "surf (proj y) \<in> S"
|
|
328 |
by (simp add: False proj_def)
|
|
329 |
then show "norm y *\<^sub>R surf (proj y) \<in> S"
|
|
330 |
by (metis dual_order.antisym le_less_linear norm_ge_zero rel_interior_subset scaleR_one
|
|
331 |
starI subset_eq \<open>norm y \<le> 1\<close>)
|
|
332 |
qed
|
|
333 |
moreover have "x \<in> (\<lambda>x. norm x *\<^sub>R surf (proj x)) ` (?CBALL)" if "x \<in> S" for x
|
|
334 |
proof (cases "x=0")
|
|
335 |
case True with that hull_inc show ?thesis by fastforce
|
|
336 |
next
|
|
337 |
case False
|
|
338 |
then have psp: "proj (surf (proj x)) = proj x"
|
|
339 |
by (metis homeomorphism_def hull_inc proj_spherI surf that)
|
|
340 |
have nxx: "norm x *\<^sub>R proj x = x"
|
|
341 |
by (simp add: False local.proj_def)
|
|
342 |
have affineI: "(1 / norm (surf (proj x))) *\<^sub>R x \<in> affine hull S"
|
|
343 |
by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_clauses(4) that)
|
|
344 |
have sproj_nz: "surf (proj x) \<noteq> 0"
|
|
345 |
by (metis False proj0_iff psp)
|
|
346 |
then have "proj x = proj (proj x)"
|
|
347 |
by (metis False nxx proj_scaleR zero_less_norm_iff)
|
|
348 |
moreover have scaleproj: "\<And>a r. r *\<^sub>R proj a = (r / norm a) *\<^sub>R a"
|
|
349 |
by (simp add: divide_inverse local.proj_def)
|
|
350 |
ultimately have "(norm (surf (proj x)) / norm x) *\<^sub>R x \<notin> rel_interior S"
|
|
351 |
by (metis (no_types) sproj_nz divide_self_if hull_inc norm_eq_zero nproj1 projI psp scaleR_one surfp_notin that)
|
|
352 |
then have "(norm (surf (proj x)) / norm x) \<ge> 1"
|
|
353 |
using starI [OF that] by (meson starI [OF that] le_less_linear norm_ge_zero zero_le_divide_iff)
|
|
354 |
then have nole: "norm x \<le> norm (surf (proj x))"
|
|
355 |
by (simp add: le_divide_eq_1)
|
|
356 |
show ?thesis
|
|
357 |
apply (rule_tac x="inverse(norm(surf (proj x))) *\<^sub>R x" in image_eqI)
|
|
358 |
apply (metis (no_types, hide_lams) mult.commute scaleproj abs_inverse abs_norm_cancel divide_inverse norm_scaleR nxx positive_imp_inverse_positive proj_scaleR psp sproj_nz zero_less_norm_iff)
|
|
359 |
apply (auto simp: divide_simps nole affineI)
|
|
360 |
done
|
|
361 |
qed
|
|
362 |
ultimately have im_cball: "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` ?CBALL = S"
|
|
363 |
by blast
|
|
364 |
have inj_cball: "inj_on (\<lambda>x. norm x *\<^sub>R surf (proj x)) ?CBALL"
|
|
365 |
proof
|
|
366 |
fix x y
|
|
367 |
assume "x \<in> ?CBALL" "y \<in> ?CBALL"
|
|
368 |
and eq: "norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
|
|
369 |
then have x: "x \<in> affine hull S" and y: "y \<in> affine hull S"
|
|
370 |
using 0 by auto
|
|
371 |
show "x = y"
|
|
372 |
proof (cases "x=0 \<or> y=0")
|
|
373 |
case True then show "x = y" using eq proj_spherI surf_nz x y by force
|
|
374 |
next
|
|
375 |
case False
|
|
376 |
with x y have speq: "surf (proj x) = surf (proj y)"
|
|
377 |
by (metis eq homeomorphism_apply2 proj_scaleR proj_spherI surf zero_less_norm_iff)
|
|
378 |
then have "norm x = norm y"
|
|
379 |
by (metis \<open>x \<in> affine hull S\<close> \<open>y \<in> affine hull S\<close> eq proj_spherI real_vector.scale_cancel_right surf_nz)
|
|
380 |
moreover have "proj x = proj y"
|
|
381 |
by (metis (no_types) False speq homeomorphism_apply2 proj_spherI surf x y)
|
|
382 |
ultimately show "x = y"
|
|
383 |
using eq eqI by blast
|
|
384 |
qed
|
|
385 |
qed
|
|
386 |
have co01: "compact ?CBALL"
|
|
387 |
by (simp add: closed_affine_hull compact_Int_closed)
|
|
388 |
show "S homeomorphic ?CBALL"
|
|
389 |
apply (subst homeomorphic_sym)
|
|
390 |
apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball])
|
|
391 |
done
|
|
392 |
qed
|
|
393 |
|
|
394 |
corollary
|
|
395 |
fixes S :: "'a::euclidean_space set"
|
|
396 |
assumes "compact S" and a: "a \<in> rel_interior S"
|
|
397 |
and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
|
|
398 |
shows starlike_compact_projective1:
|
|
399 |
"S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
|
|
400 |
and starlike_compact_projective2:
|
|
401 |
"S homeomorphic cball a 1 \<inter> affine hull S"
|
|
402 |
proof -
|
|
403 |
have 1: "compact (op+ (-a) ` S)" by (meson assms compact_translation)
|
|
404 |
have 2: "0 \<in> rel_interior (op+ (-a) ` S)"
|
|
405 |
by (simp add: a rel_interior_translation)
|
|
406 |
have 3: "open_segment 0 x \<subseteq> rel_interior (op+ (-a) ` S)" if "x \<in> (op+ (-a) ` S)" for x
|
|
407 |
proof -
|
|
408 |
have "x+a \<in> S" using that by auto
|
|
409 |
then have "open_segment a (x+a) \<subseteq> rel_interior S" by (metis star)
|
|
410 |
then show ?thesis using open_segment_translation
|
|
411 |
using rel_interior_translation by fastforce
|
|
412 |
qed
|
|
413 |
have "S - rel_interior S homeomorphic (op+ (-a) ` S) - rel_interior (op+ (-a) ` S)"
|
|
414 |
by (metis rel_interior_translation translation_diff homeomorphic_translation)
|
|
415 |
also have "... homeomorphic sphere 0 1 \<inter> affine hull (op+ (-a) ` S)"
|
|
416 |
by (rule starlike_compact_projective1_0 [OF 1 2 3])
|
|
417 |
also have "... = op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
|
|
418 |
by (metis affine_hull_translation left_minus sphere_translation translation_Int)
|
|
419 |
also have "... homeomorphic sphere a 1 \<inter> affine hull S"
|
|
420 |
using homeomorphic_translation homeomorphic_sym by blast
|
|
421 |
finally show "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S" .
|
|
422 |
|
|
423 |
have "S homeomorphic (op+ (-a) ` S)"
|
|
424 |
by (metis homeomorphic_translation)
|
|
425 |
also have "... homeomorphic cball 0 1 \<inter> affine hull (op+ (-a) ` S)"
|
|
426 |
by (rule starlike_compact_projective2_0 [OF 1 2 3])
|
|
427 |
also have "... = op+ (-a) ` (cball a 1 \<inter> affine hull S)"
|
|
428 |
by (metis affine_hull_translation left_minus cball_translation translation_Int)
|
|
429 |
also have "... homeomorphic cball a 1 \<inter> affine hull S"
|
|
430 |
using homeomorphic_translation homeomorphic_sym by blast
|
|
431 |
finally show "S homeomorphic cball a 1 \<inter> affine hull S" .
|
|
432 |
qed
|
|
433 |
|
|
434 |
corollary starlike_compact_projective_special:
|
|
435 |
assumes "compact S"
|
|
436 |
and cb01: "cball (0::'a::euclidean_space) 1 \<subseteq> S"
|
|
437 |
and scale: "\<And>x u. \<lbrakk>x \<in> S; 0 \<le> u; u < 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x \<in> S - frontier S"
|
|
438 |
shows "S homeomorphic (cball (0::'a::euclidean_space) 1)"
|
|
439 |
proof -
|
|
440 |
have "ball 0 1 \<subseteq> interior S"
|
|
441 |
using cb01 interior_cball interior_mono by blast
|
|
442 |
then have 0: "0 \<in> rel_interior S"
|
|
443 |
by (meson centre_in_ball subsetD interior_subset_rel_interior le_numeral_extra(2) not_le)
|
|
444 |
have [simp]: "affine hull S = UNIV"
|
|
445 |
using \<open>ball 0 1 \<subseteq> interior S\<close> by (auto intro!: affine_hull_nonempty_interior)
|
|
446 |
have star: "open_segment 0 x \<subseteq> rel_interior S" if "x \<in> S" for x
|
63627
|
447 |
proof
|
63130
|
448 |
fix p assume "p \<in> open_segment 0 x"
|
|
449 |
then obtain u where "x \<noteq> 0" and u: "0 \<le> u" "u < 1" and p: "u *\<^sub>R x = p"
|
63627
|
450 |
by (auto simp: in_segment)
|
63130
|
451 |
then show "p \<in> rel_interior S"
|
|
452 |
using scale [OF that u] closure_subset frontier_def interior_subset_rel_interior by fastforce
|
|
453 |
qed
|
|
454 |
show ?thesis
|
|
455 |
using starlike_compact_projective2_0 [OF \<open>compact S\<close> 0 star] by simp
|
|
456 |
qed
|
|
457 |
|
|
458 |
lemma homeomorphic_convex_lemma:
|
|
459 |
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
|
|
460 |
assumes "convex S" "compact S" "convex T" "compact T"
|
|
461 |
and affeq: "aff_dim S = aff_dim T"
|
|
462 |
shows "(S - rel_interior S) homeomorphic (T - rel_interior T) \<and>
|
|
463 |
S homeomorphic T"
|
|
464 |
proof (cases "rel_interior S = {} \<or> rel_interior T = {}")
|
|
465 |
case True
|
|
466 |
then show ?thesis
|
|
467 |
by (metis Diff_empty affeq \<open>convex S\<close> \<open>convex T\<close> aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty)
|
|
468 |
next
|
|
469 |
case False
|
|
470 |
then obtain a b where a: "a \<in> rel_interior S" and b: "b \<in> rel_interior T" by auto
|
|
471 |
have starS: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
|
|
472 |
using rel_interior_closure_convex_segment
|
|
473 |
a \<open>convex S\<close> closure_subset subsetCE by blast
|
|
474 |
have starT: "\<And>x. x \<in> T \<Longrightarrow> open_segment b x \<subseteq> rel_interior T"
|
|
475 |
using rel_interior_closure_convex_segment
|
|
476 |
b \<open>convex T\<close> closure_subset subsetCE by blast
|
|
477 |
let ?aS = "op+ (-a) ` S" and ?bT = "op+ (-b) ` T"
|
|
478 |
have 0: "0 \<in> affine hull ?aS" "0 \<in> affine hull ?bT"
|
|
479 |
by (metis a b subsetD hull_inc image_eqI left_minus rel_interior_subset)+
|
|
480 |
have subs: "subspace (span ?aS)" "subspace (span ?bT)"
|
|
481 |
by (rule subspace_span)+
|
|
482 |
moreover
|
|
483 |
have "dim (span (op + (- a) ` S)) = dim (span (op + (- b) ` T))"
|
|
484 |
by (metis 0 aff_dim_translation_eq aff_dim_zero affeq dim_span nat_int)
|
|
485 |
ultimately obtain f g where "linear f" "linear g"
|
|
486 |
and fim: "f ` span ?aS = span ?bT"
|
|
487 |
and gim: "g ` span ?bT = span ?aS"
|
|
488 |
and fno: "\<And>x. x \<in> span ?aS \<Longrightarrow> norm(f x) = norm x"
|
|
489 |
and gno: "\<And>x. x \<in> span ?bT \<Longrightarrow> norm(g x) = norm x"
|
|
490 |
and gf: "\<And>x. x \<in> span ?aS \<Longrightarrow> g(f x) = x"
|
|
491 |
and fg: "\<And>x. x \<in> span ?bT \<Longrightarrow> f(g x) = x"
|
|
492 |
by (rule isometries_subspaces) blast
|
|
493 |
have [simp]: "continuous_on A f" for A
|
|
494 |
using \<open>linear f\<close> linear_conv_bounded_linear linear_continuous_on by blast
|
|
495 |
have [simp]: "continuous_on B g" for B
|
|
496 |
using \<open>linear g\<close> linear_conv_bounded_linear linear_continuous_on by blast
|
|
497 |
have eqspanS: "affine hull ?aS = span ?aS"
|
|
498 |
by (metis a affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
|
|
499 |
have eqspanT: "affine hull ?bT = span ?bT"
|
|
500 |
by (metis b affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
|
|
501 |
have "S homeomorphic cball a 1 \<inter> affine hull S"
|
|
502 |
by (rule starlike_compact_projective2 [OF \<open>compact S\<close> a starS])
|
|
503 |
also have "... homeomorphic op+ (-a) ` (cball a 1 \<inter> affine hull S)"
|
|
504 |
by (metis homeomorphic_translation)
|
|
505 |
also have "... = cball 0 1 \<inter> op+ (-a) ` (affine hull S)"
|
|
506 |
by (auto simp: dist_norm)
|
|
507 |
also have "... = cball 0 1 \<inter> span ?aS"
|
|
508 |
using eqspanS affine_hull_translation by blast
|
|
509 |
also have "... homeomorphic cball 0 1 \<inter> span ?bT"
|
|
510 |
proof (rule homeomorphicI [where f=f and g=g])
|
|
511 |
show fim1: "f ` (cball 0 1 \<inter> span ?aS) = cball 0 1 \<inter> span ?bT"
|
|
512 |
apply (rule subset_antisym)
|
|
513 |
using fim fno apply (force simp:, clarify)
|
|
514 |
by (metis IntI fg gim gno image_eqI mem_cball_0)
|
|
515 |
show "g ` (cball 0 1 \<inter> span ?bT) = cball 0 1 \<inter> span ?aS"
|
|
516 |
apply (rule subset_antisym)
|
|
517 |
using gim gno apply (force simp:, clarify)
|
|
518 |
by (metis IntI fim1 gf image_eqI)
|
|
519 |
qed (auto simp: fg gf)
|
|
520 |
also have "... = cball 0 1 \<inter> op+ (-b) ` (affine hull T)"
|
|
521 |
using eqspanT affine_hull_translation by blast
|
|
522 |
also have "... = op+ (-b) ` (cball b 1 \<inter> affine hull T)"
|
|
523 |
by (auto simp: dist_norm)
|
|
524 |
also have "... homeomorphic (cball b 1 \<inter> affine hull T)"
|
|
525 |
by (metis homeomorphic_translation homeomorphic_sym)
|
|
526 |
also have "... homeomorphic T"
|
|
527 |
by (metis starlike_compact_projective2 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
|
|
528 |
finally have 1: "S homeomorphic T" .
|
|
529 |
|
|
530 |
have "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
|
|
531 |
by (rule starlike_compact_projective1 [OF \<open>compact S\<close> a starS])
|
|
532 |
also have "... homeomorphic op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
|
|
533 |
by (metis homeomorphic_translation)
|
|
534 |
also have "... = sphere 0 1 \<inter> op+ (-a) ` (affine hull S)"
|
|
535 |
by (auto simp: dist_norm)
|
|
536 |
also have "... = sphere 0 1 \<inter> span ?aS"
|
|
537 |
using eqspanS affine_hull_translation by blast
|
|
538 |
also have "... homeomorphic sphere 0 1 \<inter> span ?bT"
|
|
539 |
proof (rule homeomorphicI [where f=f and g=g])
|
|
540 |
show fim1: "f ` (sphere 0 1 \<inter> span ?aS) = sphere 0 1 \<inter> span ?bT"
|
|
541 |
apply (rule subset_antisym)
|
|
542 |
using fim fno apply (force simp:, clarify)
|
|
543 |
by (metis IntI fg gim gno image_eqI mem_sphere_0)
|
|
544 |
show "g ` (sphere 0 1 \<inter> span ?bT) = sphere 0 1 \<inter> span ?aS"
|
|
545 |
apply (rule subset_antisym)
|
|
546 |
using gim gno apply (force simp:, clarify)
|
|
547 |
by (metis IntI fim1 gf image_eqI)
|
|
548 |
qed (auto simp: fg gf)
|
|
549 |
also have "... = sphere 0 1 \<inter> op+ (-b) ` (affine hull T)"
|
|
550 |
using eqspanT affine_hull_translation by blast
|
|
551 |
also have "... = op+ (-b) ` (sphere b 1 \<inter> affine hull T)"
|
|
552 |
by (auto simp: dist_norm)
|
|
553 |
also have "... homeomorphic (sphere b 1 \<inter> affine hull T)"
|
|
554 |
by (metis homeomorphic_translation homeomorphic_sym)
|
|
555 |
also have "... homeomorphic T - rel_interior T"
|
|
556 |
by (metis starlike_compact_projective1 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
|
|
557 |
finally have 2: "S - rel_interior S homeomorphic T - rel_interior T" .
|
|
558 |
show ?thesis
|
|
559 |
using 1 2 by blast
|
|
560 |
qed
|
|
561 |
|
|
562 |
lemma homeomorphic_convex_compact_sets:
|
|
563 |
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
|
|
564 |
assumes "convex S" "compact S" "convex T" "compact T"
|
|
565 |
and affeq: "aff_dim S = aff_dim T"
|
|
566 |
shows "S homeomorphic T"
|
|
567 |
using homeomorphic_convex_lemma [OF assms] assms
|
|
568 |
by (auto simp: rel_frontier_def)
|
|
569 |
|
|
570 |
lemma homeomorphic_rel_frontiers_convex_bounded_sets:
|
|
571 |
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
|
|
572 |
assumes "convex S" "bounded S" "convex T" "bounded T"
|
|
573 |
and affeq: "aff_dim S = aff_dim T"
|
|
574 |
shows "rel_frontier S homeomorphic rel_frontier T"
|
|
575 |
using assms homeomorphic_convex_lemma [of "closure S" "closure T"]
|
|
576 |
by (simp add: rel_frontier_def convex_rel_interior_closure)
|
|
577 |
|
|
578 |
|
|
579 |
subsection\<open>Homeomorphisms between punctured spheres and affine sets\<close>
|
|
580 |
text\<open>Including the famous stereoscopic projection of the 3-D sphere to the complex plane\<close>
|
|
581 |
|
|
582 |
text\<open>The special case with centre 0 and radius 1\<close>
|
|
583 |
lemma homeomorphic_punctured_affine_sphere_affine_01:
|
|
584 |
assumes "b \<in> sphere 0 1" "affine T" "0 \<in> T" "b \<in> T" "affine p"
|
|
585 |
and affT: "aff_dim T = aff_dim p + 1"
|
|
586 |
shows "(sphere 0 1 \<inter> T) - {b} homeomorphic p"
|
|
587 |
proof -
|
|
588 |
have [simp]: "norm b = 1" "b\<bullet>b = 1"
|
|
589 |
using assms by (auto simp: norm_eq_1)
|
|
590 |
have [simp]: "T \<inter> {v. b\<bullet>v = 0} \<noteq> {}"
|
|
591 |
using \<open>0 \<in> T\<close> by auto
|
|
592 |
have [simp]: "\<not> T \<subseteq> {v. b\<bullet>v = 0}"
|
|
593 |
using \<open>norm b = 1\<close> \<open>b \<in> T\<close> by auto
|
|
594 |
define f where "f \<equiv> \<lambda>x. 2 *\<^sub>R b + (2 / (1 - b\<bullet>x)) *\<^sub>R (x - b)"
|
|
595 |
define g where "g \<equiv> \<lambda>y. b + (4 / (norm y ^ 2 + 4)) *\<^sub>R (y - 2 *\<^sub>R b)"
|
|
596 |
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b\<bullet>x = 0\<rbrakk> \<Longrightarrow> f (g x) = x"
|
|
597 |
unfolding f_def g_def by (simp add: algebra_simps divide_simps add_nonneg_eq_0_iff)
|
|
598 |
have no: "\<And>x. \<lbrakk>norm x = 1; b\<bullet>x \<noteq> 1\<rbrakk> \<Longrightarrow> (norm (f x))\<^sup>2 = 4 * (1 + b\<bullet>x) / (1 - b\<bullet>x)"
|
|
599 |
apply (simp add: dot_square_norm [symmetric])
|
|
600 |
apply (simp add: f_def vector_add_divide_simps divide_simps norm_eq_1)
|
|
601 |
apply (simp add: algebra_simps inner_commute)
|
|
602 |
done
|
|
603 |
have [simp]: "\<And>u::real. 8 + u * (u * 8) = u * 16 \<longleftrightarrow> u=1"
|
|
604 |
by algebra
|
|
605 |
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> g (f x) = x"
|
|
606 |
unfolding g_def no by (auto simp: f_def divide_simps)
|
|
607 |
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> norm (g x) = 1"
|
|
608 |
unfolding g_def
|
|
609 |
apply (rule power2_eq_imp_eq)
|
|
610 |
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps)
|
|
611 |
apply (simp add: algebra_simps inner_commute)
|
|
612 |
done
|
|
613 |
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> b \<bullet> g x \<noteq> 1"
|
|
614 |
unfolding g_def
|
|
615 |
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff)
|
|
616 |
apply (auto simp: algebra_simps)
|
|
617 |
done
|
|
618 |
have "subspace T"
|
|
619 |
by (simp add: assms subspace_affine)
|
|
620 |
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> g x \<in> T"
|
|
621 |
unfolding g_def
|
|
622 |
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
|
|
623 |
have "f ` {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<subseteq> {x. b\<bullet>x = 0}"
|
|
624 |
unfolding f_def using \<open>norm b = 1\<close> norm_eq_1
|
|
625 |
by (force simp: field_simps inner_add_right inner_diff_right)
|
|
626 |
moreover have "f ` T \<subseteq> T"
|
|
627 |
unfolding f_def using assms
|
|
628 |
apply (auto simp: field_simps inner_add_right inner_diff_right)
|
|
629 |
by (metis add_0 diff_zero mem_affine_3_minus)
|
|
630 |
moreover have "{x. b\<bullet>x = 0} \<inter> T \<subseteq> f ` ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T)"
|
|
631 |
apply clarify
|
|
632 |
apply (rule_tac x = "g x" in image_eqI, auto)
|
|
633 |
done
|
|
634 |
ultimately have imf: "f ` ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T) = {x. b\<bullet>x = 0} \<inter> T"
|
|
635 |
by blast
|
|
636 |
have no4: "\<And>y. b\<bullet>y = 0 \<Longrightarrow> norm ((y\<bullet>y + 4) *\<^sub>R b + 4 *\<^sub>R (y - 2 *\<^sub>R b)) = y\<bullet>y + 4"
|
|
637 |
apply (rule power2_eq_imp_eq)
|
|
638 |
apply (simp_all add: dot_square_norm [symmetric])
|
|
639 |
apply (auto simp: power2_eq_square algebra_simps inner_commute)
|
|
640 |
done
|
|
641 |
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> b \<bullet> f x = 0"
|
|
642 |
by (simp add: f_def algebra_simps divide_simps)
|
|
643 |
have [simp]: "\<And>x. \<lbrakk>x \<in> T; norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> f x \<in> T"
|
|
644 |
unfolding f_def
|
|
645 |
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
|
|
646 |
have "g ` {x. b\<bullet>x = 0} \<subseteq> {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1}"
|
|
647 |
unfolding g_def
|
|
648 |
apply (clarsimp simp: no4 vector_add_divide_simps divide_simps add_nonneg_eq_0_iff dot_square_norm [symmetric])
|
|
649 |
apply (auto simp: algebra_simps)
|
|
650 |
done
|
|
651 |
moreover have "g ` T \<subseteq> T"
|
|
652 |
unfolding g_def
|
|
653 |
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
|
|
654 |
moreover have "{x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T \<subseteq> g ` ({x. b\<bullet>x = 0} \<inter> T)"
|
|
655 |
apply clarify
|
|
656 |
apply (rule_tac x = "f x" in image_eqI, auto)
|
|
657 |
done
|
|
658 |
ultimately have img: "g ` ({x. b\<bullet>x = 0} \<inter> T) = {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T"
|
|
659 |
by blast
|
|
660 |
have aff: "affine ({x. b\<bullet>x = 0} \<inter> T)"
|
|
661 |
by (blast intro: affine_hyperplane assms)
|
|
662 |
have contf: "continuous_on ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T) f"
|
|
663 |
unfolding f_def by (rule continuous_intros | force)+
|
|
664 |
have contg: "continuous_on ({x. b\<bullet>x = 0} \<inter> T) g"
|
|
665 |
unfolding g_def by (rule continuous_intros | force simp: add_nonneg_eq_0_iff)+
|
|
666 |
have "(sphere 0 1 \<inter> T) - {b} = {x. norm x = 1 \<and> (b\<bullet>x \<noteq> 1)} \<inter> T"
|
|
667 |
using \<open>norm b = 1\<close> by (auto simp: norm_eq_1) (metis vector_eq \<open>b\<bullet>b = 1\<close>)
|
|
668 |
also have "... homeomorphic {x. b\<bullet>x = 0} \<inter> T"
|
|
669 |
by (rule homeomorphicI [OF imf img contf contg]) auto
|
|
670 |
also have "... homeomorphic p"
|
|
671 |
apply (rule homeomorphic_affine_sets [OF aff \<open>affine p\<close>])
|
|
672 |
apply (simp add: Int_commute aff_dim_affine_Int_hyperplane [OF \<open>affine T\<close>] affT)
|
|
673 |
done
|
|
674 |
finally show ?thesis .
|
|
675 |
qed
|
|
676 |
|
|
677 |
theorem homeomorphic_punctured_affine_sphere_affine:
|
|
678 |
fixes a :: "'a :: euclidean_space"
|
|
679 |
assumes "0 < r" "b \<in> sphere a r" "affine T" "a \<in> T" "b \<in> T" "affine p"
|
|
680 |
and aff: "aff_dim T = aff_dim p + 1"
|
|
681 |
shows "((sphere a r \<inter> T) - {b}) homeomorphic p"
|
|
682 |
proof -
|
|
683 |
have "a \<noteq> b" using assms by auto
|
|
684 |
then have inj: "inj (\<lambda>x::'a. x /\<^sub>R norm (a - b))"
|
|
685 |
by (simp add: inj_on_def)
|
|
686 |
have "((sphere a r \<inter> T) - {b}) homeomorphic
|
|
687 |
op+ (-a) ` ((sphere a r \<inter> T) - {b})"
|
|
688 |
by (rule homeomorphic_translation)
|
|
689 |
also have "... homeomorphic op *\<^sub>R (inverse r) ` op + (- a) ` (sphere a r \<inter> T - {b})"
|
|
690 |
by (metis \<open>0 < r\<close> homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl)
|
|
691 |
also have "... = sphere 0 1 \<inter> (op *\<^sub>R (inverse r) ` op + (- a) ` T) - {(b - a) /\<^sub>R r}"
|
|
692 |
using assms by (auto simp: dist_norm norm_minus_commute divide_simps)
|
|
693 |
also have "... homeomorphic p"
|
|
694 |
apply (rule homeomorphic_punctured_affine_sphere_affine_01)
|
|
695 |
using assms
|
|
696 |
apply (auto simp: dist_norm norm_minus_commute affine_scaling affine_translation [symmetric] aff_dim_translation_eq inj)
|
|
697 |
done
|
|
698 |
finally show ?thesis .
|
|
699 |
qed
|
|
700 |
|
|
701 |
proposition homeomorphic_punctured_sphere_affine_gen:
|
|
702 |
fixes a :: "'a :: euclidean_space"
|
|
703 |
assumes "convex S" "bounded S" and a: "a \<in> rel_frontier S"
|
|
704 |
and "affine T" and affS: "aff_dim S = aff_dim T + 1"
|
|
705 |
shows "rel_frontier S - {a} homeomorphic T"
|
|
706 |
proof -
|
|
707 |
have "S \<noteq> {}" using assms by auto
|
|
708 |
obtain U :: "'a set" where "affine U" and affdS: "aff_dim U = aff_dim S"
|
|
709 |
using choose_affine_subset [OF affine_UNIV aff_dim_geq]
|
|
710 |
by (meson aff_dim_affine_hull affine_affine_hull)
|
|
711 |
have "convex U"
|
|
712 |
by (simp add: affine_imp_convex \<open>affine U\<close>)
|
|
713 |
have "U \<noteq> {}"
|
|
714 |
by (metis \<open>S \<noteq> {}\<close> \<open>aff_dim U = aff_dim S\<close> aff_dim_empty)
|
|
715 |
then obtain z where "z \<in> U"
|
|
716 |
by auto
|
|
717 |
then have bne: "ball z 1 \<inter> U \<noteq> {}" by force
|
|
718 |
have [simp]: "aff_dim(ball z 1 \<inter> U) = aff_dim U"
|
|
719 |
using aff_dim_convex_Int_open [OF \<open>convex U\<close> open_ball] bne
|
|
720 |
by (fastforce simp add: Int_commute)
|
|
721 |
have "rel_frontier S homeomorphic rel_frontier (ball z 1 \<inter> U)"
|
|
722 |
apply (rule homeomorphic_rel_frontiers_convex_bounded_sets)
|
|
723 |
apply (auto simp: \<open>affine U\<close> affine_imp_convex convex_Int affdS assms)
|
|
724 |
done
|
|
725 |
also have "... = sphere z 1 \<inter> U"
|
|
726 |
using convex_affine_rel_frontier_Int [of "ball z 1" U]
|
|
727 |
by (simp add: \<open>affine U\<close> bne)
|
|
728 |
finally obtain h k where him: "h ` rel_frontier S = sphere z 1 \<inter> U"
|
|
729 |
and kim: "k ` (sphere z 1 \<inter> U) = rel_frontier S"
|
|
730 |
and hcon: "continuous_on (rel_frontier S) h"
|
|
731 |
and kcon: "continuous_on (sphere z 1 \<inter> U) k"
|
|
732 |
and kh: "\<And>x. x \<in> rel_frontier S \<Longrightarrow> k(h(x)) = x"
|
|
733 |
and hk: "\<And>y. y \<in> sphere z 1 \<inter> U \<Longrightarrow> h(k(y)) = y"
|
|
734 |
unfolding homeomorphic_def homeomorphism_def by auto
|
|
735 |
have "rel_frontier S - {a} homeomorphic (sphere z 1 \<inter> U) - {h a}"
|
|
736 |
proof (rule homeomorphicI [where f=h and g=k])
|
|
737 |
show h: "h ` (rel_frontier S - {a}) = sphere z 1 \<inter> U - {h a}"
|
|
738 |
using him a kh by auto metis
|
|
739 |
show "k ` (sphere z 1 \<inter> U - {h a}) = rel_frontier S - {a}"
|
|
740 |
by (force simp: h [symmetric] image_comp o_def kh)
|
|
741 |
qed (auto intro: continuous_on_subset hcon kcon simp: kh hk)
|
|
742 |
also have "... homeomorphic T"
|
|
743 |
apply (rule homeomorphic_punctured_affine_sphere_affine)
|
|
744 |
using a him
|
|
745 |
by (auto simp: affS affdS \<open>affine T\<close> \<open>affine U\<close> \<open>z \<in> U\<close>)
|
|
746 |
finally show ?thesis .
|
|
747 |
qed
|
|
748 |
|
|
749 |
|
|
750 |
lemma homeomorphic_punctured_sphere_affine:
|
|
751 |
fixes a :: "'a :: euclidean_space"
|
|
752 |
assumes "0 < r" and b: "b \<in> sphere a r"
|
|
753 |
and "affine T" and affS: "aff_dim T + 1 = DIM('a)"
|
|
754 |
shows "(sphere a r - {b}) homeomorphic T"
|
|
755 |
using homeomorphic_punctured_sphere_affine_gen [of "cball a r" b T]
|
|
756 |
assms aff_dim_cball by force
|
|
757 |
|
|
758 |
corollary homeomorphic_punctured_sphere_hyperplane:
|
|
759 |
fixes a :: "'a :: euclidean_space"
|
|
760 |
assumes "0 < r" and b: "b \<in> sphere a r"
|
|
761 |
and "c \<noteq> 0"
|
|
762 |
shows "(sphere a r - {b}) homeomorphic {x::'a. c \<bullet> x = d}"
|
|
763 |
apply (rule homeomorphic_punctured_sphere_affine)
|
|
764 |
using assms
|
|
765 |
apply (auto simp: affine_hyperplane of_nat_diff)
|
|
766 |
done
|
|
767 |
|
|
768 |
|
|
769 |
text\<open> When dealing with AR, ANR and ANR later, it's useful to know that every set
|
|
770 |
is homeomorphic to a closed subset of a convex set, and
|
|
771 |
if the set is locally compact we can take the convex set to be the universe.\<close>
|
|
772 |
|
|
773 |
proposition homeomorphic_closedin_convex:
|
|
774 |
fixes S :: "'m::euclidean_space set"
|
|
775 |
assumes "aff_dim S < DIM('n)"
|
|
776 |
obtains U and T :: "'n::euclidean_space set"
|
|
777 |
where "convex U" "U \<noteq> {}" "closedin (subtopology euclidean U) T"
|
|
778 |
"S homeomorphic T"
|
|
779 |
proof (cases "S = {}")
|
|
780 |
case True then show ?thesis
|
|
781 |
by (rule_tac U=UNIV and T="{}" in that) auto
|
|
782 |
next
|
|
783 |
case False
|
|
784 |
then obtain a where "a \<in> S" by auto
|
|
785 |
obtain i::'n where i: "i \<in> Basis" "i \<noteq> 0"
|
|
786 |
using SOME_Basis Basis_zero by force
|
|
787 |
have "0 \<in> affine hull (op + (- a) ` S)"
|
|
788 |
by (simp add: \<open>a \<in> S\<close> hull_inc)
|
|
789 |
then have "dim (op + (- a) ` S) = aff_dim (op + (- a) ` S)"
|
|
790 |
by (simp add: aff_dim_zero)
|
|
791 |
also have "... < DIM('n)"
|
|
792 |
by (simp add: aff_dim_translation_eq assms)
|
|
793 |
finally have dd: "dim (op + (- a) ` S) < DIM('n)"
|
|
794 |
by linarith
|
|
795 |
obtain T where "subspace T" and Tsub: "T \<subseteq> {x. i \<bullet> x = 0}"
|
|
796 |
and dimT: "dim T = dim (op + (- a) ` S)"
|
|
797 |
apply (rule choose_subspace_of_subspace [of "dim (op + (- a) ` S)" "{x::'n. i \<bullet> x = 0}"])
|
|
798 |
apply (simp add: dim_hyperplane [OF \<open>i \<noteq> 0\<close>])
|
|
799 |
apply (metis DIM_positive Suc_pred dd not_le not_less_eq_eq)
|
|
800 |
apply (metis span_eq subspace_hyperplane)
|
|
801 |
done
|
|
802 |
have "subspace (span (op + (- a) ` S))"
|
|
803 |
using subspace_span by blast
|
|
804 |
then obtain h k where "linear h" "linear k"
|
|
805 |
and heq: "h ` span (op + (- a) ` S) = T"
|
|
806 |
and keq:"k ` T = span (op + (- a) ` S)"
|
|
807 |
and hinv [simp]: "\<And>x. x \<in> span (op + (- a) ` S) \<Longrightarrow> k(h x) = x"
|
|
808 |
and kinv [simp]: "\<And>x. x \<in> T \<Longrightarrow> h(k x) = x"
|
|
809 |
apply (rule isometries_subspaces [OF _ \<open>subspace T\<close>])
|
|
810 |
apply (auto simp: dimT)
|
|
811 |
done
|
|
812 |
have hcont: "continuous_on A h" and kcont: "continuous_on B k" for A B
|
|
813 |
using \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_conv_bounded_linear by blast+
|
|
814 |
have ihhhh[simp]: "\<And>x. x \<in> S \<Longrightarrow> i \<bullet> h (x - a) = 0"
|
|
815 |
using Tsub [THEN subsetD] heq span_inc by fastforce
|
|
816 |
have "sphere 0 1 - {i} homeomorphic {x. i \<bullet> x = 0}"
|
|
817 |
apply (rule homeomorphic_punctured_sphere_affine)
|
|
818 |
using i
|
|
819 |
apply (auto simp: affine_hyperplane)
|
|
820 |
by (metis DIM_positive Suc_eq_plus1 add.left_neutral diff_add_cancel not_le not_less_eq_eq of_nat_1 of_nat_diff)
|
|
821 |
then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i \<bullet> x = 0} f g"
|
|
822 |
by (force simp: homeomorphic_def)
|
|
823 |
have "h ` op + (- a) ` S \<subseteq> T"
|
|
824 |
using heq span_clauses(1) span_linear_image by blast
|
|
825 |
then have "g ` h ` op + (- a) ` S \<subseteq> g ` {x. i \<bullet> x = 0}"
|
|
826 |
using Tsub by (simp add: image_mono)
|
|
827 |
also have "... \<subseteq> sphere 0 1 - {i}"
|
|
828 |
by (simp add: fg [unfolded homeomorphism_def])
|
|
829 |
finally have gh_sub_sph: "(g \<circ> h) ` op + (- a) ` S \<subseteq> sphere 0 1 - {i}"
|
|
830 |
by (metis image_comp)
|
|
831 |
then have gh_sub_cb: "(g \<circ> h) ` op + (- a) ` S \<subseteq> cball 0 1"
|
|
832 |
by (metis Diff_subset order_trans sphere_cball)
|
|
833 |
have [simp]: "\<And>u. u \<in> S \<Longrightarrow> norm (g (h (u - a))) = 1"
|
|
834 |
using gh_sub_sph [THEN subsetD] by (auto simp: o_def)
|
|
835 |
have ghcont: "continuous_on (op + (- a) ` S) (\<lambda>x. g (h x))"
|
|
836 |
apply (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force)
|
|
837 |
done
|
|
838 |
have kfcont: "continuous_on ((g \<circ> h \<circ> op + (- a)) ` S) (\<lambda>x. k (f x))"
|
|
839 |
apply (rule continuous_on_compose2 [OF kcont])
|
|
840 |
using homeomorphism_cont1 [OF fg] gh_sub_sph apply (force intro: continuous_on_subset, blast)
|
|
841 |
done
|
|
842 |
have "S homeomorphic op + (- a) ` S"
|
|
843 |
by (simp add: homeomorphic_translation)
|
|
844 |
also have Shom: "\<dots> homeomorphic (g \<circ> h) ` op + (- a) ` S"
|
|
845 |
apply (simp add: homeomorphic_def homeomorphism_def)
|
|
846 |
apply (rule_tac x="g \<circ> h" in exI)
|
|
847 |
apply (rule_tac x="k \<circ> f" in exI)
|
|
848 |
apply (auto simp: ghcont kfcont span_clauses(1) homeomorphism_apply2 [OF fg] image_comp)
|
|
849 |
apply (force simp: o_def homeomorphism_apply2 [OF fg] span_clauses(1))
|
|
850 |
done
|
|
851 |
finally have Shom: "S homeomorphic (g \<circ> h) ` op + (- a) ` S" .
|
|
852 |
show ?thesis
|
|
853 |
apply (rule_tac U = "ball 0 1 \<union> image (g o h) (op + (- a) ` S)"
|
|
854 |
and T = "image (g o h) (op + (- a) ` S)"
|
|
855 |
in that)
|
|
856 |
apply (rule convex_intermediate_ball [of 0 1], force)
|
|
857 |
using gh_sub_cb apply force
|
|
858 |
apply force
|
|
859 |
apply (simp add: closedin_closed)
|
|
860 |
apply (rule_tac x="sphere 0 1" in exI)
|
|
861 |
apply (auto simp: Shom)
|
|
862 |
done
|
|
863 |
qed
|
|
864 |
|
|
865 |
subsection\<open>Locally compact sets in an open set\<close>
|
|
866 |
|
|
867 |
text\<open> Locally compact sets are closed in an open set and are homeomorphic
|
|
868 |
to an absolutely closed set if we have one more dimension to play with.\<close>
|
|
869 |
|
|
870 |
lemma locally_compact_open_Int_closure:
|
|
871 |
fixes S :: "'a :: metric_space set"
|
|
872 |
assumes "locally compact S"
|
|
873 |
obtains T where "open T" "S = T \<inter> closure S"
|
|
874 |
proof -
|
|
875 |
have "\<forall>x\<in>S. \<exists>T v u. u = S \<inter> T \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> S \<and> open T \<and> compact v"
|
|
876 |
by (metis assms locally_compact openin_open)
|
|
877 |
then obtain t v where
|
|
878 |
tv: "\<And>x. x \<in> S
|
|
879 |
\<Longrightarrow> v x \<subseteq> S \<and> open (t x) \<and> compact (v x) \<and> (\<exists>u. x \<in> u \<and> u \<subseteq> v x \<and> u = S \<inter> t x)"
|
|
880 |
by metis
|
|
881 |
then have o: "open (UNION S t)"
|
|
882 |
by blast
|
|
883 |
have "S = \<Union> (v ` S)"
|
|
884 |
using tv by blast
|
|
885 |
also have "... = UNION S t \<inter> closure S"
|
|
886 |
proof
|
|
887 |
show "UNION S v \<subseteq> UNION S t \<inter> closure S"
|
|
888 |
apply safe
|
|
889 |
apply (metis Int_iff subsetD UN_iff tv)
|
|
890 |
apply (simp add: closure_def rev_subsetD tv)
|
|
891 |
done
|
|
892 |
have "t x \<inter> closure S \<subseteq> v x" if "x \<in> S" for x
|
|
893 |
proof -
|
|
894 |
have "t x \<inter> closure S \<subseteq> closure (t x \<inter> S)"
|
|
895 |
by (simp add: open_Int_closure_subset that tv)
|
|
896 |
also have "... \<subseteq> v x"
|
|
897 |
by (metis Int_commute closure_minimal compact_imp_closed that tv)
|
|
898 |
finally show ?thesis .
|
|
899 |
qed
|
|
900 |
then show "UNION S t \<inter> closure S \<subseteq> UNION S v"
|
|
901 |
by blast
|
|
902 |
qed
|
|
903 |
finally have e: "S = UNION S t \<inter> closure S" .
|
|
904 |
show ?thesis
|
|
905 |
by (rule that [OF o e])
|
|
906 |
qed
|
|
907 |
|
|
908 |
|
|
909 |
lemma locally_compact_closedin_open:
|
|
910 |
fixes S :: "'a :: metric_space set"
|
|
911 |
assumes "locally compact S"
|
|
912 |
obtains T where "open T" "closedin (subtopology euclidean T) S"
|
|
913 |
by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int)
|
|
914 |
|
|
915 |
|
|
916 |
lemma locally_compact_homeomorphism_projection_closed:
|
|
917 |
assumes "locally compact S"
|
|
918 |
obtains T and f :: "'a \<Rightarrow> 'a :: euclidean_space \<times> 'b :: euclidean_space"
|
|
919 |
where "closed T" "homeomorphism S T f fst"
|
|
920 |
proof (cases "closed S")
|
|
921 |
case True
|
|
922 |
then show ?thesis
|
|
923 |
apply (rule_tac T = "S \<times> {0}" and f = "\<lambda>x. (x, 0)" in that)
|
|
924 |
apply (auto simp: closed_Times homeomorphism_def continuous_intros)
|
|
925 |
done
|
|
926 |
next
|
|
927 |
case False
|
|
928 |
obtain U where "open U" and US: "U \<inter> closure S = S"
|
|
929 |
by (metis locally_compact_open_Int_closure [OF assms])
|
|
930 |
with False have Ucomp: "-U \<noteq> {}"
|
|
931 |
using closure_eq by auto
|
|
932 |
have [simp]: "closure (- U) = -U"
|
|
933 |
by (simp add: \<open>open U\<close> closed_Compl)
|
|
934 |
define f :: "'a \<Rightarrow> 'a \<times> 'b" where "f \<equiv> \<lambda>x. (x, One /\<^sub>R setdist {x} (- U))"
|
|
935 |
have "continuous_on U (\<lambda>x. (x, One /\<^sub>R setdist {x} (- U)))"
|
63301
|
936 |
apply (intro continuous_intros continuous_on_setdist)
|
|
937 |
by (simp add: Ucomp setdist_eq_0_sing_1)
|
63130
|
938 |
then have homU: "homeomorphism U (f`U) f fst"
|
|
939 |
by (auto simp: f_def homeomorphism_def image_iff continuous_intros)
|
|
940 |
have cloS: "closedin (subtopology euclidean U) S"
|
|
941 |
by (metis US closed_closure closedin_closed_Int)
|
|
942 |
have cont: "isCont ((\<lambda>x. setdist {x} (- U)) o fst) z" for z :: "'a \<times> 'b"
|
|
943 |
by (rule isCont_o continuous_intros continuous_at_setdist)+
|
|
944 |
have setdist1D: "setdist {a} (- U) *\<^sub>R b = One \<Longrightarrow> setdist {a} (- U) \<noteq> 0" for a::'a and b::'b
|
|
945 |
by force
|
|
946 |
have *: "r *\<^sub>R b = One \<Longrightarrow> b = (1 / r) *\<^sub>R One" for r and b::'b
|
|
947 |
by (metis One_non_0 nonzero_divide_eq_eq real_vector.scale_eq_0_iff real_vector.scale_scale scaleR_one)
|
|
948 |
have "f ` U = {z. (setdist {fst z} (- U) *\<^sub>R snd z) \<in> {One}}"
|
63301
|
949 |
apply (auto simp: f_def setdist_eq_0_sing_1 field_simps Ucomp)
|
63130
|
950 |
apply (rule_tac x=a in image_eqI)
|
63301
|
951 |
apply (auto simp: * setdist_eq_0_sing_1 dest: setdist1D)
|
63130
|
952 |
done
|
|
953 |
then have clfU: "closed (f ` U)"
|
|
954 |
apply (rule ssubst)
|
|
955 |
apply (rule continuous_closed_preimage_univ)
|
|
956 |
apply (auto intro: continuous_intros cont [unfolded o_def])
|
|
957 |
done
|
|
958 |
have "closed (f ` S)"
|
|
959 |
apply (rule closedin_closed_trans [OF _ clfU])
|
|
960 |
apply (rule homeomorphism_imp_closed_map [OF homU cloS])
|
|
961 |
done
|
|
962 |
then show ?thesis
|
|
963 |
apply (rule that)
|
|
964 |
apply (rule homeomorphism_of_subsets [OF homU])
|
|
965 |
using US apply auto
|
|
966 |
done
|
|
967 |
qed
|
|
968 |
|
|
969 |
lemma locally_compact_closed_Int_open:
|
|
970 |
fixes S :: "'a :: euclidean_space set"
|
|
971 |
shows
|
|
972 |
"locally compact S \<longleftrightarrow> (\<exists>U u. closed U \<and> open u \<and> S = U \<inter> u)"
|
|
973 |
by (metis closed_closure closed_imp_locally_compact inf_commute locally_compact_Int locally_compact_open_Int_closure open_imp_locally_compact)
|
|
974 |
|
|
975 |
|
|
976 |
proposition locally_compact_homeomorphic_closed:
|
|
977 |
fixes S :: "'a::euclidean_space set"
|
|
978 |
assumes "locally compact S" and dimlt: "DIM('a) < DIM('b)"
|
|
979 |
obtains T :: "'b::euclidean_space set" where "closed T" "S homeomorphic T"
|
|
980 |
proof -
|
|
981 |
obtain U:: "('a*real)set" and h
|
|
982 |
where "closed U" and homU: "homeomorphism S U h fst"
|
|
983 |
using locally_compact_homeomorphism_projection_closed assms by metis
|
|
984 |
let ?BP = "Basis :: ('a*real) set"
|
|
985 |
have "DIM('a * real) \<le> DIM('b)"
|
|
986 |
by (simp add: Suc_leI dimlt)
|
|
987 |
then obtain basf :: "'a*real \<Rightarrow> 'b" where surbf: "basf ` ?BP \<subseteq> Basis" and injbf: "inj_on basf Basis"
|
|
988 |
by (metis finite_Basis card_le_inj)
|
|
989 |
define basg:: "'b \<Rightarrow> 'a * real" where
|
|
990 |
"basg \<equiv> \<lambda>i. inv_into Basis basf i"
|
|
991 |
have bgf[simp]: "basg (basf i) = i" if "i \<in> Basis" for i
|
|
992 |
using inv_into_f_f injbf that by (force simp: basg_def)
|
|
993 |
define f :: "'a*real \<Rightarrow> 'b" where "f \<equiv> \<lambda>u. \<Sum>j\<in>Basis. (u \<bullet> basg j) *\<^sub>R j"
|
|
994 |
have "linear f"
|
|
995 |
unfolding f_def
|
|
996 |
apply (intro linear_compose_setsum linearI ballI)
|
|
997 |
apply (auto simp: algebra_simps)
|
|
998 |
done
|
|
999 |
define g :: "'b \<Rightarrow> 'a*real" where "g \<equiv> \<lambda>z. (\<Sum>i\<in>Basis. (z \<bullet> basf i) *\<^sub>R i)"
|
|
1000 |
have "linear g"
|
|
1001 |
unfolding g_def
|
|
1002 |
apply (intro linear_compose_setsum linearI ballI)
|
|
1003 |
apply (auto simp: algebra_simps)
|
|
1004 |
done
|
|
1005 |
have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
|
|
1006 |
using surbf that by auto
|
|
1007 |
have gf[simp]: "g (f x) = x" for x
|
|
1008 |
apply (rule euclidean_eqI)
|
|
1009 |
apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
|
|
1010 |
apply (simp add: Groups_Big.setsum_right_distrib [symmetric] *)
|
|
1011 |
done
|
|
1012 |
then have "inj f" by (metis injI)
|
|
1013 |
have gfU: "g ` f ` U = U"
|
|
1014 |
by (rule set_eqI) (auto simp: image_def)
|
|
1015 |
have "S homeomorphic U"
|
|
1016 |
using homU homeomorphic_def by blast
|
|
1017 |
also have "... homeomorphic f ` U"
|
|
1018 |
apply (rule homeomorphicI [OF refl gfU])
|
|
1019 |
apply (meson \<open>inj f\<close> \<open>linear f\<close> homeomorphism_cont2 linear_homeomorphism_image)
|
|
1020 |
using \<open>linear g\<close> linear_continuous_on linear_conv_bounded_linear apply blast
|
|
1021 |
apply auto
|
|
1022 |
done
|
|
1023 |
finally show ?thesis
|
|
1024 |
apply (rule_tac T = "f ` U" in that)
|
|
1025 |
apply (rule closed_injective_linear_image [OF \<open>closed U\<close> \<open>linear f\<close> \<open>inj f\<close>], assumption)
|
|
1026 |
done
|
|
1027 |
qed
|
|
1028 |
|
|
1029 |
|
|
1030 |
lemma homeomorphic_convex_compact_lemma:
|
|
1031 |
fixes s :: "'a::euclidean_space set"
|
|
1032 |
assumes "convex s"
|
|
1033 |
and "compact s"
|
|
1034 |
and "cball 0 1 \<subseteq> s"
|
|
1035 |
shows "s homeomorphic (cball (0::'a) 1)"
|
|
1036 |
proof (rule starlike_compact_projective_special[OF assms(2-3)])
|
|
1037 |
fix x u
|
|
1038 |
assume "x \<in> s" and "0 \<le> u" and "u < (1::real)"
|
|
1039 |
have "open (ball (u *\<^sub>R x) (1 - u))"
|
|
1040 |
by (rule open_ball)
|
|
1041 |
moreover have "u *\<^sub>R x \<in> ball (u *\<^sub>R x) (1 - u)"
|
|
1042 |
unfolding centre_in_ball using \<open>u < 1\<close> by simp
|
|
1043 |
moreover have "ball (u *\<^sub>R x) (1 - u) \<subseteq> s"
|
|
1044 |
proof
|
|
1045 |
fix y
|
|
1046 |
assume "y \<in> ball (u *\<^sub>R x) (1 - u)"
|
|
1047 |
then have "dist (u *\<^sub>R x) y < 1 - u"
|
|
1048 |
unfolding mem_ball .
|
|
1049 |
with \<open>u < 1\<close> have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
|
|
1050 |
by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
|
|
1051 |
with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s" ..
|
|
1052 |
with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \<in> s"
|
|
1053 |
using \<open>x \<in> s\<close> \<open>0 \<le> u\<close> \<open>u < 1\<close> [THEN less_imp_le] by (rule convexD_alt)
|
|
1054 |
then show "y \<in> s" using \<open>u < 1\<close>
|
|
1055 |
by simp
|
|
1056 |
qed
|
|
1057 |
ultimately have "u *\<^sub>R x \<in> interior s" ..
|
|
1058 |
then show "u *\<^sub>R x \<in> s - frontier s"
|
|
1059 |
using frontier_def and interior_subset by auto
|
|
1060 |
qed
|
|
1061 |
|
|
1062 |
proposition homeomorphic_convex_compact_cball:
|
|
1063 |
fixes e :: real
|
|
1064 |
and s :: "'a::euclidean_space set"
|
|
1065 |
assumes "convex s"
|
|
1066 |
and "compact s"
|
|
1067 |
and "interior s \<noteq> {}"
|
|
1068 |
and "e > 0"
|
|
1069 |
shows "s homeomorphic (cball (b::'a) e)"
|
|
1070 |
proof -
|
|
1071 |
obtain a where "a \<in> interior s"
|
|
1072 |
using assms(3) by auto
|
|
1073 |
then obtain d where "d > 0" and d: "cball a d \<subseteq> s"
|
|
1074 |
unfolding mem_interior_cball by auto
|
|
1075 |
let ?d = "inverse d" and ?n = "0::'a"
|
|
1076 |
have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
|
|
1077 |
apply rule
|
|
1078 |
apply (rule_tac x="d *\<^sub>R x + a" in image_eqI)
|
|
1079 |
defer
|
|
1080 |
apply (rule d[unfolded subset_eq, rule_format])
|
|
1081 |
using \<open>d > 0\<close>
|
|
1082 |
unfolding mem_cball dist_norm
|
|
1083 |
apply (auto simp add: mult_right_le_one_le)
|
|
1084 |
done
|
|
1085 |
then have "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
|
|
1086 |
using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s",
|
|
1087 |
OF convex_affinity compact_affinity]
|
|
1088 |
using assms(1,2)
|
|
1089 |
by (auto simp add: scaleR_right_diff_distrib)
|
|
1090 |
then show ?thesis
|
|
1091 |
apply (rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
|
|
1092 |
apply (rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
|
|
1093 |
using \<open>d>0\<close> \<open>e>0\<close>
|
|
1094 |
apply (auto simp add: scaleR_right_diff_distrib)
|
|
1095 |
done
|
|
1096 |
qed
|
|
1097 |
|
|
1098 |
corollary homeomorphic_convex_compact:
|
|
1099 |
fixes s :: "'a::euclidean_space set"
|
|
1100 |
and t :: "'a set"
|
|
1101 |
assumes "convex s" "compact s" "interior s \<noteq> {}"
|
|
1102 |
and "convex t" "compact t" "interior t \<noteq> {}"
|
|
1103 |
shows "s homeomorphic t"
|
|
1104 |
using assms
|
|
1105 |
by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
|
|
1106 |
|
63301
|
1107 |
subsection\<open>Covering spaces and lifting results for them\<close>
|
|
1108 |
|
|
1109 |
definition covering_space
|
|
1110 |
:: "'a::topological_space set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
|
|
1111 |
where
|
|
1112 |
"covering_space c p s \<equiv>
|
|
1113 |
continuous_on c p \<and> p ` c = s \<and>
|
|
1114 |
(\<forall>x \<in> s. \<exists>t. x \<in> t \<and> openin (subtopology euclidean s) t \<and>
|
|
1115 |
(\<exists>v. \<Union>v = {x. x \<in> c \<and> p x \<in> t} \<and>
|
|
1116 |
(\<forall>u \<in> v. openin (subtopology euclidean c) u) \<and>
|
|
1117 |
pairwise disjnt v \<and>
|
|
1118 |
(\<forall>u \<in> v. \<exists>q. homeomorphism u t p q)))"
|
|
1119 |
|
|
1120 |
lemma covering_space_imp_continuous: "covering_space c p s \<Longrightarrow> continuous_on c p"
|
|
1121 |
by (simp add: covering_space_def)
|
|
1122 |
|
|
1123 |
lemma covering_space_imp_surjective: "covering_space c p s \<Longrightarrow> p ` c = s"
|
|
1124 |
by (simp add: covering_space_def)
|
|
1125 |
|
|
1126 |
lemma homeomorphism_imp_covering_space: "homeomorphism s t f g \<Longrightarrow> covering_space s f t"
|
|
1127 |
apply (simp add: homeomorphism_def covering_space_def, clarify)
|
|
1128 |
apply (rule_tac x=t in exI, simp)
|
|
1129 |
apply (rule_tac x="{s}" in exI, auto)
|
|
1130 |
done
|
|
1131 |
|
|
1132 |
lemma covering_space_local_homeomorphism:
|
|
1133 |
assumes "covering_space c p s" "x \<in> c"
|
|
1134 |
obtains t u q where "x \<in> t" "openin (subtopology euclidean c) t"
|
|
1135 |
"p x \<in> u" "openin (subtopology euclidean s) u"
|
|
1136 |
"homeomorphism t u p q"
|
|
1137 |
using assms
|
|
1138 |
apply (simp add: covering_space_def, clarify)
|
|
1139 |
apply (drule_tac x="p x" in bspec, force)
|
|
1140 |
by (metis (no_types, lifting) Union_iff mem_Collect_eq)
|
|
1141 |
|
|
1142 |
|
|
1143 |
lemma covering_space_local_homeomorphism_alt:
|
|
1144 |
assumes p: "covering_space c p s" and "y \<in> s"
|
|
1145 |
obtains x t u q where "p x = y"
|
|
1146 |
"x \<in> t" "openin (subtopology euclidean c) t"
|
|
1147 |
"y \<in> u" "openin (subtopology euclidean s) u"
|
|
1148 |
"homeomorphism t u p q"
|
|
1149 |
proof -
|
|
1150 |
obtain x where "p x = y" "x \<in> c"
|
|
1151 |
using assms covering_space_imp_surjective by blast
|
|
1152 |
show ?thesis
|
|
1153 |
apply (rule covering_space_local_homeomorphism [OF p \<open>x \<in> c\<close>])
|
|
1154 |
using that \<open>p x = y\<close> by blast
|
|
1155 |
qed
|
|
1156 |
|
|
1157 |
proposition covering_space_open_map:
|
|
1158 |
fixes s :: "'a :: metric_space set" and t :: "'b :: metric_space set"
|
|
1159 |
assumes p: "covering_space c p s" and t: "openin (subtopology euclidean c) t"
|
|
1160 |
shows "openin (subtopology euclidean s) (p ` t)"
|
|
1161 |
proof -
|
|
1162 |
have pce: "p ` c = s"
|
|
1163 |
and covs:
|
|
1164 |
"\<And>x. x \<in> s \<Longrightarrow>
|
|
1165 |
\<exists>X VS. x \<in> X \<and> openin (subtopology euclidean s) X \<and>
|
|
1166 |
\<Union>VS = {x. x \<in> c \<and> p x \<in> X} \<and>
|
|
1167 |
(\<forall>u \<in> VS. openin (subtopology euclidean c) u) \<and>
|
|
1168 |
pairwise disjnt VS \<and>
|
|
1169 |
(\<forall>u \<in> VS. \<exists>q. homeomorphism u X p q)"
|
|
1170 |
using p by (auto simp: covering_space_def)
|
|
1171 |
have "t \<subseteq> c" by (metis openin_euclidean_subtopology_iff t)
|
|
1172 |
have "\<exists>T. openin (subtopology euclidean s) T \<and> y \<in> T \<and> T \<subseteq> p ` t"
|
|
1173 |
if "y \<in> p ` t" for y
|
|
1174 |
proof -
|
|
1175 |
have "y \<in> s" using \<open>t \<subseteq> c\<close> pce that by blast
|
|
1176 |
obtain U VS where "y \<in> U" and U: "openin (subtopology euclidean s) U"
|
|
1177 |
and VS: "\<Union>VS = {x. x \<in> c \<and> p x \<in> U}"
|
|
1178 |
and openVS: "\<forall>V \<in> VS. openin (subtopology euclidean c) V"
|
|
1179 |
and homVS: "\<And>V. V \<in> VS \<Longrightarrow> \<exists>q. homeomorphism V U p q"
|
|
1180 |
using covs [OF \<open>y \<in> s\<close>] by auto
|
|
1181 |
obtain x where "x \<in> c" "p x \<in> U" "x \<in> t" "p x = y"
|
|
1182 |
apply simp
|
|
1183 |
using t [unfolded openin_euclidean_subtopology_iff] \<open>y \<in> U\<close> \<open>y \<in> p ` t\<close> by blast
|
|
1184 |
with VS obtain V where "x \<in> V" "V \<in> VS" by auto
|
|
1185 |
then obtain q where q: "homeomorphism V U p q" using homVS by blast
|
|
1186 |
then have ptV: "p ` (t \<inter> V) = U \<inter> {z. q z \<in> (t \<inter> V)}"
|
|
1187 |
using VS \<open>V \<in> VS\<close> by (auto simp: homeomorphism_def)
|
|
1188 |
have ocv: "openin (subtopology euclidean c) V"
|
|
1189 |
by (simp add: \<open>V \<in> VS\<close> openVS)
|
|
1190 |
have "openin (subtopology euclidean U) {z \<in> U. q z \<in> t \<inter> V}"
|
|
1191 |
apply (rule continuous_on_open [THEN iffD1, rule_format])
|
|
1192 |
using homeomorphism_def q apply blast
|
|
1193 |
using openin_subtopology_Int_subset [of c] q t unfolding homeomorphism_def
|
|
1194 |
by (metis inf.absorb_iff2 Int_commute ocv openin_euclidean_subtopology_iff)
|
|
1195 |
then have os: "openin (subtopology euclidean s) (U \<inter> {z. q z \<in> t \<inter> V})"
|
|
1196 |
using openin_trans [of U] by (simp add: Collect_conj_eq U)
|
|
1197 |
show ?thesis
|
|
1198 |
apply (rule_tac x = "p ` (t \<inter> V)" in exI)
|
|
1199 |
apply (rule conjI)
|
|
1200 |
apply (simp only: ptV os)
|
|
1201 |
using \<open>p x = y\<close> \<open>x \<in> V\<close> \<open>x \<in> t\<close> apply blast
|
|
1202 |
done
|
|
1203 |
qed
|
|
1204 |
with openin_subopen show ?thesis by blast
|
|
1205 |
qed
|
|
1206 |
|
|
1207 |
lemma covering_space_lift_unique_gen:
|
|
1208 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
|
|
1209 |
fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
|
|
1210 |
assumes cov: "covering_space c p s"
|
|
1211 |
and eq: "g1 a = g2 a"
|
|
1212 |
and f: "continuous_on t f" "f ` t \<subseteq> s"
|
|
1213 |
and g1: "continuous_on t g1" "g1 ` t \<subseteq> c"
|
|
1214 |
and fg1: "\<And>x. x \<in> t \<Longrightarrow> f x = p(g1 x)"
|
|
1215 |
and g2: "continuous_on t g2" "g2 ` t \<subseteq> c"
|
|
1216 |
and fg2: "\<And>x. x \<in> t \<Longrightarrow> f x = p(g2 x)"
|
|
1217 |
and u_compt: "u \<in> components t" and "a \<in> u" "x \<in> u"
|
|
1218 |
shows "g1 x = g2 x"
|
|
1219 |
proof -
|
|
1220 |
have "u \<subseteq> t" by (rule in_components_subset [OF u_compt])
|
|
1221 |
def G12 \<equiv> "{x \<in> u. g1 x - g2 x = 0}"
|
|
1222 |
have "connected u" by (rule in_components_connected [OF u_compt])
|
|
1223 |
have contu: "continuous_on u g1" "continuous_on u g2"
|
|
1224 |
using \<open>u \<subseteq> t\<close> continuous_on_subset g1 g2 by blast+
|
|
1225 |
have o12: "openin (subtopology euclidean u) G12"
|
|
1226 |
unfolding G12_def
|
|
1227 |
proof (subst openin_subopen, clarify)
|
|
1228 |
fix z
|
|
1229 |
assume z: "z \<in> u" "g1 z - g2 z = 0"
|
|
1230 |
obtain v w q where "g1 z \<in> v" and ocv: "openin (subtopology euclidean c) v"
|
|
1231 |
and "p (g1 z) \<in> w" and osw: "openin (subtopology euclidean s) w"
|
|
1232 |
and hom: "homeomorphism v w p q"
|
|
1233 |
apply (rule_tac x = "g1 z" in covering_space_local_homeomorphism [OF cov])
|
|
1234 |
using \<open>u \<subseteq> t\<close> \<open>z \<in> u\<close> g1(2) apply blast+
|
|
1235 |
done
|
|
1236 |
have "g2 z \<in> v" using \<open>g1 z \<in> v\<close> z by auto
|
|
1237 |
have gg: "{x \<in> u. g x \<in> v} = {x \<in> u. g x \<in> (v \<inter> g ` u)}" for g
|
|
1238 |
by auto
|
|
1239 |
have "openin (subtopology euclidean (g1 ` u)) (v \<inter> g1 ` u)"
|
|
1240 |
using ocv \<open>u \<subseteq> t\<close> g1 by (fastforce simp add: openin_open)
|
|
1241 |
then have 1: "openin (subtopology euclidean u) {x \<in> u. g1 x \<in> v}"
|
|
1242 |
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
|
|
1243 |
have "openin (subtopology euclidean (g2 ` u)) (v \<inter> g2 ` u)"
|
|
1244 |
using ocv \<open>u \<subseteq> t\<close> g2 by (fastforce simp add: openin_open)
|
|
1245 |
then have 2: "openin (subtopology euclidean u) {x \<in> u. g2 x \<in> v}"
|
|
1246 |
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
|
|
1247 |
show "\<exists>T. openin (subtopology euclidean u) T \<and>
|
|
1248 |
z \<in> T \<and> T \<subseteq> {z \<in> u. g1 z - g2 z = 0}"
|
|
1249 |
using z
|
|
1250 |
apply (rule_tac x = "{x. x \<in> u \<and> g1 x \<in> v} \<inter> {x. x \<in> u \<and> g2 x \<in> v}" in exI)
|
|
1251 |
apply (intro conjI)
|
|
1252 |
apply (rule openin_Int [OF 1 2])
|
|
1253 |
using \<open>g1 z \<in> v\<close> \<open>g2 z \<in> v\<close> apply (force simp:, clarify)
|
|
1254 |
apply (metis \<open>u \<subseteq> t\<close> subsetD eq_iff_diff_eq_0 fg1 fg2 hom homeomorphism_def)
|
|
1255 |
done
|
|
1256 |
qed
|
|
1257 |
have c12: "closedin (subtopology euclidean u) G12"
|
|
1258 |
unfolding G12_def
|
|
1259 |
by (intro continuous_intros continuous_closedin_preimage_constant contu)
|
|
1260 |
have "G12 = {} \<or> G12 = u"
|
|
1261 |
by (intro connected_clopen [THEN iffD1, rule_format] \<open>connected u\<close> conjI o12 c12)
|
|
1262 |
with eq \<open>a \<in> u\<close> have "\<And>x. x \<in> u \<Longrightarrow> g1 x - g2 x = 0" by (auto simp: G12_def)
|
|
1263 |
then show ?thesis
|
|
1264 |
using \<open>x \<in> u\<close> by force
|
|
1265 |
qed
|
|
1266 |
|
|
1267 |
proposition covering_space_lift_unique:
|
|
1268 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
|
|
1269 |
fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
|
|
1270 |
assumes "covering_space c p s"
|
|
1271 |
"g1 a = g2 a"
|
|
1272 |
"continuous_on t f" "f ` t \<subseteq> s"
|
|
1273 |
"continuous_on t g1" "g1 ` t \<subseteq> c" "\<And>x. x \<in> t \<Longrightarrow> f x = p(g1 x)"
|
|
1274 |
"continuous_on t g2" "g2 ` t \<subseteq> c" "\<And>x. x \<in> t \<Longrightarrow> f x = p(g2 x)"
|
|
1275 |
"connected t" "a \<in> t" "x \<in> t"
|
|
1276 |
shows "g1 x = g2 x"
|
|
1277 |
using covering_space_lift_unique_gen [of c p s] in_components_self assms ex_in_conv by blast
|
|
1278 |
|
63130
|
1279 |
end
|