# HG changeset patch
# User immler
# Date 1546856974 -3600
# Node ID 1331e57b54c61d022494122470c566054c4fbc85
# Parent  d692ef26021ef69836cecd9511af43cd4acab385
moved material from Connected.thy to more appropriate places

diff -r d692ef26021e -r 1331e57b54c6 src/HOL/Analysis/Connected.thy
--- a/src/HOL/Analysis/Connected.thy	Mon Jan 07 10:22:22 2019 +0100
+++ b/src/HOL/Analysis/Connected.thy	Mon Jan 07 11:29:34 2019 +0100
@@ -605,7 +605,7 @@
 done
 
 
-text \<open>Proving a function is constant by proving that a level set is open\<close>
+subsection%unimportant \<open>Proving a function is constant by proving that a level set is open\<close>
 
 lemma continuous_levelset_openin_cases:
   fixes f :: "_ \<Rightarrow> 'b::t1_space"
@@ -634,1574 +634,8 @@
   using assms (3,4)
   by fast
 
-text \<open>Some arithmetical combinations (more to prove).\<close>
 
-lemma open_scaling[intro]:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "c \<noteq> 0"
-    and "open s"
-  shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
-  {
-    fix x
-    assume "x \<in> s"
-    then obtain e where "e>0"
-      and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
-      by auto
-    have "e * \<bar>c\<bar> > 0"
-      using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
-    moreover
-    {
-      fix y
-      assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
-      then have "norm ((1 / c) *\<^sub>R y - x) < e"
-        unfolding dist_norm
-        using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
-          assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
-      then have "y \<in> (*\<^sub>R) c ` s"
-        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
-        using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
-        using assms(1)
-        unfolding dist_norm scaleR_scaleR
-        by auto
-    }
-    ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> (*\<^sub>R) c ` s"
-      apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
-      done
-  }
-  then show ?thesis unfolding open_dist by auto
-qed
-
-lemma minus_image_eq_vimage:
-  fixes A :: "'a::ab_group_add set"
-  shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
-  by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
-
-lemma open_negations:
-  fixes S :: "'a::real_normed_vector set"
-  shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
-  using open_scaling [of "- 1" S] by simp
-
-lemma open_translation:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "open S"
-  shows "open((\<lambda>x. a + x) ` S)"
-proof -
-  {
-    fix x
-    have "continuous (at x) (\<lambda>x. x - a)"
-      by (intro continuous_diff continuous_ident continuous_const)
-  }
-  moreover have "{x. x - a \<in> S} = (+) a ` S"
-    by force
-  ultimately show ?thesis
-    by (metis assms continuous_open_vimage vimage_def)
-qed
-
-lemma open_neg_translation:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "open s"
-  shows "open((\<lambda>x. a - x) ` s)"
-  using open_translation[OF open_negations[OF assms], of a]
-  by (auto simp: image_image)
-
-lemma open_affinity:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "open S"  "c \<noteq> 0"
-  shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
-proof -
-  have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
-    unfolding o_def ..
-  have "(+) a ` (*\<^sub>R) c ` S = ((+) a \<circ> (*\<^sub>R) c) ` S"
-    by auto
-  then show ?thesis
-    using assms open_translation[of "(*\<^sub>R) c ` S" a]
-    unfolding *
-    by auto
-qed
-
-lemma interior_translation:
-  fixes S :: "'a::real_normed_vector set"
-  shows "interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (interior S)"
-proof (rule set_eqI, rule)
-  fix x
-  assume "x \<in> interior ((+) a ` S)"
-  then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
-    unfolding mem_interior by auto
-  then have "ball (x - a) e \<subseteq> S"
-    unfolding subset_eq Ball_def mem_ball dist_norm
-    by (auto simp: diff_diff_eq)
-  then show "x \<in> (+) a ` interior S"
-    unfolding image_iff
-    apply (rule_tac x="x - a" in bexI)
-    unfolding mem_interior
-    using \<open>e > 0\<close>
-    apply auto
-    done
-next
-  fix x
-  assume "x \<in> (+) a ` interior S"
-  then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
-    unfolding image_iff Bex_def mem_interior by auto
-  {
-    fix z
-    have *: "a + y - z = y + a - z" by auto
-    assume "z \<in> ball x e"
-    then have "z - a \<in> S"
-      using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
-      unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
-      by auto
-    then have "z \<in> (+) a ` S"
-      unfolding image_iff by (auto intro!: bexI[where x="z - a"])
-  }
-  then have "ball x e \<subseteq> (+) a ` S"
-    unfolding subset_eq by auto
-  then show "x \<in> interior ((+) a ` S)"
-    unfolding mem_interior using \<open>e > 0\<close> by auto
-qed
-
-subsection \<open>Continuity implies uniform continuity on a compact domain\<close>
-
-text\<open>From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of
-J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)\<close>
-
-lemma Heine_Borel_lemma:
-  assumes "compact S" and Ssub: "S \<subseteq> \<Union>\<G>" and opn: "\<And>G. G \<in> \<G> \<Longrightarrow> open G"
-  obtains e where "0 < e" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x e \<subseteq> G"
-proof -
-  have False if neg: "\<And>e. 0 < e \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x e \<subseteq> G"
-  proof -
-    have "\<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x (1 / Suc n) \<subseteq> G" for n
-      using neg by simp
-    then obtain f where "\<And>n. f n \<in> S" and fG: "\<And>G n. G \<in> \<G> \<Longrightarrow> \<not> ball (f n) (1 / Suc n) \<subseteq> G"
-      by metis
-    then obtain l r where "l \<in> S" "strict_mono r" and to_l: "(f \<circ> r) \<longlonglongrightarrow> l"
-      using \<open>compact S\<close> compact_def that by metis
-    then obtain G where "l \<in> G" "G \<in> \<G>"
-      using Ssub by auto
-    then obtain e where "0 < e" and e: "\<And>z. dist z l < e \<Longrightarrow> z \<in> G"
-      using opn open_dist by blast
-    obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < e/2"
-      using to_l apply (simp add: lim_sequentially)
-      using \<open>0 < e\<close> half_gt_zero that by blast
-    obtain N2 where N2: "of_nat N2 > 2/e"
-      using reals_Archimedean2 by blast
-    obtain x where "x \<in> ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x \<notin> G"
-      using fG [OF \<open>G \<in> \<G>\<close>, of "r (max N1 N2)"] by blast
-    then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
-      by simp
-    also have "... \<le> 1 / real (Suc (max N1 N2))"
-      apply (simp add: divide_simps del: max.bounded_iff)
-      using \<open>strict_mono r\<close> seq_suble by blast
-    also have "... \<le> 1 / real (Suc N2)"
-      by (simp add: field_simps)
-    also have "... < e/2"
-      using N2 \<open>0 < e\<close> by (simp add: field_simps)
-    finally have "dist (f (r (max N1 N2))) x < e / 2" .
-    moreover have "dist (f (r (max N1 N2))) l < e/2"
-      using N1 max.cobounded1 by blast
-    ultimately have "dist x l < e"
-      using dist_triangle_half_r by blast
-    then show ?thesis
-      using e \<open>x \<notin> G\<close> by blast
-  qed
-  then show ?thesis
-    by (meson that)
-qed
-
-lemma compact_uniformly_equicontinuous:
-  assumes "compact S"
-      and cont: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
-                        \<Longrightarrow> \<exists>d. 0 < d \<and>
-                                (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
-      and "0 < e"
-  obtains d where "0 < d"
-                  "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-proof -
-  obtain d where d_pos: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> 0 < d x e"
-     and d_dist : "\<And>x x' e f. \<lbrakk>dist x' x < d x e; x \<in> S; x' \<in> S; 0 < e; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-    using cont by metis
-  let ?\<G> = "((\<lambda>x. ball x (d x (e / 2))) ` S)"
-  have Ssub: "S \<subseteq> \<Union> ?\<G>"
-    by clarsimp (metis d_pos \<open>0 < e\<close> dist_self half_gt_zero_iff)
-  then obtain k where "0 < k" and k: "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> ?\<G>. ball x k \<subseteq> G"
-    by (rule Heine_Borel_lemma [OF \<open>compact S\<close>]) auto
-  moreover have "dist (f v) (f u) < e" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v
-  proof -
-    obtain G where "G \<in> ?\<G>" "u \<in> G" "v \<in> G"
-      using k that
-      by (metis \<open>dist v u < k\<close> \<open>u \<in> S\<close> \<open>0 < k\<close> centre_in_ball subsetD dist_commute mem_ball)
-    then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w \<in> S"
-      by auto
-    with that d_dist have "dist (f w) (f v) < e/2"
-      by (metis \<open>0 < e\<close> dist_commute half_gt_zero)
-    moreover
-    have "dist (f w) (f u) < e/2"
-      using that d_dist w by (metis \<open>0 < e\<close> dist_commute divide_pos_pos zero_less_numeral)
-    ultimately show ?thesis
-      using dist_triangle_half_r by blast
-  qed
-  ultimately show ?thesis using that by blast
-qed
-
-corollary compact_uniformly_continuous:
-  fixes f :: "'a :: metric_space \<Rightarrow> 'b :: metric_space"
-  assumes f: "continuous_on S f" and S: "compact S"
-  shows "uniformly_continuous_on S f"
-  using f
-    unfolding continuous_on_iff uniformly_continuous_on_def
-    by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])
-
-subsection%unimportant \<open>Topological stuff about the set of Reals\<close>
-
-lemma open_real:
-  fixes s :: "real set"
-  shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e --> x' \<in> s)"
-  unfolding open_dist dist_norm by simp
-
-lemma islimpt_approachable_real:
-  fixes s :: "real set"
-  shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"
-  unfolding islimpt_approachable dist_norm by simp
-
-lemma closed_real:
-  fixes s :: "real set"
-  shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e) \<longrightarrow> x \<in> s)"
-  unfolding closed_limpt islimpt_approachable dist_norm by simp
-
-lemma continuous_at_real_range:
-  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
-  shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> \<bar>f x' - f x\<bar> < e)"
-  unfolding continuous_at
-  unfolding Lim_at
-  unfolding dist_norm
-  apply auto
-  apply (erule_tac x=e in allE, auto)
-  apply (rule_tac x=d in exI, auto)
-  apply (erule_tac x=x' in allE, auto)
-  apply (erule_tac x=e in allE, auto)
-  done
-
-lemma continuous_on_real_range:
-  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
-  shows "continuous_on s f \<longleftrightarrow>
-    (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e))"
-  unfolding continuous_on_iff dist_norm by simp
-
-
-subsection%unimportant \<open>Cartesian products\<close>
-
-lemma bounded_Times:
-  assumes "bounded s" "bounded t"
-  shows "bounded (s \<times> t)"
-proof -
-  obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
-    using assms [unfolded bounded_def] by auto
-  then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"
-    by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
-  then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
-qed
-
-lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
-  by (induct x) simp
-
-lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
-  unfolding seq_compact_def
-  apply clarify
-  apply (drule_tac x="fst \<circ> f" in spec)
-  apply (drule mp, simp add: mem_Times_iff)
-  apply (clarify, rename_tac l1 r1)
-  apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
-  apply (drule mp, simp add: mem_Times_iff)
-  apply (clarify, rename_tac l2 r2)
-  apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
-  apply (rule_tac x="r1 \<circ> r2" in exI)
-  apply (rule conjI, simp add: strict_mono_def)
-  apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
-  apply (drule (1) tendsto_Pair) back
-  apply (simp add: o_def)
-  done
-
-lemma compact_Times:
-  assumes "compact s" "compact t"
-  shows "compact (s \<times> t)"
-proof (rule compactI)
-  fix C
-  assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
-  have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
-  proof
-    fix x
-    assume "x \<in> s"
-    have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
-    proof
-      fix y
-      assume "y \<in> t"
-      with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
-      then show "?P y" by (auto elim!: open_prod_elim)
-    qed
-    then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
-      and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
-      by metis
-    then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
-    with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
-      by metis
-    moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
-      by (fastforce simp: subset_eq)
-    ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
-      using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
-  qed
-  then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
-    and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"
-    unfolding subset_eq UN_iff by metis
-  moreover
-  from compactE_image[OF \<open>compact s\<close> a]
-  obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
-    by auto
-  moreover
-  {
-    from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
-      by auto
-    also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"
-      using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
-    finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .
-  }
-  ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
-    by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
-qed
-
-text\<open>Hence some useful properties follow quite easily.\<close>
-
-lemma compact_scaling:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "compact s"
-  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
-  let ?f = "\<lambda>x. scaleR c x"
-  have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
-  show ?thesis
-    using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
-    using linear_continuous_at[OF *] assms
-    by auto
-qed
-
-lemma compact_negations:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "compact s"
-  shows "compact ((\<lambda>x. - x) ` s)"
-  using compact_scaling [OF assms, of "- 1"] by auto
-
-lemma compact_sums:
-  fixes s t :: "'a::real_normed_vector set"
-  assumes "compact s"
-    and "compact t"
-  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
-proof -
-  have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
-    apply auto
-    unfolding image_iff
-    apply (rule_tac x="(xa, y)" in bexI)
-    apply auto
-    done
-  have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
-    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
-  then show ?thesis
-    unfolding * using compact_continuous_image compact_Times [OF assms] by auto
-qed
-
-lemma compact_differences:
-  fixes s t :: "'a::real_normed_vector set"
-  assumes "compact s"
-    and "compact t"
-  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
-proof-
-  have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
-    apply auto
-    apply (rule_tac x= xa in exI, auto)
-    done
-  then show ?thesis
-    using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
-qed
-
-lemma compact_translation:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "compact s"
-  shows "compact ((\<lambda>x. a + x) ` s)"
-proof -
-  have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
-    by auto
-  then show ?thesis
-    using compact_sums[OF assms compact_sing[of a]] by auto
-qed
-
-lemma compact_affinity:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "compact s"
-  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof -
-  have "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
-    by auto
-  then show ?thesis
-    using compact_translation[OF compact_scaling[OF assms], of a c] by auto
-qed
-
-text \<open>Hence we get the following.\<close>
-
-lemma compact_sup_maxdistance:
-  fixes s :: "'a::metric_space set"
-  assumes "compact s"
-    and "s \<noteq> {}"
-  shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
-proof -
-  have "compact (s \<times> s)"
-    using \<open>compact s\<close> by (intro compact_Times)
-  moreover have "s \<times> s \<noteq> {}"
-    using \<open>s \<noteq> {}\<close> by auto
-  moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
-    by (intro continuous_at_imp_continuous_on ballI continuous_intros)
-  ultimately show ?thesis
-    using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto
-qed
-
-
-subsection \<open>The diameter of a set\<close>
-
-definition%important diameter :: "'a::metric_space set \<Rightarrow> real" where
-  "diameter S = (if S = {} then 0 else SUP (x,y)\<in>S\<times>S. dist x y)"
-
-lemma diameter_empty [simp]: "diameter{} = 0"
-  by (auto simp: diameter_def)
-
-lemma diameter_singleton [simp]: "diameter{x} = 0"
-  by (auto simp: diameter_def)
-
-lemma diameter_le:
-  assumes "S \<noteq> {} \<or> 0 \<le> d"
-      and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> d"
-    shows "diameter S \<le> d"
-using assms
-  by (auto simp: dist_norm diameter_def intro: cSUP_least)
-
-lemma diameter_bounded_bound:
-  fixes s :: "'a :: metric_space set"
-  assumes s: "bounded s" "x \<in> s" "y \<in> s"
-  shows "dist x y \<le> diameter s"
-proof -
-  from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
-    unfolding bounded_def by auto
-  have "bdd_above (case_prod dist ` (s\<times>s))"
-  proof (intro bdd_aboveI, safe)
-    fix a b
-    assume "a \<in> s" "b \<in> s"
-    with z[of a] z[of b] dist_triangle[of a b z]
-    show "dist a b \<le> 2 * d"
-      by (simp add: dist_commute)
-  qed
-  moreover have "(x,y) \<in> s\<times>s" using s by auto
-  ultimately have "dist x y \<le> (SUP (x,y)\<in>s\<times>s. dist x y)"
-    by (rule cSUP_upper2) simp
-  with \<open>x \<in> s\<close> show ?thesis
-    by (auto simp: diameter_def)
-qed
-
-lemma diameter_lower_bounded:
-  fixes s :: "'a :: metric_space set"
-  assumes s: "bounded s"
-    and d: "0 < d" "d < diameter s"
-  shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
-proof (rule ccontr)
-  assume contr: "\<not> ?thesis"
-  moreover have "s \<noteq> {}"
-    using d by (auto simp: diameter_def)
-  ultimately have "diameter s \<le> d"
-    by (auto simp: not_less diameter_def intro!: cSUP_least)
-  with \<open>d < diameter s\<close> show False by auto
-qed
-
-lemma diameter_bounded:
-  assumes "bounded s"
-  shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
-    and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"
-  using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
-  by auto
-
-lemma bounded_two_points:
-  "bounded S \<longleftrightarrow> (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"
-  apply (rule iffI)
-  subgoal using diameter_bounded(1) by auto
-  subgoal using bounded_any_center[of S] by meson
-  done
-
-lemma diameter_compact_attained:
-  assumes "compact s"
-    and "s \<noteq> {}"
-  shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
-proof -
-  have b: "bounded s" using assms(1)
-    by (rule compact_imp_bounded)
-  then obtain x y where xys: "x\<in>s" "y\<in>s"
-    and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
-    using compact_sup_maxdistance[OF assms] by auto
-  then have "diameter s \<le> dist x y"
-    unfolding diameter_def
-    apply clarsimp
-    apply (rule cSUP_least, fast+)
-    done
-  then show ?thesis
-    by (metis b diameter_bounded_bound order_antisym xys)
-qed
-
-lemma diameter_ge_0:
-  assumes "bounded S"  shows "0 \<le> diameter S"
-  by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)
-
-lemma diameter_subset:
-  assumes "S \<subseteq> T" "bounded T"
-  shows "diameter S \<le> diameter T"
-proof (cases "S = {} \<or> T = {}")
-  case True
-  with assms show ?thesis
-    by (force simp: diameter_ge_0)
-next
-  case False
-  then have "bdd_above ((\<lambda>x. case x of (x, xa) \<Rightarrow> dist x xa) ` (T \<times> T))"
-    using \<open>bounded T\<close> diameter_bounded_bound by (force simp: bdd_above_def)
-  with False \<open>S \<subseteq> T\<close> show ?thesis
-    apply (simp add: diameter_def)
-    apply (rule cSUP_subset_mono, auto)
-    done
-qed
-
-lemma diameter_closure:
-  assumes "bounded S"
-  shows "diameter(closure S) = diameter S"
-proof (rule order_antisym)
-  have "False" if "diameter S < diameter (closure S)"
-  proof -
-    define d where "d = diameter(closure S) - diameter(S)"
-    have "d > 0"
-      using that by (simp add: d_def)
-    then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
-      by simp
-    have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
-      by (simp add: d_def divide_simps)
-     have bocl: "bounded (closure S)"
-      using assms by blast
-    moreover have "0 \<le> diameter S"
-      using assms diameter_ge_0 by blast
-    ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
-      using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] \<open>d > 0\<close> d_def by auto
-    then obtain x' y' where x'y': "x' \<in> S" "dist x' x < d/4" "y' \<in> S" "dist y' y < d/4"
-      using closure_approachable
-      by (metis \<open>0 < d\<close> zero_less_divide_iff zero_less_numeral)
-    then have "dist x' y' \<le> diameter S"
-      using assms diameter_bounded_bound by blast
-    with x'y' have "dist x y \<le> d / 4 + diameter S + d / 4"
-      by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)
-    then show ?thesis
-      using xy d_def by linarith
-  qed
-  then show "diameter (closure S) \<le> diameter S"
-    by fastforce
-  next
-    show "diameter S \<le> diameter (closure S)"
-      by (simp add: assms bounded_closure closure_subset diameter_subset)
-qed
-
-lemma diameter_cball [simp]:
-  fixes a :: "'a::euclidean_space"
-  shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
-proof -
-  have "diameter(cball a r) = 2*r" if "r \<ge> 0"
-  proof (rule order_antisym)
-    show "diameter (cball a r) \<le> 2*r"
-    proof (rule diameter_le)
-      fix x y assume "x \<in> cball a r" "y \<in> cball a r"
-      then have "norm (x - a) \<le> r" "norm (a - y) \<le> r"
-        by (auto simp: dist_norm norm_minus_commute)
-      then have "norm (x - y) \<le> r+r"
-        using norm_diff_triangle_le by blast
-      then show "norm (x - y) \<le> 2*r" by simp
-    qed (simp add: that)
-    have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"
-      apply (simp add: dist_norm)
-      by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
-    also have "... \<le> diameter (cball a r)"
-      apply (rule diameter_bounded_bound)
-      using that by (auto simp: dist_norm)
-    finally show "2*r \<le> diameter (cball a r)" .
-  qed
-  then show ?thesis by simp
-qed
-
-lemma diameter_ball [simp]:
-  fixes a :: "'a::euclidean_space"
-  shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
-proof -
-  have "diameter(ball a r) = 2*r" if "r > 0"
-    by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
-  then show ?thesis
-    by (simp add: diameter_def)
-qed
-
-lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
-proof -
-  have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
-    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
-  then show ?thesis
-    by simp
-qed
-
-lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
-proof -
-  have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
-    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
-  then show ?thesis
-    by simp
-qed
-
-proposition Lebesgue_number_lemma:
-  assumes "compact S" "\<C> \<noteq> {}" "S \<subseteq> \<Union>\<C>" and ope: "\<And>B. B \<in> \<C> \<Longrightarrow> open B"
-  obtains \<delta> where "0 < \<delta>" "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B \<in> \<C>. T \<subseteq> B"
-proof (cases "S = {}")
-  case True
-  then show ?thesis
-    by (metis \<open>\<C> \<noteq> {}\<close> zero_less_one empty_subsetI equals0I subset_trans that)
-next
-  case False
-  { fix x assume "x \<in> S"
-    then obtain C where C: "x \<in> C" "C \<in> \<C>"
-      using \<open>S \<subseteq> \<Union>\<C>\<close> by blast
-    then obtain r where r: "r>0" "ball x (2*r) \<subseteq> C"
-      by (metis mult.commute mult_2_right not_le ope openE field_sum_of_halves zero_le_numeral zero_less_mult_iff)
-    then have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>"
-      using C by blast
-  }
-  then obtain r where r: "\<And>x. x \<in> S \<Longrightarrow> r x > 0 \<and> (\<exists>C \<in> \<C>. ball x (2*r x) \<subseteq> C)"
-    by metis
-  then have "S \<subseteq> (\<Union>x \<in> S. ball x (r x))"
-    by auto
-  then obtain \<T> where "finite \<T>" "S \<subseteq> \<Union>\<T>" and \<T>: "\<T> \<subseteq> (\<lambda>x. ball x (r x)) ` S"
-    by (rule compactE [OF \<open>compact S\<close>]) auto
-  then obtain S0 where "S0 \<subseteq> S" "finite S0" and S0: "\<T> = (\<lambda>x. ball x (r x)) ` S0"
-    by (meson finite_subset_image)
-  then have "S0 \<noteq> {}"
-    using False \<open>S \<subseteq> \<Union>\<T>\<close> by auto
-  define \<delta> where "\<delta> = Inf (r ` S0)"
-  have "\<delta> > 0"
-    using \<open>finite S0\<close> \<open>S0 \<subseteq> S\<close> \<open>S0 \<noteq> {}\<close> r by (auto simp: \<delta>_def finite_less_Inf_iff)
-  show ?thesis
-  proof
-    show "0 < \<delta>"
-      by (simp add: \<open>0 < \<delta>\<close>)
-    show "\<exists>B \<in> \<C>. T \<subseteq> B" if "T \<subseteq> S" and dia: "diameter T < \<delta>" for T
-    proof (cases "T = {}")
-      case True
-      then show ?thesis
-        using \<open>\<C> \<noteq> {}\<close> by blast
-    next
-      case False
-      then obtain y where "y \<in> T" by blast
-      then have "y \<in> S"
-        using \<open>T \<subseteq> S\<close> by auto
-      then obtain x where "x \<in> S0" and x: "y \<in> ball x (r x)"
-        using \<open>S \<subseteq> \<Union>\<T>\<close> S0 that by blast
-      have "ball y \<delta> \<subseteq> ball y (r x)"
-        by (metis \<delta>_def \<open>S0 \<noteq> {}\<close> \<open>finite S0\<close> \<open>x \<in> S0\<close> empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
-      also have "... \<subseteq> ball x (2*r x)"
-        by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)
-      finally obtain C where "C \<in> \<C>" "ball y \<delta> \<subseteq> C"
-        by (meson r \<open>S0 \<subseteq> S\<close> \<open>x \<in> S0\<close> dual_order.trans subsetCE)
-      have "bounded T"
-        using \<open>compact S\<close> bounded_subset compact_imp_bounded \<open>T \<subseteq> S\<close> by blast
-      then have "T \<subseteq> ball y \<delta>"
-        using \<open>y \<in> T\<close> dia diameter_bounded_bound by fastforce
-      then show ?thesis
-        apply (rule_tac x=C in bexI)
-        using \<open>ball y \<delta> \<subseteq> C\<close> \<open>C \<in> \<C>\<close> by auto
-    qed
-  qed
-qed
-
-lemma diameter_cbox:
-  fixes a b::"'a::euclidean_space"
-  shows "(\<forall>i \<in> Basis. a \<bullet> i \<le> b \<bullet> i) \<Longrightarrow> diameter (cbox a b) = dist a b"
-  by (force simp: diameter_def intro!: cSup_eq_maximum L2_set_mono
-     simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
-
-subsection \<open>Separation between points and sets\<close>
-
-proposition separate_point_closed:
-  fixes s :: "'a::heine_borel set"
-  assumes "closed s" and "a \<notin> s"
-  shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
-proof (cases "s = {}")
-  case True
-  then show ?thesis by(auto intro!: exI[where x=1])
-next
-  case False
-  from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"
-    using \<open>s \<noteq> {}\<close> by (blast intro: distance_attains_inf [of s a])
-  with \<open>x\<in>s\<close> show ?thesis using dist_pos_lt[of a x] and\<open>a \<notin> s\<close>
-    by blast
-qed
-
-proposition separate_compact_closed:
-  fixes s t :: "'a::heine_borel set"
-  assumes "compact s"
-    and t: "closed t" "s \<inter> t = {}"
-  shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
-proof cases
-  assume "s \<noteq> {} \<and> t \<noteq> {}"
-  then have "s \<noteq> {}" "t \<noteq> {}" by auto
-  let ?inf = "\<lambda>x. infdist x t"
-  have "continuous_on s ?inf"
-    by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)
-  then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"
-    using continuous_attains_inf[OF \<open>compact s\<close> \<open>s \<noteq> {}\<close>] by auto
-  then have "0 < ?inf x"
-    using t \<open>t \<noteq> {}\<close> in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
-  moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
-    using x by (auto intro: order_trans infdist_le)
-  ultimately show ?thesis by auto
-qed (auto intro!: exI[of _ 1])
-
-proposition separate_closed_compact:
-  fixes s t :: "'a::heine_borel set"
-  assumes "closed s"
-    and "compact t"
-    and "s \<inter> t = {}"
-  shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
-proof -
-  have *: "t \<inter> s = {}"
-    using assms(3) by auto
-  show ?thesis
-    using separate_compact_closed[OF assms(2,1) *] by (force simp: dist_commute)
-qed
-
-proposition compact_in_open_separated:
-  fixes A::"'a::heine_borel set"
-  assumes "A \<noteq> {}"
-  assumes "compact A"
-  assumes "open B"
-  assumes "A \<subseteq> B"
-  obtains e where "e > 0" "{x. infdist x A \<le> e} \<subseteq> B"
-proof atomize_elim
-  have "closed (- B)" "compact A" "- B \<inter> A = {}"
-    using assms by (auto simp: open_Diff compact_eq_bounded_closed)
-  from separate_closed_compact[OF this]
-  obtain d'::real where d': "d'>0" "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d' \<le> dist x y"
-    by auto
-  define d where "d = d' / 2"
-  hence "d>0" "d < d'" using d' by auto
-  with d' have d: "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d < dist x y"
-    by force
-  show "\<exists>e>0. {x. infdist x A \<le> e} \<subseteq> B"
-  proof (rule ccontr)
-    assume "\<nexists>e. 0 < e \<and> {x. infdist x A \<le> e} \<subseteq> B"
-    with \<open>d > 0\<close> obtain x where x: "infdist x A \<le> d" "x \<notin> B"
-      by auto
-    from assms have "closed A" "A \<noteq> {}" by (auto simp: compact_eq_bounded_closed)
-    from infdist_attains_inf[OF this]
-    obtain y where y: "y \<in> A" "infdist x A = dist x y"
-      by auto
-    have "dist x y \<le> d" using x y by simp
-    also have "\<dots> < dist x y" using y d x by auto
-    finally show False by simp
-  qed
-qed
-
-
-subsection%unimportant \<open>Compact sets and the closure operation\<close>
-
-lemma closed_scaling:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "closed S"
-  shows "closed ((\<lambda>x. c *\<^sub>R x) ` S)"
-proof (cases "c = 0")
-  case True then show ?thesis
-    by (auto simp: image_constant_conv)
-next
-  case False
-  from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) -` S)"
-    by (simp add: continuous_closed_vimage)
-  also have "(\<lambda>x. inverse c *\<^sub>R x) -` S = (\<lambda>x. c *\<^sub>R x) ` S"
-    using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
-  finally show ?thesis .
-qed
-
-lemma closed_negations:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "closed S"
-  shows "closed ((\<lambda>x. -x) ` S)"
-  using closed_scaling[OF assms, of "- 1"] by simp
-
-lemma compact_closed_sums:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "compact S" and "closed T"
-  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-proof -
-  let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"
-  {
-    fix x l
-    assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"
-    from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> S"  "\<forall>n. snd (f n) \<in> T"
-      using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto
-    obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
-      using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
-    have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
-      using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
-      unfolding o_def
-      by auto
-    then have "l - l' \<in> T"
-      using assms(2)[unfolded closed_sequential_limits,
-        THEN spec[where x="\<lambda> n. snd (f (r n))"],
-        THEN spec[where x="l - l'"]]
-      using f(3)
-      by auto
-    then have "l \<in> ?S"
-      using \<open>l' \<in> S\<close>
-      apply auto
-      apply (rule_tac x=l' in exI)
-      apply (rule_tac x="l - l'" in exI, auto)
-      done
-  }
-  moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-    by force
-  ultimately show ?thesis
-    unfolding closed_sequential_limits
-    by (metis (no_types, lifting))
-qed
-
-lemma closed_compact_sums:
-  fixes S T :: "'a::real_normed_vector set"
-  assumes "closed S" "compact T"
-  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-proof -
-  have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-    by auto
-  then show ?thesis
-    using compact_closed_sums[OF assms(2,1)] by simp
-qed
-
-lemma compact_closed_differences:
-  fixes S T :: "'a::real_normed_vector set"
-  assumes "compact S" "closed T"
-  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-proof -
-  have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-    by force
-  then show ?thesis
-    using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
-qed
-
-lemma closed_compact_differences:
-  fixes S T :: "'a::real_normed_vector set"
-  assumes "closed S" "compact T"
-  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-proof -
-  have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
-    by auto
- then show ?thesis
-  using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
-qed
-
-lemma closed_translation:
-  fixes a :: "'a::real_normed_vector"
-  assumes "closed S"
-  shows "closed ((\<lambda>x. a + x) ` S)"
-proof -
-  have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a ` S)" by auto
-  then show ?thesis
-    using compact_closed_sums[OF compact_sing[of a] assms] by auto
-qed
-
-lemma closure_translation:
-  fixes a :: "'a::real_normed_vector"
-  shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
-proof -
-  have *: "(+) a ` (- s) = - (+) a ` s"
-    apply auto
-    unfolding image_iff
-    apply (rule_tac x="x - a" in bexI, auto)
-    done
-  show ?thesis
-    unfolding closure_interior translation_Compl
-    using interior_translation[of a "- s"]
-    unfolding *
-    by auto
-qed
-
-lemma frontier_translation:
-  fixes a :: "'a::real_normed_vector"
-  shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
-  unfolding frontier_def translation_diff interior_translation closure_translation
-  by auto
-
-lemma sphere_translation:
-  fixes a :: "'n::real_normed_vector"
-  shows "sphere (a+c) r = (+) a ` sphere c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma cball_translation:
-  fixes a :: "'n::real_normed_vector"
-  shows "cball (a+c) r = (+) a ` cball c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma ball_translation:
-  fixes a :: "'n::real_normed_vector"
-  shows "ball (a+c) r = (+) a ` ball c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-
-subsection%unimportant \<open>Closure of halfspaces and hyperplanes\<close>
-
-lemma continuous_on_closed_Collect_le:
-  fixes f g :: "'a::t2_space \<Rightarrow> real"
-  assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
-  shows "closed {x \<in> s. f x \<le> g x}"
-proof -
-  have "closed ((\<lambda>x. g x - f x) -` {0..} \<inter> s)"
-    using closed_real_atLeast continuous_on_diff [OF g f]
-    by (simp add: continuous_on_closed_vimage [OF s])
-  also have "((\<lambda>x. g x - f x) -` {0..} \<inter> s) = {x\<in>s. f x \<le> g x}"
-    by auto
-  finally show ?thesis .
-qed
-
-lemma continuous_at_inner: "continuous (at x) (inner a)"
-  unfolding continuous_at by (intro tendsto_intros)
-
-lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
-  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
-  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_hyperplane: "closed {x. inner a x = b}"
-  by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
-  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
-  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_interval_left:
-  fixes b :: "'a::euclidean_space"
-  shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
-  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_interval_right:
-  fixes a :: "'a::euclidean_space"
-  shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
-  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma continuous_le_on_closure:
-  fixes a::real
-  assumes f: "continuous_on (closure s) f"
-      and x: "x \<in> closure(s)"
-      and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"
-    shows "f(x) \<le> a"
-    using image_closure_subset [OF f]
-  using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
-  by force
-
-lemma continuous_ge_on_closure:
-  fixes a::real
-  assumes f: "continuous_on (closure s) f"
-      and x: "x \<in> closure(s)"
-      and xlo: "\<And>x. x \<in> s ==> f(x) \<ge> a"
-    shows "f(x) \<ge> a"
-  using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms
-  by force
-
-lemma Lim_component_le:
-  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
-  assumes "(f \<longlongrightarrow> l) net"
-    and "\<not> (trivial_limit net)"
-    and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
-  shows "l\<bullet>i \<le> b"
-  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
-
-lemma Lim_component_ge:
-  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
-  assumes "(f \<longlongrightarrow> l) net"
-    and "\<not> (trivial_limit net)"
-    and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
-  shows "b \<le> l\<bullet>i"
-  by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
-
-lemma Lim_component_eq:
-  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
-  assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"
-    and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
-  shows "l\<bullet>i = b"
-  using ev[unfolded order_eq_iff eventually_conj_iff]
-  using Lim_component_ge[OF net, of b i]
-  using Lim_component_le[OF net, of i b]
-  by auto
-
-text \<open>Limits relative to a union.\<close>
-
-lemma eventually_within_Un:
-  "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
-    eventually P (at x within s) \<and> eventually P (at x within t)"
-  unfolding eventually_at_filter
-  by (auto elim!: eventually_rev_mp)
-
-lemma Lim_within_union:
- "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
-  (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
-  unfolding tendsto_def
-  by (auto simp: eventually_within_Un)
-
-lemma Lim_topological:
-  "(f \<longlongrightarrow> l) net \<longleftrightarrow>
-    trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
-  unfolding tendsto_def trivial_limit_eq by auto
-
-text \<open>Continuity relative to a union.\<close>
-
-lemma continuous_on_Un_local:
-    "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
-      continuous_on s f; continuous_on t f\<rbrakk>
-     \<Longrightarrow> continuous_on (s \<union> t) f"
-  unfolding continuous_on closedin_limpt
-  by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
-
-lemma continuous_on_cases_local:
-     "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
-       continuous_on s f; continuous_on t g;
-       \<And>x. \<lbrakk>x \<in> s \<and> \<not>P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
-      \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
-  by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
-
-lemma continuous_on_cases_le:
-  fixes h :: "'a :: topological_space \<Rightarrow> real"
-  assumes "continuous_on {t \<in> s. h t \<le> a} f"
-      and "continuous_on {t \<in> s. a \<le> h t} g"
-      and h: "continuous_on s h"
-      and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
-    shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
-proof -
-  have s: "s = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
-    by force
-  have 1: "closedin (subtopology euclidean s) (s \<inter> h -` atMost a)"
-    by (rule continuous_closedin_preimage [OF h closed_atMost])
-  have 2: "closedin (subtopology euclidean s) (s \<inter> h -` atLeast a)"
-    by (rule continuous_closedin_preimage [OF h closed_atLeast])
-  have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
-    by auto
-  show ?thesis
-    apply (rule continuous_on_subset [of s, OF _ order_refl])
-    apply (subst s)
-    apply (rule continuous_on_cases_local)
-    using 1 2 s assms apply (auto simp: eq)
-    done
-qed
-
-lemma continuous_on_cases_1:
-  fixes s :: "real set"
-  assumes "continuous_on {t \<in> s. t \<le> a} f"
-      and "continuous_on {t \<in> s. a \<le> t} g"
-      and "a \<in> s \<Longrightarrow> f a = g a"
-    shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
-using assms
-by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
-
-subsubsection\<open>Some more convenient intermediate-value theorem formulations\<close>
-
-lemma connected_ivt_hyperplane:
-  assumes "connected S" and xy: "x \<in> S" "y \<in> S" and b: "inner a x \<le> b" "b \<le> inner a y"
-  shows "\<exists>z \<in> S. inner a z = b"
-proof (rule ccontr)
-  assume as:"\<not> (\<exists>z\<in>S. inner a z = b)"
-  let ?A = "{x. inner a x < b}"
-  let ?B = "{x. inner a x > b}"
-  have "open ?A" "open ?B"
-    using open_halfspace_lt and open_halfspace_gt by auto
-  moreover have "?A \<inter> ?B = {}" by auto
-  moreover have "S \<subseteq> ?A \<union> ?B" using as by auto
-  ultimately show False
-    using \<open>connected S\<close>[unfolded connected_def not_ex,
-      THEN spec[where x="?A"], THEN spec[where x="?B"]]
-    using xy b by auto
-qed
-
-lemma connected_ivt_component:
-  fixes x::"'a::euclidean_space"
-  shows "connected S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>S.  z\<bullet>k = a)"
-  using connected_ivt_hyperplane[of S x y "k::'a" a]
-  by (auto simp: inner_commute)
-
-lemma image_affinity_cbox: fixes m::real
-  fixes a b c :: "'a::euclidean_space"
-  shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b =
-    (if cbox a b = {} then {}
-     else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
-     else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
-proof (cases "m = 0")
-  case True
-  {
-    fix x
-    assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"
-    then have "x = c"
-      by (simp add: dual_order.antisym euclidean_eqI)
-  }
-  moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
-    unfolding True by (auto simp: cbox_sing)
-  ultimately show ?thesis using True by (auto simp: cbox_def)
-next
-  case False
-  {
-    fix y
-    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"
-    then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
-      by (auto simp: inner_distrib)
-  }
-  moreover
-  {
-    fix y
-    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"
-    then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"
-      by (auto simp: mult_left_mono_neg inner_distrib)
-  }
-  moreover
-  {
-    fix y
-    assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
-    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
-      unfolding image_iff Bex_def mem_box
-      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
-      apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
-      done
-  }
-  moreover
-  {
-    fix y
-    assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"
-    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
-      unfolding image_iff Bex_def mem_box
-      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
-      apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
-      done
-  }
-  ultimately show ?thesis using False by (auto simp: cbox_def)
-qed
-
-lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
-  (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
-  using image_affinity_cbox[of m 0 a b] by auto
-
-lemma islimpt_greaterThanLessThan1:
-  fixes a b::"'a::{linorder_topology, dense_order}"
-  assumes "a < b"
-  shows  "a islimpt {a<..<b}"
-proof (rule islimptI)
-  fix T
-  assume "open T" "a \<in> T"
-  from open_right[OF this \<open>a < b\<close>]
-  obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
-  with assms dense[of a "min c b"]
-  show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
-    by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
-      not_le order.strict_implies_order subset_eq)
-qed
-
-lemma islimpt_greaterThanLessThan2:
-  fixes a b::"'a::{linorder_topology, dense_order}"
-  assumes "a < b"
-  shows  "b islimpt {a<..<b}"
-proof (rule islimptI)
-  fix T
-  assume "open T" "b \<in> T"
-  from open_left[OF this \<open>a < b\<close>]
-  obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
-  with assms dense[of "max a c" b]
-  show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
-    by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
-      not_le order.strict_implies_order subset_eq)
-qed
-
-lemma closure_greaterThanLessThan[simp]:
-  fixes a b::"'a::{linorder_topology, dense_order}"
-  shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
-proof
-  have "?l \<subseteq> closure ?r"
-    by (rule closure_mono) auto
-  thus "closure {a<..<b} \<subseteq> {a..b}" by simp
-qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
-  islimpt_greaterThanLessThan2)
-
-lemma closure_greaterThan[simp]:
-  fixes a b::"'a::{no_top, linorder_topology, dense_order}"
-  shows "closure {a<..} = {a..}"
-proof -
-  from gt_ex obtain b where "a < b" by auto
-  hence "{a<..} = {a<..<b} \<union> {b..}" by auto
-  also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
-    by auto
-  finally show ?thesis .
-qed
-
-lemma closure_lessThan[simp]:
-  fixes b::"'a::{no_bot, linorder_topology, dense_order}"
-  shows "closure {..<b} = {..b}"
-proof -
-  from lt_ex obtain a where "a < b" by auto
-  hence "{..<b} = {a<..<b} \<union> {..a}" by auto
-  also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
-    by auto
-  finally show ?thesis .
-qed
-
-lemma closure_atLeastLessThan[simp]:
-  fixes a b::"'a::{linorder_topology, dense_order}"
-  assumes "a < b"
-  shows "closure {a ..< b} = {a .. b}"
-proof -
-  from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
-  also have "closure \<dots> = {a .. b}" unfolding closure_Un
-    by (auto simp: assms less_imp_le)
-  finally show ?thesis .
-qed
-
-lemma closure_greaterThanAtMost[simp]:
-  fixes a b::"'a::{linorder_topology, dense_order}"
-  assumes "a < b"
-  shows "closure {a <.. b} = {a .. b}"
-proof -
-  from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
-  also have "closure \<dots> = {a .. b}" unfolding closure_Un
-    by (auto simp: assms less_imp_le)
-  finally show ?thesis .
-qed
-
-
-subsection \<open>Homeomorphisms\<close>
-
-definition%important "homeomorphism s t f g \<longleftrightarrow>
-  (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
-  (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
-
-lemma homeomorphismI [intro?]:
-  assumes "continuous_on S f" "continuous_on T g"
-          "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
-    shows "homeomorphism S T f g"
-  using assms by (force simp: homeomorphism_def)
-
-lemma homeomorphism_translation:
-  fixes a :: "'a :: real_normed_vector"
-  shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
-unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
-
-lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
-  by (rule homeomorphismI) (auto simp: continuous_on_id)
-
-lemma homeomorphism_compose:
-  assumes "homeomorphism S T f g" "homeomorphism T U h k"
-    shows "homeomorphism S U (h o f) (g o k)"
-  using assms
-  unfolding homeomorphism_def
-  by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])
-
-lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
-  by (force simp: homeomorphism_def)
-
-definition%important homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
-    (infixr "homeomorphic" 60)
-  where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
-
-lemma homeomorphic_empty [iff]:
-     "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
-  by (auto simp: homeomorphic_def homeomorphism_def)
-
-lemma homeomorphic_refl: "s homeomorphic s"
-  unfolding homeomorphic_def homeomorphism_def
-  using continuous_on_id
-  apply (rule_tac x = "(\<lambda>x. x)" in exI)
-  apply (rule_tac x = "(\<lambda>x. x)" in exI)
-  apply blast
-  done
-
-lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
-  unfolding homeomorphic_def homeomorphism_def
-  by blast
-
-lemma homeomorphic_trans [trans]:
-  assumes "S homeomorphic T"
-      and "T homeomorphic U"
-    shows "S homeomorphic U"
-  using assms
-  unfolding homeomorphic_def
-by (metis homeomorphism_compose)
-
-lemma homeomorphic_minimal:
-  "s homeomorphic t \<longleftrightarrow>
-    (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
-           (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
-           continuous_on s f \<and> continuous_on t g)"
-   (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    by (fastforce simp: homeomorphic_def homeomorphism_def)
-next
-  assume ?rhs
-  then show ?lhs
-    apply clarify
-    unfolding homeomorphic_def homeomorphism_def
-    by (metis equalityI image_subset_iff subsetI)
- qed
-
-lemma homeomorphicI [intro?]:
-   "\<lbrakk>f ` S = T; g ` T = S;
-     continuous_on S f; continuous_on T g;
-     \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
-     \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
-unfolding homeomorphic_def homeomorphism_def by metis
-
-lemma homeomorphism_of_subsets:
-   "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
-    \<Longrightarrow> homeomorphism S' T' f g"
-apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
-by (metis subsetD imageI)
-
-lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
-  by (simp add: homeomorphism_def)
-
-lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
-  by (simp add: homeomorphism_def)
-
-lemma continuous_on_no_limpt:
-   "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
-  unfolding continuous_on_def
-  by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
-
-lemma continuous_on_finite:
-  fixes S :: "'a::t1_space set"
-  shows "finite S \<Longrightarrow> continuous_on S f"
-by (metis continuous_on_no_limpt islimpt_finite)
-
-lemma homeomorphic_finite:
-  fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
-  assumes "finite T"
-  shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
-proof
-  assume "S homeomorphic T"
-  with assms show ?rhs
-    apply (auto simp: homeomorphic_def homeomorphism_def)
-     apply (metis finite_imageI)
-    by (metis card_image_le finite_imageI le_antisym)
-next
-  assume R: ?rhs
-  with finite_same_card_bij obtain h where "bij_betw h S T"
-    by auto
-  with R show ?lhs
-    apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
-    apply (rule_tac x=h in exI)
-    apply (rule_tac x="inv_into S h" in exI)
-    apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
-    apply (metis bij_betw_def bij_betw_inv_into)
-    done
-qed
-
-text \<open>Relatively weak hypotheses if a set is compact.\<close>
-
-lemma homeomorphism_compact:
-  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
-  assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
-  shows "\<exists>g. homeomorphism s t f g"
-proof -
-  define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
-  have g: "\<forall>x\<in>s. g (f x) = x"
-    using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
-  {
-    fix y
-    assume "y \<in> t"
-    then obtain x where x:"f x = y" "x\<in>s"
-      using assms(3) by auto
-    then have "g (f x) = x" using g by auto
-    then have "f (g y) = y" unfolding x(1)[symmetric] by auto
-  }
-  then have g':"\<forall>x\<in>t. f (g x) = x" by auto
-  moreover
-  {
-    fix x
-    have "x\<in>s \<Longrightarrow> x \<in> g ` t"
-      using g[THEN bspec[where x=x]]
-      unfolding image_iff
-      using assms(3)
-      by (auto intro!: bexI[where x="f x"])
-    moreover
-    {
-      assume "x\<in>g ` t"
-      then obtain y where y:"y\<in>t" "g y = x" by auto
-      then obtain x' where x':"x'\<in>s" "f x' = y"
-        using assms(3) by auto
-      then have "x \<in> s"
-        unfolding g_def
-        using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
-        unfolding y(2)[symmetric] and g_def
-        by auto
-    }
-    ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
-  }
-  then have "g ` t = s" by auto
-  ultimately show ?thesis
-    unfolding homeomorphism_def homeomorphic_def
-    apply (rule_tac x=g in exI)
-    using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
-    apply auto
-    done
-qed
-
-lemma homeomorphic_compact:
-  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
-  shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
-  unfolding homeomorphic_def by (metis homeomorphism_compact)
-
-text\<open>Preservation of topological properties.\<close>
-
-lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
-  unfolding homeomorphic_def homeomorphism_def
-  by (metis compact_continuous_image)
-
-text\<open>Results on translation, scaling etc.\<close>
-
-lemma homeomorphic_scaling:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "c \<noteq> 0"
-  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
-  unfolding homeomorphic_minimal
-  apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
-  apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
-  using assms
-  apply (auto simp: continuous_intros)
-  done
-
-lemma homeomorphic_translation:
-  fixes s :: "'a::real_normed_vector set"
-  shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
-  unfolding homeomorphic_minimal
-  apply (rule_tac x="\<lambda>x. a + x" in exI)
-  apply (rule_tac x="\<lambda>x. -a + x" in exI)
-  using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
-    continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
-  apply auto
-  done
-
-lemma homeomorphic_affinity:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "c \<noteq> 0"
-  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof -
-  have *: "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
-  show ?thesis
-    using homeomorphic_trans
-    using homeomorphic_scaling[OF assms, of s]
-    using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a]
-    unfolding *
-    by auto
-qed
-
-lemma homeomorphic_balls:
-  fixes a b ::"'a::real_normed_vector"
-  assumes "0 < d"  "0 < e"
-  shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
-    and "(cball a d) homeomorphic (cball b e)" (is ?cth)
-proof -
-  show ?th unfolding homeomorphic_minimal
-    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
-    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
-    using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
-    done
-  show ?cth unfolding homeomorphic_minimal
-    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
-    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
-    using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
-    done
-qed
-
-lemma homeomorphic_spheres:
-  fixes a b ::"'a::real_normed_vector"
-  assumes "0 < d"  "0 < e"
-  shows "(sphere a d) homeomorphic (sphere b e)"
-unfolding homeomorphic_minimal
-    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
-    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
-    using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
-    done
-
-lemma homeomorphic_ball01_UNIV:
-  "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
-  (is "?B homeomorphic ?U")
-proof
-  have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
-    apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
-     apply (auto simp: divide_simps)
-    using norm_ge_zero [of x] apply linarith+
-    done
-  then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
-    by blast
-  have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
-    apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)
-    using that apply (auto simp: divide_simps)
-    done
-  then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
-    by (force simp: divide_simps dest: add_less_zeroD)
-  show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
-    by (rule continuous_intros | force)+
-  show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
-    apply (intro continuous_intros)
-    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
-    done
-  show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
-         x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
-    by (auto simp: divide_simps)
-  show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
-    apply (auto simp: divide_simps)
-    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
-    done
-qed
-
-proposition homeomorphic_ball_UNIV:
-  fixes a ::"'a::real_normed_vector"
-  assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
-  using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
-
-
-text \<open>Connectedness is invariant under homeomorphisms.\<close>
+subsection%unimportant \<open>Connectedness is invariant under homeomorphisms.\<close>
 
 lemma homeomorphic_connectedness:
   assumes "s homeomorphic t"
@@ -2209,806 +643,6 @@
 using assms unfolding homeomorphic_def homeomorphism_def by (metis connected_continuous_image)
 
 
-subsection%unimportant\<open>Inverse function property for open/closed maps\<close>
-
-lemma continuous_on_inverse_open_map:
-  assumes contf: "continuous_on S f"
-    and imf: "f ` S = T"
-    and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
-    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
-  shows "continuous_on T g"
-proof -
-  from imf injf have gTS: "g ` T = S"
-    by force
-  from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
-    by force
-  show ?thesis
-    by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
-qed
-
-lemma continuous_on_inverse_closed_map:
-  assumes contf: "continuous_on S f"
-    and imf: "f ` S = T"
-    and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
-    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
-  shows "continuous_on T g"
-proof -
-  from imf injf have gTS: "g ` T = S"
-    by force
-  from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
-    by force
-  show ?thesis
-    by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
-qed
-
-lemma homeomorphism_injective_open_map:
-  assumes contf: "continuous_on S f"
-    and imf: "f ` S = T"
-    and injf: "inj_on f S"
-    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
-  obtains g where "homeomorphism S T f g"
-proof
-  have "continuous_on T (inv_into S f)"
-    by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
-  with imf injf contf show "homeomorphism S T f (inv_into S f)"
-    by (auto simp: homeomorphism_def)
-qed
-
-lemma homeomorphism_injective_closed_map:
-  assumes contf: "continuous_on S f"
-    and imf: "f ` S = T"
-    and injf: "inj_on f S"
-    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
-  obtains g where "homeomorphism S T f g"
-proof
-  have "continuous_on T (inv_into S f)"
-    by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
-  with imf injf contf show "homeomorphism S T f (inv_into S f)"
-    by (auto simp: homeomorphism_def)
-qed
-
-lemma homeomorphism_imp_open_map:
-  assumes hom: "homeomorphism S T f g"
-    and oo: "openin (subtopology euclidean S) U"
-  shows "openin (subtopology euclidean T) (f ` U)"
-proof -
-  from hom oo have [simp]: "f ` U = T \<inter> g -` U"
-    using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
-  from hom have "continuous_on T g"
-    unfolding homeomorphism_def by blast
-  moreover have "g ` T = S"
-    by (metis hom homeomorphism_def)
-  ultimately show ?thesis
-    by (simp add: continuous_on_open oo)
-qed
-
-lemma homeomorphism_imp_closed_map:
-  assumes hom: "homeomorphism S T f g"
-    and oo: "closedin (subtopology euclidean S) U"
-  shows "closedin (subtopology euclidean T) (f ` U)"
-proof -
-  from hom oo have [simp]: "f ` U = T \<inter> g -` U"
-    using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
-  from hom have "continuous_on T g"
-    unfolding homeomorphism_def by blast
-  moreover have "g ` T = S"
-    by (metis hom homeomorphism_def)
-  ultimately show ?thesis
-    by (simp add: continuous_on_closed oo)
-qed
-
-
-subsection \<open>"Isometry" (up to constant bounds) of injective linear map etc\<close>
-
-lemma cauchy_isometric:
-  assumes e: "e > 0"
-    and s: "subspace s"
-    and f: "bounded_linear f"
-    and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
-    and xs: "\<forall>n. x n \<in> s"
-    and cf: "Cauchy (f \<circ> x)"
-  shows "Cauchy x"
-proof -
-  interpret f: bounded_linear f by fact
-  have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real
-  proof -
-    from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
-      using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
-      by auto
-    have "norm (x n - x N) < d" if "n \<ge> N" for n
-    proof -
-      have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
-        using subspace_diff[OF s, of "x n" "x N"]
-        using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
-        using normf[THEN bspec[where x="x n - x N"]]
-        by auto
-      also have "norm (f (x n - x N)) < e * d"
-        using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
-      finally show ?thesis
-        using \<open>e>0\<close> by simp
-    qed
-    then show ?thesis by auto
-  qed
-  then show ?thesis
-    by (simp add: Cauchy_altdef2 dist_norm)
-qed
-
-lemma complete_isometric_image:
-  assumes "0 < e"
-    and s: "subspace s"
-    and f: "bounded_linear f"
-    and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
-    and cs: "complete s"
-  shows "complete (f ` s)"
-proof -
-  have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
-    if as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g" for g
-  proof -
-    from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
-      using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
-    then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
-    then have "f \<circ> x = g" by (simp add: fun_eq_iff)
-    then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
-      using cs[unfolded complete_def, THEN spec[where x=x]]
-      using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
-      by auto
-    then show ?thesis
-      using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
-      by (auto simp: \<open>f \<circ> x = g\<close>)
-  qed
-  then show ?thesis
-    unfolding complete_def by auto
-qed
-
-proposition injective_imp_isometric:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes s: "closed s" "subspace s"
-    and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
-  shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
-proof (cases "s \<subseteq> {0::'a}")
-  case True
-  have "norm x \<le> norm (f x)" if "x \<in> s" for x
-  proof -
-    from True that have "x = 0" by auto
-    then show ?thesis by simp
-  qed
-  then show ?thesis
-    by (auto intro!: exI[where x=1])
-next
-  case False
-  interpret f: bounded_linear f by fact
-  from False obtain a where a: "a \<noteq> 0" "a \<in> s"
-    by auto
-  from False have "s \<noteq> {}"
-    by auto
-  let ?S = "{f x| x. x \<in> s \<and> norm x = norm a}"
-  let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
-  let ?S'' = "{x::'a. norm x = norm a}"
-
-  have "?S'' = frontier (cball 0 (norm a))"
-    by (simp add: sphere_def dist_norm)
-  then have "compact ?S''" by (metis compact_cball compact_frontier)
-  moreover have "?S' = s \<inter> ?S''" by auto
-  ultimately have "compact ?S'"
-    using closed_Int_compact[of s ?S''] using s(1) by auto
-  moreover have *:"f ` ?S' = ?S" by auto
-  ultimately have "compact ?S"
-    using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
-  then have "closed ?S"
-    using compact_imp_closed by auto
-  moreover from a have "?S \<noteq> {}" by auto
-  ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"
-    using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
-  then obtain b where "b\<in>s"
-    and ba: "norm b = norm a"
-    and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
-    unfolding *[symmetric] unfolding image_iff by auto
-
-  let ?e = "norm (f b) / norm b"
-  have "norm b > 0"
-    using ba and a and norm_ge_zero by auto
-  moreover have "norm (f b) > 0"
-    using f(2)[THEN bspec[where x=b], OF \<open>b\<in>s\<close>]
-    using \<open>norm b >0\<close> by simp
-  ultimately have "0 < norm (f b) / norm b" by simp
-  moreover
-  have "norm (f b) / norm b * norm x \<le> norm (f x)" if "x\<in>s" for x
-  proof (cases "x = 0")
-    case True
-    then show "norm (f b) / norm b * norm x \<le> norm (f x)"
-      by auto
-  next
-    case False
-    with \<open>a \<noteq> 0\<close> have *: "0 < norm a / norm x"
-      unfolding zero_less_norm_iff[symmetric] by simp
-    have "\<forall>x\<in>s. c *\<^sub>R x \<in> s" for c
-      using s[unfolded subspace_def] by simp
-    with \<open>x \<in> s\<close> \<open>x \<noteq> 0\<close> have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
-      by simp
-    with \<open>x \<noteq> 0\<close> \<open>a \<noteq> 0\<close> show "norm (f b) / norm b * norm x \<le> norm (f x)"
-      using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
-      unfolding f.scaleR and ba
-      by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
-  qed
-  ultimately show ?thesis by auto
-qed
-
-proposition closed_injective_image_subspace:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"
-  shows "closed(f ` s)"
-proof -
-  obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
-    using injective_imp_isometric[OF assms(4,1,2,3)] by auto
-  show ?thesis
-    using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)
-    unfolding complete_eq_closed[symmetric] by auto
-qed
-
-
-subsection%unimportant \<open>Some properties of a canonical subspace\<close>
-
-lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
-  by (auto simp: subspace_def inner_add_left)
-
-lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"
-  (is "closed ?A")
-proof -
-  let ?D = "{i\<in>Basis. P i}"
-  have "closed (\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0})"
-    by (simp add: closed_INT closed_Collect_eq continuous_on_inner
-        continuous_on_const continuous_on_id)
-  also have "(\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0}) = ?A"
-    by auto
-  finally show "closed ?A" .
-qed
-
-lemma dim_substandard:
-  assumes d: "d \<subseteq> Basis"
-  shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
-proof (rule dim_unique)
-  from d show "d \<subseteq> ?A"
-    by (auto simp: inner_Basis)
-  from d show "independent d"
-    by (rule independent_mono [OF independent_Basis])
-  have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
-  proof -
-    have "finite d"
-      by (rule finite_subset [OF d finite_Basis])
-    then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
-      by (simp add: span_sum span_clauses)
-    also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
-      by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
-    finally show "x \<in> span d"
-      by (simp only: euclidean_representation)
-  qed
-  then show "?A \<subseteq> span d" by auto
-qed simp
-
-text \<open>Hence closure and completeness of all subspaces.\<close>
-lemma ex_card:
-  assumes "n \<le> card A"
-  shows "\<exists>S\<subseteq>A. card S = n"
-proof (cases "finite A")
-  case True
-  from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
-  moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
-    by (auto simp: bij_betw_def intro: subset_inj_on)
-  ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
-    by (auto simp: bij_betw_def card_image)
-  then show ?thesis by blast
-next
-  case False
-  with \<open>n \<le> card A\<close> show ?thesis by force
-qed
-
-lemma closed_subspace:
-  fixes s :: "'a::euclidean_space set"
-  assumes "subspace s"
-  shows "closed s"
-proof -
-  have "dim s \<le> card (Basis :: 'a set)"
-    using dim_subset_UNIV by auto
-  with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"
-    by auto
-  let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
-  have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>
-      inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
-    using dim_substandard[of d] t d assms
-    by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)
-  then obtain f where f:
-      "linear f"
-      "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
-      "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
-    by blast
-  interpret f: bounded_linear f
-    using f by (simp add: linear_conv_bounded_linear)
-  have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x
-    using f.zero d f(3)[THEN inj_onD, of x 0] by auto
-  moreover have "closed ?t" by (rule closed_substandard)
-  moreover have "subspace ?t" by (rule subspace_substandard)
-  ultimately show ?thesis
-    using closed_injective_image_subspace[of ?t f]
-    unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
-qed
-
-lemma complete_subspace: "subspace s \<Longrightarrow> complete s"
-  for s :: "'a::euclidean_space set"
-  using complete_eq_closed closed_subspace by auto
-
-lemma closed_span [iff]: "closed (span s)"
-  for s :: "'a::euclidean_space set"
-  by (simp add: closed_subspace subspace_span)
-
-lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
-  for s :: "'a::euclidean_space set"
-proof -
-  have "?dc \<le> ?d"
-    using closure_minimal[OF span_superset, of s]
-    using closed_subspace[OF subspace_span, of s]
-    using dim_subset[of "closure s" "span s"]
-    by simp
-  then show ?thesis
-    using dim_subset[OF closure_subset, of s]
-    by simp
-qed
-
-
-subsection%unimportant \<open>Affine transformations of intervals\<close>
-
-lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"
-  for m :: "'a::linordered_field"
-  by (simp add: field_simps)
-
-
-subsection \<open>Banach fixed point theorem (not really topological ...)\<close>
-
-theorem banach_fix:
-  assumes s: "complete s" "s \<noteq> {}"
-    and c: "0 \<le> c" "c < 1"
-    and f: "f ` s \<subseteq> s"
-    and lipschitz: "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
-  shows "\<exists>!x\<in>s. f x = x"
-proof -
-  from c have "1 - c > 0" by simp
-
-  from s(2) obtain z0 where z0: "z0 \<in> s" by blast
-  define z where "z n = (f ^^ n) z0" for n
-  with f z0 have z_in_s: "z n \<in> s" for n :: nat
-    by (induct n) auto
-  define d where "d = dist (z 0) (z 1)"
-
-  have fzn: "f (z n) = z (Suc n)" for n
-    by (simp add: z_def)
-  have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * d" for n :: nat
-  proof (induct n)
-    case 0
-    then show ?case
-      by (simp add: d_def)
-  next
-    case (Suc m)
-    with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
-      using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
-    then show ?case
-      using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
-      by (simp add: fzn mult_le_cancel_left)
-  qed
-
-  have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * d * (1 - c ^ n)" for n m :: nat
-  proof (induct n)
-    case 0
-    show ?case by simp
-  next
-    case (Suc k)
-    from c have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>
-        (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
-      by (simp add: dist_triangle)
-    also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
-      by simp
-    also from Suc have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
-      by (simp add: field_simps)
-    also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
-      by (simp add: power_add field_simps)
-    also from c have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
-      by (simp add: field_simps)
-    finally show ?case by simp
-  qed
-
-  have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e" if "e > 0" for e
-  proof (cases "d = 0")
-    case True
-    from \<open>1 - c > 0\<close> have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x
-      by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
-    with c cf_z2[of 0] True have "z n = z0" for n
-      by (simp add: z_def)
-    with \<open>e > 0\<close> show ?thesis by simp
-  next
-    case False
-    with zero_le_dist[of "z 0" "z 1"] have "d > 0"
-      by (metis d_def less_le)
-    with \<open>1 - c > 0\<close> \<open>e > 0\<close> have "0 < e * (1 - c) / d"
-      by simp
-    with c obtain N where N: "c ^ N < e * (1 - c) / d"
-      using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
-    have *: "dist (z m) (z n) < e" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat
-    proof -
-      from c \<open>n \<ge> N\<close> have *: "c ^ n \<le> c ^ N"
-        using power_decreasing[OF \<open>n\<ge>N\<close>, of c] by simp
-      from c \<open>m > n\<close> have "1 - c ^ (m - n) > 0"
-        using power_strict_mono[of c 1 "m - n"] by simp
-      with \<open>d > 0\<close> \<open>0 < 1 - c\<close> have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
-        by simp
-      from cf_z2[of n "m - n"] \<open>m > n\<close>
-      have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
-        by (simp add: pos_le_divide_eq[OF \<open>1 - c > 0\<close>] mult.commute dist_commute)
-      also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
-        using mult_right_mono[OF * order_less_imp_le[OF **]]
-        by (simp add: mult.assoc)
-      also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
-        using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
-      also from c \<open>d > 0\<close> \<open>1 - c > 0\<close> have "\<dots> = e * (1 - c ^ (m - n))"
-        by simp
-      also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>e > 0\<close> have "\<dots> \<le> e"
-        using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
-      finally show ?thesis by simp
-    qed
-    have "dist (z n) (z m) < e" if "N \<le> m" "N \<le> n" for m n :: nat
-    proof (cases "n = m")
-      case True
-      with \<open>e > 0\<close> show ?thesis by simp
-    next
-      case False
-      with *[of n m] *[of m n] and that show ?thesis
-        by (auto simp: dist_commute nat_neq_iff)
-    qed
-    then show ?thesis by auto
-  qed
-  then have "Cauchy z"
-    by (simp add: cauchy_def)
-  then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"
-    using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
-
-  define e where "e = dist (f x) x"
-  have "e = 0"
-  proof (rule ccontr)
-    assume "e \<noteq> 0"
-    then have "e > 0"
-      unfolding e_def using zero_le_dist[of "f x" x]
-      by (metis dist_eq_0_iff dist_nz e_def)
-    then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
-      using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
-    then have N':"dist (z N) x < e / 2" by auto
-    have *: "c * dist (z N) x \<le> dist (z N) x"
-      unfolding mult_le_cancel_right2
-      using zero_le_dist[of "z N" x] and c
-      by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
-    have "dist (f (z N)) (f x) \<le> c * dist (z N) x"
-      using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
-      using z_in_s[of N] \<open>x\<in>s\<close>
-      using c
-      by auto
-    also have "\<dots> < e / 2"
-      using N' and c using * by auto
-    finally show False
-      unfolding fzn
-      using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
-      unfolding e_def
-      by auto
-  qed
-  then have "f x = x" by (auto simp: e_def)
-  moreover have "y = x" if "f y = y" "y \<in> s" for y
-  proof -
-    from \<open>x \<in> s\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"
-      using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp
-    with c and zero_le_dist[of x y] have "dist x y = 0"
-      by (simp add: mult_le_cancel_right1)
-    then show ?thesis by simp
-  qed
-  ultimately show ?thesis
-    using \<open>x\<in>s\<close> by blast
-qed
-
-lemma banach_fix_type:
-  fixes f::"'a::complete_space\<Rightarrow>'a"
-  assumes c:"0 \<le> c" "c < 1"
-      and lipschitz:"\<forall>x. \<forall>y. dist (f x) (f y) \<le> c * dist x y"
-  shows "\<exists>!x. (f x = x)"
-  using assms banach_fix[OF complete_UNIV UNIV_not_empty assms(1,2) subset_UNIV, of f]
-  by auto
-
-
-subsection \<open>Edelstein fixed point theorem\<close>
-
-theorem edelstein_fix:
-  fixes s :: "'a::metric_space set"
-  assumes s: "compact s" "s \<noteq> {}"
-    and gs: "(g ` s) \<subseteq> s"
-    and dist: "\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
-  shows "\<exists>!x\<in>s. g x = x"
-proof -
-  let ?D = "(\<lambda>x. (x, x)) ` s"
-  have D: "compact ?D" "?D \<noteq> {}"
-    by (rule compact_continuous_image)
-       (auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)
-
-  have "\<And>x y e. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 < e \<Longrightarrow> dist y x < e \<Longrightarrow> dist (g y) (g x) < e"
-    using dist by fastforce
-  then have "continuous_on s g"
-    by (auto simp: continuous_on_iff)
-  then have cont: "continuous_on ?D (\<lambda>x. dist ((g \<circ> fst) x) (snd x))"
-    unfolding continuous_on_eq_continuous_within
-    by (intro continuous_dist ballI continuous_within_compose)
-       (auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)
-
-  obtain a where "a \<in> s" and le: "\<And>x. x \<in> s \<Longrightarrow> dist (g a) a \<le> dist (g x) x"
-    using continuous_attains_inf[OF D cont] by auto
-
-  have "g a = a"
-  proof (rule ccontr)
-    assume "g a \<noteq> a"
-    with \<open>a \<in> s\<close> gs have "dist (g (g a)) (g a) < dist (g a) a"
-      by (intro dist[rule_format]) auto
-    moreover have "dist (g a) a \<le> dist (g (g a)) (g a)"
-      using \<open>a \<in> s\<close> gs by (intro le) auto
-    ultimately show False by auto
-  qed
-  moreover have "\<And>x. x \<in> s \<Longrightarrow> g x = x \<Longrightarrow> x = a"
-    using dist[THEN bspec[where x=a]] \<open>g a = a\<close> and \<open>a\<in>s\<close> by auto
-  ultimately show "\<exists>!x\<in>s. g x = x"
-    using \<open>a \<in> s\<close> by blast
-qed
-
-
-lemma cball_subset_cball_iff:
-  fixes a :: "'a :: euclidean_space"
-  shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
-    (is "?lhs \<longleftrightarrow> ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-  proof (cases "r < 0")
-    case True
-    then show ?rhs by simp
-  next
-    case False
-    then have [simp]: "r \<ge> 0" by simp
-    have "norm (a - a') + r \<le> r'"
-    proof (cases "a = a'")
-      case True
-      then show ?thesis
-        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
-        by (force simp: SOME_Basis dist_norm)
-    next
-      case False
-      have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
-        by (simp add: algebra_simps)
-      also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
-        by (simp add: algebra_simps)
-      also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"
-        by (simp add: abs_mult_pos field_simps)
-      finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>"
-        by linarith
-      from \<open>a \<noteq> a'\<close> show ?thesis
-        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>]
-        by (simp add: dist_norm scaleR_add_left)
-    qed
-    then show ?rhs
-      by (simp add: dist_norm)
-  qed
-next
-  assume ?rhs
-  then show ?lhs
-    by (auto simp: ball_def dist_norm)
-      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
-qed
-
-lemma cball_subset_ball_iff: "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
-  (is "?lhs \<longleftrightarrow> ?rhs")
-  for a :: "'a::euclidean_space"
-proof
-  assume ?lhs
-  then show ?rhs
-  proof (cases "r < 0")
-    case True then
-    show ?rhs by simp
-  next
-    case False
-    then have [simp]: "r \<ge> 0" by simp
-    have "norm (a - a') + r < r'"
-    proof (cases "a = a'")
-      case True
-      then show ?thesis
-        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
-        by (force simp: SOME_Basis dist_norm)
-    next
-      case False
-      have False if "norm (a - a') + r \<ge> r'"
-      proof -
-        from that have "\<bar>r' - norm (a - a')\<bar> \<le> r"
-          by (simp split: abs_split)
-            (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
-        then show ?thesis
-          using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
-          by (simp add: dist_norm field_simps)
-            (simp add: diff_divide_distrib scaleR_left_diff_distrib)
-      qed
-      then show ?thesis by force
-    qed
-    then show ?rhs by (simp add: dist_norm)
-  qed
-next
-  assume ?rhs
-  then show ?lhs
-    by (auto simp: ball_def dist_norm)
-      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
-qed
-
-lemma ball_subset_cball_iff: "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
-  (is "?lhs = ?rhs")
-  for a :: "'a::euclidean_space"
-proof (cases "r \<le> 0")
-  case True
-  then show ?thesis
-    using dist_not_less_zero less_le_trans by force
-next
-  case False
-  show ?thesis
-  proof
-    assume ?lhs
-    then have "(cball a r \<subseteq> cball a' r')"
-      by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
-    with False show ?rhs
-      by (fastforce iff: cball_subset_cball_iff)
-  next
-    assume ?rhs
-    with False show ?lhs
-      using ball_subset_cball cball_subset_cball_iff by blast
-  qed
-qed
-
-lemma ball_subset_ball_iff:
-  fixes a :: "'a :: euclidean_space"
-  shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
-        (is "?lhs = ?rhs")
-proof (cases "r \<le> 0")
-  case True then show ?thesis
-    using dist_not_less_zero less_le_trans by force
-next
-  case False show ?thesis
-  proof
-    assume ?lhs
-    then have "0 < r'"
-      by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
-    then have "(cball a r \<subseteq> cball a' r')"
-      by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
-    then show ?rhs
-      using False cball_subset_cball_iff by fastforce
-  next
-  assume ?rhs then show ?lhs
-    apply (auto simp: ball_def)
-    apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
-    using dist_not_less_zero order.strict_trans2 apply blast
-    done
-  qed
-qed
-
-
-lemma ball_eq_ball_iff:
-  fixes x :: "'a :: euclidean_space"
-  shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
-        (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-  proof (cases "d \<le> 0 \<or> e \<le> 0")
-    case True
-      with \<open>?lhs\<close> show ?rhs
-        by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
-  next
-    case False
-    with \<open>?lhs\<close> show ?rhs
-      apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
-      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
-      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
-      done
-  qed
-next
-  assume ?rhs then show ?lhs
-    by (auto simp: set_eq_subset ball_subset_ball_iff)
-qed
-
-lemma cball_eq_cball_iff:
-  fixes x :: "'a :: euclidean_space"
-  shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
-        (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-  proof (cases "d < 0 \<or> e < 0")
-    case True
-      with \<open>?lhs\<close> show ?rhs
-        by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
-  next
-    case False
-    with \<open>?lhs\<close> show ?rhs
-      apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
-      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
-      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
-      done
-  qed
-next
-  assume ?rhs then show ?lhs
-    by (auto simp: set_eq_subset cball_subset_cball_iff)
-qed
-
-lemma ball_eq_cball_iff:
-  fixes x :: "'a :: euclidean_space"
-  shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
-    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
-    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
-    using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
-    done
-next
-  assume ?rhs then show ?lhs by auto
-qed
-
-lemma cball_eq_ball_iff:
-  fixes x :: "'a :: euclidean_space"
-  shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0"
-  using ball_eq_cball_iff by blast
-
-lemma finite_ball_avoid:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "open S" "finite X" "p \<in> S"
-  shows "\<exists>e>0. \<forall>w\<in>ball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
-proof -
-  obtain e1 where "0 < e1" and e1_b:"ball p e1 \<subseteq> S"
-    using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto
-  obtain e2 where "0 < e2" and "\<forall>x\<in>X. x \<noteq> p \<longrightarrow> e2 \<le> dist p x"
-    using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto
-  hence "\<forall>w\<in>ball p (min e1 e2). w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)" using e1_b by auto
-  thus "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> \<open>e1>0\<close>
-    apply (rule_tac x="min e1 e2" in exI)
-    by auto
-qed
-
-lemma finite_cball_avoid:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "open S" "finite X" "p \<in> S"
-  shows "\<exists>e>0. \<forall>w\<in>cball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
-proof -
-  obtain e1 where "e1>0" and e1: "\<forall>w\<in>ball p e1. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
-    using finite_ball_avoid[OF assms] by auto
-  define e2 where "e2 \<equiv> e1/2"
-  have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto
-  then have "cball p e2 \<subseteq> ball p e1" by (subst cball_subset_ball_iff,auto)
-  then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto
-qed
-
 subsection\<open>Various separability-type properties\<close>
 
 lemma univ_second_countable:
diff -r d692ef26021e -r 1331e57b54c6 src/HOL/Analysis/Elementary_Metric_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy	Mon Jan 07 10:22:22 2019 +0100
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy	Mon Jan 07 11:29:34 2019 +0100
@@ -44,7 +44,7 @@
 qed
 
 
-subsection \<open>Combination of Elementary and Abstract Topology\<close>
+subsection \<open>Combination of Elementary and Abstract Topology (TODO: this might be a separate theory?)\<close>
 
 lemma closedin_limpt:
   "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
@@ -257,6 +257,145 @@
 by metis
 
 
+subsubsection%unimportant \<open>Continuity relative to a union.\<close>
+
+lemma continuous_on_Un_local:
+    "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
+      continuous_on s f; continuous_on t f\<rbrakk>
+     \<Longrightarrow> continuous_on (s \<union> t) f"
+  unfolding continuous_on closedin_limpt
+  by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
+
+lemma continuous_on_cases_local:
+     "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
+       continuous_on s f; continuous_on t g;
+       \<And>x. \<lbrakk>x \<in> s \<and> \<not>P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
+      \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
+  by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
+
+lemma continuous_on_cases_le:
+  fixes h :: "'a :: topological_space \<Rightarrow> real"
+  assumes "continuous_on {t \<in> s. h t \<le> a} f"
+      and "continuous_on {t \<in> s. a \<le> h t} g"
+      and h: "continuous_on s h"
+      and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
+    shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
+proof -
+  have s: "s = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
+    by force
+  have 1: "closedin (subtopology euclidean s) (s \<inter> h -` atMost a)"
+    by (rule continuous_closedin_preimage [OF h closed_atMost])
+  have 2: "closedin (subtopology euclidean s) (s \<inter> h -` atLeast a)"
+    by (rule continuous_closedin_preimage [OF h closed_atLeast])
+  have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
+    by auto
+  show ?thesis
+    apply (rule continuous_on_subset [of s, OF _ order_refl])
+    apply (subst s)
+    apply (rule continuous_on_cases_local)
+    using 1 2 s assms apply (auto simp: eq)
+    done
+qed
+
+lemma continuous_on_cases_1:
+  fixes s :: "real set"
+  assumes "continuous_on {t \<in> s. t \<le> a} f"
+      and "continuous_on {t \<in> s. a \<le> t} g"
+      and "a \<in> s \<Longrightarrow> f a = g a"
+    shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
+using assms
+by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
+
+
+subsubsection%unimportant\<open>Inverse function property for open/closed maps\<close>
+
+lemma continuous_on_inverse_open_map:
+  assumes contf: "continuous_on S f"
+    and imf: "f ` S = T"
+    and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
+    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+  shows "continuous_on T g"
+proof -
+  from imf injf have gTS: "g ` T = S"
+    by force
+  from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
+    by force
+  show ?thesis
+    by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
+qed
+
+lemma continuous_on_inverse_closed_map:
+  assumes contf: "continuous_on S f"
+    and imf: "f ` S = T"
+    and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
+    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
+  shows "continuous_on T g"
+proof -
+  from imf injf have gTS: "g ` T = S"
+    by force
+  from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = T \<inter> g -` U" for U
+    by force
+  show ?thesis
+    by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
+qed
+
+lemma homeomorphism_injective_open_map:
+  assumes contf: "continuous_on S f"
+    and imf: "f ` S = T"
+    and injf: "inj_on f S"
+    and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
+  obtains g where "homeomorphism S T f g"
+proof
+  have "continuous_on T (inv_into S f)"
+    by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
+  with imf injf contf show "homeomorphism S T f (inv_into S f)"
+    by (auto simp: homeomorphism_def)
+qed
+
+lemma homeomorphism_injective_closed_map:
+  assumes contf: "continuous_on S f"
+    and imf: "f ` S = T"
+    and injf: "inj_on f S"
+    and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
+  obtains g where "homeomorphism S T f g"
+proof
+  have "continuous_on T (inv_into S f)"
+    by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
+  with imf injf contf show "homeomorphism S T f (inv_into S f)"
+    by (auto simp: homeomorphism_def)
+qed
+
+lemma homeomorphism_imp_open_map:
+  assumes hom: "homeomorphism S T f g"
+    and oo: "openin (subtopology euclidean S) U"
+  shows "openin (subtopology euclidean T) (f ` U)"
+proof -
+  from hom oo have [simp]: "f ` U = T \<inter> g -` U"
+    using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
+  from hom have "continuous_on T g"
+    unfolding homeomorphism_def by blast
+  moreover have "g ` T = S"
+    by (metis hom homeomorphism_def)
+  ultimately show ?thesis
+    by (simp add: continuous_on_open oo)
+qed
+
+lemma homeomorphism_imp_closed_map:
+  assumes hom: "homeomorphism S T f g"
+    and oo: "closedin (subtopology euclidean S) U"
+  shows "closedin (subtopology euclidean T) (f ` U)"
+proof -
+  from hom oo have [simp]: "f ` U = T \<inter> g -` U"
+    using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
+  from hom have "continuous_on T g"
+    unfolding homeomorphism_def by blast
+  moreover have "g ` T = S"
+    by (metis hom homeomorphism_def)
+  ultimately show ?thesis
+    by (simp add: continuous_on_closed oo)
+qed
+
+
 subsection \<open>Open and closed balls\<close>
 
 definition%important ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
@@ -734,6 +873,149 @@
   using assms by (fast intro: tendsto_le tendsto_intros)
 
 
+subsection \<open>Continuity\<close>
+
+text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
+
+proposition continuous_within_eps_delta:
+  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
+  unfolding continuous_within and Lim_within  by fastforce
+
+corollary continuous_at_eps_delta:
+  "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
+  using continuous_within_eps_delta [of x UNIV f] by simp
+
+lemma continuous_at_right_real_increasing:
+  fixes f :: "real \<Rightarrow> real"
+  assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"
+  shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
+  apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
+  apply (intro all_cong ex_cong, safe)
+  apply (erule_tac x="a + d" in allE, simp)
+  apply (simp add: nondecF field_simps)
+  apply (drule nondecF, simp)
+  done
+
+lemma continuous_at_left_real_increasing:
+  assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"
+  shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"
+  apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
+  apply (intro all_cong ex_cong, safe)
+  apply (erule_tac x="a - d" in allE, simp)
+  apply (simp add: nondecF field_simps)
+  apply (cut_tac x="a - d" and y=x in nondecF, simp_all)
+  done
+
+text\<open>Versions in terms of open balls.\<close>
+
+lemma continuous_within_ball:
+  "continuous (at x within s) f \<longleftrightarrow>
+    (\<forall>e > 0. \<exists>d > 0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
+  (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  {
+    fix e :: real
+    assume "e > 0"
+    then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
+      using \<open>?lhs\<close>[unfolded continuous_within Lim_within] by auto
+    {
+      fix y
+      assume "y \<in> f ` (ball x d \<inter> s)"
+      then have "y \<in> ball (f x) e"
+        using d(2)
+        apply (auto simp: dist_commute)
+        apply (erule_tac x=xa in ballE, auto)
+        using \<open>e > 0\<close>
+        apply auto
+        done
+    }
+    then have "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e"
+      using \<open>d > 0\<close>
+      unfolding subset_eq ball_def by (auto simp: dist_commute)
+  }
+  then show ?rhs by auto
+next
+  assume ?rhs
+  then show ?lhs
+    unfolding continuous_within Lim_within ball_def subset_eq
+    apply (auto simp: dist_commute)
+    apply (erule_tac x=e in allE, auto)
+    done
+qed
+
+lemma continuous_at_ball:
+  "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+    apply auto
+    apply (erule_tac x=e in allE, auto)
+    apply (rule_tac x=d in exI, auto)
+    apply (erule_tac x=xa in allE)
+    apply (auto simp: dist_commute)
+    done
+next
+  assume ?rhs
+  then show ?lhs
+    unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+    apply auto
+    apply (erule_tac x=e in allE, auto)
+    apply (rule_tac x=d in exI, auto)
+    apply (erule_tac x="f xa" in allE)
+    apply (auto simp: dist_commute)
+    done
+qed
+
+text\<open>Define setwise continuity in terms of limits within the set.\<close>
+
+lemma continuous_on_iff:
+  "continuous_on s f \<longleftrightarrow>
+    (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
+  unfolding continuous_on_def Lim_within
+  by (metis dist_pos_lt dist_self)
+
+lemma continuous_within_E:
+  assumes "continuous (at x within s) f" "e>0"
+  obtains d where "d>0"  "\<And>x'. \<lbrakk>x'\<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+  using assms apply (simp add: continuous_within_eps_delta)
+  apply (drule spec [of _ e], clarify)
+  apply (rule_tac d="d/2" in that, auto)
+  done
+
+lemma continuous_onI [intro?]:
+  assumes "\<And>x e. \<lbrakk>e > 0; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
+  shows "continuous_on s f"
+apply (simp add: continuous_on_iff, clarify)
+apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+done
+
+text\<open>Some simple consequential lemmas.\<close>
+
+lemma continuous_onE:
+    assumes "continuous_on s f" "x\<in>s" "e>0"
+    obtains d where "d>0"  "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+  using assms
+  apply (simp add: continuous_on_iff)
+  apply (elim ballE allE)
+  apply (auto intro: that [where d="d/2" for d])
+  done
+
+text\<open>The usual transformation theorems.\<close>
+
+lemma continuous_transform_within:
+  fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
+  assumes "continuous (at x within s) f"
+    and "0 < d"
+    and "x \<in> s"
+    and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
+  shows "continuous (at x within s) g"
+  using assms
+  unfolding continuous_within
+  by (force intro: Lim_transform_within)
+
+
 subsection \<open>Closure and Limit Characterization\<close>
 
 lemma closure_approachable:
@@ -814,6 +1096,7 @@
   qed
 qed
 
+
 subsection \<open>Boundedness\<close>
 
   (* FIXME: This has to be unified with BSEQ!! *)
@@ -945,67 +1228,16 @@
     by (auto intro!: boundedI)
 qed
 
-
-subsection \<open>Consequences for Real Numbers\<close>
-
-lemma closed_contains_Inf:
-  fixes S :: "real set"
-  shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
-  by (metis closure_contains_Inf closure_closed)
-
-lemma closed_subset_contains_Inf:
-  fixes A C :: "real set"
-  shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C"
-  by (metis closure_contains_Inf closure_minimal subset_eq)
-
-lemma atLeastAtMost_subset_contains_Inf:
-  fixes A :: "real set" and a b :: real
-  shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
-  by (rule closed_subset_contains_Inf)
-     (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
-
-lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
-  by (simp add: bounded_iff)
-
-lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"
-  by (auto simp: bounded_def bdd_above_def dist_real_def)
-     (metis abs_le_D1 abs_minus_commute diff_le_eq)
-
-lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
-  by (auto simp: bounded_def bdd_below_def dist_real_def)
-     (metis abs_le_D1 add.commute diff_le_eq)
-
-lemma bounded_has_Sup:
-  fixes S :: "real set"
-  assumes "bounded S"
-    and "S \<noteq> {}"
-  shows "\<forall>x\<in>S. x \<le> Sup S"
-    and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
-proof
-  show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
-    using assms by (metis cSup_least)
-qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
-
-lemma Sup_insert:
-  fixes S :: "real set"
-  shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
-  by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
-
-lemma bounded_has_Inf:
-  fixes S :: "real set"
-  assumes "bounded S"
-    and "S \<noteq> {}"
-  shows "\<forall>x\<in>S. x \<ge> Inf S"
-    and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
-proof
-  show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
-    using assms by (metis cInf_greatest)
-qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
-
-lemma Inf_insert:
-  fixes S :: "real set"
-  shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
-  by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
+lemma bounded_Times:
+  assumes "bounded s" "bounded t"
+  shows "bounded (s \<times> t)"
+proof -
+  obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
+    using assms [unfolded bounded_def] by auto
+  then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"
+    by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
+  then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
+qed
 
 
 subsection \<open>Compactness\<close>
@@ -1029,6 +1261,23 @@
   shows "T \<inter> ball x r \<noteq> {}"
   using assms centre_in_ball closure_iff_nhds_not_empty by blast
 
+lemma compact_sup_maxdistance:
+  fixes s :: "'a::metric_space set"
+  assumes "compact s"
+    and "s \<noteq> {}"
+  shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
+proof -
+  have "compact (s \<times> s)"
+    using \<open>compact s\<close> by (intro compact_Times)
+  moreover have "s \<times> s \<noteq> {}"
+    using \<open>s \<noteq> {}\<close> by auto
+  moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
+    by (intro continuous_at_imp_continuous_on ballI continuous_intros)
+  ultimately show ?thesis
+    using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto
+qed
+
+
 subsubsection\<open>Totally bounded\<close>
 
 lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N \<longrightarrow> dist (s m) (s n) < e)"
@@ -1141,6 +1390,403 @@
   using compact_imp_bounded unfolding compact_eq_Bolzano_Weierstrass .
 
 
+subsection \<open>Banach fixed point theorem\<close>
+  
+theorem banach_fix:\<comment> \<open>TODO: rename to \<open>Banach_fix\<close>\<close>
+  assumes s: "complete s" "s \<noteq> {}"
+    and c: "0 \<le> c" "c < 1"
+    and f: "f ` s \<subseteq> s"
+    and lipschitz: "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
+  shows "\<exists>!x\<in>s. f x = x"
+proof -
+  from c have "1 - c > 0" by simp
+
+  from s(2) obtain z0 where z0: "z0 \<in> s" by blast
+  define z where "z n = (f ^^ n) z0" for n
+  with f z0 have z_in_s: "z n \<in> s" for n :: nat
+    by (induct n) auto
+  define d where "d = dist (z 0) (z 1)"
+
+  have fzn: "f (z n) = z (Suc n)" for n
+    by (simp add: z_def)
+  have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * d" for n :: nat
+  proof (induct n)
+    case 0
+    then show ?case
+      by (simp add: d_def)
+  next
+    case (Suc m)
+    with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
+      using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
+    then show ?case
+      using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
+      by (simp add: fzn mult_le_cancel_left)
+  qed
+
+  have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * d * (1 - c ^ n)" for n m :: nat
+  proof (induct n)
+    case 0
+    show ?case by simp
+  next
+    case (Suc k)
+    from c have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>
+        (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
+      by (simp add: dist_triangle)
+    also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
+      by simp
+    also from Suc have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
+      by (simp add: field_simps)
+    also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
+      by (simp add: power_add field_simps)
+    also from c have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
+      by (simp add: field_simps)
+    finally show ?case by simp
+  qed
+
+  have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e" if "e > 0" for e
+  proof (cases "d = 0")
+    case True
+    from \<open>1 - c > 0\<close> have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x
+      by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
+    with c cf_z2[of 0] True have "z n = z0" for n
+      by (simp add: z_def)
+    with \<open>e > 0\<close> show ?thesis by simp
+  next
+    case False
+    with zero_le_dist[of "z 0" "z 1"] have "d > 0"
+      by (metis d_def less_le)
+    with \<open>1 - c > 0\<close> \<open>e > 0\<close> have "0 < e * (1 - c) / d"
+      by simp
+    with c obtain N where N: "c ^ N < e * (1 - c) / d"
+      using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
+    have *: "dist (z m) (z n) < e" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat
+    proof -
+      from c \<open>n \<ge> N\<close> have *: "c ^ n \<le> c ^ N"
+        using power_decreasing[OF \<open>n\<ge>N\<close>, of c] by simp
+      from c \<open>m > n\<close> have "1 - c ^ (m - n) > 0"
+        using power_strict_mono[of c 1 "m - n"] by simp
+      with \<open>d > 0\<close> \<open>0 < 1 - c\<close> have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
+        by simp
+      from cf_z2[of n "m - n"] \<open>m > n\<close>
+      have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
+        by (simp add: pos_le_divide_eq[OF \<open>1 - c > 0\<close>] mult.commute dist_commute)
+      also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
+        using mult_right_mono[OF * order_less_imp_le[OF **]]
+        by (simp add: mult.assoc)
+      also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
+        using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
+      also from c \<open>d > 0\<close> \<open>1 - c > 0\<close> have "\<dots> = e * (1 - c ^ (m - n))"
+        by simp
+      also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>e > 0\<close> have "\<dots> \<le> e"
+        using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
+      finally show ?thesis by simp
+    qed
+    have "dist (z n) (z m) < e" if "N \<le> m" "N \<le> n" for m n :: nat
+    proof (cases "n = m")
+      case True
+      with \<open>e > 0\<close> show ?thesis by simp
+    next
+      case False
+      with *[of n m] *[of m n] and that show ?thesis
+        by (auto simp: dist_commute nat_neq_iff)
+    qed
+    then show ?thesis by auto
+  qed
+  then have "Cauchy z"
+    by (simp add: cauchy_def)
+  then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"
+    using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
+
+  define e where "e = dist (f x) x"
+  have "e = 0"
+  proof (rule ccontr)
+    assume "e \<noteq> 0"
+    then have "e > 0"
+      unfolding e_def using zero_le_dist[of "f x" x]
+      by (metis dist_eq_0_iff dist_nz e_def)
+    then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
+      using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
+    then have N':"dist (z N) x < e / 2" by auto
+    have *: "c * dist (z N) x \<le> dist (z N) x"
+      unfolding mult_le_cancel_right2
+      using zero_le_dist[of "z N" x] and c
+      by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
+    have "dist (f (z N)) (f x) \<le> c * dist (z N) x"
+      using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
+      using z_in_s[of N] \<open>x\<in>s\<close>
+      using c
+      by auto
+    also have "\<dots> < e / 2"
+      using N' and c using * by auto
+    finally show False
+      unfolding fzn
+      using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
+      unfolding e_def
+      by auto
+  qed
+  then have "f x = x" by (auto simp: e_def)
+  moreover have "y = x" if "f y = y" "y \<in> s" for y
+  proof -
+    from \<open>x \<in> s\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"
+      using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp
+    with c and zero_le_dist[of x y] have "dist x y = 0"
+      by (simp add: mult_le_cancel_right1)
+    then show ?thesis by simp
+  qed
+  ultimately show ?thesis
+    using \<open>x\<in>s\<close> by blast
+qed
+
+
+subsection \<open>Edelstein fixed point theorem\<close>
+
+theorem edelstein_fix:\<comment> \<open>TODO: rename to \<open>Edelstein_fix\<close>\<close>
+  fixes s :: "'a::metric_space set"
+  assumes s: "compact s" "s \<noteq> {}"
+    and gs: "(g ` s) \<subseteq> s"
+    and dist: "\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
+  shows "\<exists>!x\<in>s. g x = x"
+proof -
+  let ?D = "(\<lambda>x. (x, x)) ` s"
+  have D: "compact ?D" "?D \<noteq> {}"
+    by (rule compact_continuous_image)
+       (auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)
+
+  have "\<And>x y e. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 < e \<Longrightarrow> dist y x < e \<Longrightarrow> dist (g y) (g x) < e"
+    using dist by fastforce
+  then have "continuous_on s g"
+    by (auto simp: continuous_on_iff)
+  then have cont: "continuous_on ?D (\<lambda>x. dist ((g \<circ> fst) x) (snd x))"
+    unfolding continuous_on_eq_continuous_within
+    by (intro continuous_dist ballI continuous_within_compose)
+       (auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)
+
+  obtain a where "a \<in> s" and le: "\<And>x. x \<in> s \<Longrightarrow> dist (g a) a \<le> dist (g x) x"
+    using continuous_attains_inf[OF D cont] by auto
+
+  have "g a = a"
+  proof (rule ccontr)
+    assume "g a \<noteq> a"
+    with \<open>a \<in> s\<close> gs have "dist (g (g a)) (g a) < dist (g a) a"
+      by (intro dist[rule_format]) auto
+    moreover have "dist (g a) a \<le> dist (g (g a)) (g a)"
+      using \<open>a \<in> s\<close> gs by (intro le) auto
+    ultimately show False by auto
+  qed
+  moreover have "\<And>x. x \<in> s \<Longrightarrow> g x = x \<Longrightarrow> x = a"
+    using dist[THEN bspec[where x=a]] \<open>g a = a\<close> and \<open>a\<in>s\<close> by auto
+  ultimately show "\<exists>!x\<in>s. g x = x"
+    using \<open>a \<in> s\<close> by blast
+qed
+
+subsection \<open>The diameter of a set\<close>
+
+definition%important diameter :: "'a::metric_space set \<Rightarrow> real" where
+  "diameter S = (if S = {} then 0 else SUP (x,y)\<in>S\<times>S. dist x y)"
+
+lemma diameter_empty [simp]: "diameter{} = 0"
+  by (auto simp: diameter_def)
+
+lemma diameter_singleton [simp]: "diameter{x} = 0"
+  by (auto simp: diameter_def)
+
+lemma diameter_le:
+  assumes "S \<noteq> {} \<or> 0 \<le> d"
+      and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> d"
+    shows "diameter S \<le> d"
+using assms
+  by (auto simp: dist_norm diameter_def intro: cSUP_least)
+
+lemma diameter_bounded_bound:
+  fixes s :: "'a :: metric_space set"
+  assumes s: "bounded s" "x \<in> s" "y \<in> s"
+  shows "dist x y \<le> diameter s"
+proof -
+  from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
+    unfolding bounded_def by auto
+  have "bdd_above (case_prod dist ` (s\<times>s))"
+  proof (intro bdd_aboveI, safe)
+    fix a b
+    assume "a \<in> s" "b \<in> s"
+    with z[of a] z[of b] dist_triangle[of a b z]
+    show "dist a b \<le> 2 * d"
+      by (simp add: dist_commute)
+  qed
+  moreover have "(x,y) \<in> s\<times>s" using s by auto
+  ultimately have "dist x y \<le> (SUP (x,y)\<in>s\<times>s. dist x y)"
+    by (rule cSUP_upper2) simp
+  with \<open>x \<in> s\<close> show ?thesis
+    by (auto simp: diameter_def)
+qed
+
+lemma diameter_lower_bounded:
+  fixes s :: "'a :: metric_space set"
+  assumes s: "bounded s"
+    and d: "0 < d" "d < diameter s"
+  shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
+proof (rule ccontr)
+  assume contr: "\<not> ?thesis"
+  moreover have "s \<noteq> {}"
+    using d by (auto simp: diameter_def)
+  ultimately have "diameter s \<le> d"
+    by (auto simp: not_less diameter_def intro!: cSUP_least)
+  with \<open>d < diameter s\<close> show False by auto
+qed
+
+lemma diameter_bounded:
+  assumes "bounded s"
+  shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
+    and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"
+  using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
+  by auto
+
+lemma bounded_two_points:
+  "bounded S \<longleftrightarrow> (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"
+  apply (rule iffI)
+  subgoal using diameter_bounded(1) by auto
+  subgoal using bounded_any_center[of S] by meson
+  done
+
+lemma diameter_compact_attained:
+  assumes "compact s"
+    and "s \<noteq> {}"
+  shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
+proof -
+  have b: "bounded s" using assms(1)
+    by (rule compact_imp_bounded)
+  then obtain x y where xys: "x\<in>s" "y\<in>s"
+    and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
+    using compact_sup_maxdistance[OF assms] by auto
+  then have "diameter s \<le> dist x y"
+    unfolding diameter_def
+    apply clarsimp
+    apply (rule cSUP_least, fast+)
+    done
+  then show ?thesis
+    by (metis b diameter_bounded_bound order_antisym xys)
+qed
+
+lemma diameter_ge_0:
+  assumes "bounded S"  shows "0 \<le> diameter S"
+  by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)
+
+lemma diameter_subset:
+  assumes "S \<subseteq> T" "bounded T"
+  shows "diameter S \<le> diameter T"
+proof (cases "S = {} \<or> T = {}")
+  case True
+  with assms show ?thesis
+    by (force simp: diameter_ge_0)
+next
+  case False
+  then have "bdd_above ((\<lambda>x. case x of (x, xa) \<Rightarrow> dist x xa) ` (T \<times> T))"
+    using \<open>bounded T\<close> diameter_bounded_bound by (force simp: bdd_above_def)
+  with False \<open>S \<subseteq> T\<close> show ?thesis
+    apply (simp add: diameter_def)
+    apply (rule cSUP_subset_mono, auto)
+    done
+qed
+
+lemma diameter_closure:
+  assumes "bounded S"
+  shows "diameter(closure S) = diameter S"
+proof (rule order_antisym)
+  have "False" if "diameter S < diameter (closure S)"
+  proof -
+    define d where "d = diameter(closure S) - diameter(S)"
+    have "d > 0"
+      using that by (simp add: d_def)
+    then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
+      by simp
+    have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
+      by (simp add: d_def divide_simps)
+     have bocl: "bounded (closure S)"
+      using assms by blast
+    moreover have "0 \<le> diameter S"
+      using assms diameter_ge_0 by blast
+    ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
+      using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] \<open>d > 0\<close> d_def by auto
+    then obtain x' y' where x'y': "x' \<in> S" "dist x' x < d/4" "y' \<in> S" "dist y' y < d/4"
+      using closure_approachable
+      by (metis \<open>0 < d\<close> zero_less_divide_iff zero_less_numeral)
+    then have "dist x' y' \<le> diameter S"
+      using assms diameter_bounded_bound by blast
+    with x'y' have "dist x y \<le> d / 4 + diameter S + d / 4"
+      by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)
+    then show ?thesis
+      using xy d_def by linarith
+  qed
+  then show "diameter (closure S) \<le> diameter S"
+    by fastforce
+  next
+    show "diameter S \<le> diameter (closure S)"
+      by (simp add: assms bounded_closure closure_subset diameter_subset)
+qed
+
+proposition Lebesgue_number_lemma:
+  assumes "compact S" "\<C> \<noteq> {}" "S \<subseteq> \<Union>\<C>" and ope: "\<And>B. B \<in> \<C> \<Longrightarrow> open B"
+  obtains \<delta> where "0 < \<delta>" "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B \<in> \<C>. T \<subseteq> B"
+proof (cases "S = {}")
+  case True
+  then show ?thesis
+    by (metis \<open>\<C> \<noteq> {}\<close> zero_less_one empty_subsetI equals0I subset_trans that)
+next
+  case False
+  { fix x assume "x \<in> S"
+    then obtain C where C: "x \<in> C" "C \<in> \<C>"
+      using \<open>S \<subseteq> \<Union>\<C>\<close> by blast
+    then obtain r where r: "r>0" "ball x (2*r) \<subseteq> C"
+      by (metis mult.commute mult_2_right not_le ope openE field_sum_of_halves zero_le_numeral zero_less_mult_iff)
+    then have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>"
+      using C by blast
+  }
+  then obtain r where r: "\<And>x. x \<in> S \<Longrightarrow> r x > 0 \<and> (\<exists>C \<in> \<C>. ball x (2*r x) \<subseteq> C)"
+    by metis
+  then have "S \<subseteq> (\<Union>x \<in> S. ball x (r x))"
+    by auto
+  then obtain \<T> where "finite \<T>" "S \<subseteq> \<Union>\<T>" and \<T>: "\<T> \<subseteq> (\<lambda>x. ball x (r x)) ` S"
+    by (rule compactE [OF \<open>compact S\<close>]) auto
+  then obtain S0 where "S0 \<subseteq> S" "finite S0" and S0: "\<T> = (\<lambda>x. ball x (r x)) ` S0"
+    by (meson finite_subset_image)
+  then have "S0 \<noteq> {}"
+    using False \<open>S \<subseteq> \<Union>\<T>\<close> by auto
+  define \<delta> where "\<delta> = Inf (r ` S0)"
+  have "\<delta> > 0"
+    using \<open>finite S0\<close> \<open>S0 \<subseteq> S\<close> \<open>S0 \<noteq> {}\<close> r by (auto simp: \<delta>_def finite_less_Inf_iff)
+  show ?thesis
+  proof
+    show "0 < \<delta>"
+      by (simp add: \<open>0 < \<delta>\<close>)
+    show "\<exists>B \<in> \<C>. T \<subseteq> B" if "T \<subseteq> S" and dia: "diameter T < \<delta>" for T
+    proof (cases "T = {}")
+      case True
+      then show ?thesis
+        using \<open>\<C> \<noteq> {}\<close> by blast
+    next
+      case False
+      then obtain y where "y \<in> T" by blast
+      then have "y \<in> S"
+        using \<open>T \<subseteq> S\<close> by auto
+      then obtain x where "x \<in> S0" and x: "y \<in> ball x (r x)"
+        using \<open>S \<subseteq> \<Union>\<T>\<close> S0 that by blast
+      have "ball y \<delta> \<subseteq> ball y (r x)"
+        by (metis \<delta>_def \<open>S0 \<noteq> {}\<close> \<open>finite S0\<close> \<open>x \<in> S0\<close> empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
+      also have "... \<subseteq> ball x (2*r x)"
+        by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)
+      finally obtain C where "C \<in> \<C>" "ball y \<delta> \<subseteq> C"
+        by (meson r \<open>S0 \<subseteq> S\<close> \<open>x \<in> S0\<close> dual_order.trans subsetCE)
+      have "bounded T"
+        using \<open>compact S\<close> bounded_subset compact_imp_bounded \<open>T \<subseteq> S\<close> by blast
+      then have "T \<subseteq> ball y \<delta>"
+        using \<open>y \<in> T\<close> dia diameter_bounded_bound by fastforce
+      then show ?thesis
+        apply (rule_tac x=C in bexI)
+        using \<open>ball y \<delta> \<subseteq> C\<close> \<open>C \<in> \<C>\<close> by auto
+    qed
+  qed
+qed
+
+
 subsection \<open>Metric spaces with the Heine-Borel property\<close>
 
 text \<open>
@@ -1308,7 +1954,7 @@
 qed
 
 
-subsubsection \<open>Completeness\<close>
+subsection \<open>Completeness\<close>
 
 proposition (in metric_space) completeI:
   assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f \<longlonglongrightarrow> l"
@@ -1566,8 +2212,22 @@
   using frontier_subset_closed compact_eq_bounded_closed
   by blast
 
-
-subsubsection \<open>Properties of Balls and Spheres\<close>
+lemma continuous_closed_imp_Cauchy_continuous:
+  fixes S :: "('a::complete_space) set"
+  shows "\<lbrakk>continuous_on S f; closed S; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
+  apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
+  by (meson LIMSEQ_imp_Cauchy complete_def)
+
+lemma banach_fix_type:
+  fixes f::"'a::complete_space\<Rightarrow>'a"
+  assumes c:"0 \<le> c" "c < 1"
+      and lipschitz:"\<forall>x. \<forall>y. dist (f x) (f y) \<le> c * dist x y"
+  shows "\<exists>!x. (f x = x)"
+  using assms banach_fix[OF complete_UNIV UNIV_not_empty assms(1,2) subset_UNIV, of f]
+  by auto
+
+
+subsection \<open>Properties of Balls and Spheres\<close>
 
 lemma compact_cball[simp]:
   fixes x :: "'a::heine_borel"
@@ -1589,7 +2249,7 @@
   by blast
 
 
-subsubsection \<open>Distance from a Set\<close>
+subsection \<open>Distance from a Set\<close>
 
 lemma distance_attains_sup:
   assumes "compact s" "s \<noteq> {}"
@@ -1625,6 +2285,7 @@
   with that show ?thesis by fastforce
 qed
 
+
 subsection \<open>Infimum Distance\<close>
 
 definition%important "infdist x A = (if A = {} then 0 else INF a\<in>A. dist x a)"
@@ -1857,134 +2518,90 @@
 qed
 
 
-subsection \<open>Continuity\<close>
-
-text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
-
-proposition continuous_within_eps_delta:
-  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
-  unfolding continuous_within and Lim_within  by fastforce
-
-corollary continuous_at_eps_delta:
-  "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
-  using continuous_within_eps_delta [of x UNIV f] by simp
-
-lemma continuous_at_right_real_increasing:
-  fixes f :: "real \<Rightarrow> real"
-  assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"
-  shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
-  apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
-  apply (intro all_cong ex_cong, safe)
-  apply (erule_tac x="a + d" in allE, simp)
-  apply (simp add: nondecF field_simps)
-  apply (drule nondecF, simp)
-  done
-
-lemma continuous_at_left_real_increasing:
-  assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"
-  shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"
-  apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
-  apply (intro all_cong ex_cong, safe)
-  apply (erule_tac x="a - d" in allE, simp)
-  apply (simp add: nondecF field_simps)
-  apply (cut_tac x="a - d" and y=x in nondecF, simp_all)
-  done
-
-text\<open>Versions in terms of open balls.\<close>
-
-lemma continuous_within_ball:
-  "continuous (at x within s) f \<longleftrightarrow>
-    (\<forall>e > 0. \<exists>d > 0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
-  (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  {
-    fix e :: real
-    assume "e > 0"
-    then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
-      using \<open>?lhs\<close>[unfolded continuous_within Lim_within] by auto
-    {
-      fix y
-      assume "y \<in> f ` (ball x d \<inter> s)"
-      then have "y \<in> ball (f x) e"
-        using d(2)
-        apply (auto simp: dist_commute)
-        apply (erule_tac x=xa in ballE, auto)
-        using \<open>e > 0\<close>
-        apply auto
-        done
-    }
-    then have "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e"
-      using \<open>d > 0\<close>
-      unfolding subset_eq ball_def by (auto simp: dist_commute)
-  }
-  then show ?rhs by auto
+subsection \<open>Separation between Points and Sets\<close>
+
+proposition separate_point_closed:
+  fixes s :: "'a::heine_borel set"
+  assumes "closed s" and "a \<notin> s"
+  shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
+proof (cases "s = {}")
+  case True
+  then show ?thesis by(auto intro!: exI[where x=1])
 next
-  assume ?rhs
-  then show ?lhs
-    unfolding continuous_within Lim_within ball_def subset_eq
-    apply (auto simp: dist_commute)
-    apply (erule_tac x=e in allE, auto)
-    done
-qed
-
-lemma continuous_at_ball:
-  "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
-    apply auto
-    apply (erule_tac x=e in allE, auto)
-    apply (rule_tac x=d in exI, auto)
-    apply (erule_tac x=xa in allE)
-    apply (auto simp: dist_commute)
-    done
-next
-  assume ?rhs
-  then show ?lhs
-    unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
-    apply auto
-    apply (erule_tac x=e in allE, auto)
-    apply (rule_tac x=d in exI, auto)
-    apply (erule_tac x="f xa" in allE)
-    apply (auto simp: dist_commute)
-    done
+  case False
+  from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"
+    using \<open>s \<noteq> {}\<close> by (blast intro: distance_attains_inf [of s a])
+  with \<open>x\<in>s\<close> show ?thesis using dist_pos_lt[of a x] and\<open>a \<notin> s\<close>
+    by blast
 qed
 
-text\<open>Define setwise continuity in terms of limits within the set.\<close>
-
-lemma continuous_on_iff:
-  "continuous_on s f \<longleftrightarrow>
-    (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
-  unfolding continuous_on_def Lim_within
-  by (metis dist_pos_lt dist_self)
-
-lemma continuous_within_E:
-  assumes "continuous (at x within s) f" "e>0"
-  obtains d where "d>0"  "\<And>x'. \<lbrakk>x'\<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-  using assms apply (simp add: continuous_within_eps_delta)
-  apply (drule spec [of _ e], clarify)
-  apply (rule_tac d="d/2" in that, auto)
-  done
-
-lemma continuous_onI [intro?]:
-  assumes "\<And>x e. \<lbrakk>e > 0; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
-  shows "continuous_on s f"
-apply (simp add: continuous_on_iff, clarify)
-apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
-done
-
-text\<open>Some simple consequential lemmas.\<close>
-
-lemma continuous_onE:
-    assumes "continuous_on s f" "x\<in>s" "e>0"
-    obtains d where "d>0"  "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-  using assms
-  apply (simp add: continuous_on_iff)
-  apply (elim ballE allE)
-  apply (auto intro: that [where d="d/2" for d])
-  done
+proposition separate_compact_closed:
+  fixes s t :: "'a::heine_borel set"
+  assumes "compact s"
+    and t: "closed t" "s \<inter> t = {}"
+  shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
+proof cases
+  assume "s \<noteq> {} \<and> t \<noteq> {}"
+  then have "s \<noteq> {}" "t \<noteq> {}" by auto
+  let ?inf = "\<lambda>x. infdist x t"
+  have "continuous_on s ?inf"
+    by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)
+  then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"
+    using continuous_attains_inf[OF \<open>compact s\<close> \<open>s \<noteq> {}\<close>] by auto
+  then have "0 < ?inf x"
+    using t \<open>t \<noteq> {}\<close> in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
+  moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
+    using x by (auto intro: order_trans infdist_le)
+  ultimately show ?thesis by auto
+qed (auto intro!: exI[of _ 1])
+
+proposition separate_closed_compact:
+  fixes s t :: "'a::heine_borel set"
+  assumes "closed s"
+    and "compact t"
+    and "s \<inter> t = {}"
+  shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
+proof -
+  have *: "t \<inter> s = {}"
+    using assms(3) by auto
+  show ?thesis
+    using separate_compact_closed[OF assms(2,1) *] by (force simp: dist_commute)
+qed
+
+proposition compact_in_open_separated:
+  fixes A::"'a::heine_borel set"
+  assumes "A \<noteq> {}"
+  assumes "compact A"
+  assumes "open B"
+  assumes "A \<subseteq> B"
+  obtains e where "e > 0" "{x. infdist x A \<le> e} \<subseteq> B"
+proof atomize_elim
+  have "closed (- B)" "compact A" "- B \<inter> A = {}"
+    using assms by (auto simp: open_Diff compact_eq_bounded_closed)
+  from separate_closed_compact[OF this]
+  obtain d'::real where d': "d'>0" "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d' \<le> dist x y"
+    by auto
+  define d where "d = d' / 2"
+  hence "d>0" "d < d'" using d' by auto
+  with d' have d: "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d < dist x y"
+    by force
+  show "\<exists>e>0. {x. infdist x A \<le> e} \<subseteq> B"
+  proof (rule ccontr)
+    assume "\<nexists>e. 0 < e \<and> {x. infdist x A \<le> e} \<subseteq> B"
+    with \<open>d > 0\<close> obtain x where x: "infdist x A \<le> d" "x \<notin> B"
+      by auto
+    from assms have "closed A" "A \<noteq> {}" by (auto simp: compact_eq_bounded_closed)
+    from infdist_attains_inf[OF this]
+    obtain y where y: "y \<in> A" "infdist x A = dist x y"
+      by auto
+    have "dist x y \<le> d" using x y by simp
+    also have "\<dots> < dist x y" using y d x by auto
+    finally show False by simp
+  qed
+qed
+
+
+subsection \<open>Uniform Continuity\<close>
 
 lemma uniformly_continuous_onE:
   assumes "uniformly_continuous_on s f" "0 < e"
@@ -2069,34 +2686,99 @@
     unfolding uniformly_continuous_on_def by blast
 qed
 
-lemma continuous_closed_imp_Cauchy_continuous:
-  fixes S :: "('a::complete_space) set"
-  shows "\<lbrakk>continuous_on S f; closed S; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
-  apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
-  by (meson LIMSEQ_imp_Cauchy complete_def)
-
-text\<open>The usual transformation theorems.\<close>
-
-lemma continuous_transform_within:
-  fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
-  assumes "continuous (at x within s) f"
-    and "0 < d"
-    and "x \<in> s"
-    and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
-  shows "continuous (at x within s) g"
-  using assms
-  unfolding continuous_within
-  by (force intro: Lim_transform_within)
-
-subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
-
-lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
-  fixes g :: "_::metric_space \<Rightarrow> _"
-  assumes "uniformly_continuous_on s g"
-  shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
-  using assms unfolding uniformly_continuous_on_sequentially
-  unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
-  by (auto intro: tendsto_zero)
+
+subsection \<open>Continuity on a Compact Domain Implies Uniform Continuity\<close>
+
+text\<open>From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of
+J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)\<close>
+
+lemma Heine_Borel_lemma:
+  assumes "compact S" and Ssub: "S \<subseteq> \<Union>\<G>" and opn: "\<And>G. G \<in> \<G> \<Longrightarrow> open G"
+  obtains e where "0 < e" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x e \<subseteq> G"
+proof -
+  have False if neg: "\<And>e. 0 < e \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x e \<subseteq> G"
+  proof -
+    have "\<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x (1 / Suc n) \<subseteq> G" for n
+      using neg by simp
+    then obtain f where "\<And>n. f n \<in> S" and fG: "\<And>G n. G \<in> \<G> \<Longrightarrow> \<not> ball (f n) (1 / Suc n) \<subseteq> G"
+      by metis
+    then obtain l r where "l \<in> S" "strict_mono r" and to_l: "(f \<circ> r) \<longlonglongrightarrow> l"
+      using \<open>compact S\<close> compact_def that by metis
+    then obtain G where "l \<in> G" "G \<in> \<G>"
+      using Ssub by auto
+    then obtain e where "0 < e" and e: "\<And>z. dist z l < e \<Longrightarrow> z \<in> G"
+      using opn open_dist by blast
+    obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < e/2"
+      using to_l apply (simp add: lim_sequentially)
+      using \<open>0 < e\<close> half_gt_zero that by blast
+    obtain N2 where N2: "of_nat N2 > 2/e"
+      using reals_Archimedean2 by blast
+    obtain x where "x \<in> ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x \<notin> G"
+      using fG [OF \<open>G \<in> \<G>\<close>, of "r (max N1 N2)"] by blast
+    then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
+      by simp
+    also have "... \<le> 1 / real (Suc (max N1 N2))"
+      apply (simp add: divide_simps del: max.bounded_iff)
+      using \<open>strict_mono r\<close> seq_suble by blast
+    also have "... \<le> 1 / real (Suc N2)"
+      by (simp add: field_simps)
+    also have "... < e/2"
+      using N2 \<open>0 < e\<close> by (simp add: field_simps)
+    finally have "dist (f (r (max N1 N2))) x < e / 2" .
+    moreover have "dist (f (r (max N1 N2))) l < e/2"
+      using N1 max.cobounded1 by blast
+    ultimately have "dist x l < e"
+      using dist_triangle_half_r by blast
+    then show ?thesis
+      using e \<open>x \<notin> G\<close> by blast
+  qed
+  then show ?thesis
+    by (meson that)
+qed
+
+lemma compact_uniformly_equicontinuous:
+  assumes "compact S"
+      and cont: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
+                        \<Longrightarrow> \<exists>d. 0 < d \<and>
+                                (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
+      and "0 < e"
+  obtains d where "0 < d"
+                  "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+proof -
+  obtain d where d_pos: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> 0 < d x e"
+     and d_dist : "\<And>x x' e f. \<lbrakk>dist x' x < d x e; x \<in> S; x' \<in> S; 0 < e; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+    using cont by metis
+  let ?\<G> = "((\<lambda>x. ball x (d x (e / 2))) ` S)"
+  have Ssub: "S \<subseteq> \<Union> ?\<G>"
+    by clarsimp (metis d_pos \<open>0 < e\<close> dist_self half_gt_zero_iff)
+  then obtain k where "0 < k" and k: "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> ?\<G>. ball x k \<subseteq> G"
+    by (rule Heine_Borel_lemma [OF \<open>compact S\<close>]) auto
+  moreover have "dist (f v) (f u) < e" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v
+  proof -
+    obtain G where "G \<in> ?\<G>" "u \<in> G" "v \<in> G"
+      using k that
+      by (metis \<open>dist v u < k\<close> \<open>u \<in> S\<close> \<open>0 < k\<close> centre_in_ball subsetD dist_commute mem_ball)
+    then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w \<in> S"
+      by auto
+    with that d_dist have "dist (f w) (f v) < e/2"
+      by (metis \<open>0 < e\<close> dist_commute half_gt_zero)
+    moreover
+    have "dist (f w) (f u) < e/2"
+      using that d_dist w by (metis \<open>0 < e\<close> dist_commute divide_pos_pos zero_less_numeral)
+    ultimately show ?thesis
+      using dist_triangle_half_r by blast
+  qed
+  ultimately show ?thesis using that by blast
+qed
+
+corollary compact_uniformly_continuous:
+  fixes f :: "'a :: metric_space \<Rightarrow> 'b :: metric_space"
+  assumes f: "continuous_on S f" and S: "compact S"
+  shows "uniformly_continuous_on S f"
+  using f
+    unfolding continuous_on_iff uniformly_continuous_on_def
+    by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])
+
 
 subsection%unimportant\<open> Theorems relating continuity and uniform continuity to closures\<close>
 
@@ -2389,8 +3071,6 @@
 apply (rule_tac x="e/2" in exI, force+)
   done
 
-subsection \<open>With abstract Topology\<close>
-
 lemma Times_in_interior_subtopology:
   fixes U :: "('a::metric_space \<times> 'b::metric_space) set"
   assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
@@ -2729,5 +3409,112 @@
   using continuous_at_avoid[of x f a] assms(4)
   by auto
 
+subsection \<open>Consequences for Real Numbers\<close>
+
+lemma closed_contains_Inf:
+  fixes S :: "real set"
+  shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
+  by (metis closure_contains_Inf closure_closed)
+
+lemma closed_subset_contains_Inf:
+  fixes A C :: "real set"
+  shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C"
+  by (metis closure_contains_Inf closure_minimal subset_eq)
+
+lemma atLeastAtMost_subset_contains_Inf:
+  fixes A :: "real set" and a b :: real
+  shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
+  by (rule closed_subset_contains_Inf)
+     (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
+
+lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
+  by (simp add: bounded_iff)
+
+lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"
+  by (auto simp: bounded_def bdd_above_def dist_real_def)
+     (metis abs_le_D1 abs_minus_commute diff_le_eq)
+
+lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
+  by (auto simp: bounded_def bdd_below_def dist_real_def)
+     (metis abs_le_D1 add.commute diff_le_eq)
+
+lemma bounded_has_Sup:
+  fixes S :: "real set"
+  assumes "bounded S"
+    and "S \<noteq> {}"
+  shows "\<forall>x\<in>S. x \<le> Sup S"
+    and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
+proof
+  show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
+    using assms by (metis cSup_least)
+qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
+
+lemma Sup_insert:
+  fixes S :: "real set"
+  shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
+  by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
+
+lemma bounded_has_Inf:
+  fixes S :: "real set"
+  assumes "bounded S"
+    and "S \<noteq> {}"
+  shows "\<forall>x\<in>S. x \<ge> Inf S"
+    and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
+proof
+  show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
+    using assms by (metis cInf_greatest)
+qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
+
+lemma Inf_insert:
+  fixes S :: "real set"
+  shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
+  by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
+
+lemma open_real:
+  fixes s :: "real set"
+  shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e --> x' \<in> s)"
+  unfolding open_dist dist_norm by simp
+
+lemma islimpt_approachable_real:
+  fixes s :: "real set"
+  shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"
+  unfolding islimpt_approachable dist_norm by simp
+
+lemma closed_real:
+  fixes s :: "real set"
+  shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e) \<longrightarrow> x \<in> s)"
+  unfolding closed_limpt islimpt_approachable dist_norm by simp
+
+lemma continuous_at_real_range:
+  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
+  shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> \<bar>f x' - f x\<bar> < e)"
+  unfolding continuous_at
+  unfolding Lim_at
+  unfolding dist_norm
+  apply auto
+  apply (erule_tac x=e in allE, auto)
+  apply (rule_tac x=d in exI, auto)
+  apply (erule_tac x=x' in allE, auto)
+  apply (erule_tac x=e in allE, auto)
+  done
+
+lemma continuous_on_real_range:
+  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
+  shows "continuous_on s f \<longleftrightarrow>
+    (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e))"
+  unfolding continuous_on_iff dist_norm by simp
+
+lemma continuous_on_closed_Collect_le:
+  fixes f g :: "'a::t2_space \<Rightarrow> real"
+  assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
+  shows "closed {x \<in> s. f x \<le> g x}"
+proof -
+  have "closed ((\<lambda>x. g x - f x) -` {0..} \<inter> s)"
+    using closed_real_atLeast continuous_on_diff [OF g f]
+    by (simp add: continuous_on_closed_vimage [OF s])
+  also have "((\<lambda>x. g x - f x) -` {0..} \<inter> s) = {x\<in>s. f x \<le> g x}"
+    by auto
+  finally show ?thesis .
+qed
 
 end
\ No newline at end of file
diff -r d692ef26021e -r 1331e57b54c6 src/HOL/Analysis/Elementary_Normed_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy	Mon Jan 07 10:22:22 2019 +0100
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy	Mon Jan 07 11:29:34 2019 +0100
@@ -1022,6 +1022,14 @@
 
 subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
 
+lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
+  fixes g :: "_::metric_space \<Rightarrow> _"
+  assumes "uniformly_continuous_on s g"
+  shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
+  using assms unfolding uniformly_continuous_on_sequentially
+  unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
+  by (auto intro: tendsto_zero)
+
 lemma uniformly_continuous_on_dist[continuous_intros]:
   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
   assumes "uniformly_continuous_on s f"
@@ -1119,4 +1127,513 @@
   "bounded_linear f \<Longrightarrow> continuous_on s f"
   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
 
+subsection%unimportant \<open>Arithmetic Preserves Topological Properties\<close>
+
+lemma open_scaling[intro]:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "c \<noteq> 0"
+    and "open s"
+  shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
+proof -
+  {
+    fix x
+    assume "x \<in> s"
+    then obtain e where "e>0"
+      and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
+      by auto
+    have "e * \<bar>c\<bar> > 0"
+      using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
+    moreover
+    {
+      fix y
+      assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
+      then have "norm ((1 / c) *\<^sub>R y - x) < e"
+        unfolding dist_norm
+        using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
+          assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
+      then have "y \<in> (*\<^sub>R) c ` s"
+        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
+        using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
+        using assms(1)
+        unfolding dist_norm scaleR_scaleR
+        by auto
+    }
+    ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> (*\<^sub>R) c ` s"
+      apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
+      done
+  }
+  then show ?thesis unfolding open_dist by auto
+qed
+
+lemma minus_image_eq_vimage:
+  fixes A :: "'a::ab_group_add set"
+  shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
+  by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
+
+lemma open_negations:
+  fixes S :: "'a::real_normed_vector set"
+  shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
+  using open_scaling [of "- 1" S] by simp
+
+lemma open_translation:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "open S"
+  shows "open((\<lambda>x. a + x) ` S)"
+proof -
+  {
+    fix x
+    have "continuous (at x) (\<lambda>x. x - a)"
+      by (intro continuous_diff continuous_ident continuous_const)
+  }
+  moreover have "{x. x - a \<in> S} = (+) a ` S"
+    by force
+  ultimately show ?thesis
+    by (metis assms continuous_open_vimage vimage_def)
+qed
+
+lemma open_neg_translation:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "open s"
+  shows "open((\<lambda>x. a - x) ` s)"
+  using open_translation[OF open_negations[OF assms], of a]
+  by (auto simp: image_image)
+
+lemma open_affinity:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "open S"  "c \<noteq> 0"
+  shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
+proof -
+  have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
+    unfolding o_def ..
+  have "(+) a ` (*\<^sub>R) c ` S = ((+) a \<circ> (*\<^sub>R) c) ` S"
+    by auto
+  then show ?thesis
+    using assms open_translation[of "(*\<^sub>R) c ` S" a]
+    unfolding *
+    by auto
+qed
+
+lemma interior_translation:
+  fixes S :: "'a::real_normed_vector set"
+  shows "interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (interior S)"
+proof (rule set_eqI, rule)
+  fix x
+  assume "x \<in> interior ((+) a ` S)"
+  then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
+    unfolding mem_interior by auto
+  then have "ball (x - a) e \<subseteq> S"
+    unfolding subset_eq Ball_def mem_ball dist_norm
+    by (auto simp: diff_diff_eq)
+  then show "x \<in> (+) a ` interior S"
+    unfolding image_iff
+    apply (rule_tac x="x - a" in bexI)
+    unfolding mem_interior
+    using \<open>e > 0\<close>
+    apply auto
+    done
+next
+  fix x
+  assume "x \<in> (+) a ` interior S"
+  then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
+    unfolding image_iff Bex_def mem_interior by auto
+  {
+    fix z
+    have *: "a + y - z = y + a - z" by auto
+    assume "z \<in> ball x e"
+    then have "z - a \<in> S"
+      using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
+      unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
+      by auto
+    then have "z \<in> (+) a ` S"
+      unfolding image_iff by (auto intro!: bexI[where x="z - a"])
+  }
+  then have "ball x e \<subseteq> (+) a ` S"
+    unfolding subset_eq by auto
+  then show "x \<in> interior ((+) a ` S)"
+    unfolding mem_interior using \<open>e > 0\<close> by auto
+qed
+
+lemma compact_scaling:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "compact s"
+  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
+proof -
+  let ?f = "\<lambda>x. scaleR c x"
+  have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
+  show ?thesis
+    using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
+    using linear_continuous_at[OF *] assms
+    by auto
+qed
+
+lemma compact_negations:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "compact s"
+  shows "compact ((\<lambda>x. - x) ` s)"
+  using compact_scaling [OF assms, of "- 1"] by auto
+
+lemma compact_sums:
+  fixes s t :: "'a::real_normed_vector set"
+  assumes "compact s"
+    and "compact t"
+  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
+proof -
+  have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
+    apply auto
+    unfolding image_iff
+    apply (rule_tac x="(xa, y)" in bexI)
+    apply auto
+    done
+  have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
+    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+  then show ?thesis
+    unfolding * using compact_continuous_image compact_Times [OF assms] by auto
+qed
+
+lemma compact_differences:
+  fixes s t :: "'a::real_normed_vector set"
+  assumes "compact s"
+    and "compact t"
+  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
+proof-
+  have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
+    apply auto
+    apply (rule_tac x= xa in exI, auto)
+    done
+  then show ?thesis
+    using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
+qed
+
+lemma compact_translation:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "compact s"
+  shows "compact ((\<lambda>x. a + x) ` s)"
+proof -
+  have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
+    by auto
+  then show ?thesis
+    using compact_sums[OF assms compact_sing[of a]] by auto
+qed
+
+lemma compact_affinity:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "compact s"
+  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
+proof -
+  have "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
+    by auto
+  then show ?thesis
+    using compact_translation[OF compact_scaling[OF assms], of a c] by auto
+qed
+
+lemma closed_scaling:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "closed S"
+  shows "closed ((\<lambda>x. c *\<^sub>R x) ` S)"
+proof (cases "c = 0")
+  case True then show ?thesis
+    by (auto simp: image_constant_conv)
+next
+  case False
+  from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) -` S)"
+    by (simp add: continuous_closed_vimage)
+  also have "(\<lambda>x. inverse c *\<^sub>R x) -` S = (\<lambda>x. c *\<^sub>R x) ` S"
+    using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
+  finally show ?thesis .
+qed
+
+lemma closed_negations:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "closed S"
+  shows "closed ((\<lambda>x. -x) ` S)"
+  using closed_scaling[OF assms, of "- 1"] by simp
+
+lemma compact_closed_sums:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "compact S" and "closed T"
+  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+proof -
+  let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"
+  {
+    fix x l
+    assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"
+    from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> S"  "\<forall>n. snd (f n) \<in> T"
+      using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto
+    obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
+      using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
+    have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
+      using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
+      unfolding o_def
+      by auto
+    then have "l - l' \<in> T"
+      using assms(2)[unfolded closed_sequential_limits,
+        THEN spec[where x="\<lambda> n. snd (f (r n))"],
+        THEN spec[where x="l - l'"]]
+      using f(3)
+      by auto
+    then have "l \<in> ?S"
+      using \<open>l' \<in> S\<close>
+      apply auto
+      apply (rule_tac x=l' in exI)
+      apply (rule_tac x="l - l'" in exI, auto)
+      done
+  }
+  moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+    by force
+  ultimately show ?thesis
+    unfolding closed_sequential_limits
+    by (metis (no_types, lifting))
+qed
+
+lemma closed_compact_sums:
+  fixes S T :: "'a::real_normed_vector set"
+  assumes "closed S" "compact T"
+  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+proof -
+  have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+    by auto
+  then show ?thesis
+    using compact_closed_sums[OF assms(2,1)] by simp
+qed
+
+lemma compact_closed_differences:
+  fixes S T :: "'a::real_normed_vector set"
+  assumes "compact S" "closed T"
+  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+proof -
+  have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+    by force
+  then show ?thesis
+    using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
+qed
+
+lemma closed_compact_differences:
+  fixes S T :: "'a::real_normed_vector set"
+  assumes "closed S" "compact T"
+  shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+proof -
+  have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
+    by auto
+ then show ?thesis
+  using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
+qed
+
+lemma closed_translation:
+  fixes a :: "'a::real_normed_vector"
+  assumes "closed S"
+  shows "closed ((\<lambda>x. a + x) ` S)"
+proof -
+  have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a ` S)" by auto
+  then show ?thesis
+    using compact_closed_sums[OF compact_sing[of a] assms] by auto
+qed
+
+lemma closure_translation:
+  fixes a :: "'a::real_normed_vector"
+  shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
+proof -
+  have *: "(+) a ` (- s) = - (+) a ` s"
+    by (auto intro!: image_eqI[where x="x - a" for x])
+  show ?thesis
+    unfolding closure_interior translation_Compl
+    using interior_translation[of a "- s"]
+    unfolding *
+    by auto
+qed
+
+lemma frontier_translation:
+  fixes a :: "'a::real_normed_vector"
+  shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
+  unfolding frontier_def translation_diff interior_translation closure_translation
+  by auto
+
+lemma sphere_translation:
+  fixes a :: "'n::real_normed_vector"
+  shows "sphere (a+c) r = (+) a ` sphere c r"
+  by (auto simp: dist_norm algebra_simps intro!: image_eqI[where x="x - a" for x])
+
+lemma cball_translation:
+  fixes a :: "'n::real_normed_vector"
+  shows "cball (a+c) r = (+) a ` cball c r"
+  by (auto simp: dist_norm algebra_simps intro!: image_eqI[where x="x - a" for x])
+
+lemma ball_translation:
+  fixes a :: "'n::real_normed_vector"
+  shows "ball (a+c) r = (+) a ` ball c r"
+  by (auto simp: dist_norm algebra_simps intro!: image_eqI[where x="x - a" for x])
+
+
+subsection%unimportant\<open>Homeomorphisms\<close>
+
+lemma homeomorphic_scaling:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "c \<noteq> 0"
+  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
+  unfolding homeomorphic_minimal
+  apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
+  apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
+  using assms
+  apply (auto simp: continuous_intros)
+  done
+
+lemma homeomorphic_translation:
+  fixes s :: "'a::real_normed_vector set"
+  shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
+  unfolding homeomorphic_minimal
+  apply (rule_tac x="\<lambda>x. a + x" in exI)
+  apply (rule_tac x="\<lambda>x. -a + x" in exI)
+  using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
+    continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
+  apply auto
+  done
+
+lemma homeomorphic_affinity:
+  fixes s :: "'a::real_normed_vector set"
+  assumes "c \<noteq> 0"
+  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
+proof -
+  have *: "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
+  show ?thesis
+    using homeomorphic_trans
+    using homeomorphic_scaling[OF assms, of s]
+    using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a]
+    unfolding *
+    by auto
+qed
+
+lemma homeomorphic_balls:
+  fixes a b ::"'a::real_normed_vector"
+  assumes "0 < d"  "0 < e"
+  shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
+    and "(cball a d) homeomorphic (cball b e)" (is ?cth)
+proof -
+  show ?th unfolding homeomorphic_minimal
+    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
+    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
+    using assms
+    apply (auto intro!: continuous_intros
+      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
+    done
+  show ?cth unfolding homeomorphic_minimal
+    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
+    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
+    using assms
+    apply (auto intro!: continuous_intros
+      simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
+    done
+qed
+
+lemma homeomorphic_spheres:
+  fixes a b ::"'a::real_normed_vector"
+  assumes "0 < d"  "0 < e"
+  shows "(sphere a d) homeomorphic (sphere b e)"
+unfolding homeomorphic_minimal
+    apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
+    apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
+    using assms
+    apply (auto intro!: continuous_intros
+      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
+    done
+
+lemma homeomorphic_ball01_UNIV:
+  "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
+  (is "?B homeomorphic ?U")
+proof
+  have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
+    apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
+     apply (auto simp: divide_simps)
+    using norm_ge_zero [of x] apply linarith+
+    done
+  then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
+    by blast
+  have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
+    apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)
+    using that apply (auto simp: divide_simps)
+    done
+  then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
+    by (force simp: divide_simps dest: add_less_zeroD)
+  show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
+    by (rule continuous_intros | force)+
+  show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
+    apply (intro continuous_intros)
+    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
+    done
+  show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
+         x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
+    by (auto simp: divide_simps)
+  show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
+    apply (auto simp: divide_simps)
+    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
+    done
+qed
+
+proposition homeomorphic_ball_UNIV:
+  fixes a ::"'a::real_normed_vector"
+  assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
+  using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
+
+
+subsection%unimportant \<open>Completeness of "Isometry" (up to constant bounds)\<close>
+
+lemma cauchy_isometric:\<comment> \<open>TODO: rename lemma to \<open>Cauchy_isometric\<close>\<close>
+  assumes e: "e > 0"
+    and s: "subspace s"
+    and f: "bounded_linear f"
+    and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
+    and xs: "\<forall>n. x n \<in> s"
+    and cf: "Cauchy (f \<circ> x)"
+  shows "Cauchy x"
+proof -
+  interpret f: bounded_linear f by fact
+  have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real
+  proof -
+    from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
+      using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
+      by auto
+    have "norm (x n - x N) < d" if "n \<ge> N" for n
+    proof -
+      have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
+        using subspace_diff[OF s, of "x n" "x N"]
+        using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
+        using normf[THEN bspec[where x="x n - x N"]]
+        by auto
+      also have "norm (f (x n - x N)) < e * d"
+        using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
+      finally show ?thesis
+        using \<open>e>0\<close> by simp
+    qed
+    then show ?thesis by auto
+  qed
+  then show ?thesis
+    by (simp add: Cauchy_altdef2 dist_norm)
+qed
+
+lemma complete_isometric_image:
+  assumes "0 < e"
+    and s: "subspace s"
+    and f: "bounded_linear f"
+    and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
+    and cs: "complete s"
+  shows "complete (f ` s)"
+proof -
+  have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
+    if as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g" for g
+  proof -
+    from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
+      using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
+    then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
+    then have "f \<circ> x = g" by (simp add: fun_eq_iff)
+    then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
+      using cs[unfolded complete_def, THEN spec[where x=x]]
+      using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
+      by auto
+    then show ?thesis
+      using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
+      by (auto simp: \<open>f \<circ> x = g\<close>)
+  qed
+  then show ?thesis
+    unfolding complete_def by auto
+qed
+
+
 end
\ No newline at end of file
diff -r d692ef26021e -r 1331e57b54c6 src/HOL/Analysis/Elementary_Topology.thy
--- a/src/HOL/Analysis/Elementary_Topology.thy	Mon Jan 07 10:22:22 2019 +0100
+++ b/src/HOL/Analysis/Elementary_Topology.thy	Mon Jan 07 11:29:34 2019 +0100
@@ -19,6 +19,123 @@
   using openI by auto
 
 
+subsubsection%unimportant \<open>Archimedean properties and useful consequences\<close>
+
+text\<open>Bernoulli's inequality\<close>
+proposition Bernoulli_inequality:
+  fixes x :: real
+  assumes "-1 \<le> x"
+    shows "1 + n * x \<le> (1 + x) ^ n"
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
+  case (Suc n)
+  have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
+    by (simp add: algebra_simps)
+  also have "... = (1 + x) * (1 + n*x)"
+    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
+  also have "... \<le> (1 + x) ^ Suc n"
+    using Suc.hyps assms mult_left_mono by fastforce
+  finally show ?case .
+qed
+
+corollary Bernoulli_inequality_even:
+  fixes x :: real
+  assumes "even n"
+    shows "1 + n * x \<le> (1 + x) ^ n"
+proof (cases "-1 \<le> x \<or> n=0")
+  case True
+  then show ?thesis
+    by (auto simp: Bernoulli_inequality)
+next
+  case False
+  then have "real n \<ge> 1"
+    by simp
+  with False have "n * x \<le> -1"
+    by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
+  then have "1 + n * x \<le> 0"
+    by auto
+  also have "... \<le> (1 + x) ^ n"
+    using assms
+    using zero_le_even_power by blast
+  finally show ?thesis .
+qed
+
+corollary real_arch_pow:
+  fixes x :: real
+  assumes x: "1 < x"
+  shows "\<exists>n. y < x^n"
+proof -
+  from x have x0: "x - 1 > 0"
+    by arith
+  from reals_Archimedean3[OF x0, rule_format, of y]
+  obtain n :: nat where n: "y < real n * (x - 1)" by metis
+  from x0 have x00: "x- 1 \<ge> -1" by arith
+  from Bernoulli_inequality[OF x00, of n] n
+  have "y < x^n" by auto
+  then show ?thesis by metis
+qed
+
+corollary real_arch_pow_inv:
+  fixes x y :: real
+  assumes y: "y > 0"
+    and x1: "x < 1"
+  shows "\<exists>n. x^n < y"
+proof (cases "x > 0")
+  case True
+  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
+  from real_arch_pow[OF ix, of "1/y"]
+  obtain n where n: "1/y < (1/x)^n" by blast
+  then show ?thesis using y \<open>x > 0\<close>
+    by (auto simp add: field_simps)
+next
+  case False
+  with y x1 show ?thesis
+    by (metis less_le_trans not_less power_one_right)
+qed
+
+lemma forall_pos_mono:
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
+  by (metis real_arch_inverse)
+
+lemma forall_pos_mono_1:
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
+  apply (rule forall_pos_mono)
+  apply auto
+  apply (metis Suc_pred of_nat_Suc)
+  done
+
+subsubsection%unimportant \<open>Affine transformations of intervals\<close>
+
+lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"
+  for m :: "'a::linordered_field"
+  by (simp add: field_simps)
+
+
+
 subsection \<open>Topological Basis\<close>
 
 context topological_space
@@ -1112,6 +1229,23 @@
 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   by (simp add: filter_eq_iff)
 
+lemma Lim_topological:
+  "(f \<longlongrightarrow> l) net \<longleftrightarrow>
+    trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
+  unfolding tendsto_def trivial_limit_eq by auto
+
+lemma eventually_within_Un:
+  "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
+    eventually P (at x within s) \<and> eventually P (at x within t)"
+  unfolding eventually_at_filter
+  by (auto elim!: eventually_rev_mp)
+
+lemma Lim_within_union:
+ "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
+  (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
+  unfolding tendsto_def
+  by (auto simp: eventually_within_Un)
+
 
 subsection \<open>Limits\<close>
 
@@ -1971,6 +2105,73 @@
   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
 
 
+subsection%unimportant \<open>Cartesian products\<close>
+
+lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
+  unfolding seq_compact_def
+  apply clarify
+  apply (drule_tac x="fst \<circ> f" in spec)
+  apply (drule mp, simp add: mem_Times_iff)
+  apply (clarify, rename_tac l1 r1)
+  apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
+  apply (drule mp, simp add: mem_Times_iff)
+  apply (clarify, rename_tac l2 r2)
+  apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
+  apply (rule_tac x="r1 \<circ> r2" in exI)
+  apply (rule conjI, simp add: strict_mono_def)
+  apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
+  apply (drule (1) tendsto_Pair) back
+  apply (simp add: o_def)
+  done
+
+lemma compact_Times:
+  assumes "compact s" "compact t"
+  shows "compact (s \<times> t)"
+proof (rule compactI)
+  fix C
+  assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
+  have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
+  proof
+    fix x
+    assume "x \<in> s"
+    have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
+    proof
+      fix y
+      assume "y \<in> t"
+      with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
+      then show "?P y" by (auto elim!: open_prod_elim)
+    qed
+    then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
+      and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
+      by metis
+    then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
+    with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
+      by metis
+    moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
+      by (fastforce simp: subset_eq)
+    ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
+      using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
+  qed
+  then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
+    and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"
+    unfolding subset_eq UN_iff by metis
+  moreover
+  from compactE_image[OF \<open>compact s\<close> a]
+  obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
+    by auto
+  moreover
+  {
+    from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
+      by auto
+    also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"
+      using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
+    finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .
+  }
+  ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
+    by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
+qed
+
+
 subsection \<open>Continuity\<close>
 
 lemma continuous_at_imp_continuous_within:
@@ -2096,5 +2297,296 @@
     using T_def by (auto elim!: eventually_mono)
 qed
 
+subsection \<open>Homeomorphisms\<close>
+
+definition%important "homeomorphism s t f g \<longleftrightarrow>
+  (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
+  (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
+
+lemma homeomorphismI [intro?]:
+  assumes "continuous_on S f" "continuous_on T g"
+          "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
+    shows "homeomorphism S T f g"
+  using assms by (force simp: homeomorphism_def)
+
+lemma homeomorphism_translation:
+  fixes a :: "'a :: real_normed_vector"
+  shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
+unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
+
+lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
+  by (rule homeomorphismI) (auto simp: continuous_on_id)
+
+lemma homeomorphism_compose:
+  assumes "homeomorphism S T f g" "homeomorphism T U h k"
+    shows "homeomorphism S U (h o f) (g o k)"
+  using assms
+  unfolding homeomorphism_def
+  by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])
+
+lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
+  by (force simp: homeomorphism_def)
+
+definition%important homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
+    (infixr "homeomorphic" 60)
+  where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
+
+lemma homeomorphic_empty [iff]:
+     "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
+  by (auto simp: homeomorphic_def homeomorphism_def)
+
+lemma homeomorphic_refl: "s homeomorphic s"
+  unfolding homeomorphic_def homeomorphism_def
+  using continuous_on_id
+  apply (rule_tac x = "(\<lambda>x. x)" in exI)
+  apply (rule_tac x = "(\<lambda>x. x)" in exI)
+  apply blast
+  done
+
+lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
+  unfolding homeomorphic_def homeomorphism_def
+  by blast
+
+lemma homeomorphic_trans [trans]:
+  assumes "S homeomorphic T"
+      and "T homeomorphic U"
+    shows "S homeomorphic U"
+  using assms
+  unfolding homeomorphic_def
+by (metis homeomorphism_compose)
+
+lemma homeomorphic_minimal:
+  "s homeomorphic t \<longleftrightarrow>
+    (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
+           (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
+           continuous_on s f \<and> continuous_on t g)"
+   (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    by (fastforce simp: homeomorphic_def homeomorphism_def)
+next
+  assume ?rhs
+  then show ?lhs
+    apply clarify
+    unfolding homeomorphic_def homeomorphism_def
+    by (metis equalityI image_subset_iff subsetI)
+ qed
+
+lemma homeomorphicI [intro?]:
+   "\<lbrakk>f ` S = T; g ` T = S;
+     continuous_on S f; continuous_on T g;
+     \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
+     \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
+unfolding homeomorphic_def homeomorphism_def by metis
+
+lemma homeomorphism_of_subsets:
+   "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
+    \<Longrightarrow> homeomorphism S' T' f g"
+apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
+by (metis subsetD imageI)
+
+lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
+  by (simp add: homeomorphism_def)
+
+lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
+  by (simp add: homeomorphism_def)
+
+lemma continuous_on_no_limpt:
+   "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
+  unfolding continuous_on_def
+  by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
+
+lemma continuous_on_finite:
+  fixes S :: "'a::t1_space set"
+  shows "finite S \<Longrightarrow> continuous_on S f"
+by (metis continuous_on_no_limpt islimpt_finite)
+
+lemma homeomorphic_finite:
+  fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
+  assumes "finite T"
+  shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
+proof
+  assume "S homeomorphic T"
+  with assms show ?rhs
+    apply (auto simp: homeomorphic_def homeomorphism_def)
+     apply (metis finite_imageI)
+    by (metis card_image_le finite_imageI le_antisym)
+next
+  assume R: ?rhs
+  with finite_same_card_bij obtain h where "bij_betw h S T"
+    by auto
+  with R show ?lhs
+    apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
+    apply (rule_tac x=h in exI)
+    apply (rule_tac x="inv_into S h" in exI)
+    apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
+    apply (metis bij_betw_def bij_betw_inv_into)
+    done
+qed
+
+text \<open>Relatively weak hypotheses if a set is compact.\<close>
+
+lemma homeomorphism_compact:
+  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
+  assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
+  shows "\<exists>g. homeomorphism s t f g"
+proof -
+  define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
+  have g: "\<forall>x\<in>s. g (f x) = x"
+    using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
+  {
+    fix y
+    assume "y \<in> t"
+    then obtain x where x:"f x = y" "x\<in>s"
+      using assms(3) by auto
+    then have "g (f x) = x" using g by auto
+    then have "f (g y) = y" unfolding x(1)[symmetric] by auto
+  }
+  then have g':"\<forall>x\<in>t. f (g x) = x" by auto
+  moreover
+  {
+    fix x
+    have "x\<in>s \<Longrightarrow> x \<in> g ` t"
+      using g[THEN bspec[where x=x]]
+      unfolding image_iff
+      using assms(3)
+      by (auto intro!: bexI[where x="f x"])
+    moreover
+    {
+      assume "x\<in>g ` t"
+      then obtain y where y:"y\<in>t" "g y = x" by auto
+      then obtain x' where x':"x'\<in>s" "f x' = y"
+        using assms(3) by auto
+      then have "x \<in> s"
+        unfolding g_def
+        using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
+        unfolding y(2)[symmetric] and g_def
+        by auto
+    }
+    ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
+  }
+  then have "g ` t = s" by auto
+  ultimately show ?thesis
+    unfolding homeomorphism_def homeomorphic_def
+    apply (rule_tac x=g in exI)
+    using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
+    apply auto
+    done
+qed
+
+lemma homeomorphic_compact:
+  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
+  shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
+  unfolding homeomorphic_def by (metis homeomorphism_compact)
+
+text\<open>Preservation of topological properties.\<close>
+
+lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
+  unfolding homeomorphic_def homeomorphism_def
+  by (metis compact_continuous_image)
+
+
+subsection%unimportant \<open>On Linorder Topologies\<close>
+
+lemma islimpt_greaterThanLessThan1:
+  fixes a b::"'a::{linorder_topology, dense_order}"
+  assumes "a < b"
+  shows  "a islimpt {a<..<b}"
+proof (rule islimptI)
+  fix T
+  assume "open T" "a \<in> T"
+  from open_right[OF this \<open>a < b\<close>]
+  obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
+  with assms dense[of a "min c b"]
+  show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
+    by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
+      not_le order.strict_implies_order subset_eq)
+qed
+
+lemma islimpt_greaterThanLessThan2:
+  fixes a b::"'a::{linorder_topology, dense_order}"
+  assumes "a < b"
+  shows  "b islimpt {a<..<b}"
+proof (rule islimptI)
+  fix T
+  assume "open T" "b \<in> T"
+  from open_left[OF this \<open>a < b\<close>]
+  obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
+  with assms dense[of "max a c" b]
+  show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
+    by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
+      not_le order.strict_implies_order subset_eq)
+qed
+
+lemma closure_greaterThanLessThan[simp]:
+  fixes a b::"'a::{linorder_topology, dense_order}"
+  shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
+proof
+  have "?l \<subseteq> closure ?r"
+    by (rule closure_mono) auto
+  thus "closure {a<..<b} \<subseteq> {a..b}" by simp
+qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
+  islimpt_greaterThanLessThan2)
+
+lemma closure_greaterThan[simp]:
+  fixes a b::"'a::{no_top, linorder_topology, dense_order}"
+  shows "closure {a<..} = {a..}"
+proof -
+  from gt_ex obtain b where "a < b" by auto
+  hence "{a<..} = {a<..<b} \<union> {b..}" by auto
+  also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
+    by auto
+  finally show ?thesis .
+qed
+
+lemma closure_lessThan[simp]:
+  fixes b::"'a::{no_bot, linorder_topology, dense_order}"
+  shows "closure {..<b} = {..b}"
+proof -
+  from lt_ex obtain a where "a < b" by auto
+  hence "{..<b} = {a<..<b} \<union> {..a}" by auto
+  also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
+    by auto
+  finally show ?thesis .
+qed
+
+lemma closure_atLeastLessThan[simp]:
+  fixes a b::"'a::{linorder_topology, dense_order}"
+  assumes "a < b"
+  shows "closure {a ..< b} = {a .. b}"
+proof -
+  from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
+  also have "closure \<dots> = {a .. b}" unfolding closure_Un
+    by (auto simp: assms less_imp_le)
+  finally show ?thesis .
+qed
+
+lemma closure_greaterThanAtMost[simp]:
+  fixes a b::"'a::{linorder_topology, dense_order}"
+  assumes "a < b"
+  shows "closure {a <.. b} = {a .. b}"
+proof -
+  from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
+  also have "closure \<dots> = {a .. b}" unfolding closure_Un
+    by (auto simp: assms less_imp_le)
+  finally show ?thesis .
+qed
+
 
 end
\ No newline at end of file
diff -r d692ef26021e -r 1331e57b54c6 src/HOL/Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Mon Jan 07 10:22:22 2019 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Mon Jan 07 11:29:34 2019 +0100
@@ -31,6 +31,239 @@
 qed
 
 
+subsection%unimportant\<open>Balls in Euclidean Space\<close>
+
+lemma cball_subset_cball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
+    (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+  proof (cases "r < 0")
+    case True
+    then show ?rhs by simp
+  next
+    case False
+    then have [simp]: "r \<ge> 0" by simp
+    have "norm (a - a') + r \<le> r'"
+    proof (cases "a = a'")
+      case True
+      then show ?thesis
+        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
+        by (force simp: SOME_Basis dist_norm)
+    next
+      case False
+      have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
+        by (simp add: algebra_simps)
+      also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
+        by (simp add: algebra_simps)
+      also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"
+        by (simp add: abs_mult_pos field_simps)
+      finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>"
+        by linarith
+      from \<open>a \<noteq> a'\<close> show ?thesis
+        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>]
+        by (simp add: dist_norm scaleR_add_left)
+    qed
+    then show ?rhs
+      by (simp add: dist_norm)
+  qed
+next
+  assume ?rhs
+  then show ?lhs
+    by (auto simp: ball_def dist_norm)
+      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
+qed
+
+lemma cball_subset_ball_iff: "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+  for a :: "'a::euclidean_space"
+proof
+  assume ?lhs
+  then show ?rhs
+  proof (cases "r < 0")
+    case True then
+    show ?rhs by simp
+  next
+    case False
+    then have [simp]: "r \<ge> 0" by simp
+    have "norm (a - a') + r < r'"
+    proof (cases "a = a'")
+      case True
+      then show ?thesis
+        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
+        by (force simp: SOME_Basis dist_norm)
+    next
+      case False
+      have False if "norm (a - a') + r \<ge> r'"
+      proof -
+        from that have "\<bar>r' - norm (a - a')\<bar> \<le> r"
+          by (simp split: abs_split)
+            (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
+        then show ?thesis
+          using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
+          by (simp add: dist_norm field_simps)
+            (simp add: diff_divide_distrib scaleR_left_diff_distrib)
+      qed
+      then show ?thesis by force
+    qed
+    then show ?rhs by (simp add: dist_norm)
+  qed
+next
+  assume ?rhs
+  then show ?lhs
+    by (auto simp: ball_def dist_norm)
+      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
+qed
+
+lemma ball_subset_cball_iff: "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
+  (is "?lhs = ?rhs")
+  for a :: "'a::euclidean_space"
+proof (cases "r \<le> 0")
+  case True
+  then show ?thesis
+    using dist_not_less_zero less_le_trans by force
+next
+  case False
+  show ?thesis
+  proof
+    assume ?lhs
+    then have "(cball a r \<subseteq> cball a' r')"
+      by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
+    with False show ?rhs
+      by (fastforce iff: cball_subset_cball_iff)
+  next
+    assume ?rhs
+    with False show ?lhs
+      using ball_subset_cball cball_subset_cball_iff by blast
+  qed
+qed
+
+lemma ball_subset_ball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
+        (is "?lhs = ?rhs")
+proof (cases "r \<le> 0")
+  case True then show ?thesis
+    using dist_not_less_zero less_le_trans by force
+next
+  case False show ?thesis
+  proof
+    assume ?lhs
+    then have "0 < r'"
+      by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
+    then have "(cball a r \<subseteq> cball a' r')"
+      by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
+    then show ?rhs
+      using False cball_subset_cball_iff by fastforce
+  next
+  assume ?rhs then show ?lhs
+    apply (auto simp: ball_def)
+    apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
+    using dist_not_less_zero order.strict_trans2 apply blast
+    done
+  qed
+qed
+
+
+lemma ball_eq_ball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+  proof (cases "d \<le> 0 \<or> e \<le> 0")
+    case True
+      with \<open>?lhs\<close> show ?rhs
+        by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
+  next
+    case False
+    with \<open>?lhs\<close> show ?rhs
+      apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
+      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
+      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
+      done
+  qed
+next
+  assume ?rhs then show ?lhs
+    by (auto simp: set_eq_subset ball_subset_ball_iff)
+qed
+
+lemma cball_eq_cball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+  proof (cases "d < 0 \<or> e < 0")
+    case True
+      with \<open>?lhs\<close> show ?rhs
+        by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
+  next
+    case False
+    with \<open>?lhs\<close> show ?rhs
+      apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
+      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
+      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
+      done
+  qed
+next
+  assume ?rhs then show ?lhs
+    by (auto simp: set_eq_subset cball_subset_cball_iff)
+qed
+
+lemma ball_eq_cball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
+    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
+    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
+    using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
+    done
+next
+  assume ?rhs then show ?lhs by auto
+qed
+
+lemma cball_eq_ball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0"
+  using ball_eq_cball_iff by blast
+
+lemma finite_ball_avoid:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "open S" "finite X" "p \<in> S"
+  shows "\<exists>e>0. \<forall>w\<in>ball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
+proof -
+  obtain e1 where "0 < e1" and e1_b:"ball p e1 \<subseteq> S"
+    using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto
+  obtain e2 where "0 < e2" and "\<forall>x\<in>X. x \<noteq> p \<longrightarrow> e2 \<le> dist p x"
+    using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto
+  hence "\<forall>w\<in>ball p (min e1 e2). w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)" using e1_b by auto
+  thus "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> \<open>e1>0\<close>
+    apply (rule_tac x="min e1 e2" in exI)
+    by auto
+qed
+
+lemma finite_cball_avoid:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "open S" "finite X" "p \<in> S"
+  shows "\<exists>e>0. \<forall>w\<in>cball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
+proof -
+  obtain e1 where "e1>0" and e1: "\<forall>w\<in>ball p e1. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
+    using finite_ball_avoid[OF assms] by auto
+  define e2 where "e2 \<equiv> e1/2"
+  have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto
+  then have "cball p e2 \<subseteq> ball p e1" by (subst cball_subset_ball_iff,auto)
+  then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto
+qed
+
+
 subsection \<open>Boxes\<close>
 
 abbreviation One :: "'a::euclidean_space"
@@ -537,6 +770,65 @@
     by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
 qed
 
+lemma image_affinity_cbox: fixes m::real
+  fixes a b c :: "'a::euclidean_space"
+  shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b =
+    (if cbox a b = {} then {}
+     else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
+     else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
+proof (cases "m = 0")
+  case True
+  {
+    fix x
+    assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"
+    then have "x = c"
+      by (simp add: dual_order.antisym euclidean_eqI)
+  }
+  moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
+    unfolding True by (auto simp: cbox_sing)
+  ultimately show ?thesis using True by (auto simp: cbox_def)
+next
+  case False
+  {
+    fix y
+    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"
+    then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
+      by (auto simp: inner_distrib)
+  }
+  moreover
+  {
+    fix y
+    assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"
+    then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"
+      by (auto simp: mult_left_mono_neg inner_distrib)
+  }
+  moreover
+  {
+    fix y
+    assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
+    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
+      unfolding image_iff Bex_def mem_box
+      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
+      apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
+      done
+  }
+  moreover
+  {
+    fix y
+    assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"
+    then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
+      unfolding image_iff Bex_def mem_box
+      apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
+      apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
+      done
+  }
+  ultimately show ?thesis using False by (auto simp: cbox_def)
+qed
+
+lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
+  (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
+  using image_affinity_cbox[of m 0 a b] by auto
+
 
 subsection \<open>General Intervals\<close>
 
@@ -756,7 +1048,8 @@
   using bounded_bilinear_inner assms
   by (rule bounded_bilinear.continuous_on)
 
-subsection \<open>Openness of halfspaces.\<close>
+
+subsection%unimportant \<open>Openness of halfspaces.\<close>
 
 lemma open_halfspace_lt: "open {x. inner a x < b}"
   by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
@@ -781,9 +1074,110 @@
   shows "open {x. x <e a}" "open {x. a <e x}"
   by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
 
+subsection%unimportant \<open>Closure of halfspaces and hyperplanes\<close>
+
+lemma continuous_at_inner: "continuous (at x) (inner a)"
+  unfolding continuous_at by (intro tendsto_intros)
+
+lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
+  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
+  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_hyperplane: "closed {x. inner a x = b}"
+  by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
+  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
+  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_interval_left:
+  fixes b :: "'a::euclidean_space"
+  shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
+  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma closed_interval_right:
+  fixes a :: "'a::euclidean_space"
+  shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
+  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
+
+lemma continuous_le_on_closure:
+  fixes a::real
+  assumes f: "continuous_on (closure s) f"
+      and x: "x \<in> closure(s)"
+      and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"
+    shows "f(x) \<le> a"
+    using image_closure_subset [OF f]
+  using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
+  by force
+
+lemma continuous_ge_on_closure:
+  fixes a::real
+  assumes f: "continuous_on (closure s) f"
+      and x: "x \<in> closure(s)"
+      and xlo: "\<And>x. x \<in> s ==> f(x) \<ge> a"
+    shows "f(x) \<ge> a"
+  using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms
+  by force
+
+
+subsection%unimportant\<open>Some more convenient intermediate-value theorem formulations\<close>
+
+lemma connected_ivt_hyperplane:
+  assumes "connected S" and xy: "x \<in> S" "y \<in> S" and b: "inner a x \<le> b" "b \<le> inner a y"
+  shows "\<exists>z \<in> S. inner a z = b"
+proof (rule ccontr)
+  assume as:"\<not> (\<exists>z\<in>S. inner a z = b)"
+  let ?A = "{x. inner a x < b}"
+  let ?B = "{x. inner a x > b}"
+  have "open ?A" "open ?B"
+    using open_halfspace_lt and open_halfspace_gt by auto
+  moreover have "?A \<inter> ?B = {}" by auto
+  moreover have "S \<subseteq> ?A \<union> ?B" using as by auto
+  ultimately show False
+    using \<open>connected S\<close>[unfolded connected_def not_ex,
+      THEN spec[where x="?A"], THEN spec[where x="?B"]]
+    using xy b by auto
+qed
+
+lemma connected_ivt_component:
+  fixes x::"'a::euclidean_space"
+  shows "connected S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>S.  z\<bullet>k = a)"
+  using connected_ivt_hyperplane[of S x y "k::'a" a]
+  by (auto simp: inner_commute)
+
 
 subsection \<open>Limit Component Bounds\<close>
 
+lemma Lim_component_le:
+  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes "(f \<longlongrightarrow> l) net"
+    and "\<not> (trivial_limit net)"
+    and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
+  shows "l\<bullet>i \<le> b"
+  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
+
+lemma Lim_component_ge:
+  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes "(f \<longlongrightarrow> l) net"
+    and "\<not> (trivial_limit net)"
+    and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
+  shows "b \<le> l\<bullet>i"
+  by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
+
+lemma Lim_component_eq:
+  fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"
+    and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
+  shows "l\<bullet>i = b"
+  using ev[unfolded order_eq_iff eventually_conj_iff]
+  using Lim_component_ge[OF net, of b i]
+  using Lim_component_le[OF net, of i b]
+  by auto
+
 lemma open_box[intro]: "open (box a b)"
 proof -
   have "open (\<Inter>i\<in>Basis. ((\<bullet>) i) -` {a \<bullet> i <..< b \<bullet> i})"
@@ -1192,6 +1586,67 @@
 qed
 
 
+subsection%unimportant \<open>Diameter\<close>
+
+lemma diameter_cball [simp]:
+  fixes a :: "'a::euclidean_space"
+  shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
+proof -
+  have "diameter(cball a r) = 2*r" if "r \<ge> 0"
+  proof (rule order_antisym)
+    show "diameter (cball a r) \<le> 2*r"
+    proof (rule diameter_le)
+      fix x y assume "x \<in> cball a r" "y \<in> cball a r"
+      then have "norm (x - a) \<le> r" "norm (a - y) \<le> r"
+        by (auto simp: dist_norm norm_minus_commute)
+      then have "norm (x - y) \<le> r+r"
+        using norm_diff_triangle_le by blast
+      then show "norm (x - y) \<le> 2*r" by simp
+    qed (simp add: that)
+    have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"
+      apply (simp add: dist_norm)
+      by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
+    also have "... \<le> diameter (cball a r)"
+      apply (rule diameter_bounded_bound)
+      using that by (auto simp: dist_norm)
+    finally show "2*r \<le> diameter (cball a r)" .
+  qed
+  then show ?thesis by simp
+qed
+
+lemma diameter_ball [simp]:
+  fixes a :: "'a::euclidean_space"
+  shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
+proof -
+  have "diameter(ball a r) = 2*r" if "r > 0"
+    by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
+  then show ?thesis
+    by (simp add: diameter_def)
+qed
+
+lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
+proof -
+  have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
+    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
+  then show ?thesis
+    by simp
+qed
+
+lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
+proof -
+  have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
+    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
+  then show ?thesis
+    by simp
+qed
+
+lemma diameter_cbox:
+  fixes a b::"'a::euclidean_space"
+  shows "(\<forall>i \<in> Basis. a \<bullet> i \<le> b \<bullet> i) \<Longrightarrow> diameter (cbox a b) = dist a b"
+  by (force simp: diameter_def intro!: cSup_eq_maximum L2_set_mono
+     simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
+
+
 subsection%unimportant\<open>Relating linear images to open/closed/interior/closure\<close>
 
 proposition open_surjective_linear_image:
@@ -1294,6 +1749,202 @@
   shows "interior(uminus ` S) = image uminus (interior S)"
   by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)
 
+subsection%unimportant \<open>"Isometry" (up to constant bounds) of Injective Linear Map\<close>
+
+proposition injective_imp_isometric:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes s: "closed s" "subspace s"
+    and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
+  shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
+proof (cases "s \<subseteq> {0::'a}")
+  case True
+  have "norm x \<le> norm (f x)" if "x \<in> s" for x
+  proof -
+    from True that have "x = 0" by auto
+    then show ?thesis by simp
+  qed
+  then show ?thesis
+    by (auto intro!: exI[where x=1])
+next
+  case False
+  interpret f: bounded_linear f by fact
+  from False obtain a where a: "a \<noteq> 0" "a \<in> s"
+    by auto
+  from False have "s \<noteq> {}"
+    by auto
+  let ?S = "{f x| x. x \<in> s \<and> norm x = norm a}"
+  let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
+  let ?S'' = "{x::'a. norm x = norm a}"
+
+  have "?S'' = frontier (cball 0 (norm a))"
+    by (simp add: sphere_def dist_norm)
+  then have "compact ?S''" by (metis compact_cball compact_frontier)
+  moreover have "?S' = s \<inter> ?S''" by auto
+  ultimately have "compact ?S'"
+    using closed_Int_compact[of s ?S''] using s(1) by auto
+  moreover have *:"f ` ?S' = ?S" by auto
+  ultimately have "compact ?S"
+    using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
+  then have "closed ?S"
+    using compact_imp_closed by auto
+  moreover from a have "?S \<noteq> {}" by auto
+  ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"
+    using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
+  then obtain b where "b\<in>s"
+    and ba: "norm b = norm a"
+    and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
+    unfolding *[symmetric] unfolding image_iff by auto
+
+  let ?e = "norm (f b) / norm b"
+  have "norm b > 0"
+    using ba and a and norm_ge_zero by auto
+  moreover have "norm (f b) > 0"
+    using f(2)[THEN bspec[where x=b], OF \<open>b\<in>s\<close>]
+    using \<open>norm b >0\<close> by simp
+  ultimately have "0 < norm (f b) / norm b" by simp
+  moreover
+  have "norm (f b) / norm b * norm x \<le> norm (f x)" if "x\<in>s" for x
+  proof (cases "x = 0")
+    case True
+    then show "norm (f b) / norm b * norm x \<le> norm (f x)"
+      by auto
+  next
+    case False
+    with \<open>a \<noteq> 0\<close> have *: "0 < norm a / norm x"
+      unfolding zero_less_norm_iff[symmetric] by simp
+    have "\<forall>x\<in>s. c *\<^sub>R x \<in> s" for c
+      using s[unfolded subspace_def] by simp
+    with \<open>x \<in> s\<close> \<open>x \<noteq> 0\<close> have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
+      by simp
+    with \<open>x \<noteq> 0\<close> \<open>a \<noteq> 0\<close> show "norm (f b) / norm b * norm x \<le> norm (f x)"
+      using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
+      unfolding f.scaleR and ba
+      by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
+  qed
+  ultimately show ?thesis by auto
+qed
+
+proposition closed_injective_image_subspace:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"
+  shows "closed(f ` s)"
+proof -
+  obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
+    using injective_imp_isometric[OF assms(4,1,2,3)] by auto
+  show ?thesis
+    using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)
+    unfolding complete_eq_closed[symmetric] by auto
+qed
+
+
+subsection%unimportant \<open>Some properties of a canonical subspace\<close>
+
+lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
+  by (auto simp: subspace_def inner_add_left)
+
+lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"
+  (is "closed ?A")
+proof -
+  let ?D = "{i\<in>Basis. P i}"
+  have "closed (\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0})"
+    by (simp add: closed_INT closed_Collect_eq continuous_on_inner
+        continuous_on_const continuous_on_id)
+  also have "(\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0}) = ?A"
+    by auto
+  finally show "closed ?A" .
+qed
+
+lemma dim_substandard:
+  assumes d: "d \<subseteq> Basis"
+  shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
+proof (rule dim_unique)
+  from d show "d \<subseteq> ?A"
+    by (auto simp: inner_Basis)
+  from d show "independent d"
+    by (rule independent_mono [OF independent_Basis])
+  have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
+  proof -
+    have "finite d"
+      by (rule finite_subset [OF d finite_Basis])
+    then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
+      by (simp add: span_sum span_clauses)
+    also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
+      by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
+    finally show "x \<in> span d"
+      by (simp only: euclidean_representation)
+  qed
+  then show "?A \<subseteq> span d" by auto
+qed simp
+
+text \<open>Hence closure and completeness of all subspaces.\<close>
+lemma ex_card:
+  assumes "n \<le> card A"
+  shows "\<exists>S\<subseteq>A. card S = n"
+proof (cases "finite A")
+  case True
+  from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
+  moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
+    by (auto simp: bij_betw_def intro: subset_inj_on)
+  ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
+    by (auto simp: bij_betw_def card_image)
+  then show ?thesis by blast
+next
+  case False
+  with \<open>n \<le> card A\<close> show ?thesis by force
+qed
+
+lemma closed_subspace:
+  fixes s :: "'a::euclidean_space set"
+  assumes "subspace s"
+  shows "closed s"
+proof -
+  have "dim s \<le> card (Basis :: 'a set)"
+    using dim_subset_UNIV by auto
+  with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"
+    by auto
+  let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+  have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>
+      inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
+    using dim_substandard[of d] t d assms
+    by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)
+  then obtain f where f:
+      "linear f"
+      "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
+      "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
+    by blast
+  interpret f: bounded_linear f
+    using f by (simp add: linear_conv_bounded_linear)
+  have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x
+    using f.zero d f(3)[THEN inj_onD, of x 0] by auto
+  moreover have "closed ?t" by (rule closed_substandard)
+  moreover have "subspace ?t" by (rule subspace_substandard)
+  ultimately show ?thesis
+    using closed_injective_image_subspace[of ?t f]
+    unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
+qed
+
+lemma complete_subspace: "subspace s \<Longrightarrow> complete s"
+  for s :: "'a::euclidean_space set"
+  using complete_eq_closed closed_subspace by auto
+
+lemma closed_span [iff]: "closed (span s)"
+  for s :: "'a::euclidean_space set"
+  by (simp add: closed_subspace subspace_span)
+
+lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
+  for s :: "'a::euclidean_space set"
+proof -
+  have "?dc \<le> ?d"
+    using closure_minimal[OF span_superset, of s]
+    using closed_subspace[OF subspace_span, of s]
+    using dim_subset[of "closure s" "span s"]
+    by simp
+  then show ?thesis
+    using dim_subset[OF closure_subset, of s]
+    by simp
+qed
+
+
 no_notation
   eucl_less (infix "<e" 50)