--- 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:
--- 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
--- 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
--- 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
--- 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)